Research Article Analytical Modeling and Contradictions in ...downloads.hindawi.com/journals/jpe/2015/895786.pdf · Research Article Analytical Modeling and Contradictions in Limestone

Research ArticleAnalytical Modeling and Contradictions inLimestone Reservoirs Breccias Vugs and Fractures

Nelson Barros-Galvis1 Pedro Villasentildeor1 and Fernando Samaniego2

1Programa Doctoral Direccion de Investigacion y Posgrado Instituto Mexicano del PetroleoEje Central Lazaro Cardenas Norte 152 07730 City of Mexico DF Mexico2Secretarıa de Posgrado Facultad de Ingenierıa Universidad Autonoma de Mexico Avenida Universidad 300004515 City of Mexico DF Mexico

Correspondence should be addressed to Nelson Barros-Galvis nelbarrosgalvisgmailcom

Received 24 September 2014 Revised 13 February 2015 Accepted 23 February 2015

Academic Editor Lixin Cheng

Copyright copy 2015 Nelson Barros-Galvis et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Modeling of limestone reservoirs is traditionally developed applying tectonic fractures concepts or planar discontinuities and hasbeen simulated dynamically without considering nonplanar discontinuities as sedimentary breccias vugs fault breccias and impactbreccias assuming that all these nonplanar discontinuities are tectonic fractures causing confusion and contradictions in reservoirscharacterization The differences in geometry and connectivity in each discontinuity affect fluid flow generating the challenge todevelop specific analytical models that describe quantitatively hydrodynamic behavior in breccias vugs and fractures focusingon oil flow in limestone reservoirs This paper demonstrates the differences between types of discontinuities that affect limestonereservoirs and recommends that all discontinuities should be included in simulation and static-dynamic characterization becausethey impact fluid flow To demonstrate these differences different analytic models are developed Findings of this work are based onobservations of cores outcrops and tomography and are validated with field data The explanations and mathematical modelingdeveloped here could be used as diagnostic tools to predict fluid velocity and fluid flow in limestone reservoirs improving thecomplex reservoirs static-dynamic characterization

1 Introduction

A significant portion of the world oil reservoirs is found incarbonates [1 2] early identification of the types of geologicaldiscontinuities that are dominant in limestone rock is oneof the principal problems for advanced development andefficient exploitation of the reservoirs because of impact inthe reservoir oil flow These complex limestone reservoirsare associated with geological events (tectonic fractures sed-imentary breccias impact breccias vugs and fault breccias)that require to be understood dynamically and geologically toproperly derive an analytical model that would describe flowbehavior

The phenomenology of fluid flow in limestone carbonatereservoirs depends on type of discontinuities connecteddiscontinuities have a dependence on at least diagenesis

structural history and lithology A key concept is that pla-nar or nonplanar discontinuities can lead to different flowresponse during oil production Moreover limestone car-bonate reservoirs are modeled confused and associatedwith planar discontinuities named tectonic fracture networkscausing contradictions Processes that generate nonplanardiscontinuities are usually excluded and models do not rep-resent the reservoir reality when based on an equivalent flowmedium A case well-known is Cantarell field which is theeighth largest oil field in the world with fractures brecciasand vugs [3ndash5] located in the Campeche Sound YucatanPlatform in Mexico This reservoir is modeled using tectonicfracture networks without considering vugs fault and impactbreccias and caverns [6ndash8] in effect these tectonic fracturesare used to represent all types of discontinuities

Hindawi Publishing CorporationJournal of Petroleum EngineeringVolume 2015 Article ID 895786 28 pageshttpdxdoiorg1011552015895786

2 Journal of Petroleum Engineering

In 1972 Neale andNader [9] described the flow dynamicsin a vuggy porous medium using the creeping Navier-Stokesand the Darcy equations in the surrounding homogeneousand isotropic medium Koenraad and Bakker in 1981 [10]proposed a theoretical study about fracturevug collapsebreccias and brecciated karst based on geological infor-mation Wu et al in 2011 [11] proposed numerical modelsmultiphase and single-phase flow for vuggy reservoirs usingbalance mass equations Darcy and Forchheimer equationsand Hagen Poiseuille tube flow In 1999 McKeown et al[12] developed a numerical groundwater flow model forSellafield in which they modeled hydrogeological Brockramfault breccia using Darcyrsquos equation applied to a limestonewith porosity between 1 and 10 percent obtaining a rangeof hydraulic conductivity Gudmundsson [13] studied fluidflow behavior using Darcy equation applied to breccias andfaults

Sedimentary breccias are related to debris flow Thephysics of debris flow published by Iverson in 1997 [14]analyzed flows of dry granular solids and solid-fluidmixturesand described the grains behavior throughmomentummassand energy balances Enos [15] described debris reservoirand sedimentary breccias in Tamabra Limestone of thePoza Rica field in Mexico with field ranges of porosityof 37ndash97 and permeability of 001ndash700md with depthranging between 1980 and 2700m and cumulative pro-duction to July 1983 of 198 billion oil barrels There aremany impact breccias but associated with oil reservoir themost known is KT limit in Cantarell located in CampecheBay Yucatan Platform in Mexico related to the Chicxulubimpact [16] From the point of view of fluid flow Mayret al [17] in 2008 estimated rocks hydraulic permeabilityassociated with the impact crater Chicxulub using NuclearMagnetic Resonance Kozeny-Carman equation and thefractal PaRis-model obtaining low hydraulic permeabil-ity

The purpose of this paper is to demonstrate how qualita-tive and quantitative differences between tectonic fracturesfault breccias sedimentary breccias vugs and impact brec-cias affect the exploitation strategy and advanced develop-ment of carbonate reservoirs Our hypothesis considers thatall the discontinuities in carbonate reservoirs are distinctand cannot be called or analyzed as tectonic fractures with-out knowing their genesis and discontinuity impact in thereservoir or only focusing in fluid flow To demonstrate thishypothesis we understood the rocks nature and used sub-surface geological analogs (outcrops) cores computerizedtomography and analytical modeling making use of fluidflow kinematics

2 First Contradiction Tectonic Fracture orNonplanar Discontinuity

Unfortunately whatever the discontinuity (planar or non-planar) it is considered as a natural fracture Usuallyoil reservoirs with discontinuities are ldquonaturally fracturedrdquo

Figure 1 Limestone core with planar tectonic fractures EarlyCretaceous Samaria-Luna Region TAB Mexico

causing confusion in reservoirs simulation Our discussionbegins with the following the word fracture derived fromthe Latin fractus means ldquobrokenrdquo and its definitions are asfollows

ldquoA natural fracture is a macroscopic planar discontinuitywhich results from stresses that exceed the rupture strengthof the rockrdquo [18]

ldquoA reservoir fracture is a naturally occurringmacroscopicplanar discontinuity in rock due to deformation or physicaldiagenesisrdquo [19]

ldquoA fracture is any planar or subplanar discontinuity that isvery narrow in one dimension compared to the other two andforms as a result of external (eg tectonic) or internal (thermalor residual) stressrdquo [20]

These definitions converge in that fractures result fromstresses and deformation on the rock mechanical diagene-sis that generates volume reduction by compaction duringburial Whether natural tectonic fractures result as the effectof stresses and mechanical compaction why are sedimentarybreccia impact breccia fault breccia and vug considerednatural fractures regardless of the genesis of these geologicalevents This is the first contradiction

The discontinuities observed in limestone reservoirs areplanar or nonplanarThe observation of the core of limestonereservoirs in Figure 1 shows a planar natural fracture withinclination it is a preferential path for oil flow its aperture hasbeen developed on the plane of weakness (with low cohesion)as consequence of stresses on the rock Although the corewas badly preserved its natural fracture presents an apertureincrease but it is natural

However many reservoirs normally contain nonplanardiscontinuities caused by diagenesis The dissolution pro-cesses (chemical diagenesis) as result of expelled watergenerate vugs caverns and clasts dissolution This type ofdiscontinuities affects the fluid flow and creates an intercon-nected system through poresTheFigure 2(a) presents Clado-coropsis that could act as barrier because these fragmentshave not been fragmented or dissolved Also Figure 2(b)shows a core with fragmented Cladocoropsis grainstonedue to chemical diagenesis creating a secundary porosityand permeability connecting pores system (primary andsecondary porosity)

Reservoir engineers have assumed discontinuities as atype of pores regardless of their origin or genesis and tectonic

Journal of Petroleum Engineering 3

5 cm(a) (b)

Figure 2 Core slabs of Cladocoropsis (a) Mud supported lime-stone (b) Fragmented Cladocoropsis grainstone Ghawar fieldSaudi Arabia [23]

fractures are considered as structural discontinuities impact-ing fluid flow [21] So the discontinuities or heterogeneitiessuch as vugs tectonic fractures and porosity associated withbreccias can be interpreted with an equivalent medium [22]In this case ldquoall is tectonic fracturesrdquo regardless of theirgenesis creating confusion When we compare and observeFigure 1 with Figure 2 it can be inferred that in dissimilarmedia oil flows have different velocities From a modelingpoint of view the discontinuities (planar or nonplanar) inthe reservoirs should be related to their genesis and fluidcharacteristics flow to understand the phenomenology oflimestone systems

3 Second Contradiction How Do DifferentDiscontinuities Impact

Despite the fact that there exist different kinds of discontinu-ities such as tectonic fractures vugs caves collapse brecciassedimentary breccias fault breccias and impact brecciasin transient pressure analysis and reservoir simulation allof them are considered as fractures or fissures In conse-quence they are represented as planes in the reservoir withorientation intensity or fracturing frequency with specificgeometry and with fluid dynamical properties This study isbased on open and connected discontinuities

The discontinuities clearly affect fluid flow in the reser-voir However each discontinuity creates a particular hydro-carbons flow behavior specifically in limestone reservoirs Itis essential to understand them to properly study its influenceon the production behavior of the reservoirs

The phenomenology of discontinuities begins with theirgeometry In principle a planar geometry describes aparabolic flow profile for conditions of laminar flow fortectonic fractures the fluid velocity is associated with theaperture and roughness of the discontinuity which is linkedto the permeability The pressure profile changes are due to

friction tortuosity the complex interconnected system andcross flow This phenomenon is present in tectonic fractures

A nonplanar discontinuity shows nonparabolic flow pro-file because of high oil velocity and heterogeneousmediaThefluid high velocity is directly linked with geometry shapeaperture roughness diameter discontinuities distributionirregular surfaces and pressure In the case of vugs andcaverns their diameters or apertures are large and generatezones of high permeability or superhighway production

Discontinuities (planar or nonplanar) can be describedthrough visible geological features related to geometryspecifically cross section flow area which has a direct impactin fluid flow This can be understood using the flow concept119902 = 119860119906 A nonplanar discontinuity area is greater than aplanar discontinuity area in effect flow increases becausepores may be connected and their size are increased due todissolution In addition when the flow area is increasedoil velocity 119906 increases too This will be demonstrated usinganalytical models in this study Moreover it can be observedwhen vugs and caverns are compared with tectonic fracturesthat they have a greater permeability In summary fluid flowin a planar discontinuity (fracture) is different to flow in anonplanar discontinuity (vugs breccias and caverns) This isthe second contradiction considering that hydrodynamics isintrinsic to geometry or flow area because there is a mass andmomentum transfer process oil flow in each discontinuity isunique and cannot be traditionally considered as fracturesThe understanding and comparison of these dynamic andgeological differences in limestone reservoirs allow the accu-racy of fluid flow characteristics and approach their optimumexploitation strategy

Although discontinuities generate highly heterogeneousand anisotropic systems a complex reservoir could evidencedifferent types of geological characteristics production andpressure behavior might identify dominant discontinuitiesand to achieve a reasonable modeling it should be consid-ered that they might in some cases perform as seal or flowbarriers

However before carrying out pressure transient test andreservoir dynamic simulations it is necessary to character-ize these discontinuities and establish convergence betweengeology andfluid flow features using cores outcrops seismicwell logs tomography pressures profiles production andmathematical modeling that describe the flow kinematic inreservoir fluid flow

It is becoming increasingly difficult to ignore the differ-ences between types of discontinuities in which all of themare called fractures by simplicityTherefore there is a need tobe explicit about the phenomenology of discontinuities

4 Analogs Reservoirs and Outcrops fora Flow Analytical Model

Reservoir analog based on outcrop studies provides defini-tions of geometry and heterogeneities at interwell scales andcharacterizes reservoirs [24] A subsurface reservoir modelusing outcrop analog [25] is employed for an understandingof geological parameters associated with reservoir and at


Figure 3 Tectonic fractures system outcrop AB Canada

outcrop rock (porosity permeability and heterogeneities)with the objective of characterizing it

Outcrops were at geological conditions of subsurfacefortunately they are exposed and wemay describe a reservoirwith similar features In this paper we used outcrops tounderstand physics of each discontinuity (vugs breccias andfractures) and tomography to know how is the fluid flowthrough the rock As a result of understanding that rock wedeveloped an analytical model

5 Geological and Tomography Features ofTectonic Fractures

Limestone reservoirs constitute a portion of the basinTectonic fractures are planar discontinuities in reservoirsGenerally the genesis of fracturing is tectonism and it isrelated to local folds faults or a regional system [26] Thereare different fractures types shear fractures or slip surfaceextension fractures such as joints fissures and veins andcontraction or closing fractures as stylolites [20] Openfractures can behave as hydraulic conductors and impact theproductivity of the reservoir

Tectonic fractures have been observed in carbonate coresbut it is difficult to obtain complete cores due to its brittlenature (Figure 1) Tectonic fractures can be open deformedand mineral-filled In spite of deformation it is evident thattectonic fractures are planar discontinuities that could storeand produce hydrocarbons In outcrops these fractures arestudied because they explain paleostresses (Figure 3) Alsothese surface systems are used as an analogical model inreservoir characterization Figure 3 shows different planarfractures systems as a result of paleostresses in the rockand it is observed that each fracture has its aperture thatmay be interconnected inclined parallel orthogonal openor closed This outcrop was buried at subsurface conditionsduring a geological age nowadays it is exposed In Figure 3diverse types of natural fractures can be observed

X-ray computerized tomography (CT) is a nondestructivetechnique that allows visualization of the internal struc-ture of rocks determined mainly by variations in densityand atomic composition [27] One well-known advantageof tomography is fracture morphology description insidethe rock This implies that open fractures are observed asplanar discontinuities with a specific physical behavior Inconsequence the dominant parameter in tectonic fractures is

aperture Aperture is associated with permeability parabolicflow profile maximum and mean velocity and the flow rate

A study illustrated how width or aperture impacts frac-ture permeability using an equation flow [28] It was shownthat a single fracture of 025mm aperture has the equivalentpermeability of 188m of unfractured rock with a uniformpermeability of 10md and a 127mm aperture fracture isequivalent to 173m of rock with a permeability of 1000md

Tomography was used to show internal fractures and toverify theirmorphology Useful calcareous coreswithout dis-solutions or visible discontinuities are considered as matrixrock without fractures In many cases narrow subplanarfractures can be observed using tomography or would beinferred if the density in the core internal structure showschanges

A sample core in a limestone reservoir (159 cm in lengthand 10 cm in diameter) was obtainedThe scan of a limestonecore (Figure 4) at 3mm spacing showed that rock containsnarrow subplanar fracturesThey are open deformed andormineral-filled In images 16 to 21 an open fracture with greataperture can be seen and in images 30 to 42 other fracturescan be observed but with small aperture Moreover doinga visual inspection of core does not observe any fracturesthat may be connected (see slides 30 to 53 in Figure 4)Approximately these limited length fractures have 1 to 2mmapertures and would allow hydrocarbons flow in effect theyalso provide effective porosity because fracture connectsspaces in this rock

6 Analytical Model for the Profile Velocity ofTectonic Fractures

Fluid flow is defined by a velocity vector field and a pres-sure scalar field In order to understand how permeabilityinfluences fluid flow in fractures we analyze flow and fluidkinematics equations Kinematic equations are applied toan isothermal low viscosity irrotational and single-phasefluid flow and are referred to as potential flows and streamfunction Applicability of kinematic equations is limited toReynolds numbers Re more than unity because the inviscidfluid flow theory corresponds to high flow values and smallfluid viscosity Calcareous reservoirs heterogeneities mayproduce high velocity flow

At larger Reynolds numbers kinematic equations aresuggested for Newtonian liquids of minimal viscosity andgases

Fluid kinematics describes fluidmotion using streamlinesand potential function with respect to a plane and a flowequation describes a close relationship between fractureaperture pressure and velocity and flow rate using clas-sical methods in fluid mechanics Kinematics of fluid flowwould show a particular behavior of flow lines for eachdiscontinuity in this case for tectonic fractures (Figure 5(a))Classical methods in fluid mechanics are illustrated by ananalytical model for velocity profile in parallel plates [29]Two symmetrically inclined parallel surfaces with respect tothe 119909-axis are shown in Figure 5(b) Fluid motion is in the119909 direction and flow velocity distribution in the 119910 directionfor steady state of an incompressible fluid of constant density


1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53

Figure 4 Scan of limestone core with dark shading associatedwith low density andwhite with high density regions with evidentmacroscopicfractures Early Cretaceous Samaria-Luna Region TAB Mexico

is indicated in this figure pressure variation is a function of119909 coordinate and the surface longitudes are larger than theaperture

Figure 5 shows top and lateral views in an open frac-ture simulated with two inclined plates that may be fixedFigure 5(a) presents straight and parallel flow lines andFigure 5(b) shows a parabolic velocity profile Our contribu-tion is the application of the Couette flow exact solution fornatural fracture to obtain specific solution that describes flow

profile for natural fissure When a plate is moving namelysystem may be interpreted as a stress-sensitive system (useCouette flow) Fixed plates may be understood nonstresssensitive (use Poiseuille flow) So Couette equation is ageneral case with respect to Poiseuille equation

Fundamentally Newtonian fluid flow of a single phasethrough a fractured rock is governed by the Navier-Stokesequations [30 31] To simplify the discussion wewill considera ldquoone-dimensionalrdquo fracture and use the Navier-Stokes


x

z

Flowline

Aperture

(a)

H

120573

u

x

U

y

a

(b)

Figure 5 Top and lateral views of fluid flow in a tectonic fracture (a) Flow lines in an open fracture (top view) (b) Flow profile between twoinclined parallel plates (lateral view) [29]

equations exact solution known as Couette flow and thisdiscussion about laminar flow between platesmay be detailedin [29] Using Navier-Stokes equations

120588(120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 119908

120597119906

120597119911)

= minus120597119901

120597119909+ 120583(

1205972119906

1205971199092+1205972119906

1205971199102+1205972119906

1205971199112) + 120574 sen120573

(1)

Considering the previously stated fracture flow problemregarding Figure 5(a) (1) can be simplified as follows

0 = minus120597119901

120597119909+ 120583(

1205972119906

1205971199102) + 120574 sen120573 (2)

In Figure 5(b) sen120573 = 119889119867119889119909 and applying in (2)

1205972119906

1205971199102=1

120583(120597119901

120597119909+ 120574119867) (3)

Boundary conditions are 119906 = 0 for 119910 = 0 In the limit when119910 nearly reaches the fracture aperture 119886 and fluid velocity (119906)is close to the terminal upper plate velocity (119880) then 119906 = 119880for 119910 = 119886 and 12059721199061205971199102 = 120582 where 120582 = constant

To obtain the solution of (3) it is necessary to integrateand apply boundary conditions This solution is 119906(119910) =

(1205822)1199102+ 119860119910 + 119861 which is a parabola The constants 119860 119861

and 120582 are obtained by integration and finally result in thefollowing equation

119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909+119880

119886119910 (4)

Equation (4) describes the Couette flow because there is alinear plate movement and can be used for stress-sensitivetectonic fractures However (4) can be written as

119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909 (5)

Equation (5) corresponds to the steady flow pressure distri-bution through two inclined parallel surfaces when 119880 = 0

(Poiseuille flow) it can be used to derive an expression forthe rate flow of nonstress-sensitive tectonic fractures using119902 = int 119906119889119860 where 119860 = 119886 sdot 1 gives

119902 = int

119886

0

1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909119889119910 = minus

1198863

12120583

120597 (119901 + 120574119867)

120597119909

(6)

Equation (6) is known asTheCubic Lawwhich has been usedin fractured rocks [30] and the mean velocity (119906) is obtainedusing 119906 = 119902119860 substituting (6) gives

119906 = minus1198862

12120583

120597 (119901 + 120574119867)

120597119909 (7)

Deriving with respect to 119910 equation (5) substituting 119910 =

1198862 and applying the maximum andminimum criterion themaximum velocity can be obtained

119906max = minus1198862

8120583

120597 (119901 + 120574119867)

120597119909to 119910 = 119886

2 (8)

Equations described based on the classical methods in fluidmechanics have been developed to calculate flow rate intectonic fractures The negative sign in (6) (7) and (8)is related to flow in the direction of the negative pressuregradient Although the cubic equation was derived for tec-tonic fractures it is used for diverse kinds of discontinuities(breccias vugs and channels) yet

For natural fractures our contribution is the applicationof the Couette flow general solution that describes flowprofile in a natural fissure But it will be compared with flowkinematics equations to know the range of velocity or whenmaximum and mean flow velocity can be used

7 Kinematic Analytical Modeling forTectonic Fractures

For flow visualization three types of flow lines are usedThreetypes of fluid element trajectories are defined streamlines


y

x

(a) (b) (c)

Figure 6 Streamlines in uniform rectilinear flow in tectonic fractures with parallel walls (a) horizontal direction (b) inclined direction and(c) vertical direction

pathlines and streaklines They are all equivalent for steadyflow but differ conceptually for unsteady flows

Streamlines are lines tangents to the velocity vector andperpendicular at lines of constant potential called equipoten-tial lines a pathline is a line traced out in time by a given fluidparticle as it flows and a streakline is a line traced out by aneutrally buoyantmarker fluidwhich is continuously injectedinto a flow field at a fixed point in space [32]

To demonstrate the difference between the types ofdiscontinuities streamlines are used These are associatedwith stream analytical function (Ψ) and potential analyticalfunction (Φ) In tectonic fractures their streamlines areparallel and would tend to be uniform (Figure 6)

The theory of complex variable guarantees that Laplacersquosequation for the velocity potentialnabla2Φ = 0 and for the streamfunction nabla2Ψ = 0 is satisfied and can be solved Then

119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)

Equations (9) are recognized as the Cauchy-Riemann equa-tions for Φ(119909 119910) and Ψ(119909 119910) and the analytical expressionsused for uniform flow in Cartesian and polar coordinates arethe following

Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)

Equations (10) and (12) are Laplacersquos equation solutionsThusthey are satisfied to nabla2Φ = 0 and nabla

2Ψ = 0 For this

demonstration in one-dimension is used the stream functionΨ(119909 119910) as follows

nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)

Following a similar procedure for velocity potentialΦ(119909 119910)

nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)

Thus Laplacersquos equations for the velocity potential nabla2Φ = 0

and for the stream function nabla2Ψ = 0 have been satisfied

8 Third Contradiction Darcyrsquos Flowor Couette General Flow for PlanarDiscontinuity (Tectonic Fracture)

For stress-sensitive system we recommend Couette flow andfor nonstress-sensitive system use Poiseuille flow Initiallyit has been referenced the use of Darcyrsquos flow to describebehavior fluid flow in vugs [9 11] fault [12] fault breccias[12 13] and fractures [33]

The application ofDarcyrsquos flow is recommendedwhen therange of value of Reynolds number (Re) is between zero andunity Last information was reported by Muskat [34] and isapplied for low laminar velocity namely in porous mediaalthough it is also applied for tectonic fractures or tectonicfracturedmedia without considering value of Reynolds num-ber Our third contradiction is based on calculating Re andif this number is greater that unity then The Cubic Law forplanar discontinuity (tectonic fracture) is necessary in which


Figure 7 Fault breccia outcrop AB Canada

it derived Couette flow So Darcyrsquos Law should be used withcaution

9 Geological and Tomography Features ofFault Breccias

Fault breccias consist of planar discontinuities associatedwith faulting These breccias are incohesive characterized bypredominant angular to subangular fragments and internalfractures of cataclastic rock present in a fault zone Fragmentsare slickensided with variable slip directions diverse sizesand greater than 30 percent visibility Cataclastic rocks haveinternal planar structure are incohesive when faulting occursabove depths of 1 to 4 km or upper crustal fault zones wherebrittle deformation increases breccia formation processesbecause of stresses and tectonic activity

Fault breccia zones enhance permeability [35] In effectfault breccias are a superhighway production in limestonereservoirs High permeability is linked with unconsolidatedfaulted rock that can be a channel for the hydrocarbons flow

A fault breccia is presented in Figure 7 which showsan outcrop of limestone rock with subangular fragmentsembedded in the matrix with absence of primary cohe-sion and erosion its fault breccia zone is approximately7 meters which implies huge movement rock mass andstresses (Figure 7) This outcrop is a point of comparison forfault breccia present in limestone reservoir because it wasclearly buried rock in the past geological time In effect faultbreccias in limestone reservoirs are production paths withapproximate 7ndash10 meters (width)

Limestone reservoirs show fault breccias associated withfaulting these channels have been described as horizontalinclined and vertical flux surfaces which can be regarded asplanar discontinuities A sample core (10 cm in diameter) wasobtained in a limestone reservoir of the Campeche SoundFigure 8 shows a limestone core with fault breccia its geom-etry corresponds to successive flux planes between barriersThese planes are connected networks in the whole mediumand generate interconductivity associated with permeabilityand effective porosity

On the other hand computerized tomography (CT) wasused to study the internalmorphology of fault breccias Lime-stone cores with fault breccias present fragments embeddedin the matrix We used information of the Integrated Ocean

Figure 8 Fault breccia core Late Cretaceous Campeche SoundMarine Region Mexico

Drilling Program [36] Tomography images of limestone rockshow the contrast between matrix and fragments that can beobserved based on the different CT numbers

Normally fault breccia fragments are denser than thematrix which indicates higher bulk density and low poros-ity and are identified with high CT numbers Moreoverfragments origin should be studied considering their agelithology CT numbers total minerals composition andphysical properties to explain their density and porosityVoids have minor CT numbers with respect to matrix andfragments

Fault and drilling-induced breccias show high contrastbetween fragments and matrix because of voids that CThas identified Fault breccias present low contrast betweenfragments and matrix with voids too These empty spacesallow the oil flow through the rock in effect fragments actas barriers and fluid movement is linked to voids betweenmatrix and fragments This can be observed in Figure 9

Observing Figures 7 8 and 9 the following observationscan be listed

(1) Embedded clasts in a fault breccia are subangular andangular creating a huge flow area

(2) Fault breccias are consequence of stresses in the lime-stone rock with planar and connected discontinuitiesthat contain embedded clasts

(3) Fracturing and faulting intensity obeys stresses inten-sity

(4) In cores the proportion of brecciated rock is greaterthan matrix

(5) In an outcrop the brecciated rock has dimensions ofmeters

(6) Fault breccias can be described with multiple con-nected planes and embedded clasts

These observations are useful for the development of ananalytical model that will follow next


L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)

Figure 9 Representative appearance of fault breccia in computerized tomography (CT) images (a) Slice of fault breccia cut perpendicular tocore axis (interval 316-C0004D-28R-2 length 4513 cm) (b) Same as (a) with CT numbers of matrix and fragment Note the small contrastin CT numbers between matrix and fragment (1162 versus 1294) [36]

Figure 10 Streamlines through an angular fragment

10 Kinematic Analytical Modeling forFault Breccias

In order to understand fluid flow in fault breccias their kine-matics should be studied According to geological evidencesand computerized tomography these breccias show angularclasts with similar lithology which act as barriers they arepoorly sorted and vary in size (1mmndash1m) A representativescheme could be as shown in Figure 10 in which clasts maybe grain-supported or floating and have a matrix or cementto be responsible for close or open discontinuities

Fractures contained in a fault breccia have a size apertureof millimeters and zone influences of fault breccias arecentimeters or metersThe unfilled spaces between clasts in abrecciated zone are preferential ways or a complex networkof discontinuities for fluid flow

Flow modeling between unfilled spaces and clasts couldbe developed using the flow theory near a blunt nose orRankine half-body This premise is based on geometricalsimilarity between the angular clasts and Rankine half-body considering aerofoil shapes particularly under flowconditions where the viscosity effects could be minimal [37]It is appropriate to use and to adapt the implications andequations of potential flow and streamlines to describe flowthrough fault breccias considering the observations previ-ously discussed regarding the physical phenomenon Thisproblem is solved in cylindrical and spherical coordinatesWhen cylindrical coordinates are used physical condition is

related to the radial geometry of clast and a radial functionis used as a potential function When spherical coordinatesare used the angular geometry of clast is considered and apotential function is represented using a cosine function todescribe this flow problem Then using Laplace equation incylindrical coordinates (119903 119909 120579)

1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)

The solution for (15) is a function as given by

Φ = 119890119888119909119891 (119903) (16)

For our physical phenomenon119891(119903) is a radial function due toclasts changing radially and 119888 is an integration constant thatdepends on the porous media (limestone) The velocities 119906

119903

and 119906119909are given as

119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)

Equation (15) is a Partial Differential Equation (PDE) andsubstituting (16) into (15) gives an Ordinary DifferentialEquation (ODE)

1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)

Equation (18) is the Bessel differential equation of order zeroand its general solution is [38]

119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)

where 1198881and 1198882are constants and 119869

119900and119884119900are theBessel func-

tion of order zero of the first and second kinds respectively Aseries development for the first kind of Bessel function 119869

119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)


The particular solution of Laplacersquos equation given by (16) is

Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)

Solution of analytical model has a constant (119888) that is associ-ated with material (limestone) and its roughness

On the other hand Laplacersquos equation can also be solvedfor a flow described in spherical coordinates (119903 120579 0) If Φ =

Φ(119903 120579) then there is a solution Φ = 119903119899119891(119911) where 119891(119911) is

function 119911 = cos 120579 and 119899 is an integer [38] Through theprevious transformation the following Laplace equation canbe expressed as follows

sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)

The velocities 119906119903and 119906

120579are given as 119906

119903= 120597Φ120597119903 and 119906

120579=

120597Φ120597120579 Deriving the solutionΦ previously statedwith respectto 119903 and substituting it in Laplacersquos equation given by (22)

(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)

Equation (23) is the Legendre differential equation which isa second-order ODE and its solution for an integer 119899 is givenby the Legendre polynomials 119875

119899(119911) Then the solution for

(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)

Legendrersquos polynomial of order 119899 (0 and 1) is

1198750(119909) = 1

1198751(119909) = 119909

(25)

Equation (23) has a term 119899(119899+1)119891(119911) which indicates a linearcombination [24] A general solution has an additional term

Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants

The differential equation given by (22) is linear thensolutions can be superposed to produce more complexsolutions

Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)

Equation (27) is a solution for Legendrersquos equation Accordingto Legendrersquos polynomial of order 119899 with 119875

119899= 1 this

potential function can be used in stable steady irrotationalflow and spherical coordinates and can represent some fluidflowproblemsThis equation describes uniformflow a sourceand a sink flow a flowdue to a doublet a flow around a spherea line-distributed source and a flow near a blunt nose

119860119899and 119861

119899play an important role in the solution because

it is possible to superpose the geological events that influence

the fault breccias In addition this solution does not dependon porous media (limestone) or experimental constantnamely it considers fluid flow only For example for uniformflow (applied to planar discontinuities whose conditions areparallel irrotational flow) the 119860

119899and 119861

119899parameters in (27)

simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)

Then the potential the stream function and the radial andangular velocities can be expressed as

Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)

For a source and sink flow (applied to discontinuity withembedded clasts whose conditions are slow and irrotationalflow)

If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)

Then the velocity potential the stream function and theradial and angular velocities can be expressed as

Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)

Similarly the last expressions can be deduced using (27) (thesolution of Legendrersquos equation) and Cauchy equations

For flow near a blunt nose or Rankine half-body whichcan be modeled with the superposition of uniform flow and


ux

u

uy

120579u

Figure 11 Flow kinematics through fault breccias using Rankinehalf-body

a source (applied to planar discontinuity with embeddedclasts) as illustrated in Figure 11

Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)

Equations (32) (33) (34) and (35) can be used to describe theflow kinematics in fault brecciasThese analytical expressionsapply to a steady irrotational and axisymmetric flow Thisproblem can be considered as an irrotational flow becauseof the slow laminar movement of a viscous fluid on a planarsurface [38] Moreover two classical methods have beenproposed and adapted to describe fluid flow through faultbreccias The first method uses a solution of the Besselequation with an experimental constant and second methodemploys a solution of the Legendre equation

11 Geological and TomographyFeatures of Chicxulub Impact Breccias andCantarell Reservoir

Impact breccias are a consequence of the impact of asteroidsmeteorites and comets to the Earth Meteorites are cosmicfragments rich in iron and nickel which have generatedcraters on the earth surface Impact melt breccias and suevitecan contain material from the melting of target rocks Inimpact breccias brecciated target rocks melt fragments andallochthonous fallback breccia can be observed

Impact breccias have been observed in cores and out-crops with embedded fragments in the ejected matrix due tothe impact A core sequence was recovered in the Yaxcopoil-1 (Yax-1) borehole located 40 km southwest of Merida andapproximately 60 km from the center of the Chicxulub struc-ture between 79465 and 894m a 100m thick impact (sue-vite) breccia and impactites overlie a 617m thick sequenceof horizontally layered shallow-water lagoonal to subtidalCretaceous limestone dolomites and anhydrites [39 40] Incontrast Grajales-Nishimura et al [41 42] Rebolledo-VieyraandUrrutia-Fucugauchi [43] Kring et al [44]Wittman et al

[45] Dressler et al [46] Tuchscherer et al [47] and Stoffleret al [48] used outcrops and core sequence from the Yax-1borehole but there are differences with respect to lithologyage and stratigraphic column Most of the authors coincideclosely with the mark of Chicxulub regarding namely itssuevitic impact breccia This impact breccia consists primar-ily of clasts from the underlying shallow-water lithologiessome crystalline rock of continental basement origin anddevitrified glass fragments and spherules [43 49 50]

Representative core samples of Yaxcopoil-1 were ana-lyzed for the estimation of hydraulic permeability ForTertiary limestones and suevites their bulk permeabilitiesrange between 10ndash14 and 10ndash19 [m2] and their porositiesare between 008 and 035 For Lowermost Suevite UnitCretaceous anhydrites and dolomites (900ndash1350m) theirpermeabilities range between 10minus15 and 10minus23 [m2] and theirporosities are between 01 and 015The previous permeabilityand porosity values do not include macroscopic events suchas faults and tectonic fractures [17]

Three decades ago the impact crater discussion wasbegun In 1975 Lopez-Ramos [51 52] and Penfield andCamargo in 1981 [53] reported 210 km diameter circularstructure with total magnetic-field data In 1991 Hildebrandet al [54] suggested that a buried 180 km diameter circularstructure (the Chicxulub impact crater) was located on theYucatan Peninsula Mexico which was formed 65 millionyears ago [46] proposing it as candidate for the KPgboundary impact site

In the offshore zone of the western margin of YucatanPlatform or Campeche Bay is located the largest oil fieldin Mexico and has produced more than 18000 millionbarrels of oil and 10000 billion of cubic feet of gas [55]Outcrops analogs petrographic analysis well logs and coresdescription indicate that Cantarell Oil Field is geneticallyrelated to the Chicxulub meteorite-impact [4] This geneticlink is found for the Cretaceous-Tertiary (KT) in Cantarelloil field that can also been seen in the Guayal (TabascoRegion) and Chiapas outcrops [41]

Impact breccia clasts are impact melt fragments with rip-up morphology that are consolidated and embedded in thematrix and can be observed in Figure 12 showing similargeometrical characteristics Figure 12(a) shows a CampecheSound core with embedded clasts and rip-up morphologyand Figure 12(b) shows an analog outcrop with rip-up mor-phology clasts too Considering Figure 12 it can be inferredthat embedded clasts act as nonflow barriers Moreoverimpact breccias contain ejected material which generatenonplanar discontinuities which can be observed in coresand in outcrop samples (Figure 12)

The Cantarell reservoir includes the Upper Cretaceous(Upper Breccia) that is associated with the Chicxulub eventwith total average porosity and permeabilities ranging from8 to 10 and 800 to 5000md respectively with averagethickness of 11 to 105 meters thinning to the south-westshowing dissolution and dolomitization and the Upper Juras-sic (Tithonian) with dolomitized limestone The field oilproduction is associated with tectonic fractures that providehigh permeability [4 8 56 57]


(a) (b)

Figure 12 Core and outcrop of impact breccia embedded clasts (a) Limestone core of the Campeche Sound with rip-up morphology clastsLate Cretaceous Campeche Sound Marine Region Cantarell Complex Mexico [58] (b) Analog limestone outcrop with rip-up morphologyclasts Guayal outcrop TAB Mexico [41]

Computerized tomography is a tool to observe thedifferences between matrix and ejected clasts and describeinternal texture of rock Figure 13 indicates that clasts arenonflow barriers in impact breccias and generate nonpla-nar discontinuities Figure 13(a) shows different textures inbreccia sequence of Yax-1 well clasts or nonflow barriersare present it indicates that there is a range of porositiesand permeabilities in limestone rocks Low porosity andpermeability are related to fine ejected material Figure 13(b)shows other impact breccias (Bosumtwi) in three dimensionsthe nonflow barriers (high-density clasts) can be observedwith rip-up geometry generating nonplanar discontinuitiesand they are embedded in matrix rock (low-density)

Observing Figures 12 and 13 the following can beinferred

(1) Embedded clasts in an impact breccia have rip-upgeometry creating a nonflow barrier in the porousmedia (matrix)

(2) Impact breccias are a consequence of the impactof asteroids meteorites and comets in the Earthgenerating nonplanar discontinuities that containembedded clasts

(3) Porosity and permeability in an impact breccia areassociated with ejected material (clasts) providing atype of primary porosity in the limestone

(4) In the core the proportion of matrix rock is greaterthan embedded clasts

(5) In the outcrop and reservoirs brecciated rock dimen-sions are related to impact size

(6) Impact breccias can be described with multipleembedded clasts (high-density) that act as nonflowbarriers into connected matrix (low-density)

The previous inferences will be used in the present paper forthe development of an analytical model

12 Kinematic Analytical Modeling forImpact Breccias

When an impact breccia is observed without the presence ofother geological events as fault breccias tectonic fracturesvugs sedimentary breccias dissolution and dolomitizationits permeability would be ultralow [17] and can have a lowto moderate porosity (1ndash8) [4] In consequence impactbreccia can act as seal and have fluid storage but its oilproduction depends on tectonic fractures fault brecciasdissolution and vugs It is very highly observed in theCantarell field

Because of its low permeability and porosity fluid flowvelocity in impact breccias is slow and can be considered asirrotational flow in a porous media with primary porosityIn addition to low permeability rip-up geometry observedin tomography and geological evidences and clasts act asbarriers for fluid flow in porous media Figure 14 illustratesfluid flow with nonflow barriers and rip-up geometry for animpact breccia

The impact breccia flow may be studied in terms ofthe streamlines taking into consideration the axial flowsymmetry To solve this problem we apply and adapt the flowthrough a sphere considered by Stokes then it is possible touse the Laplace Biharmonic equation to describe this flow[59 60]

nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)

For spherical coordinates (36) can be expressed by

Ψ (119903 120579) = 0 (37)


Carbonates Redepositedsuevite

Suevite Chocolate brownmelt breccia

Carbonates

Redepositedsuevite

Suevite

Chocolate brownmelt breccia

Sueviticbreccia

Sueviticbreccia

Green monomictbreccia


Polymict meltbreccia


(a)

1 cm

(b)

Figure 13 Samples of CT slices through the Bosumtwi and Chicx-ulub impacts (a) Breccia sequence in Yaxcopoil-1 borehole Coreimages obtained with the Core Scan System for the breccias sec-tions illustrating the different textures [61] (b) Three-dimensionalreconstruction of the clasts population of the Bosumtwi sueviteTheimage on the left shows the sample with color-coded clasts with thematrix (groundmass) rendered partly transparentThe center imageshows only the high-density clasts with the low-density clasts (andthe groundmass) rendered transparent and the image on the rightshows only the rock edges and the low-density clasts [27]

The boundary conditions for (37) are given by

119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1

21199061199032sin2120579 for 119903 997888rarr infin (38c)

Figure 14 Streamlines for the flow through in rip-up fragments inan impact breccia

ao

u

Figure 15 Streamlines for a rip-up clast Impact breccia

The boundary conditions given by (38a) and (38b) describefluid adherence on a clast with rip-upmorphology in a porousmedia Figure 15 represents this fluid adherence at a clast withsimilar rip-up morphology

Boundary condition given by (38c) describes the stream-lines before-after a breccia clast which implies a velocitychange in fluid flow because the clast is a barrier namely for119903 rarr infin the final clast velocity is obtained In accordancewith this boundary condition is a general solution for LaplaceBiharmonic equation expressed as follows

Ψ (119903 120579) = 119891 (119903) sin2120579 (39)

This solution proposes radius and angle change of clastHowever it is necessary to study the Stokes solution and testif it can be adapted to oil flow in an impact breccia Equation(39) contains 119891(119903) which is a radial function related to theclast radius Substituting (39) in (37)

[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)

Considering streamlines with trajectory angle of 90∘ then(40) can be expressed as

(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)

Equation (41) is a fourth-order differential homogeneouslinear equation and its solution is of type 119891(119903) = 119888119903119899 where


119899 can have values (minus1 1 2 3) and 119888 includes integrationconstants (119860 119861 119862119863) such that

119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)

Deriving (39) with respect to angle 120579 120597Ψ120597120579 =

2119891(119903) sin 120579 cos 120579 and substituting it in (38a)

119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)

Substituting (42) in (43)

119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)

119906120579can be expressed as

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)

Substituting (42) in (39) and deriving the resulting equation

120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)

Then substituting (46) in (45)

119906120579= minus(minus

119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)

Applying boundary conditions given by (38a) and (38b) to(44) and (47)

119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)

Now applying the boundary condition described by (38c) tovelocity 119906

120579when rarr infin

minus119906 sin 120579 = minus(minus1198601199033+119861

119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)

Considering in 119906119903(44) that 119903 rarr infin then

119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)

If 119903 rarr infin clasts should be smaller in layer thickness (twoorders of magnitude) Moreover initial fluid velocity changeswhen fluid impactswith clast (barrier) but fluid velocitymustbe different from zero to apply the mass and momentumconservation principles

Thus119863 = 0 because119863 isin R and 119906 isin R From (50)

119862 =119906

2 119863 = 0 (51)

If 119906119903= 119906120579= 0 ((38a) and (38b)) then the previously

described procedure regarding velocity 119906119903indicates a stagna-

tion pointFollowing a similar solution for119906

120579 the following equation

can be derived

0 = minus(minus119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)

Substituting119862 = 1199062 and119863 = 0 and 120579 = minus90∘ (perpendicularstreamline that describe streamline around clast) in (52)

0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)

Substituting 119862 = 1199062 and 119863 = 0 and 120579 = 0∘ (radius localizedat 120579 = 0∘) in (54)

0 = (2119860

1199033+2119861

119903+ 119906) (55)

Solving the system of (53) and (55) constants 119860 and 119861 areexpressed as follows

119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)

Then through the expressions (values) for the parameters 119860119861 119862 and119863 (44) (47) and (39) can be written as

119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)

The potential velocity is obtained using 119906120579= 1119903 (120597Φ120597120579)

and integration Then Φ can be derived as

Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)

As impact breccias clasts do not have spherical geometry andtheir rip-up morphology shows subrounded sides and otherangular sides in our analytical model we consider symmetry


and an angle between 0∘ and 90∘ Moreover we propose thatthe range of 120579 may be 0∘ lt 120579 lt 180

∘ and rate change withrespect to the angle can be expressed as

119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)

Equations (60) (61) and (62) describe the changes in stream-line velocities in impact breccias and could be used to studyclasts with a variety of angles or subrounded side In effect asstated the Stokes solution was adapted to impact breccias

13 Fourth Contradiction Fluid Flow ofthe Cantarell Reservoir Modeled withoutConsiderer Chicxulub Impact

Several authors document the influence of the Chicxulubimpact in the Campeche Sound and discussed if the Chicx-ulub impact breccias are seal production zone [4 41 42] orthe impact is a geophysical survey reference as part of an oilexploration program [16 53 58 61]

On the other hand the Cantarell field is modeled asa tectonic fractures reservoir [6ndash8 56 62] As this paperis focused on oil flow in limestone reservoirs then thereis a question why is the Cantarell reservoir modeled withtectonic fractures without considering the fluid flow throughimpact breccias Or the impact breccia oil does not flow Acontribution of this paper is related to describing fluid flowthrough an impact breccia In absence of vugs fractures andfault breccias impact breccia has moderate porosity and lowpermeability which indicates low flow velocity unique in thistype of breccia Clearly we modeled impact breccia withoutfractures or other geological events

14 Geological and Tomography Features ofSedimentary Breccias

Sedimentary breccias are a type of clastic sedimentaryrock with subangular to subrounded clasts These rocks aredeposited by transport and fast moving of a body of sedimentparticles by water andor air These breccias are observed indebris flows mud flows and mass flows such as landslidesand talus

According to type of clasts sedimentary breccias can bemonomict oligomict or polymict Clasts may be extrafor-mational or intraformational matrix-supported or clast-supported The morphology of fragments clasts has beenreworked by transport Moreover rocks with rounded clastsare known as conglomerates and they are different frombreccias

Sedimentary breccias contain clasts embedded in thematrix These breccias do not have linear or internal planar

Figure 16 Sedimentary breccia outcrop with reworked clasts LaPopa basin NL Mexico

L2

W

W = clast widthL = clast length

Figure 17 Matrix-clast interface in sedimentary breccia core LateCretaceous Cardenas Field TAB Mexico [63] Note W = clastwidth and L = clast length

structure resulting from lithification and consolidation ofrocks and they have been seen in outcrops (Figure 16)Figure 16 shows reworked clasts embedded in rock matrixwith subrounded sides which indicates a short clasts trans-port Long transport implies that clasts are rounded and canbe modeled as ellipsoids

In principle sedimentary breccias affect carbonate reser-voirs and may enhance the total porosity Fluid flow can besignificant in the matrix-clast interface due to clast beingimpermeable and acting as flow barriers Figure 17 shows asedimentary breccia corewith embedded clasts in a limestonematrix with subrounded and subangular side

Brecciation often results in enhanced porosity butwith poor interconnection of pores A specific example is


0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast

Debris flow sediment

Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)

Figure 18 Tomography (CT) image of sedimentary breccia (a) Core (b) Tomography (c) Lithological description Interval 316-C00040ndash80 cm [36]

the Cretaceous Debris Reservoir Poza Rica Field VERMexico [15]

On the other hand we used information of the Inte-grated Ocean Drilling Program that had sedimentary brecciatomography images [36] Denser fragments embedded hada contrast with the matrix indicating high bulk density andlow porosity In many cases clasts can have high density andultralow porosity

In Figure 18 tomography shows dark shading that corre-sponds to low density and white to high density regions thisfigure presents poorly sorted elongated and impermeableclasts with low porosity in addition to Debris flow sedimentsin different layers These events indicate that clasts areoligomict or polymict and extraformational that act as flowbarriers in limestone reservoirs

Observing Figures 17 and 18 the following can be in-ferred

(1) Embedded clasts in a sedimentary breccia have sub-rounded reworked and elongated geometry creatinga nonflow barrier in porous media (matrix)

(2) Clasts are poorly sorted and dispersed(3) Sedimentary breccias are consequence of Debris flow

generating nonplanar discontinuities that containembedded clasts

(4) Porosity and permeability in a sedimentary brecciaare associated with matrix embedded clast andinterface matrix-clast producing a type of primaryporosity in the limestone rock


Figure 19 Streamlines in sedimentary breccia with reworked clasts

(5) In the core matrix rock portion is greater thanembedded clasts

(6) The presence of Debris flow sediment in differentlayers indicates that there are diverse processes of sed-imentation and that rock was exposed in the surfacenamely it was an outcrop in another geological age atsubsurface conditions

(7) In outcrops and reservoirs sedimentary brecciadimensions are related to Debris flow

These observations are stated with the objective of thedevelopment of an analytical model

15 Kinematic Analytical Modeling forSedimentary Breccia

Fluid kinematics could describe fluid flow in this type ofrocks A representative scheme is shown in Figure 19 withstreamlines for sedimentary breccia clasts where reworkedclasts may be grain-supported or matrix-supported

Modeling betweenmatrix-clast interfaces could be devel-oped using flow around an elliptical body like the Rankinesolids due to reworked clast The Rankine methodology issimilar to that previously applied to fault breccias to describefluid kinematics therefore the flow around an elliptical bodyshould satisfy Laplacersquos equation [64]

Considering uniform flow the Rankine solution isobtained using the superposition of a linear source and alinear sink of equal strength combined with uniform flow(Figure 19) given by

Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)

Conceptually (63) and (64) express that a sink and a sourceare equivalent but geometry shows differences they havedifferent angles with respect to streamlines (Ψ) and are sep-arated from distance 119897 Distance between origin and sourceis 1198972 due to their symmetry This is observed in Figure 20that shows different angles and radius in a source and sinkfor Rankinersquos solid The combined flow (Ψ

119879) is represented

by the stream function From geological point of view clast

Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ

Figure 20 Source and sink angles in the Rankine body [64]

geometry indicates angular rate and oil flows around theimpermeable clast generating a nonplanar discontinuity

Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where

1205791= tanminus1 (

1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)

Considering (66) and (67) the combined flow (Ψ119879) is given

by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus tanminus1 (

1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)

Figure 21 shows two stagnation points associated with asource and sink the length of the Rankine body is (119909

119879) 119909119879=

1199091+ 1199092 and the width of the Rankine body (119908) is related to

the maximum value of 119910 on the contour of the body in the119910-axis direction

Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular

coordinates using (66) horizontal and vertical velocities aregiven by

119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y

ymaxStagnation pointStagnation point

minus1 +1

Sink SourceO

x1 x2

Figure 21 Streamlines for the Rankine solid with sink and source[64]

The total velocity is given by 119906 = radic1199061199092 + 1199061199102 and potentialvelocity is

Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)

Equation (70a) can also be expressed substituting (67) for 1205791

and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)

To determine the width of the Rankine body the maximumvalue of 119910 (119910max) in Figure 21 should be estimated at 119909 =

0 (V119910max

= 0)Geological information could provide the width and

length of clasts embedded in cores of a sedimentary brecciawhich would be assumed as length and width of the Rankinebody


Vugs are a type of pores in carbonate rocks Choquette andPray (1970) [65] presented a classification based on the porespace genesis Lucia [66] proposed a classification focusedon petrophysical properties and on distribution of pore sizeswithin the rock

Vuggy pore space is classified into touching-vug poresand separate-vug pores Touching-vug pores as tectonicfractures breccias and caverns are nonfabric-selective intheir origin Separate-vug pores are defined as pore spacelarger than the particle size and fabric-selective in their originand are interconnected only through interparticle pore spacesuch as moldic intrafossil intragrain and shelter pores [66]

According to Lucia [66] vugs compared to separate-vug pores are secondary solution pores that are not fabric-selective in their origin with irregular sizes and shapes whichcould be interconnected [65]

In absence of tectonic fractures vugs are nonfabric-selective pore spaces or discontinuities from various sizes

Figure 22 Limestone reservoir core with irregular vugs LateCretaceous Campeche Sound Marine Region Mexico

with irregular shape as a result of chemical dissolutionthat could be interconnected or separated Figure 22 shows acore with vugs created by chemical dissolution and irregularshapes Normally in a double porosity reservoir vugs interactwith the matrix

Permeability of vugular porous media depends on itsinterconnection of the pore space In this study vugs areconsidered as nonplanar discontinuities that may be sepa-rated and nonfabric-selective and interact with the matrix ofcarbonate rocks

Vuggy porosity can be produced by the interaction ofmeteoric waters with rock by grains dissolution fossils andmatrix pores by deep-burial fluids and by exposition of rocksurfaces in humid climates

Vugs geometry is irregular Moreover spherical cavitieshave been used to describe them [9 67ndash69]

In limestone outcrops vugs are interconnected only withmatrix vuggy pore space also can be regarded as intercon-nected with matrix and other vuggy systems (Figure 23)Figure 23 shows a chemical dissolution process (diagenesis)with multiple size vugs similar to Figure 22 then core andoutcrops could be used as analogs

Tomography images of an oil impregnated core obtainedfrom a vuggy limestone reservoir were taken to determinethe internal characteristics geometry and connectivity of thepore space

The obtained one hundred and thirteen images describethe geometry and irregular shapes of the vugs Figure 24shows an oil saturated core with a length of 36 cm withvisible and irregular vugs as result of a chemical diagenesisand CT slices were taken every 3mm of distance through thesample (Figure 25)

CT slices for the vuggy core of Figure 24 are presentedin Figure 25 Figure 25 shows slices with vugs (black color)matrix (yellow color) filled or recrystallized cavities (greencolor) and embedded clasts (orange color) Cavities withblack color are big pore spaces or void spaces saturated withoil When the slices are compared vugs connectivity changesare observed If we see a vug with black color in an image itdisappears in subsequent images In contrast connected vugscan be observed through different slices

Considering core axisymmetry there are permeablezones with isolated porosity and others with interconnectedporosity Isolated zones interchange fluid with matrix but


Figure 23 Zoomed photo of vugs in an outcrop Guayal outcropTAB Mexico

36 cm

Figure 24Oil impregnated core of a vuggy limestone reservoir LateCretaceous the Campeche Sound Marine Region Mexico

connected region presents matrix-vug and vugs system flowMoreover dissolution is the dominant process that enhancesconnection vugs

Figure 25 CT images of an oil impregnated vuggy limestone coreLate Cretaceous Campeche Sound core Marine Region Mexico

Observing Figures 22 23 24 and 25 the following can beinferred

(1) Limestone vugs geometry is irregular and sphericalfor outcrop as for core

(2) In the core the vugs proportion is equivalent to thematrix proportion

(3) The vugs distribution is approximately uniform(4) Vugs may be connected or isolated depending on

interface vug-matrix(5) The presence of vugs indicates chemical diagenesis

specially dissolution due to meteoric water

Considering these observations the analytical model pro-posed by Neale and Nader [9] may be used although wewill propose expressions for angular radial and potentialvelocities using Neale and Naderrsquos analytical model


u

Matrix

Vug

Figure 26 Lateral view of a composite porous medium with vugsmatrix and a fluid flow field

Matrix

Vug

u

Figure 27 Proposed solution for a composite porous media withvugs matrix and a fluid flow field

17 Kinematic Analytical Modeling for Vugs

Vugs are irregular spaces like spheroids and ellipsoids thatinteract with the matrix

To predict the flow field within a vug we considerthe mathematical solution developed by Neale and Nader[9] which was used to describe the pressure distributionwithin an isotropic and homogeneous porous medium witha spherical cavity

Considerations employed to derive the mathematicalsolution were as follows (1) uniformly vuggy medium withmonosized cavities (2) spherical cavity geometry (3) sur-rounding homogeneous and isotropic porous media (4)steady-state flow and (5) incompressible fluid

The problem and solution described by Neale and Nader[9] are represented as shown in Figures 26 and 27

To develop the analytical solution for the flow problemdescribed a composite porous medium (matrix and vugs)was considered and the creeping Navier-Stokes and theDarcy equations were used Combining these two equationsthe Laplace Biharmonic equation in spherical coordinate can

be obtained and solved with its boundary conditions thesolution is given by

Ψ (alefsym 120579) =119896119906lowast

2[119860alefsym2+ 119861alefsym4] sin2120579 0 lt alefsym lt 119883 (71a)

Ψlowast(alefsym 120579) =

119896119906lowast

2[119862alefsymminus1+ 119863alefsym

2] sin2120579 0 lt alefsym lt infin (71b)

using the normalized radial coordinate and the normalizedradius of the cavity where

119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)

Equations (71a) and (71b) with their constants (119860 119861 119862119863119883 120590 alefsym) predict the streamlines behavior in vugs andporousmedia However the authors Neale andNadar did notdescribe angular radial velocities or potential function

Considering (71a) and (71b) with their constants wedeveloped the angular radial velocities and potential func-tion which are given by

119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)

Defining potential flow as Φ = intr0119906119903119889119903 then

Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)

Equations (73) and (74) provide a description of the fluid flowin a composite porous media

18 Fifth Contradiction Liquids RetentionParadox for Vugs

Regarding vugs there is a physical phenomenon relatedto oil flow and their geometry because they are concaveand convex cavities with irregular shapes which creates oilentrapping This phenomenon has been called ldquodead zonerdquo[70] or ldquothe stagnation porosityrdquo [71] however this problem is


well-known in fluidized bed reactors and their design in thechemical engineering area [60]

During the production oil entrapping is a result oftwo fluid dynamic processes Initially oil can be saturatingthe cavities and its flow obeys a pressure gradient and abalance between viscous and inertial forces thus this is adynamic flow process When the first process concludesoil flow follows a diffusion process with slower velocitiesgenerating a second process with static characteristics andfluid entrapping The described phenomenon is related tothe cavity geometry and vuggy pore space this problem isassociated with static-dynamic liquid retention in vugs andshould be called liquids retention paradox

It is a paradox because liquid in the cavities may betrapped in stagnation or dead zones however fluid flow willalways occur in the vuggy porosity (secondary) producing achange in fluid velocity In consequence stagnation zones donot exist because there is always movement of fluid

19 Application Examples and Discussion

In this section examples are presented to illustrate theproposed kinematics models in the analysis of limestonereservoirs with different discontinuities

191 Behavior of Fluid Velocities To examine the differencesbetween the types of discontinuities we used analyticalkinematics models to calculate and compare fluid velocitiesKey data and parameter values employed are presented inTable 1Note that quantitativemodels should be implementedwith geological characterization for a better prediction

Additionally to quantify fluid velocities in this study wetook into consideration a pressure drop a discontinuity anglewith respect to streamline aperture clasts angle and sink andsource for sedimentary breccia which are included in Table 2

Table 3 shows obtained velocities and flow rates values fordiverse kinds of discontinuities The goal is to compare theseparameters

These models and their results would be applicable to oillimestone reservoirs In this example the calculated valuesof Table 3 illustrate that discontinuities related to fluid floware dissimilar and that flow velocities and volumetric flowcan be different The results suggest that discontinuities ina limestone reservoir may not yield the same level of oilproduction This implies that it is necessary to identify andverify the dominant discontinuity using static and dynamiccharacterization

Note that the calculated values of Table 3 were obtainedusing (5) (6) (12) (34) (35) (57) (58) and (69) which implythe described constants for all of the discontinuities presentedin this paper For sedimentary breccias it is necessary toconvert Cartesian coordinates to radial coordinates usingtrigonometrical functions The total velocity and volumetricflow are given by 119906 = radic1199061199092 + 1199061199102 and 119902 = 119906119860 respectivelyEquation (12) (The Cubic Law) is used for tectonic fracture

Calculated values show the differences for each type ofdiscontinuity Horizontal tectonic fracture presents equiva-lent velocities (radial and angular) because streamlines are

Table 1 Data and parameters of limestone reservoir A

Parameters Symbol ValuesInitial reservoir pressure 119901

119894 3450 times 107 PaBubble-point pressure 119901

119887 1717 times 107 PaReservoir depth 119863 2726mFormation thickness ℎ 25mPrimary porosity 120593

1 0015 (fraction)Primary permeability 119896

1 395 times 10minus17m2

Secondary porosity 1205932 015 (fraction)

Secondary permeability 1198962 158 times 10minus10m2

Oil viscosity 120583119900 000038 Pasdots

Reservoir temperature 119879 12185∘C

Oil specific gravity (156∘C) 120574119900

08156(dimensionless)

Oil specific gravity 120574119900

0745(dimensionless)

Oil specific gravity at 12185∘C was determined by 120574119900 = 120574119900(156∘C)1 + 120575(119879minus60) where 120575 = exp(00106 times ∘API minus 805) [72] Oil API gravity was 42∘API

Table 2 Additional data for the calculation of fluid velocities

Simulation properties ValuesPressure drop 2 PaDiscontinuity angle 45∘ (degrees)Clast angle 45∘ (degrees)Aperture (fracture) 0005mVug radius 002mDistance (vug-matrix) 004mClast radius 002mSink and source 1m2sInitial velocity 000639

Table 3 Calculated parameters values

Discontinuities Parameters119906 (ms) 119906

119903(ms) 119906

120579(ms) 119902 (m3s)

Fracture 00064 00045 00045 00399Fault Breccia 00066 00048 00045 00413Impact breccia 00013 00003 00012 00078Vug 00190 00046 00184 01186Sedimentary breccia 00046 00033 00033 00290The positive sign in fluid velocities represents its direction from of origin in119909-axis or 119910-axis direction

parallel and volumetric flow depends on fracture apertureFault breccia has high velocity and flow because it dependson elongated and subangular clast Also vugs have superhighvelocity and flow because they are connected cavities withoutflow barriers

In contrast impact and sedimentary breccias have lowvelocity and volumetric flow due to clast geometry (rip-upand ellipsoidal) creating low radial velocity In additionclasts are flow barriers and fluid flow is related to interac-tion matrix-clast and their permeabilities that are normally


very low because they behave as primary permeability andporosity This implies that proportion matrix is greater thanthe proportion clast

192 Carbonate Reservoir Characteristics Cardenas FieldApplication According to the proposed geological model forthe Cardenas Field which is an anticline with a fault systemcontrolled by stratigraphic traps rather than structural trapsIn accordance with the characteristics of primary porosity(15ndash17) secondary porosity (5ndash11) primary permeability(395 times 10minus17ndash329 times 10minus17m2) and secondary permeability(196 times 10minus10ndash89 times 10minus10) production layers are subdividedinto different breccias intervals in the reservoir Figure 28(a)shows correlation through longitudinal breccias intervals forthe Cardenas Field wells moreover breccias sequence witha chemical diagenesis (dolomites and vugs) and limestonelayers were a criterion for the selection of wells It is shown inthe cross section (Figure 28(a)) Oil production in the Carde-nas reservoir comes from lateral interconnected sedimentarybreccias and vugs [63]

Interconnected vugs by microfractures provide high per-meability and porosity On the other hand sedimentarybreccias are related to salt tectonics and generate permeabilityand porosity which are low

Application of the flowkinematicmodels (vugs sedimen-tary breccia models) to the Cardenas Field may be used as adiagnosis tool for the fluid velocities prediction and in thediscontinuity characterization In this case there are wellswith a similar pressure behavior (Figure 28(b)) that producefrom breccia intervals with vugs and microfractures

In Figures 28(a) and 28(b) we observe a cross section 1-11015840 with eight wells and their similar pressure history InFigure 28(b) 3D seismic section shows wells layers correla-tion and confirms their lateral continuity Also the sectioncorrelation shows clearly a connected thickness of DebrisflowAs an application we have chosen threewells Cardenas-109 Cardenas-129 and Cardenas-308 with geologic het-erogeneities related to sedimentary breccias and vugs Thegoal is to compare fluid velocities and characteristics flowin sedimentary breccias and vugs and validate them withproduction data Wells data and parameters are given inTables 4 5 and 6

Figure 29 shows wells Cardenas-109 Cardenas-129 andCardenas 308 In accordance with this figure well Cardenas-109 has a high oil cumulative production (gt20MMBls) due togeological events juxtaposition of sedimentary breccias andvugs Vugular porosity was created by dissolution processesassociated with pressure and temperature drop and circula-tion of high corrosive hydrothermal fluids and its brecciasdeposits were exposed to subaerial erosion after they affectedby dissolution and dolomitization In addition oil cumulativeproduction for well Cardenas-129 is associated with vugularporosity although its described production (12ndash20MMBls)in Figure 29 is smaller than for Cardenas-109Well Cardenas-308 geological description is related to sedimentary brecciaintervals Its production (6ndash12MMBls) is low with respect tothe other two wells (Figure 29) but it can be considered that

Table 4 Data and parameters for the Cardenas Field wellCardenas-109

Parameter Symbol ValueInitial reservoir pressure 119901

119894 6154 times 106 PaPressure drop Δ119901 20 PaDepth 119863 3969mFormation thickness ℎ 270mPrimary porosity 120593






Reservoir temperature 119879 124∘CClast radius 119903 008mVug radius 119877 003mSink and source 119876 1m2s


0720(dimensionless)

Table 5Data and parameters for theCardenas Field well Cardenas-129




1 329 times 10minus17 m2


Secondary permeability 1198962 92 times 10minus10 m2


Reservoir temperature 119879 1245∘CClast radius 119903 008mVug radius R 001mSink and source 119876 1m2s


0720(dimensionless)

there is a high oil storage in the breccia intervals namely oilstorage is localized in its primary porosity

According to Figures 28(a) 28(b) and 29 diverse vol-umetric flow and velocities should be calculated becausegeological events for the Cardenas Field are different andjuxtaposed Juxtaposed geological events imply a high volu-metric flow and fluid velocity

We applied the kinematic equations as a semiquantitativediagnosis tool to determinate fluid velocities andfluid flow forthe three studywells Used equations for sedimentary brecciavugs and superposed flow (sedimentary breccia and vugs)are (57) (58) and (69) with their geometrical parametersThe fluid velocities can help in the interpretation of thetype of geological discontinuity moreover it is necessary


C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129

Closed producer interval

Maximum flooding surface

Producer interval

Unconformity

Breccias intervals

KiC-4KiC-3KiC-2KiC-1KiB-8KiB-7KiB-6KiB-5

KiB-4KiB-3KiB-2KiB-1KiA-4KiA-3KiA-2KiA-1

Section 1-1998400

Closed producing wellProducing in lower cretaceousProducing in breccias of cycle KiC

(i) Dolomudstone

(ii) Dolomites

(iii) Argillaceous dolomites

(iv) Dark dolomites (very argillaceous)

(v) Carbonated dolomites

Dolomites

Limestones

(i) Limestones

(ii) Argillaceous limestones

(iii) Argillaceous mudstones

(iv) Argillaceous dolomitic limestones

(v) Dark dolomitic limestones (very argillaceous)

Breccias sequence of cycle KiC

(Interval KiC1 to KiC4)

C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued


Pressure history10000

8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)

Figure 28 (a) Section 1-11015840 producing levels correlation through breccias intervals longitudinal to the northeastern reservoir Matchingpressure curve declination among wells is shown (b) 3D seismic section and matching pressure curve among wells [63]

Table 6Data andparameters for theCardenas Fieldwell Cardenas-308








Reservoir temperature 119879 1241∘CClast radius 119903 008mVugs radius 119877 0001mSink and source 119876 1m2s


0720(dimensionless)

to considerer static characterization for estimate values ofvelocities and rates with reduced uncertainty

Kinematics equations validation is developed consideringa low fluid velocity in the sedimentary breccia of wellCardenas-308 because clasts are barriers during fluid flowbut porosity and permeability of the sedimentary brecciamay

5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040

gt20MMBls12ndash20MMBls

6ndash12MMBls0ndash6 MMBls

Figure 29Oil cumulative production inwells of theCardenas Fieldwells C-109 C-129 and C-308 [63]

store oilThis indicates that path for fluid flowwill be slow andtortuous and then its velocity and flow are low This premisewas corroborated in Table 7

Well Cardenas-129 is studied considering a high oilvelocity because vugs are cavities that do not have flowbarrierand they are normally connected with limestone matrix


Table 7 Calculated velocities and rates values for the Cardenas Field wells C-109 C-129 and C-308

Discontinuities Wells Velocities and volumetric flows119906 (ms) 119906

119903(ms) 119906

120579(ms) 119902 (m3s)

Breccia + vugs C-109 00350 00129 00314 25505Vugs C-129 00146 00035 00142 21311Sedimentary breccia C-308 00096 00068 00068 8269

The porosity in this reservoir layer is a preferential way forfluid flow When there are connected vugs with oil storage inthe rock production conditions are excellent due to oil freepath In contrast well Cardenas-129 oil production should begreater thanwell Cardenas-308 oil productionThis claimwasdemonstrated in Table 7

Well Cardenas-109 presents complex geological eventsjuxtaposition Sedimentary breccias store fluid and vugs storeand transport oil through porous media due to their inter-connection in this case porosity (storage) and secondarypermeability (transport) Then the already stated eventsjuxtaposition is applied to the geological events superposi-tion which is a characteristic of analytical models developedin this paper because our equations are lineal and theyare solutions of the Laplace equation In consequence themaximum oil production in this well must be obtained andthis is demonstrated in Table 7

Table 7 shows the obtained results with input parametersof wells Cardenas-109 Cardenas-129 and Cardenas-308Chosen wells for models validation have diverse produc-tion in consequence fluid velocity and flow volumetric aredifferent and they are related to geological event as vugssedimentary breccia and their juxtaposition

The wells production is associated with phenomenologyof complex limestone reservoirThe key point here is that sed-imentary breccias contain embedded clasts that are barriersfor fluid flow nevertheless they can store oil The degree ofcomplexity of clasts interactions with fluid depends on clastdiameters and their spacing In the case of vugs it depends ontheir interconnection When different geological events arejuxtaposed and interconnected as in well Cardenas-109 oilproduction is high

The Cardenas Field has been chosen because we knowunderstand and have geological control of reservoir whichwas developed in a doctoral thesis Data for the field appli-cation previously discussed was mainly borrowed from thePhD dissertation of one of the authors of the present paper[63]

20 Conclusions

This paper has presented a discussion on the effects of planarand nonplanar discontinuities in fluid flow of carbonatereservoirsTheproposedmathematicalmodels have provideda simple method to identify the flow characteristics offault breccias vugs tectonic fractures impact breccias andsedimentary breccias

From the results of the study the conclusions are as fol-lows

(1) It has been demonstrated through the analyticalmodels of this paper tomography and observationsof outcrops and cores that planar and nonplanardiscontinuities in limestone reservoir show distinc-tive representative flow characteristics Fault brecciastectonics fractures sedimentary breccias vugs andimpact breccias have flow and geological patterns thataffect oil production and generate differences in staticand dynamic characteristics that affect flow behavior

(2) It was demonstrative that discontinuities (vugs faultbreccia and tectonic fractures) are superhigh pro-duction ways and others (impact and sedimentarybreccias) are storage zone In effect each discontinu-ity is different and cannot be called or analyzed astectonic fractures unknowing geological genesis andflow patterns

(3) It has been shown that the unknowing of the origin ofdiscontinuities causes confusions and contradictionsfor static-dynamic characterization simulation andexploitation strategy of limestone reservoirs This isone of the most important conclusions of this study

(4) Fluid velocity and volumetric flow for vugs (nonpla-nar discontinuity) are the greatest obtained valuesbetween all of discontinuities because they are cavitiesthat do not have barrier flow In consequence alimestone reservoir well with connected vugs willhave a high oil production

(5) In the absence of fault breccias vugs sedimentarybreccias and tectonic fractures impact breccias inlimestone reservoirs present low fluid velocity andvolumetric flow because they have flow barriers(ejected clasts) that act as a type of primary porositydepending on their values of porosity and low perme-ability In effect these impact breccias would not havehigh oil production In this case impact brecciasmustbe superposed with an additional geological event fora high volumetric flow

(6) Oil production of limestone reservoirs with juxtapo-sition of geological events (breccias fractures andvugs) is high obeying a dominant geological eventin fluid production (vugs fault breccias and tectonicfractures) and other dominant geological events influid storage (sedimentary breccias and impact brec-cia)


Symbols

119909 119910 119911 Reference axis in Cartesian coordinates(119903 120579) Radial and angular coordinates119906 Fluid velocity in direction 119909 [ms]V Fluid velocity in direction 119910 [ms]119908 Fluid velocity in direction 119911 [ms]ℎ Vertical distance [m]120583 Liquid viscosity [Pasdots]119905 Time [s]120588 Density [kgm3]120573 Inclination angle [grades]119886 Fracture aperture [m]119886119900 Impact clast radius [m]

119901 Pressure [Pa]120574 Fluid specific gravity [dimensionless]119902 Volumetric flow [m3s]119860 Area [m2]119880 Terminal velocity of upper plate [ms]119880infin Horizontal flow of uniform velocity [ms]

119906 Mean flow velocity [ms]119906max Maximum fluid velocity [ms]119906119903 Radial velocity [ms]

119906120579 Angular velocity [ms]

119876 Linear sink or source [m2s]119909 Horizontal distance [m]Φ Velocity potential [m2s]Ψ Stream function [m2s]119903 Clast radius [m]120579 Clast angle [degrees]1205791 Source angle [degrees]

1205792 Source angle [degrees]

119896 Permeability of porous medium [m2]120590 Slip factor120593 Porosity of porous medium [fraction]119877 Radius of spherical cavity [m]alefsym Normalized radial coordinate119883 Normalized radius of spherical cavitylowast Average pertaining to a porous medium

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge with gratitude thesupport and the collaboration by Norma Garcia The authorsthank Juan Jose Valencia Islas Erick Luna and ArmandoPineda for sharing their recycled outcrops cores imagesphotos and comments Also they appreciate Jerry Luciarsquoscomments

References

[1] Z Wenzhi H Suyun L Wei W Tongshan and L YongxinldquoPetroleumgeological features and exploration prospect of deep

marine carbonate rocks in China onshore a further discussionrdquoNatural Gas Industry vol 34 no 4 pp 1ndash9 2014

[2] EManriqueMGurfinkel andVMuci ldquoEnhanced oil recoveryfield experiences in carbonate reservoirs in the United Statesrdquoin Proceedings of the 25th Annual Workshop amp SymposiumCollaborative Project on Enhanced Oil Recovery InternationalEnergy Agency Stavanger Norway September 2004

[3] J M Grajales D J Moran P Padilla et al ldquoThe Lomas TristesBreccia a KT impact-related breccia from southern MexicordquoGeological Society of America Abstracts with Programs vol 28p A-183 1996

[4] G Murillo-Muneton J M Grajales-Nishimura E Cedillo-Pardo J Garcıa-Hernandez and S Garcıa-Hernandez ldquoStrati-graphic architecture and sedimentology of the main oil-producing stratigraphic interval at the Cantarell Oil Field theKT boundary sedimentary successionrdquo in Proceedings of theSPE International Petroleum Conference and Exhibition PaperSPE 74431 pp 643ndash649 Villahermosa Mexico February 2002

[5] G Levresse J Bourdet J Tritlla J Pironon and A C ChavezldquoEvolucion de los Fluidos Acuosos e Hidrocarburos en unCampo Petrolero Afectado por Diapiros Salinosrdquo in Plays yYacimientos de Aceite y Gas en Rocas Carbonatadas pp 15ndash17AMGP Ciudad del Carmen Mexico 2006

[6] S Rivas-Gomez J Cruz-Hernandez J A Gonzalez-Guevara etal ldquoBlock size and fracture permeability in naturally fracturedreservoirsrdquo in Proceedings of the 10th International PetroleumExhibition and Conference Paper SPE 78502 Abu Dhabi UAEOctober 2002

[7] E Manceau E Delamaide J C Sabathier et al ldquoImplementingconvection in a reservoir simulator a key feature in adequatelymodeling the exploitation of the cantarell complexrdquo SPE Reser-voir Evaluation amp Engineering vol 4 no 2 pp 128ndash134 2001SPE 71303-PA

[8] L Cruz J Sheridan E Aguirre and E Celis ldquoRelative con-tribution to fluid flow from natural fractures in the cantarellfield Mexicordquo in Proceedings of the Latin American andCaribbean Petroleum Engineering Conference Paper SPE-122182-MS Cartagena de Indias Colo USA May-June 2009

[9] G H Neale and W K Nader ldquoThe permeability of a uniformlyvuggy porousrdquo SPE Journal vol 13 no 2 pp 69ndash74 1974

[10] J W Koenraad and M Bakker ldquoFracture and vuggy porosityrdquoin Proceedings of the 56th Annual Fall Technical Conference andExhibition and Conference Paper SPE 10332 San Antonio TexUSA October 1981

[11] Y-S Wu Y Di Z Kang and P Fakcharoenphol ldquoA multiple-continuum model for simulating single-phase and multi-phase flow in naturally fractured vuggy reservoirsrdquo Journal ofPetroleum Science and Engineering vol 78 no 1 pp 13ndash22 2011

[12] C McKeown R S Haszeldine and G D Couples ldquoMath-ematical modelling of groundwater flow at Sellafield UKrdquoEngineering Geology vol 52 no 3-4 pp 231ndash250 1999

[13] A Gudmundsson Rock Fractures in Geological Processes chap-ters 15 16 Cambridge University Press Cambridge UK 2011

[14] R M Iverson ldquoThe physics of debris flowsrdquo Reviews of Geo-physics vol 35 no 3 pp 245ndash296 1997

[15] P Enos ldquoCretaceous debris reservoirs poza rica field VeracruzMexicordquo in Carbonate Petroleum Reservoirs P O Roehl and PW Carbonate Eds Casebooks in Earth Sciences chapter 28pp 455ndash469 Springer New York NY USA 1985

[16] J Urrutia-Fucugauchi L Perez-Cruz and A Camargo-Zanoguera ldquoOil exploration in the SouthernGulf ofMexico and


theChicxulub impactrdquoGeologyToday vol 29 no 5 pp 182ndash1892013

[17] S I Mayr H Burkhardt Y Popov and A Wittmann ldquoEsti-mation of hydraulic permeability considering the micro mor-phology of rocks of the borehole YAXCOPOIL-1 (Impact craterChicxulub Mexico)rdquo International Journal of Earth Sciencesvol 97 no 2 pp 385ndash399 2008

[18] D W Stearns Fractured Reservoirs Schools Notes AAPG GreatFalls Mont USA 1992

[19] R A Nelson Geologic Analysis of Naturally Fractured Reser-voirs Gulf Publishing Company Houston Tex USA 2ndedition 2001

[20] H Fossen Structural Geology chapter 7 Cambridge UniversityPress New York NY USA 1st edition 2010

[21] A Alhuthali S Lyngra D Widjaja U F Al-Otaibi and LGodail ldquoA holistic approach to detect and characterize fracturesin a mature middle eastern oil fieldrdquo Saudi Aramco Journal ofTechnology 2011

[22] J Lee J B Rollins and J P Spivey Pressure Transient Testingvol 9 chapter 1 SPE Richardson Tex USA 2003

[23] J Voelker A reservoir characterization of Arab-D Super-K asa discrete fracture network flow system Ghawar Field SaudiArabia [PhD dissertation] Stanford University Stanford CalifUSA 2004

[24] G A McMechan G C Gaynor and R B Szerbiak ldquoUse ofground-penetrating radar for 3-D sedimentological characteri-zation of clastic reservoir analogsrdquoGeophysics vol 62 no 3 pp786ndash796 1997

[25] J K Pringle J A Howell D Hodgetts A R Westerman andDMHodgson ldquoVirtual outcropmodels of petroleum reservoiranalogues a review of the current state-of-the-artrdquo First Breakvol 24 no 3 pp 33ndash42 2006

[26] D W Stearns and M Friedman ldquoReservoirs in fracturedrock geologic exploration methodsrdquo in M 16 Stratigraphic Oiland Gas FieldsmdashClassification Exploration Methods and CaseHistories pp 82ndash106 AAPG Special Volumes 1972

[27] F Mees R Swennen M van Geet and P JacobsApplications ofX-ray Computed Tomography in the GeosciencesTheGeologicalSociety London UK 1st edition 2003

[28] W Baker ldquoFlow in fissured formationrdquo in Proceedings of the 5thWorld Petroleum Congress Sec IIE pp 379ndash393 1955

[29] M Potter D Wiggert and B Ramadan Mechanics of FluidsCengage Learning Boston Mass USA 4th edition 2012

[30] P AWitherspoon J S YWang K Iwai and J E Gale ldquoValidityof cubic law for fluid flow in a deformable rock fracturerdquoWaterResources Research vol 16 no 6 pp 1016ndash1024 1980

[31] RWZimmerman and I Yeo ldquoFluid flow in rock fractures fromthe navier-stokes equations to the cubic lawrdquo in Dynamics ofFluids in Fractured Rock B Faybishenko P A Witherspoonand S M Benson Eds American Geophysical Union Wash-ington DC USA 2000

[32] I Currie Fundamental Mechanics of Fluids Marcel DekkerNew York NY USA 3rd edition 2003

[33] I I Bogdanov V V Mourzenko J-F Thovert and P M AdlerldquoPressure drawdown well tests in fractured porous mediardquoWater Resources Research vol 39 no 1 2003

[34] M Muskat The Flow of Homogeneous Fluids through PorousMedia JW Edward Ann Arbor Mich USA 1st edition 1946

[35] N H Woodcock and K Mort ldquoClassification of fault brecciasand related fault rocks Rapid communicationrdquo GeologicalMagazine vol 145 no 3 pp 435ndash440 2008

[36] M Kinoshita H Tobin J Ashi et al Expedition 316 Site C0004Expedition 316 Scientists Proceedings of the Integrated OceanDrilling Program vol 314-315-316 Integrated Ocean DrillingProgram Management International Washington DC USA2009

[37] T E Faber Fluid Dynamics for Physicists Cambridge UniversityPress Cambridge UK 1st edition 1995

[38] E Levi Mecanica de los Fluidos Introduccion Teorica a laHidraulica Moderna Universidad Autonoma de Mexico Mex-ico City Mexico 1st edition 1965

[39] G Keller T Adatte W Stinnesbeck et al ldquoChixculub impactpredates the K-T boundary mass extinctionrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 101 no 11 pp 3753ndash3758 2004

[40] G Keller T Adatte W Stinnesbeck et al ldquoMore evidencethat the Chicxulub impact predates the KT mass extinctionrdquoMeteoritics amp Planetary Science vol 39 no 7 pp 1127ndash11442004

[41] J M Grajales Nishimura E Cedillo-Pardo C Rosales-Domınguez et al ldquoChicxulub impact the origin of reservoirand seal facies in the southeastern Mexico oil fieldsrdquo Geologyvol 28 no 4 pp 307ndash310 2000

[42] J M Grajales-Nishimura G Murillo-Muneton C Rosales-Domınguez et al ldquoTheCretaceous-Paleogene boundary Chicx-ulub impact its effect on carbonate sedimentation on thewestern margin of the Yucatan platform and nearby areasrdquoAAPG Memoir no 90 pp 315ndash335 2009

[43] M Rebolledo-Vieyra and J Urrutia-Fucugauchi ldquoMagne-tostratigraphy of the impact breccias and post-impact car-bonates from borehole Yaxcopoil-1 Chicxulub impact craterYucatan Mexicordquo Meteoritics amp Planetary Science vol 39 no6 pp 821ndash829 2004

[44] D A Kring F Horz L Zurcher and J Urrutia ldquoImpactlithologies and their emplacement in the Chicxulub impactcrater initial results from the Chicxulub Scientific DrillingProject Yaxcopoil Mexicordquo Meteoritics amp Planetary Sciencevol 39 no 6 pp 879ndash897 2004

[45] A Wittman T Kenkmann R T Schmitt L Hecht and DStoffler ldquoImpact-related dike breccia lithologies in the ICDPdrill core Yaxcopoil-1 Chicxulub impact structure MexicordquoMeteoritics amp Planetary Science vol 39 no 6 pp 931ndash954 2004

[46] B O Dressler V L Sharpton J Morgan et al ldquoInvestigating a65-Ma-old smoking gun deep drilling of the Chicxulub impactstructurerdquo Eos Transactions AGU vol 84 pp 125ndash131 2003

[47] MG TuchschererMU Reimold CKoeberl andR LGibsonldquoMajor and trace element characteristics of impactites from theYaxcopoil-1 borehole Chicxulub structure MexicordquoMeteoriticsamp Planetary Science vol 39 no 6 pp 955ndash978 2004

[48] D Stoffler N A Artemieva B A Ivanov et al ldquoOrigin andemplacement of the impact formations at Chicxulub Mexicoas revealed by the ICDP deep drilling at Yaxcopoil-1 and bynumerical modelingrdquo Meteoritics amp Planetary Science vol 39no 7 pp 1035ndash1067 2004

[49] W Stinnesbeck G Keller T Adatte M Harting U Kramarand D Stuben ldquoYucatan subsurface stratigraphy based on theYaxcopoil-1 drill holerdquo in Proceedings of the EGSAGUEUGJoint Assembly Abstract 10868 Geophysical ResearchAbstracts 5 Nice France April 2003

[50] W Stinnesbeck G Keller T Adatte et al ldquoYaxcopoil-1 and theChicxulub impactrdquo International Journal of Earth Sciences (GeolRundsch) vol 93 no 6 pp 1042ndash1065 2004


[51] E Lopez-Ramos ldquoGeological summary of the Yucatan Penin-sulardquo in The Ocean Basins and Margins Volume 3 The Gulf ofMexico and the Caribbean A E M Nairn and F G Stehli Edspp 257ndash282 Plenum Press New York NY USA 1975

[52] E Lopez-Ramos Geologıa de Mexico Universidad NacionalAutonoma de Mexico Mexico City Mexico 3rd edition 1983

[53] G T Penfield and Z Camargo ldquoDefinition of a major igneouszone in the central Yucatan platform with aeromagnetics andgravityrdquo in Technical Program Abstracts and Biographies (Soci-ety of Exploration Geophysicists) p 37 Society of ExplorationGeophysicists Los Angeles Calif USA 1981

[54] A R Hildebrand G T Penfield D A Kring et al ldquoChicxulubCrater a possible CretaceousTertiary boundary impact crateron the Yucatan Peninsula Mexicordquo Geology vol 19 no 9 pp867ndash871 1991

[55] Pemex Exploracion y Produccion Las Reservas de Hidrocarburosen Mexico Mexico DF Mexico 2014

[56] A Cervantes and L Montes ldquoThe stratigraphic-sedimentologymodel of upper cretaceous to oil exploration field lsquoCampecheOrientersquordquo in Proceedings of the SPE Latin American andCaribbean Petroleum Engineering Conference Paper SPE169461-MS Maracaibo Venezuela May 2014

[57] R Barton K Bird J G Hernandez et al ldquoHigh-impactreservoirsrdquo Oilfield Review vol 21 no 4 pp 14ndash29 2009

[58] E Ortuno ldquoCaracterısticas de la brecha hıbrida (esteril BP0 otransicional) y su potencial como roca yacimiento en la sondade campecherdquo in Congreso Mexicano del Petroleo Ciudad deMexico Mexico September 2012

[59] Z U A Warsi Fluid Dynamics Theoretical and ComputationalApproaches CRC Press New York NY USA 2nd edition 1999

[60] R B Bird W E Stewart and E N Lightfoot Transport Phe-nomena John Wiley amp Sons New York NY USA 2nd edition2002

[61] J Urrutia-Fucugauchi A Camargo-Zanoguera L Perez-Cruzand G Perez-Cruz ldquoThe chicxulub multi-ring impact craterYucatan carbonate platform Gulf of MexicordquoGeofısica Interna-cional vol 50 no 1 pp 99ndash127 2011

[62] F Shen A Pino J Hernandez A V Bustos and J R Aven-dano ldquoCharacterization and modeling study of the carbonate-fractured reservoir in the cantarell field Mexicordquo in Proceedingsof the SPE Annual Technical Conference and Exhibition PaperSPE 115907 Denver Colo USA September 2008

[63] P Villasenor-Rojas Structural evolution and sedimentologicaland diagenetic controls in the lower cretaceous reservoirs of theCardenas Field Mexico [PhD dissertation] Ecole DoctoraleSciences et Ingenierie De lrsquoUniversite de Cergy-Pontoise 2003

[64] J F Douglas J M Gasiorek J A Swaffield et al FluidMechanics Pearson Prentice Hall New York NY USA 5thedition 2005

[65] P W Choquette and L C Pray ldquoGeologic Nomenclature andclassification of porosity in sedimentary carbonatesrdquo AmericanAssociation of Petroleum Geologists Bulletin vol 54 no 2 pp207ndash250 1970

[66] F J Lucia Carbonate Reservoir Characterization an IntegratedApproach Springer Berlin Germany 2nd edition 2007

[67] A Moctezuma Deplacements immiscibles dans des carbonatesvacuolaires experimentations et modelisation [These de Doc-torat] Institut du Physique du Globe de Paris Paris France2003

[68] A de Swaan O ldquoAnalytical solutions for determining natu-rally fractured reservoir properties by well testingrdquo Society ofPetroleum Engineers Journal vol 16 no 3 pp 117ndash122 1976

[69] E R Rangel-German and A R Kovscek ldquoMatrix-fractureshape factors and multiphase-flow properties of fracturedporous mediardquo in Proceedings of the SPE Latin American andCaribbean Petroleum Engineering Conference SPE 95105-MSRio de Janeiro Brazil June 2005

[70] C Perez-Rosales ldquoDetermination of geometrical character-isitics of porous mediardquo Journal of Petroleum Technology no 8pp 413ndash416 1969

[71] R Martinez-Angeles L Hernandez-Escobedo and C Perez-Rosales ldquo3D quantification of vugs and fractures networks inlimestone coresrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition pp 3833ndash3848 San Antonio TexasUSA October 2002

[72] V L StreeterHandbook of Fluid Dynamics McGraw-Hill BookNew York NY USA 1st edition 1961

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



In 1972 Neale andNader [9] described the flow dynamicsin a vuggy porous medium using the creeping Navier-Stokesand the Darcy equations in the surrounding homogeneousand isotropic medium Koenraad and Bakker in 1981 [10]proposed a theoretical study about fracturevug collapsebreccias and brecciated karst based on geological infor-mation Wu et al in 2011 [11] proposed numerical modelsmultiphase and single-phase flow for vuggy reservoirs usingbalance mass equations Darcy and Forchheimer equationsand Hagen Poiseuille tube flow In 1999 McKeown et al[12] developed a numerical groundwater flow model forSellafield in which they modeled hydrogeological Brockramfault breccia using Darcyrsquos equation applied to a limestonewith porosity between 1 and 10 percent obtaining a rangeof hydraulic conductivity Gudmundsson [13] studied fluidflow behavior using Darcy equation applied to breccias andfaults

Sedimentary breccias are related to debris flow Thephysics of debris flow published by Iverson in 1997 [14]analyzed flows of dry granular solids and solid-fluidmixturesand described the grains behavior throughmomentummassand energy balances Enos [15] described debris reservoirand sedimentary breccias in Tamabra Limestone of thePoza Rica field in Mexico with field ranges of porosityof 37ndash97 and permeability of 001ndash700md with depthranging between 1980 and 2700m and cumulative pro-duction to July 1983 of 198 billion oil barrels There aremany impact breccias but associated with oil reservoir themost known is KT limit in Cantarell located in CampecheBay Yucatan Platform in Mexico related to the Chicxulubimpact [16] From the point of view of fluid flow Mayret al [17] in 2008 estimated rocks hydraulic permeabilityassociated with the impact crater Chicxulub using NuclearMagnetic Resonance Kozeny-Carman equation and thefractal PaRis-model obtaining low hydraulic permeabil-ity

The purpose of this paper is to demonstrate how qualita-tive and quantitative differences between tectonic fracturesfault breccias sedimentary breccias vugs and impact brec-cias affect the exploitation strategy and advanced develop-ment of carbonate reservoirs Our hypothesis considers thatall the discontinuities in carbonate reservoirs are distinctand cannot be called or analyzed as tectonic fractures with-out knowing their genesis and discontinuity impact in thereservoir or only focusing in fluid flow To demonstrate thishypothesis we understood the rocks nature and used sub-surface geological analogs (outcrops) cores computerizedtomography and analytical modeling making use of fluidflow kinematics

2 First Contradiction Tectonic Fracture orNonplanar Discontinuity

Unfortunately whatever the discontinuity (planar or non-planar) it is considered as a natural fracture Usuallyoil reservoirs with discontinuities are ldquonaturally fracturedrdquo

Figure 1 Limestone core with planar tectonic fractures EarlyCretaceous Samaria-Luna Region TAB Mexico

causing confusion in reservoirs simulation Our discussionbegins with the following the word fracture derived fromthe Latin fractus means ldquobrokenrdquo and its definitions are asfollows

ldquoA natural fracture is a macroscopic planar discontinuitywhich results from stresses that exceed the rupture strengthof the rockrdquo [18]

ldquoA reservoir fracture is a naturally occurringmacroscopicplanar discontinuity in rock due to deformation or physicaldiagenesisrdquo [19]

ldquoA fracture is any planar or subplanar discontinuity that isvery narrow in one dimension compared to the other two andforms as a result of external (eg tectonic) or internal (thermalor residual) stressrdquo [20]

These definitions converge in that fractures result fromstresses and deformation on the rock mechanical diagene-sis that generates volume reduction by compaction duringburial Whether natural tectonic fractures result as the effectof stresses and mechanical compaction why are sedimentarybreccia impact breccia fault breccia and vug considerednatural fractures regardless of the genesis of these geologicalevents This is the first contradiction

The discontinuities observed in limestone reservoirs areplanar or nonplanarThe observation of the core of limestonereservoirs in Figure 1 shows a planar natural fracture withinclination it is a preferential path for oil flow its aperture hasbeen developed on the plane of weakness (with low cohesion)as consequence of stresses on the rock Although the corewas badly preserved its natural fracture presents an apertureincrease but it is natural

However many reservoirs normally contain nonplanardiscontinuities caused by diagenesis The dissolution pro-cesses (chemical diagenesis) as result of expelled watergenerate vugs caverns and clasts dissolution This type ofdiscontinuities affects the fluid flow and creates an intercon-nected system through poresTheFigure 2(a) presents Clado-coropsis that could act as barrier because these fragmentshave not been fragmented or dissolved Also Figure 2(b)shows a core with fragmented Cladocoropsis grainstonedue to chemical diagenesis creating a secundary porosityand permeability connecting pores system (primary andsecondary porosity)

Reservoir engineers have assumed discontinuities as atype of pores regardless of their origin or genesis and tectonic


5 cm(a) (b)
































1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53







x

z

Flowline

Aperture

(a)

H

120573

u

x

U

y

a

(b)



120588(120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 119908

120597119906

120597119911)

= minus120597119901

120597119909+ 120583(

1205972119906

1205971199092+1205972119906

1205971199102+1205972119906

1205971199112) + 120574 sen120573

(1)


0 = minus120597119901

120597119909+ 120583(

1205972119906

1205971199102) + 120574 sen120573 (2)


1205972119906

1205971199102=1

120583(120597119901

120597119909+ 120574119867) (3)





119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909+119880

119886119910 (4)


119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909 (5)



119902 = int

119886

0

1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909119889119910 = minus

1198863

12120583

120597 (119901 + 120574119867)

120597119909

(6)


119906 = minus1198862

12120583

120597 (119901 + 120574119867)

120597119909 (7)



119906max = minus1198862

8120583

120597 (119901 + 120574119867)

120597119909to 119910 = 119886

2 (8)






y

x

(a) (b) (c)






119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)


Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)


2Ψ = 0 For this


nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)


nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)




























L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5 cm(a) (b)
































1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53







x

z

Flowline

Aperture

(a)

H

120573

u

x

U

y

a

(b)



120588(120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 119908

120597119906

120597119911)

= minus120597119901

120597119909+ 120583(

1205972119906

1205971199092+1205972119906

1205971199102+1205972119906

1205971199112) + 120574 sen120573

(1)


0 = minus120597119901

120597119909+ 120583(

1205972119906

1205971199102) + 120574 sen120573 (2)


1205972119906

1205971199102=1

120583(120597119901

120597119909+ 120574119867) (3)





119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909+119880

119886119910 (4)


119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909 (5)



119902 = int

119886

0

1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909119889119910 = minus

1198863

12120583

120597 (119901 + 120574119867)

120597119909

(6)


119906 = minus1198862

12120583

120597 (119901 + 120574119867)

120597119909 (7)



119906max = minus1198862

8120583

120597 (119901 + 120574119867)

120597119909to 119910 = 119886

2 (8)






y

x

(a) (b) (c)






119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)


Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)


2Ψ = 0 For this


nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)


nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)




























L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53







x

z

Flowline

Aperture

(a)

H

120573

u

x

U

y

a

(b)



120588(120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 119908

120597119906

120597119911)

= minus120597119901

120597119909+ 120583(

1205972119906

1205971199092+1205972119906

1205971199102+1205972119906

1205971199112) + 120574 sen120573

(1)


0 = minus120597119901

120597119909+ 120583(

1205972119906

1205971199102) + 120574 sen120573 (2)


1205972119906

1205971199102=1

120583(120597119901

120597119909+ 120574119867) (3)





119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909+119880

119886119910 (4)


119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909 (5)



119902 = int

119886

0

1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909119889119910 = minus

1198863

12120583

120597 (119901 + 120574119867)

120597119909

(6)


119906 = minus1198862

12120583

120597 (119901 + 120574119867)

120597119909 (7)



119906max = minus1198862

8120583

120597 (119901 + 120574119867)

120597119909to 119910 = 119886

2 (8)






y

x

(a) (b) (c)






119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)


Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)


2Ψ = 0 For this


nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)


nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)




























L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

31 32 33 34 35 36

37 38 39 40 41 42

43 44 45 46 47 48

49 50 51 52 53







x

z

Flowline

Aperture

(a)

H

120573

u

x

U

y

a

(b)



120588(120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 119908

120597119906

120597119911)

= minus120597119901

120597119909+ 120583(

1205972119906

1205971199092+1205972119906

1205971199102+1205972119906

1205971199112) + 120574 sen120573

(1)


0 = minus120597119901

120597119909+ 120583(

1205972119906

1205971199102) + 120574 sen120573 (2)


1205972119906

1205971199102=1

120583(120597119901

120597119909+ 120574119867) (3)





119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909+119880

119886119910 (4)


119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909 (5)



119902 = int

119886

0

1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909119889119910 = minus

1198863

12120583

120597 (119901 + 120574119867)

120597119909

(6)


119906 = minus1198862

12120583

120597 (119901 + 120574119867)

120597119909 (7)



119906max = minus1198862

8120583

120597 (119901 + 120574119867)

120597119909to 119910 = 119886

2 (8)






y

x

(a) (b) (c)






119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)


Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)


2Ψ = 0 For this


nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)


nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)




























L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

z

Flowline

Aperture

(a)

H

120573

u

x

U

y

a

(b)



120588(120597119906

120597119905+ 119906

120597119906

120597119909+ V

120597119906

120597119910+ 119908

120597119906

120597119911)

= minus120597119901

120597119909+ 120583(

1205972119906

1205971199092+1205972119906

1205971199102+1205972119906

1205971199112) + 120574 sen120573

(1)


0 = minus120597119901

120597119909+ 120583(

1205972119906

1205971199102) + 120574 sen120573 (2)


1205972119906

1205971199102=1

120583(120597119901

120597119909+ 120574119867) (3)





119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909+119880

119886119910 (4)


119906 (119910) =1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909 (5)



119902 = int

119886

0

1

2120583(1199102minus 119886119910)

120597 (119901 + 120574119867)

120597119909119889119910 = minus

1198863

12120583

120597 (119901 + 120574119867)

120597119909

(6)


119906 = minus1198862

12120583

120597 (119901 + 120574119867)

120597119909 (7)



119906max = minus1198862

8120583

120597 (119901 + 120574119867)

120597119909to 119910 = 119886

2 (8)






y

x

(a) (b) (c)






119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)


Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)


2Ψ = 0 For this


nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)


nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)




























L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y

x

(a) (b) (c)






119906 =120597Φ

120597119909=120597Ψ

120597119910

V =120597Φ

120597119910= minus

120597Ψ

120597119909

(9)


Ψ = 119906119910 Φ = 119906119909 (10)

119906 = 119880infin V = 0 (11)

119906119903= 119906 cos120573 119906

120579= 119906 sen120573 (12)


2Ψ = 0 For this


nabla2Ψ = 0

120597Ψ

120597119909= 0

1205972Ψ

1205971199092= 0

(13)


nabla2Φ = 0

120597Φ

120597119909= 119906

1205972Φ

1205971199092= 0

(14)




























L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










L

CT n

umbe

rs

3750

2000

250

2 cm

(a)

CT n

umbe

rs

3750

2000

250

L

2 cm

(b)








1205972Φ

1205971199092+1205972Φ

1205971199032+1

119903

120597Φ

120597119903= 0 (15)


Φ = 119890119888119909119891 (119903) (16)


119903


119906119903=120597Φ

120597119903 119906

119909=120597Φ

120597119909 (17)


1198892119891 (119903)

1198891199032

+1

119903

119889119891 (119903)

119889119903+ 1198882119891 (119903) = 0 (18)


119891(119903) = 1198881119869119900(119888119903) + 119888

2119884119900(119888119903) (19)




119900(119888119903)

can be written as

119869119900(119896119903) = 1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

+ sdot sdot sdot

(20)



Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Φ = 119890119888119909119869119900(119888119903)

= 119890119888119909[1 minus (

119888119903

2) +

1

(2)2(119888119903

2)

4

minus1

(3)2(119888119903

2)

6

]

(21)





sin 120579 120597120597119903(1199032 120597Φ

120597119903) +

120597

120597120579(sin 120579120597Φ

120597120579) = 0 (22)



119903= 120597Φ120597119903 and 119906

120579=


(1 minus 1199112)1198892119891 (119911)

1198891199112

minus 2119911119889119891 (119911)

119889119911+ 119899 (119899 + 1) 119891 (119911) = 0 (23)



(23) is given by

Φ (119903 120579) = 119903119899119875119899(cos 120579) (24)


1198750(119909) = 1

1198751(119909) = 119909

(25)


Φ (119903 120579) = (119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (26)

where 119860119899and 119861

119899are constants


Φ (119903 120579) =

infin

sum

119899=0

(119860119899119903119899+119861119899

119903119899+1)119875119899(cos 120579) (27)


119899= 1 this


119860119899and 119861




119899and 119861


simplify as

If 119861119899= 0 forall119899

119860119899= 0 for 119899 = 1

119860119899= 119906 for 119899 = 1

(28)


Φ (119903 120579) = 119906119903 (cos 120579)

Ψ (119903 120579) =1

21199061199032sen2120579

119906119903=120597Φ

120597119903= 119906 cos 120579

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579

(29)


If 119860119899= 0 forall119899

119861119899= 0 for 119899 = 0

119861119899= 0 for 119899 = 0

(30)


Φ (119903 120579) = minus119876

4120587119903

Ψ (119903 120579) = minus119876

4120587(1 + cos 120579)

119906119903=120597Φ

120597119903=

119876

41205871199032

119906120579=1

119903

120597Φ

120597120579= 0

(31)




ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ux

u

uy

120579u



Ψ (119903 120579) =1

21199061199032sen2120579 minus 119876

4120587(1 + cos 120579) (32)

Φ (119903 120579) = 119906119903 (cos 120579) minus 119876

4120587119903 (33)

119906119903=120597Φ

120597119903= 119906 (cos 120579) + 119876

41205871199032 (34)

119906120579=1

119903

120597Φ

120597120579= minus119906 sen 120579 (35)












(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) (b)















nabla4Ψ = nabla

2(nabla2Ψ) = 0 (36)


Ψ (119903 120579) = 0 (37)




Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Carbonates

Redepositedsuevite

Suevite


Sueviticbreccia

Sueviticbreccia





(a)

1 cm

(b)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 0 for 119903 = 119886

119900 (38a)

119906120579= minus

1

119903 sin 120579120597Ψ

120597119903= 0 for 119903 = 119886

119900 (38b)

Ψ =1



ao

u




Ψ (119903 120579) = 119891 (119903) sin2120579 (39)


[sin21205791205972119891 (119903)

1205971199032

minus 2sin 120579119891 (119903)

1199032]

2

= 0

[sin2120579119891 (119903) ( 1205972

1205971199032minus2

1199032)]

2

= 0

(40)


(1205972

1205971199032minus2

1199032)(

1205972

1205971199032minus2

1199032)119891 (119903) = 0 (41)




119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119891 (119903) =119860

119903+ 119861119903 + 119862119903

2+ 1198631199033 (42)



119906119903=

1

1199032 sin 120579120597Ψ

120597120579= 119891 (119903)

2

1199032cos 120579 (43)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 (44)


119906120579= minus

1

119903 sin 120579120597Ψ

120597119903 (45)


120597Ψ

120597119903= sin2120579 (minus119860

1199032+ 119861 + 2119862119903 + 3119863119903

2) (46)



119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (47)


119906119903= 2 (

119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579 = 0

0 = (119860

119886119900

3+119861

119886119900

+ 119862 + 119863119886119900) cos 120579


119860

1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 = 0

0 = (minus119860

119886119900

3+119861

119886119900

+ 2119862 + 3119863119886119900) sin 120579

(48)




119903+ 2119862 + 3119863119903) sin 120579

119906 = (2119862 + 3119863 lowastinfin) sin 120579(49)


119906119903= 119906 cos 120579 = 2 (119860

1199033+119861

119903+ 119862 + 119863119903) cos 120579

119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863 lowastinfin) cos 120579

119906 = 2 (119862 + 119863 lowastinfin)

(50)



119862 =119906

2 119863 = 0 (51)





can be derived


1199033+119861

119903+ 2119862 + 3119863119903) sin 120579 (52)


0 = (minus119860

1199033+119861

119903+ 119906) (53)

For 119906119903from (44)

0 = 119906 cos 120579 = 2 (1198601199033+119861

119903+ 119862 + 119863119903) cos 120579 (54)


0 = (2119860

1199033+2119861

119903+ 119906) (55)


119860 =1199033119906

4 119861 =

minus3119903119906

4 (56)


119906119903= 119906(1 minus

3119886119900

2119903+119886119900

3

21199033) cos 120579 (57)

119906120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) sin 120579 (58)

Ψ =1199032

2119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin2120579 (59)



Φ = 119906(119903 minus3119886119900

4minus119886119900

3

41199033) cos 120579 (59b)





119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119889Ψ

119889120579= 1199032119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 cos 120579 (60)

119889119906120579

119889120579= 119906(minus1 +

3119886119900

4119903+119886119900

3

41199033) cos 120579 (61)

119889119906119903

119889120579= minus119906(1 minus

3119886119900

2119903+119886119900

3

21199033) sin 120579 (62)










L2

W







0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

10

20

30

40

50

60

70

80

(cm

)

(a) (b)

Coherent layer

Trace fossil

Clast


Coherent layer


Coherent layer

Coarse sediment

Coherent layer


Coherent layer

CC

D

C

D

C

C

C

DD

C

(c)




















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















Ψsource (119903 120579) =119876

21205871205791 (63)

Ψsink (119903 120579) = minus119876

21205871205792 (64)

Ψuniform (119903 120579) = 119880119910 (65)




Source Sink

l

x998400

y998400

12057911205792

r1r2

x

y

Ψ



Ψ119879(119903 120579) =

119876

21205871205791minus119876

21205871205792+ 119880119910 (66)

where


1199101015840

1199091015840 minus (1198972))


1199101015840

1199091015840 + (1198972))

(67)


by

Ψ119879=119876

2120587tanminus1 (

1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(68a)

Ψ119879=119876

2120587[tanminus1 (

1199101015840


1199101015840

1199091015840 + (1198972))] + 119880119910

(68b)


119879) 119909119879=



Defining 119906119909= 120597Ψ119879120597119910 and 119906

119910= minus120597Ψ

119879120597119909 in rectangular


119906119909=

Q2120587

[1199091015840minus (1198972)

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199091015840+ (1198972)

(1199091015840 + (1198972))2

+ 11991010158402] + 119880

119906119910= minus

Q2120587

[1199101015840

(1199091015840 minus (1198972))2

+ 11991010158402minus

1199101015840

(1199091015840 + (1198972))2

+ 11991010158402]

(69)


y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y


minus1 +1

Sink SourceO

x1 x2



Φ119879=

Q21205871205791minus

Q21205871205792+ 119880119910 (70a)


and 1205792

Φ119879=

Q2120587

tanminus1 (1199101015840

1199091015840 minus (1198972)) minus

119876

2120587tanminus1 (

1199101015840

1199091015840 + (1198972)) + 119880119910

(70b)


0 (V119910max




















36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











36 cm












u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










u

Matrix

Vug


Matrix

Vug

u









Ψ (alefsym 120579) =119896119906lowast



119896119906lowast




119860 =6 (alefsym2+ 5120590)

1198832 + 10120590 + 20

119861 =3

1198832 + 10120590 + 20

119862 =21198833(1198832+ 10120590 minus 10)

1198832 + 10120590 + 20

119863 = 1

alefsym =119903

radic119896

119883 =119877

radic119896

120590 = 120590 (120593) 0 lt 120593 lt 1

(72)



119906119903=1

119903

120597Ψ

120597120579= (

91199033+ 30120590119903119896

1198772 + 10120590119896 + 20119896)119906lowast sin 120579 cos 120579

119906120579= minus

120597Ψ

120597119903= minus(

361199033+ 60120590119903119896

1198772 + 10120590119896 + 20119896)119906lowastsin2120579

(73)


Φ = (120583 sin 120579 cos 120579

1198772 + 10120590119896 + 20119896)(

91199034

4+ 15120590119903

31198963) (74)





























0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































0745(dimensionless)






119903(ms) 119906

120579(ms) 119902 (m3s)





















0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























0720(dimensionless)











0720(dimensionless)





C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










C-358

C-139 C-338 C-119 C-318 C-109 C-308

C-129



Producer interval

Unconformity

Breccias intervals



Section 1-1998400


(i) Dolomudstone

(ii) Dolomites




Dolomites

Limestones

(i) Limestones







C-171

C-152

C-134AC-132

C-151

C-112C-114B

C-104A

C-153C-155

C-131C-133

C-111A

C-102AC-124

C-144C-142

C-135

C-113C-103

C-105C-125

C-123

C-115C-117A

C-163AC-161A

C-162C-164 C-182

C-107B

C-101B

C-122C-121

C-358C-338

C-318C-308C-119

C-129

C-127C-147

C-145C-165

C-201

C-291

C-294C-184

C-141C-143

C-181

C-109

C-139C-137

0 1 2

450 455

(km)

1995

1990

N 1

1998400

(a)

Figure 28 Continued



8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











8000

6000

4000

2000

Botto

n pr

essu

re(p

si)

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

Time (years)

C-358C-338C-318C-308C-109C-119C-129C-139

3500

3750

4000

4250

TWT

(ms)

C-358

C-338

C-318

C-308

C-109

C-119

C-129

C-139

3D seismic section

(b)












0720(dimensionless)



5221

5240

5260

5280

5300

5320

5340

5360

5380

5400

5420

5440

5460

5480

3220

3200

3180

3160

3140

3120

3100

3080

3060

3040









119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119903(ms) 119906

120579(ms) 119902 (m3s)







20 Conclusions










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Symbols










Acknowledgments


References















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Analytical Modeling and Contradictions in ...downloads.hindawi.com/journals/jpe/2015/895786.pdf · Research Article Analytical Modeling and Contradictions in Limestone