Improved estimation of mineral and fluid volumetric ...calculate volumetric mineral and fluid concentrations together with their uncertainty (Busch et al., 1987). Even though constrained

Improved estimation of mineral and fluid volumetric concentrationsfrom well logs in thinly bedded and invaded formations

Zoya Heidari1, Carlos Torres-Verdín2, and William E. Preeg3

ABSTRACT

Calculation of mineral and fluid volumetric concentrationsfrom well logs is one of the most important outcomes of forma-tion evaluation. Conventional estimation methods assume linearor quasi-linear relationships between volumetric concentrationsof solid/fluid constituents and well logs. Experience shows,however, that the relationship between neutron porosity logsand mineral concentrations is generally nonlinear. More impor-tantly, linear estimation methods do not explicitly account forshoulder-bed and/or invasion effects on well logs, nor do theyaccount for differences in the volume of investigation of themeasurements involved in the estimation. The latter deficienciesof linear estimation methods can cause appreciable errors in thecalculation of porosity and hydrocarbon pore volume. We inves-tigated three nonlinear inversion methods for assessment of vo-lumetric concentrations of mineral and fluid constituents ofrocks from multiple well logs. All three of these methods ac-

counted for the general nonlinear relationship between welllogs, mineral concentrations, and fluid saturations. The firstmethod accounted for the combined effects of invasion andshoulder beds on well logs. The second method also accountedfor shoulder-bed effects but was intended for cases where mud-filtrate invasion is negligible or radially deep. Finally, the thirdmethod was designed specifically for analysis of thick bedswhere mud-filtrate invasion is either negligible or radially deep.Numerical synthetic examples of application indicated that non-linear inversion of multiple well logs is a reliable method toquantify complex mineral and fluid compositions in the pre-sence of thin beds and invasion. Comparison of results againstthose obtained with conventional multimineral estimation meth-ods confirmed the advantage of nonlinear inversion of multiplewell logs in quantifying thinly bedded invaded formations withvariable and complex lithology, such as those often encounteredin carbonate formations.

INTRODUCTION

Lithology identification and quantification have been of great inter-est to formation-evaluation specialists for decades. Well-site geologicdescription of drilling rock cuttings is the first indication of mineralconstituents. Core analysis aided by mud logs is a common approachto diagnose lithology. Earlier detection and quantification methodsbased on well logs included the use of density-sonic, density-photo-electric factor, neutron-sonic, neutron-density, and matrix identification(MID) crossplots (Clavier and Rust, 1976; Schlumberger, 2005).Estimation of porosity and fluid saturations is the cornerstone

of formation evaluation, because it affects the calculation of hydro-

carbon pore volume. In complex lithologies, reliable assessments ofvolumetric mineral/fluid concentrations from well logs, core data,and mud logs are necessary for accurate estimation of porosity andhydrocarbon saturation. Practically all commercial software avail-able for detection and estimation of mineral volumetric concentra-tions assume linear or quasi-linear relationships between formationproperties and well logs, and do not account for shoulder-bed and/orinvasion effects on the estimation. Such conditions often contributeto erroneous estimations of porosity and hydrocarbon pore volume,especially in thinly bedded and mixed carbonate sequences.Neutron-capture spectroscopy measurements are the basis

of yet another approach to quantify mineralogy. Volumetric

Manuscript received by the Editor 10 November 2011; revised manuscript received 22 January 2012; published online 10 May 2012; corrected versionpublished online 15 May 2012.

1Formerly with The University of Texas at Austin, Department of Petroleum and Geosystems Engineering, Austin, Texas, USA; presently with Texas A&MUniversity, Harold Vance Department of Petroleum Engineering, College Station, Texas, USA. E-mail: [email protected].

2The University of Texas at Austin, Department of Petroleum and Geosystems, Engineering, Austin, Texas, USA. E-mail: [email protected] consultant, Austin, Texas, USA. E-mail: [email protected].

© 2012 Society of Exploration Geophysicists. All rights reserved.

WA79

GEOPHYSICS, VOL. 77, NO. 3 (MAY-JUNE 2012); P. WA79–WA98, 10 FIGS., 7 TABLES.10.1190/GEO2011-0454.1

Downloaded 17 Aug 2012 to 128.83.167.155. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

concentrations of minerals (including clay minerals) are estimatedfrom elemental spectroscopy measurements (Herron and Herron,1996; Herron et al., 2002). This method can be applied to open-and cased-hole environments. Herron et al. (2002) integrated neu-tron-capture spectroscopy measurements with electrical resistivity,neutron porosity, and density logs to quantify porosity, water satura-tion, and irreducible water saturation, and to diagnose presenceof gas.Numerical methods have also been developed to diagnose and

quantify lithofacies based on artificial neural networks, fuzzy logic,and neuro-fuzzy models (Cuddy, 2000; Gonçalves et al., 1995).These methods require substantial training with core measurementsto warrant reliable and accurate estimations and can easily fail in thepresence of complex mineralogy and thinly bedded rock sequences.Stochastic Bayesian inversion of well logs is another approach toestimate petrophysical properties (Woodruff et al., 2010), with theadded advantage of concomitant estimation of uncertainty of inver-sion products.Deterministic linear inversion of well logs is the most common

numerical method used to quantify lithology in the presence ofmultiple minerals (Mayer and Sibbit, 1980; Quirein et al., 1986;Doveton, 1994; Rabaute et al., 2003). The implicit assumptionof linear estimation methods is that measurements such as density,photoelectric factor (PEF), neutron porosity, and sonic transienttime are linear functions of the volumetric concentrations of theassumed rock mineral and fluid constituents. Volumetric concentra-tions of mineral and fluid constituents are then obtained depth-by-depth by minimizing the difference between well logs and theirlinear prediction, formally expressed as

min kAx − bk22; (1)

subject to

0 ≤ xi ≤ 1; (2)

and

Xncþ2

i¼1

xi ¼ 1; (3)

where x is the n-size vector of volumetric mineral and fluid con-centrations, given by

x ¼ ½C1; C2; : : : ; Cnc; Csh;ϕs�T; (4)

where the superscript T indicates transposition, Ci is volumetricconcentration of the assumed mineral constituents, Csh is volu-metric concentration of shale, ϕs is nonshale porosity, and nc isthe predefined number of mineral constituents. In equation 1, matrixA is expressed as

A ¼

2664ρb;1 ρb;2 : : : ρb;nc ρb;sh ρfluidϕN;1 ϕN;2 : : : ϕN;nc ϕN;sh ϕN;fluid

Δt1 Δt2 : : : Δtnc Δtsh ΔtfluidU1 U2 : : : Unc

Ush Ufluid

3775; (5)

and vector b as

b ¼ ½ ρb ϕN Δt U �T; (6)

where b is the vector of available well-log measurements, ϕN isneutron porosity, ρb is bulk density, U is volumetric photoelectriccross section, and Δt is sonic interval transient time. The entries ofmatrix A are defined from well-log measurements acquired in rocksor fluids with “pure” elemental compositions.We wish to emphasize that the above constrained linear estima-

tion method presupposes knowledge of the number and type ofmineral and fluid components prior to performing the estimation.In Rabaute et al. (2003), for instance, the authors use a clusteringalgorithm to determine existing mineral/fluid components beforeperforming the inversion. It is also pertinent to note that, eventhough the aforementioned estimation methods are linear in nature,they can be solved with nonlinear minimization algorithms, as donein the case of Rabaute et al. (2003).Fluid saturations also affect the assessment of porosity and volu-

metric concentrations of mineral constituents. Resistivity measure-ments are commonly used in combination with the described linearestimation method to assess fluid saturations based on the estimatedporosity and a predefined porosity-saturation-resistivity model.Saturation estimates obtained this way are updated iteratively usingresistivity measurements and then entered into matrixA (equation 5)to define fluid properties.Doveton (1994) introduced a robust linear inversion algorithm

to estimate unknown properties based on equations 1–6. Inmost practical cases of lithology quantification, the number ofunknowns (i.e., minerals and petrophysical properties such as por-osity and water saturation) included in equation 1 is greater than thenumber of data (i.e., the number of well logs). The latter conditiongives rise to an underdetermined estimation whose solution requiresoutside (a priori) constraints and/or subjective selections of relation-ships among unknown properties. Another complication of linearinversion methods is the lack of reliable well-log responses acrosspure lithologies and fluids, which are necessary to define the entriesof matrix A in equation 5. Well-log responses across pure litholo-gies and fluids cannot be drawn from tabulated values because someof them may vary according to local conditions of sedimentationand diagenesis. Users of linear inversion methods commonly definethese entries by trial and error, albeit usually guided by mud logsand core measurements. However, it is commonly the case thatsmall perturbations in the entries of matrix A lead to sizable per-turbations in the calculated mineral/fluid volumetric concentrations.Experience is necessary to parse petrophysical acceptable solutionsfrom those that are not.Several commercial software packages use the above method to

evaluate volumetric concentrations of mineral constituents, poros-ity, and water saturation. Mayer and Sibbit (1980) introduced per-haps the first commercial software for linearized inversion. Theyalso incorporated water saturation into the inversion to estimatefluid saturations together with volumetric concentrations of mineralconstituents. Other commercial methods (and software) iterativelyestimate mineral concentrations using the linear inversion methodgiven by equation 1 in sequence with the calculation of fluid satura-tions until reaching convergence between the two calculations.Quirein et al. (1986) suggested a probabilistic approach to esti-mate mineral and fluid concentrations based on equation 1 thatcombined various porosity-saturation-resistivity models. Alter-native approaches make use of statistical estimation methods to

WA80 Heidari et al.


calculate volumetric mineral and fluid concentrations together withtheir uncertainty (Busch et al., 1987).Even though constrained linear inversion is a fast and efficient

method that yields accurate estimations in many simple situations,it can fail in the presence of complex mineral compositions. A sig-nificant drawback of linear inversion methods is that the assumptionof a linear relationship between properties and mineral volumetricconcentrations is often invalid in the presence of light hydrocarbonand clay. For instance, neutron porosity measurements are usuallycorrected for presence of dispersed shale via the formula (Thomasand Stieber, 1975)

ϕshN ¼ ϕN − CshϕN;sh; (7)

where ϕN;sh is neutron porosity in a pure shale zone and ϕNsh isneutron porosity corrected for shale. Equation 7 is valid withrelative errors lower than 20% in water-saturated zones. The errorincreases, however, for decreasing values of shale porosity andnonshale porosity; please note that all the error values reportedin this paper are relative errors. Furthermore, equation 7 is no longervalid in saline formations or in the presence of halite. In such situa-tions, it is necessary to correct neutron porosity for connate-watersalinity before applying the linear shale-correction equation.Although errors associated with equation 7 might be acceptablein water-bearing formations, the linear correction usually fails in thepresence of gas, where the equation needs to consider neutronporosity and density measurements with, for instance, the nonlinearapproximation advanced by Gaymard and Poupon (1968):

ϕs ≅

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðϕsh

N Þ2 þ ðϕshD Þ2

2

r; (8)

where ϕshD is density porosity corrected for shale, expressed as

ϕshD ¼ ϕD − CshϕD;sh; (9)

where ϕD;sh is density porosity in a pure shale zone, and ϕD is den-sity porosity. Equation 8 quantifies nonshale porosity with relativeerrors lower than 3% within the shale-porosity range of 0.00–0.20.Therefore, the assumption of linear relationship between nonshaleporosity and volumetric concentration of shale is no longer valid.Another important remark concerning linear estimation methods

is that Wyllie’s slowness mixing formula (Wyllie et al., 1956) isthe sonic model commonly assumed in equations 1–6. Thisformula is not accurate in soft and unconsolidated sediments, organ-ic shale, and fractured formations to name but a few importantcases. Specialized effective-medium theories need to be invokedto reliably relate sonic transient times with fluids and mineralcompositions (Mavko et al., 2009).It is also known that linear inversion methods can break down

under conditions such as the presence of iron in the rock matrixor in gas-bearing formations. More importantly, linear inversionmethods are implemented depth-by-depth and hence do not expli-citly account for shoulder-bed effects on the well logs included inthe estimation.Joint inversion of electrical resistivity, gamma-ray (GR), and den-

sity logs was previously implemented to improve the petrophysicalassessment of thinly bedded siliciclastic formations (Liu et al.,2007; Sanchez-Ramirez et al., 2010). The task remaining is to im-plement similar methods and to include additional well logs in the

estimation of volumetric mineral/fluid concentrations in carbonateand complex mixed sedimentary sequences that include thin bedsand in formations that are subject to invasion.The objective of this paper is to introduce, cross validate, and

benchmark new, automatic nonlinear methods to estimate volu-metric mineral/fluid concentrations from multiple well logs inthe presence of complex matrix composition, mud-filtrate invasion,and bed-boundary effects. These new methods are possible becauseof the availability of newly developed algorithms for the rapidnumerical simulation of nuclear logs (Mendoza et al., 2010). Suchwell-log simulation methods explicitly quantify the generally non-linear relationship between rock/fluid properties and nuclear logs,thereby accounting for differences in the volume of investigation ofthe measurements involved in the estimation, as well as the presenceof shoulder beds and invasion.We introduce three methods for the assessment of petrophysical

and compositional properties of hydrocarbon-bearing formationsexhibiting complex lithology. The first method takes into accountthe effects of radial variations of fluid saturation (in cases of water-base mud [WBM] or oil-base mud [OBM]) and salt concentration(in cases of WBM) on well logs due to mud-filtrate invasion. It alsoaccounts for shoulder-bed effects on well logs and nonlinear deter-ministic relationships between volumetric concentrations of miner-al/fluid constituents. This method assumes a multilayer reservoirmodel previously constructed with the detection of bed boundariesfrom density, PEF, and GR logs. Subsequently, nonlinear inversionis initialized with a multilayer petrophysical model that consists ofvolumetric mineral/fluid concentrations combined with resistivity-saturation equations. Nonlinear inversion combined with numericalsimulation of mud-filtrate invasion minimizes the difference be-tween well logs and their numerical simulations across the depthzone of interest by making progressive adjustments to fluid andmineral concentrations in each layer.The second method quantifies petrophysical/compositional prop-

erties in thinly bedded formations exhibiting complex lithology,where the effect of mud-filtrate invasion is negligible. This methodestimates bed physical properties (layer-by-layer density, neutronmigration length, PEF, conductivity/resistivity, and uranium (Ur),thorium (Th), and potassium (K) concentrations) based on theseparate inversion of well logs. Separate linear inversion of density,PEF, and GR-spectroscopy logs is implemented with a recentlyintroduced method for fast modeling of nuclear logs, which usesprecalculated flux sensitivity functions (FSFs). The final step ofthe second method implements joint inversion of estimated proper-ties to assess mineral and fluid concentrations.The third method estimates petrophysical/compositional proper-

ties in formations with complex lithology and thick beds, where theeffect of mud-filtrate invasion is negligible. We quantify the non-linear deterministic relationship between volumetric concentrationsof mineral/fluid constituents and well logs via Schlumberger’s com-mercial software, SNUPAR (McKeon and Scott, 1989) to assessneutron migration length and PEF. Chemical formulas and volu-metric/weight concentrations of mineral/fluid constituents are inputto SNUPAR. Similar to conventional petrophysical/compositionalinterpretation, however, this method does not account for eithershoulder-bed or mud-filtrate invasion effects on well logs.The construction of three nonlinear inversion methods is neces-

sary to perform estimation in an efficient and stable way; the choiceof method depends on the assumed degree of complexity in the

Mineral fluid concentrations from logs WA81


formations under consideration. In the following sections, we de-scribe and successfully implement the three nonlinear inversionmethods. Examples of application consider three challenging syn-thetic formations constructed to replicate actual field situations.

METHOD

Inputs to the three estimation methods introduced in this paperare conventional well logs such as density, neutron porosity, PEF,GR/GR-spectroscopy, and electrical resistivity/conductivity logs.The possibility also exists of considering subsets of these logs.For instance, one could ignore PEF in cases where it is not reliabledue to presence of barite in the mud. The methods described in thispaper are intended for estimating volumetric concentrations ofmineral and fluid constituents. However, they are reliable in asses-sing weight concentrations of mineral constituents as well.Depending on the number of minerals in the formation, the num-

ber of equality constraints, and the number of reliable well logs, theestimation problem could be overdetermined or underdetermined.Most well-logging problems involve underdetermined estimations,in which the number of unknowns is greater than the numberof logs and equality constraints, whereby there are multiplesolutions that honor well logs and petrophysical constraints. In

under determined estimations, the choice of initial guess isimportant. The following sections detail the assumed rock model,the three nonlinear interpretation methods, and the strategiesadopted for initial-guess construction. Figure 1 is a flowchart de-scribing the processes of numerical simulation of well logs and non-linear inversion to estimate multilayer petrophysical properties andvolumetric/weight concentrations of mineral constituents.

Rock model

The rock model consists of different mineral components in thematrix, shale, and nonshale porosity (which includes water andhydrocarbon). Water in nonshale porosity can be movable or irre-ducible. Hydrocarbon may also be movable or residual. Eventhough the specific application considered in this paper assumesdispersed shale (Poupon et al., 1970), the formulation can be easilymodified for the case of laminated shale.The relationship among the volumetric concentrations of all

matrix components is expressed as (Mezzatesta et al., 2006)

Xnci¼1

Ci þ Csh þ ϕs ¼ 1; (10)

with volumetric concentration of shale defined as

Figure 1. Workflows for the three nonlinear joint inversion methods introduced in this paper. Inversion methods estimate unknown petro-physical properties and volumetric/weight concentrations of rock mineral constituents in the presence of mud-filtrate invasion, thin beds, andcomplex lithology. Nonlinear inversion progressively improves the agreement between well logs and their numerical simulations.

WA82 Heidari et al.


Csh ¼Vsh

Vr

; (11)

where Vr is rock volume including fluids and Vsh is volume of wetshale. Figure 2 illustrates the assumed volumetric description ofmatrix, shale, and matrix fluids. We calculate bulk density ρb basedon volumetric concentrations of mineral components via theequation

ρb ¼Xnci¼1

ðCiρiÞ þ ρfϕs þ ρshCsh; (12)

where ρi is density of the corresponding mineral, ρf is fluid density,and ρsh is shale density. Wet shale consists of clay and silt, whichincludes clay-bound water. Fluid density and shale density aregiven by

ρf ¼ Swρw þ ð1 − SwÞρh; (13)

and

ρsh ¼ ρsiltð1 − Ccl − ϕshÞ þ ρclCcl þ ρwϕsh; (14)

respectively, where Sw is total water saturation, ϕsh is shale-porosity, ρw is water density, ρh is hydrocarbon density, ρcl is claydensity, ρsilt is silt density, and Ccl is volumetric concentration ofdry clay, defined as

Ccl ¼Vcl

Vsh

; (15)

where Vcl is clay volume.A pertinent mixing law can be used to define the physical rela-

tionship between compressional- and shear-wave sonic logs andmineral concentrations and their dry bulk moduli (compressionaland shear), as well as the corresponding effect of fluid componentsand their saturations (Mavko et al., 2009). Such an approachenables the input of sonic measurements in the inversion, in addi-tion to resistivity and nuclear logs.

Method 1: Nonlinear multilayer joint inversion of welllogs in the presence of mud-filtrate invasion

This method accounts for the effects of mud-filtrate invasion andshoulder beds on well logs. After data quality control, applying ne-cessary corrections for depth shift, defining bed boundaries, andconstructing the multilayer reservoir model, the first step of non-linear inversion is the construction of an initial guess of layer-by-layer petrophysical properties. Such initial guess can be selectedfrom (1) results obtained from conventional/linear multimineral sol-vers, (2) core/X-ray diffraction (XRD) data, or (3) resultsobtained from depth-by-depth nonlinear inversion of well logs(method 3).Nonlinear inversion is performed by minimizing the quadratic

cost function

CðxÞ ¼ kWd½dðxÞ − dm�k22 þ α2kxk22; (16)

subject to

0 ≤ xj ≤ 1; (17)

and

Xnci¼1

Ci þ Csh þ ϕs ¼ 1; (18)

where Wd is a data weighting matrix, dðxÞ is the vector of numeri-cally simulated well logs, dm is the vector of available well logs, α isa regularization (stabilization) parameter, and x is the vector ofpetrophysical properties and volumetric concentrations of mineralconstituents for all beds, given by

x ¼ ½x1; x2; : : : ; xnb �T; (19)

where the superscript T indicates transposition and xi is given by

xi ¼ ½C1;i; C2;i; : : : ; Cnc;i; Csh;i;ϕs;i; Sw;i�T; (20)

where nb is the number of beds. The vector of numerically simu-lated logs is given by

d ¼ ½ϕN; ρb;PEF;GR;σ;Ur;Th;K�T; (21)

where vectors ϕN, ρb, PEF, GR, σ, Ur, Th, and K include nsp mea-surement points for neutron porosity, density, PEF, GR, apparentconductivity, Ur, Th, and K logs; σ is a vector that includes allthe available apparent electrical conductivity logs (i.e., inverse ofapparent resistivity logs with variable radial lengths of investiga-tion). The size of vector d depends on the sampling interval andnumber of well logs included in the inversion. Appendix A providesadditional details about the approach used in this paper to minimizeequation 16.Volumetric concentration of shale consists of shale porosity,

volumetric concentration of clay, and volumetric concentrationof silt, which are obtained from inversion results applied topure shale zones and thereafter set constant or assumed to beknown in synthetic cases. We assume that shale porosity andvolumetric concentration of clay in shale within permeable zonesare equal to those of adjacent shale zones. This is a customary

Figure 2. Petrophysical rock model assumed in this paper. Volumeof rock Vr consists of solid matrix, wet shale Vsh, and nonshaleporosity ϕs. Nonshale porosity includes matrix, water, andhydrocarbon.



assumption made in well-log interpretation when detailed knowl-edge of clay type and concentration is not available.Experience shows that, for this application, the cost function

defined by equation 16 is often flat around the minimum, therebygiving rise to nonunique results. Moreover, there might be severalminima due to having more unknown properties than well logs andequality constraints (underdetermined estimation problem). Ac-cordingly, the stabilization parameter σ included in the quadraticcost function is intended to reduce nonuniqueness in the presenceof noisy, inadequate, and/or incomplete data; it controls the relativeimportance between fitting the data (first additive in the cost func-tion) and finding a stable and smooth solution (second additive termin the cost function, here defaulted to the quadratic norm of thesolution). During the first iterations, a large value of α expeditesthe convergence by giving more importance to the agreementbetween well logs (data) and their numerical simulations. The valueof α decreases gradually to guarantee a stable (nonoscillatory) solu-tion at the end of the estimation. In the presence of thin beds, α con-trols the degree of smoothness of estimated properties betweenadjacent beds; high and low values of α give rise to smooth andhighly oscillatory variations of properties between adjacent beds,respectively.The data weighting matrix Wd is included in equation 16 to con-

trol the importance of each well log in the joint inversion. If all welllogs are assigned the same importance, then the data weightingmatrix scales the well logs to the same order of magnitude. Thedata weighting matrix Wd is given by

Wd ¼

26664

h1

kdm;1k2 Insp×nspi

: : : 0

..

. . .. ..

.

0 · · ·h

1kdm;nl

k2

Insp×nsp

i

37775; (22)

where I is the unity matrix, nl is the number of well logs, and nsp isthe number of sampling points in each well log. In the presence ofnoisy well logs, the influence of a given log on inversion results canbe reduced by assigning a relatively small value to the correspond-ing entry of the data weighting matrix.We minimize the cost function defined in equation 16 with

Levenberg-Marquardt’s method (Marquardt, 1963). Accordingly,entries of the Jacobian matrix are calculated numerically (finitedifferences) at every linear iteration, and the stabilization parameteris selected with Hansen’s (1994) L-curve strategy. At everyiteration, unknown properties are updated based on the calculatedJacobian matrix and the difference between well logs and theirnumerical simulations. The minimization comes to a halt when(1) the relative difference between the norms of data residualsyielded by two subsequent iterations is less than 0.01% or (2) afterreaching a prescribed maximum number of iterations.The entries of the Jacobian matrix for this problem are given by

Jij ¼∂di∂xj

; 1 ≤ i ≤ nl × nsp; 1 ≤ j ≤ nu × nb; (23)

Figure 3. Synthetic case 1, consisting of three thick layers of pure limestone, pure dolomite, and pure quartz. Comparison of final simulatedwell logs (dash-dotted black line), input well logs (solid line), and simulated well logs for the initial guess (dashed line). The left-hand panelshows the spatial distribution (radial and vertical directions) of water saturation for a fully water-saturated formation. Results are shown forarray-induction resistivity (second left-hand panel), PEF (third left-hand panel), GR (fourth left-hand panel), density and neutron porosity(water-filled limestone porosity units, fifth left-hand panel) logs. The two right-hand panels compare final volumetric concentrations of mineralconstituents, porosity, and water saturation obtained from nonlinear joint (method 3) and conventional linear inversion of resistivity, density,neutron, GR, and PEF logs.

WA84 Heidari et al.


where nu is the number of unknowns in the assumed multilayer for-mation. For the specific situation in which input well logs includedensity, neutron porosity, deep electrical conductivity, PEF, and GR(five well logs), the logging sampling rate is 0.08 m (0.25 ft), andthere are six unknown petrophysical properties in each bed. Then ina 30.48 m (100 ft) reservoir with approximately 50 beds, the dimen-sions of the Jacobian matrix will be 2000 × 300. This large matrix isusually ill-conditioned. Furthermore, accurate numerical calcula-tion of the entries of the Jacobian matrix requires 1þ nu × nb for-ward numerical simulations for each well log. For instance, in theexample of a 30.48-m (100-ft) multilayer formation, 300 forwardnumerical simulations will be required for each log. In reality, how-ever, many of the entries of the Jacobian matrix are negligible orextremely small because well logs originate from highly localizedvolumes of investigation. Thus, the Jacobian matrix can be calcu-lated locally to avoid unnecessary calculations. On a related subject,Wang et al. (2009) introduced a domain decomposition method forfast 2D resistivity inversion. They divided the entire depth intervalinto overlapping depth subdomains, assuming that variations of for-mation resistivity in each subdomain did not affect resistivity mea-surements acquired in other depth subdomains. The inverseproblem was then solved in each subdomain separately, with a smal-ler Jacobian matrix.In addition to electrical resistivity measurements, we include nu-

clear logs and numerical simulation of mud-filtrate invasion aspart of the inverse problem. CPU time in such a problem increasescompared to the case described by Wang et al. (2009). Thus, to ap-proach the inverse problem in a time-efficient manner, in this paperwe approximate the Jacobian matrix by assuming that the effect ofsmall perturbations on formation physical properties in each bed isrelatively small on well logs acquired in other beds. This assump-tion remains an efficient approximation for the calculation of theJacobian matrix because it invokes the minimum possible perturba-tion on formation petrophysical properties that affects well logs.Such an approximation of the Jacobian matrix reduces the requirednumber of forward numerical simulations for each well log to1þ nu. The remaining entries of the Jacobian matrix are set to zero.Consequently, we decompose the original inverse problem intodepth-by-depth inverse problems during each iteration. Then weaverage the estimated depth-by-depth updated formation petro-physical properties in each bed to calculate the entry for the nextiteration of nonlinear inversion.The following sections describe the specific strategies adopted

for simulation of mud-filtrate invasion and numerical simulationof nuclear and electrical resistivity logs.

Mud-filtrate invasion

Mud-filtrate invasion is simulated with a 1D radial compositionalfluid-flow algorithm included in UTAPWeLS software (developedby The University of Texas at Austin’s Joint Industry Consortiumon Formation Evaluation; Abdollah Pour, 2008). This algorithmsimulates WBM and OBM invasion and calculates rates of mud-filtrate invasion resulting from mud-cake buildup. We assume thatvertical variations of fluid saturation due to capillary equilibrium arenegligible across short depth intervals and simulate invasionseparately within each petrophysical layer. Petrophysical layers in-clude radial numerical grids used for finite-difference simulationof fluid-flow equations. Inputs to the simulation of mud-filtrateinvasion are in situ fluid and mud properties, rock-fluid properties,

nonshale porosity, absolute permeability, initial fluid saturation,mud-cake properties, and invasion properties such as time ofmud-filtrate invasion and overbalance pressure (or rate of mud-filtrate invasion). Rocks are assumed to be water-wet. In the caseof OBM invasion, mud filtrate can cause partial wettability altera-tion that formally should be taken into account in the simulation ofinvasion (Salazar et al., 2011). Saturation-dependent capillary pres-sure and relative permeability, referred to as rock-fluid properties,are described with Brooks-Corey’s parametric equations (Corey,1994). Outputs from the fluid-flow simulator are layer-by-layerradial distributions of fluid saturation.We emphasize that simulating the process of mud-filtrate inva-

sion is optional in the nonlinear inversion method. Experienceshows that this option is necessary in cases where the radialprofile of fluid saturation due to invasion is smooth and apparentresistivity logs with different radial lengths of investigation donot “stack.”

Numerical simulation of well logs

Numerical simulation of well logs requires the construction ofnumerical grids used for assimilation of radially varying propertieswithin invaded rock formations (such as water saturation and saltconcentration resulting from the process of mud-filtrate invasion) inthe estimation. We then calculate grid-by-grid electrical conductiv-ity values for simulation of electrical resistivity logs, density valuesfor density logs (equation 12), migration length for neutron porositylogs, photoelectric factor for PEF logs, and volumetric concentra-tions of Ur, Th, and K for the numerical simulation of GR/GR-spectroscopy logs.Neutron porosity and PEF logs are simulated from neutron mi-

gration length and photoelectric factor, respectively, defined at eachfinite-difference grid. We use SNUPAR commercial software(McKeon and Scott, 1989) to calculate migration length and photo-electric factor from chemical compositions and their correspondingvolumetric concentrations. Volumetric concentrations of Ur, Th,and K are also calculated based on volumetric concentrations ofmineral components in the formation.Nuclear logs are simulated with the fast linear iterative refine-

ment method developed by Mendoza et al. (2010), which explicitlyincorporates borehole and environmental conditions as well as geo-metric configuration of nuclear tools. The CPU time required by thelinear iterative refinement method to simulate nuclear logs is hun-dreds of times shorter than the CPU time required by alter-native Monte-Carlo simulation methods. This unique advantageof Mendoza et al.’s (2010) method over Monte-Carlo simulationmethods makes it feasible to implement nonlinear inversion ofnuclear logs within practical CPU times.For calculation of the spatial distribution of electrical resistivity

from the spatial distributions of water saturation and salt concen-tration, we adopt pertinent shaly-sand descriptions such as thedual-water model (Clavier et al., 1977). The calculated distributionsof electrical resistivity (radial and vertical directions) are thenused as input for the numerical simulation of array-inductionand/or dual laterolog resistivity logs (AIT and DLT, marks ofSchlumberger).We emphasize that the well-log interpretation techniques intro-

duced in this paper explicitly take into account tool geometric con-figurations in the numerical simulation of well logs. This strategy



effectively accounts for differences in the volume of investigation ofeach log involved in the interpretation.

Method 2: Fast nonlinear joint inversion of bedphysical properties

Even though method 1 is accurate in the presence of mud-filtrateinvasion and thin beds, it is not always computationally efficient. Toquantify the effects of mud-filtrate invasion and shoulder beds onthe Jacobian matrix, one needs to simulate mud-filtrate invasion andits effect on well logs for every unknown formation property.Repeating this process at every iteration renders the inversionextremely slow. If the effect of mud-filtrate invasion on well logsis negligible, then the correction for shoulder-bed effects can be im-plemented in a more efficient manner.The second inversion method was designed to exclusively take

into account the effect of shoulder beds on logs when performingthe estimation of layer-by-layer mineral and fluid constituents,thereby improving the interpretation of thinly bedded formations.This method consists of five sequential steps: (1) preanalysis ofavailable well logs and correction of depth shifts; (2) detectionof bed boundaries based on all the available well logs or boreholeimages; (3) separate inversion of density, neutron porosity, electricalresistivity, PEF, and GR-spectroscopy logs to estimate layer-by-layer physical properties such as density, migration length,PEF, electrical conductivity, and Ur, Th, and K concentrations, re-spectively; (4) construction of a multilayer petrophysical modelbased on an initial guess of petrophysical properties and volumetricconcentrations of minerals and fluids; and (5) implementation of anonlinear joint inversion algorithm on the estimated bed petrophy-sical properties to estimate layer-by-layer petrophysical andcompositional properties. This latter step is approached withSNUPAR commercial software (McKeon and Scott, 1989).Method 2 only requires specific chemical mineral/fluid constitu-

ents in addition to well logs to estimate volumetric/weight concen-trations of mineral and fluids. Mineral properties are calculatedusing SNUPAR based on input chemical formulas for minerals/fluids. However, most of conventional lithology evaluators requirecalibration with XRD data for pure mineral properties. The follow-ing sections describe the techniques implemented for separateinversion of well logs and nonlinear joint inversion of bed physicalproperties via method 2.

Separate inversion of well logs to estimate bedphysical properties

The first step in well-log inversion algorithms is the detection ofbed boundaries. PEF and density logs are good candidates for bed-boundary detection because of their relatively high vertical resolu-tion. Inflection points in well logs are assumed to be indicative ofbed boundaries. Conventional methods usually assume center-bedor average log values within each petrophysical bed as the actualbed physical property. These approaches are not reliable in the pre-sence of thin beds wherein shoulder-bed effects are not negligible.For the case of density and PEF measurements acquired in bedsthicker than 0.30 m (1 ft), shoulder-bed effects are usually negligi-ble at the center of the bed, whereby one can safely use the center-bed value as the actual bed property.For separate inversion of density, PEF, and GR-spectroscopy

logs, we make use of a linear inversion algorithm based on precal-

culated, Monte-Carlo-derived FSFs for density, PEF, and GR mea-surements. Mendoza et al. (2010) successfully applied the sameapproach to logging-while-drilling (LWD) density measurementsacquired in high-angle wells to estimate bed densities. Appendix Bprovides additional details about the linear inversion of density,PEF, and GR-spectroscopy logs. The main assumption in theFSF-based linear inversion algorithm is that the precalculated FSFsremain constant across different formation properties. Although thisassumption is usually valid for density, PEF, and GR logs, sensi-tivity functions for neutron porosity and resistivity measurementsvary with formation properties. Consequently, the linear inversionalgorithm is applicable to neither neutron porosity nor electricalresistivity measurements.Inversion of GR-spectroscopy, density, and PEF is performed in

one-step linear inversion. For separate inversion of neutron porosityand electrical resistivity measurements, we introduce a nonlinearnumerical inversion method detailed in Appendix C. The methoditeratively updates estimates of layer-by-layer migration lengthand electrical conductivity. Inputs to the inversion algorithm aremigration length and apparent-conductivity logs instead of neutronporosity and apparent-resistivity logs to improve the sensitivity ofmeasurements to bed physical properties, and thereby to improvethe rate of convergence of nonlinear inversion. Nonlinear inversionis initialized with readings of well logs taken at the center of thecorresponding bed.

Joint inversion of bed physical properties to estimate porosityand volumetric/weight concentrations of mineral/fluidconstituents

After separate inversion of well logs, estimating bed physicalproperties, and constructing an initial multilayer petrophysical mod-el, SNUPAR is used for the layer-by-layer joint inversion of bedphysical properties to assess total porosity, fluid saturations, andvolumetric concentrations of mineral constituents. The inversionis initialized with guess values for porosity and volumetric concen-trations of minerals/fluids at each depth. SNUPAR then calculatescorresponding values of migration length and PEF. Electricalconductivity is calculated separately using a porosity-saturation-resistivity model, which is assumed valid in the formation of interest(e.g., dual water, Waxman-Smits, etc.). Density (equation 12) andUr, Th, and K concentrations are also calculated based on linearcorrelations between volumetric concentrations of minerals withdensity and Ur, Th, and K concentrations of pure minerals. Theinversion iteratively reduces the difference between numericallycalculated and previously estimated values of migration length, den-sity, PEF, electrical conductivity, and concentrations of Th, Ur, andK by updating the volumetric/weight concentrations of minerals/fluids. Appendix D provides additional details about the nonlinearjoint inversion algorithm.

Method 3: Nonlinear joint inversion of individuallayer properties

Method 3 does not take into account shoulder-bed effects on welllogs and assumes that physical properties are equal to center-bedreadings of well logs. This assumption is reliable in formations withthick beds and significantly decreases CPU time in the estimation.This method assumes that bed physical properties (density,migration length, PEF, electrical conductivity, and Ur, Th, and K

WA86 Heidari et al.


concentrations) are equal to center-bed values obtained from welllogs. Subsequently, similar to method 2, SNUPAR is used for thelayer-by-layer joint inversion of bed physical properties to assesstotal porosity, fluid saturations, and volumetric concentrations ofmineral constituents in every bed, wherein each bed is treatedseparately and independently. The main difference between thelayer-by-layer nonlinear joint inversion approaches in methods 2and 3 are the values input to the inversion. In method 2, inputsto the inversion are the results from separate inversion of well logs,whereas in method 3, inputs are center-bed property values calcu-lated from well logs. Method 3 can also be applied at every log-sampling point instead of at every layer. Appendix D providesadditional details about the nonlinear joint inversion algorithm usedto estimate bed petrophysical and compositional properties.

Initial guess of layer-by-layer unknown properties

The joint inversion algorithms described above are implementedwith three possible choices of initial guess: (1) parsimonious (depthconstant) initial guess, (2) initial guess based on results obtainedfrom conventional petrophysical interpretation of well logs and con-ventional/linear multimineral solvers, and (3) initial guess based onXRD/core data. Results obtained from depth-by-depth nonlinear in-version of well logs can also be used as initial guess for method 2,whereas results obtained from method 2 become suitable to initia-lize method 3. The latter choices of the initial guess expedite theconvergence rate and decrease CPU times for methods 2 and 3.We suggest choosing XRD/core data as an initial guess in forma-tions where mineral compositions are complex and/or nonunique-ness of results is prevalent. Results obtained from depth-by-depthnonlinear inversion of well logs or from conventional/linear multi-mineral solvers are good choices for an initial guess when XRD/core data are not available.

Synthetic case 1: Comparison of porosity estimatesobtained with nonlinear and conventional linearinversion methods

The first synthetic case is intended to quantify the reliability andaccuracy of nonlinear inversion to estimate petrophysical propertiesin gas-and water-saturated beds in the absence of mud-filtrate in-vasion. We assume that formations are clay-free, comprised of thicklayers of pure limestone, pure dolomite, and pure quartz. First, weassume that the formation is fully saturated with saline water. Inputwell logs for nonlinear and linear inversion methods are array-in-duction apparent resistivity, neutron porosity, density, PEF, and GRlogs sampled at 0.08 m (0.25 ft). Table 1 summarizes the assumedmatrix and petrophysical properties.This estimation problem is evendetermined because the number

of unknown properties (porosity, saturation, volumetric concentra-tions of shale, quartz, limestone, and dolomite) is equal to the num-ber of well logs (density, neutron porosity, GR, PEF, and apparentelectrical resistivity) plus the number of equality constraints (equa-tion 18). We use method 3 (individual layer-by-layer estimation) forestimation in this case due to the presence of thick beds andnegligible shoulder-bed effects in the center of beds. The initialguess for inversion is chosen close to shale properties to test thestability of the method. We invoke a dual-water porosity-saturation-resistivity model (Clavier et al., 1977) in nonlinear and linear inver-sion methods applied to this example. Figure 3 compares input elec-

trical resistivity and nuclear logs to those used to initialize theinversion (calculated from the initial guess of properties) and simu-lated from final inversion products. The separation between neutronporosity and density logs in the water-saturated limestone zone isdue to connate water salinity. Panels in the same figure describeestimates of volumetric concentrations of mineral constituents, non-shale porosity, and fluid saturations obtained with both nonlinearand conventional linear estimation methods. In the case of fullywater-saturated beds, both linear and nonlinear inversion methodsyield accurate estimates of porosity and volumetric concentrationsof mineral constituents. However, we observe small discrepanciesbetween results obtained with conventional linear and nonlinear in-version methods near bed boundaries. The reason for this behavioris that commercial software does not explicitly account forshoulder-bed effects in the estimation when performing depth-by-depth inversion.Next, we replace water with gas in the same formations to inves-

tigate the effect of gas on nonlinear inversion and conventional lin-ear results. Figure 4 compares input electrical resistivity and nuclearlogs to those used to initialize the inversion and simulated from finalinversion products. Panels in the same figure describe the final es-timates of volumetric concentrations of mineral constituents, non-shale porosity, and fluid saturations obtained with both nonlinearand linear methods. For conventional linear mineralogy evaluation,we adopt the same matrix for pure-component responses calibratedfor the water-saturated example.We remark that, to obtain accurate estimations from commercial

software in the presence of gas, it was necessary to set the volu-metric concentration of shale fixed and equal to its actual value(without this setting, results obtained with linear commercial soft-ware remained inaccurate in the presence of gas). Furthermore, ap-parent resistivity measurements were not explicitly included in thelinear joint inversion of well logs, but rather used to estimate watersaturation after concluding the inversion. Conversely, volumetricconcentration of shale remained unknown, and apparent resistivitylogs were explicitly included in the estimation of properties withnonlinear joint inversion.

Table 1. Synthetic case 1: Summary of assumed Archie’sparameters and matrix, fluid, and formation properties.

Variable Value Units

Archie’s factor, a 1.00 ( )

Archie’s porosity exponent, m 2.00 ( )

Archie’s saturation exponent, n 2.00 ( )

Salt concentration of connate water 160 kppm NaCl

Water density 1.00 g∕cm3

Water viscosity 1.00 cp

In situ gas density 0.19 g∕cm3

In situ gas viscosity 0.02 cp

Formation temperature 230 °F

Volumetric concentration of shale 0.00 ( )

Wellbore radius 14.54 cm

Formation maximum invasion time 0.00 day



When setting volumetric concentration of shale as unknown inboth methods, linear inversion does not properly resolve mineralconcentrations in the dolomite layer. Estimates for volumetric con-centration of quartz, calcite, and shale in the pure dolomite layerobtained with commercial software are 0.28, 0.58, and 0.06, respec-tively. We also observe relative errors of 43% and 55% in theassessment of porosity and water saturation, respectively. The mainreason for such relatively high errors is the nonuniqueness of re-sults. Indeed, by eliminating volumetric concentration of shale inthe set of unknown variables, we decrease nonuniqueness of resultswith commercial software and, consequently, decrease the corre-sponding errors.

Synthetic case 2: Effect of layer thickness on inversionresults in a gas-bearing formation

The second synthetic case is intended to appraise the reliabilityand accuracy of nonlinear inversion when estimating formationproperties in the presence of thin beds. Target layers include in situgas and irreducible water, which have been subject to WBM filtrateinvasion for three days. Solid constituents of permeable beds in-clude quartz, limestone, dolomite, and shale and are separatedby intermediate pure-shale layers. Inversion results are obtainedfor bed thicknesses of 3.05, 0.61, 0.30, and 0.15 m (10, 2, 1,and 0.5 ft, respectively). Well logs are sampled at a rate of0.08 m (0.25 ft). The porosity-saturation-resistivity model usedin this synthetic case is the dual-water model. Table 2 summarizesthe matrix and petrophysical properties assumed for this synthetic

case, whereas Table 3 describes the corresponding assumed rock-fluid properties. Clay composition is exclusively chlorite.Synthetic case 2 involves an evendetermined estimation problem.

Method 1 is used in this case due to the presence of thin beds andmud-filtrate invasion. Nonlinear inversion is initialized with shaleproperties assigned to all beds to evaluate the stability and reliabilityof results. In this synthetic case, we assume that formation proper-ties are known in pure-shale beds. Final inversion products are com-pared to equivalent results obtained with linear inversionimplemented with commercial software.Figure 5 describes estimates of total porosity, total fluid satura-

tion, and volumetric concentrations of quartz, limestone, dolomite,and shale for all permeable layers obtained with nonlinear inver-sion. Error (uncertainty) bars for inverted properties in each layerwere calculated by adding 5% zero-mean Gaussian random noise toall well logs input to the inversion. We observe that the uncertaintyof inversion products due to noisy data increases with decreasinglayer thickness. Figure 6 compares input apparent resistivity andnuclear logs to those used to initialize the inversion and simulatedfrom final inversion products. The same figure compares porosity,water saturation, volumetric concentration of shale, and volumetricconcentrations of mineral constituents estimated with both non-linear and conventional linear inversion methods. Table 4 sum-marizes the results obtained from nonlinear and conventionallinear inversion methods for formations with different thicknessesbut similar petrophysical properties.Table 5 describes the relative errors of inversion products

obtained with both nonlinear and conventional linear inversion

Figure 4. Synthetic case 1, consisting of three thick layers of pure limestone, pure dolomite, and pure quartz. Comparison of final simulatedwell logs (dash-dotted black line), assumed input well logs (solid line), and simulated well logs for the initial guess (dashed line). The left-handpanel shows the spatial distribution (radial and vertical directions) of water saturation for a formation saturated with gas and irreducible water.Results are shown for array-induction resistivity (second left-hand panel), PEF (third left-hand panel), GR (fourth left-hand panel), density andneutron porosity (water-filled limestone porosity units, fifth left-hand panel) logs. The two right-hand panels compare final volumetric con-centrations of mineral constituents, porosity, and water saturation obtained from nonlinear joint (method 3) and conventional linear inversion ofresistivity, density, neutron, GR, and PEF logs.

WA88 Heidari et al.


methods with respect to actual values. Differences between non-linear inversion results and actual values are negligible. On the otherhand, the discrepancy between inversion products obtained withconventional linear inversion and actual values increases with de-creasing bed thickness. Additionally, conventional linear inversionunderestimates hydrocarbon pore volume in all the beds due tomud-filtrate invasion effects. Conversely, inverted products ob-tained with nonlinear inversion differ by less than 3% with respectto actual values when bed thickness is larger than 0.15 m (0.5 ft).Prior to mud-filtrate invasion, permeable beds are saturated with gasand irreducible water. Even though, in this case, nonlinear inversionstill provides accurate estimates of unknown properties, uncertaintybars calculated with 5% Gaussian noise increase by more than100% when compared to the case of no invasion. Indeed, presenceof mud-filtrate invasion decreases the uncertainty of estimated for-mation petrophysical properties as long as the estimation includesthe simulation of the process of invasion. Neglecting the effect ofmud-filtrate invasion in the inversion, however, seriously increasesestimation errors (which is also the case with linear inversion resultsobtained with commercial software).Next, we compare estimations obtained with both nonlinear and

conventional linear inversion methods, assuming the same bed se-quence but with permeable beds fully saturated with saline water.Figure 7 compares inversion products obtained with linear and non-linear inversion to actual properties. Nonlinear inversion yields re-sults with errors smaller than 1%. On the other hand, conventionallinear inversion correctly identifies mineral constituents for the caseof water-saturated beds. However, due to the presence of salty con-nate water, volumetric mineral concentrations obtained with linearinversion exhibit errors above 20% for the thickest bed. Eventhough gas is no longer present in target beds, shoulder-bed effectson well logs increase the error in the volumetric mineral concentra-tions estimated with conventional linear inversion.To investigate the effect of bed-boundary locations in the estima-

tion of unknown properties, we perturbed bed boundaries by � onelog-sampling interval (i.e., 0.08 m [0.25 ft]) in synthetic case 2. Inthe thickest bed, estimation errors due to bed-boundary perturbationare below 5%. However, in beds thinner than 0.61 m (2 ft), errorsincrease to 30% or higher. These exercises emphasize the impor-tance of proper selection of bed boundaries and depth-matchingof well logs to secure reliable inversion results across thin beds.

Synthetic case 3: Thinly bedded gas-bearing formation

The third synthetic case was constructed based on actual fieldmeasurements and is intended to appraise nonlinear inversion re-sults for the case of successive thin beds. It comprises a thinlybedded and shaly carbonate formation and two hydraulically sepa-rated depth intervals. The top zone is a gas-bearing formation, whilethe bottom zone is water-saturated with the same volumetric con-centrations of mineral constituents and petrophysical properties asthe top zone. Drilling mud is WBM and well logs are sampled at arate of 0.08 m (0.25 ft). The porosity-saturation-resistivity modelused in this synthetic case is the dual-water model. Table 6 describesthe assumed matrix and petrophysical properties. Rock-fluidproperties and clay type are similar to those of synthetic case 2(Table 3). Table 7 summarizes the assumed values of porosity, watersaturation, volumetric concentration of shale, and volumetricconcentrations of mineral constituents. We assume that shale prop-

erties are reliable because they were obtained from inversion per-formed across pure-shale depth intervals.Synthetic case 3 also involves an evendetermined estimation pro-

blem. We initialize the nonlinear inversion with a parsimoniousguess close to shale properties and compare the results of nonlinearinversion to those of conventional linear inversion implementedwith commercial software. Figures 8 and 9 describe the spatial dis-tribution of water saturation, actual model properties, and finalestimates obtained with nonlinear inversion after and beforemud-filtrate invasion, respectively. We use methods 2 and 1 inthe cases of estimation before and after mud-filtrate invasion, re-spectively. The same figures compare final layer-by-layer porosity,water saturation, volumetric concentration of shale, and volumetric

Table 2. Synthetic case 2: Summary of assumed Archie’sparameters and matrix, mud, fluid, and formationproperties.





Connate water salt concentration 160 kppm NaCl

Mud-filtrate salt concentration 7.5 kppm NaCl

Bound water resistivity at 230°F 0.018 ohm-m





Mud-filtrate density 1.00 g∕cm3

Mud-filtrate viscosity 1.00 cp


Initial formation pressure 25.16 MPa

Mud hydrostatic pressure 26.20 MPa

Mud-cake reference permeability 0.03 mD

Mud-cake reference porosity 0.30 ( )

Mud solid fraction 0.06 ( )

Mud-cake maximum thickness 1.02 cm

Mud-cake compressibility exponent 0.40 ( )

Mud-cake exponent multiplier 0.10 ( )

Absolute permeability 100 mD

Nonshale porosity 0.15 ( )

Shale porosity 0.12 ( )

Volumetric concentration of quartz 0.30 ( )

Volumetric concentration of limestone 0.15 ( )

Volumetric concentration of dolomite 0.25 ( )

Volumetric concentration of shale 0.15 ( )

Volumetric concentration of clay 0.60 ( )

Initial water saturation 0.15 ( )

Wet shale density 2.64 g∕cm3


Formation maximum invasion time 3.00 days



Figure 5. Synthetic case 2: Comparison of model(actual) values (black solid line), initial guess(green dashed line), and final estimates (reddash-dotted line) of volumetric concentrations ofmineral constituents, porosity, and water satura-tion with corresponding uncertainty bars (calcu-lated with 5% additive zero-mean Gaussianrandom perturbations of input well logs, includingresistivity, density, neutron porosity, GR, andPEF). Panels from left to right show total porosity,volumetric concentration of quartz, volumetricconcentration of limestone, volumetric concentra-tion of dolomite, total water saturation, and volu-metric concentration of shale.

Figure 6. Synthetic case 2 consisting of four nonshale layers with varying thickness in the range of 0.15 m to 3.05 m (0.5 ft to 10 ft) with equalmineral composition and petrophysical properties in all layers. Comparison of final simulated well logs (dash-dotted black line), input well logs(solid line), and simulated well logs for the initial guess (dashed line). The left-hand panel shows the spatial distribution (radial and verticaldirections) of water saturation after three days of WBM-filtrate invasion. Results are shown for array-induction resistivity (second left-handpanel), PEF (third left-hand panel), GR (fourth left-hand panel), density and neutron porosity (water-filled limestone porosity units, fifth left-hand panel) logs. The two right-hand panels compare final volumetric concentrations of mineral constituents, porosity, and water saturationobtained from nonlinear joint (method 1) and conventional linear inversion of resistivity, density, neutron, GR, and PEF logs.

Table 3. Synthetic case 2: Summary of assumed rock-fluid properties (Brooks-Corey’s parameters).

P0c (MPa:Darcy1∕2) ep ( ) k0rh ( ) eh ( ) k0rw ( ) ew ( ) Swr ( ) Shr ( )

0.027 15.0 0.30 3.00 0.90 3.00 0.08 0.15

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concentrations of mineral constituents obtained with nonlinear andconventional linear inversion methods to those of the actual model.We also tested a depth-by-depth implementation of method 3 to thecase of before invasion. Figure 10 compares the actual model andthe corresponding estimates of volumetric concentrations of mineralconstituents, porosity, and water saturation obtained from commer-cial software, method 2, and the depth-by-depth implementation ofmethod 1 for synthetic case 3 before mud-filtrate invasion.Prior to mud-filtrate invasion, array-induction apparent resistivity

logs overlap (“stack”) and there is crossover between neutronporosity and density logs in the gas-bearing zone (Figure 9). How-ever, shoulder-bed effects in thin beds occasionally mask the cross-over, which may lead to underestimation of gas reserves. After threedays of mud-filtrate invasion, a smooth radial variation of watersaturation ensues, which gives rise to separation between array-induction resistivity logs with different radial lengths of investiga-tion (Figure 8). Because radial length of invasion is deeper than0.61 m (2 ft) and larger than the volume of investigation of nuclear

logs, the crossover between neutron porosity and density logsvanishes. The first advantage of nonlinear over conventional linearinversion is its ability to correct for magnified shoulder-bed effectsdue to successive thin beds. Another advantage of nonlinear inver-sion is the explicit assimilation of invasion effects on well logs inthe estimation of properties. The effect of invasion on apparent re-sistivity logs becomes important when the radial invasion front iseither smooth or shallow.Nonlinear inversion yields errors lower than 5% for porosity and

lower than 3% for water saturation in the gas-saturated zone prior tomud-filtrate invasion. After mud-filtrate invasion, errors decrease to2% for porosity, and 3% for water saturation. Overall, the corre-sponding errors in the water-saturated zone decrease comparedto those in gas-saturated zones.Conventional linear inversion yields acceptable estimates of por-

osity, water saturation, and volumetric concentrations of mineralconstituents in the water-saturated zone (errors lower than 30%for water saturation). In the presence of thin beds, however, we

Table 4. Synthetic case 2: Model (actual) properties, nonlinear inversion (NL, Method 1), and conventional linear inversion Lresults for nonshale porosity ϕs, nonshale water saturation Sws, and volumetric concentrations of shale Csh, quartz CQ, limestoneCL, and dolomite CD for each layer.

Layer thickness (m) ϕs ( ) Sws ( ) Csh ( ) CQ ( ) CL ( ) CD ( )

3.05 (10 ft) Actual 0.15 0.15 0.15 0.30 0.15 0.25

NL 0.150 0.150 0.150 0.301 0.152 0.247

L 0.152 0.246 0.134 0.268 0.143 0.300

0.61 (2 ft) Actual 0.15 0.15 0.15 0.30 0.15 0.25

NL 0.150 0.150 0.150 0.301 0.151 0.249

L 0.198 0.313 0.136 0.000 0.000 0.669

0.30 (1 ft) Actual 0.15 0.15 0.15 0.30 0.15 0.25

NL 0.150 0.151 0.150 0.303 0.154 0.243

L 0.183 0.447 0.260 0.000 0.000 0.556

0.15 (0.5 ft) Actual 0.15 0.15 0.15 0.30 0.15 0.25

NL 0.150 0.151 0.150 0.302 0.153 0.245

L 0.110 1.000 0.600 0.141 0.147 0.000

Table 5. Synthetic case 2: Relative errors % for estimations obtained with conventional linear L and nonlinear inversion (NL,method 1) algorithms for nonshale porosity ϕs, nonshale water saturation Sws, and volumetric concentrations of shale Csh,quartz CQ, limestone CL, and dolomite CD for each layer.

Layer thickness (m) ϕs ( ) Sws ( ) Csh ( ) CQ ( ) CL ( ) CD ( )

3.05 (10 ft) NL 0.0 0.0 0.0 0.3 1.3 1.2

L 1.3 64.0 10.7 10.7 4.7 20.0

0.61 (2 ft) NL 0.0 0.0 0.0 0.3 0.7 0.4

L 32.0 108.7 9.3 100.0 100.0 167.6

0.30 (1 ft) NL 0.0 0.7 0.0 1.0 2.6 2.8

L 22.0 198.0 73.3 100.0 100.0 122.4

0.15 (0.5 ft) NL 0.0 0.7 0.0 0.7 2.0 2.0

L 26.7 566.7 300.0 53.0 2.0 100.0



observe significant errors due to shoulder-bed effects. Furthermore,errors as high as 40% arise when quantifying mineral composi-tion across the gas-bearing zone, which deleteriously influenceestimates of porosity and water saturation. Both shoulder-bedand gas effects give rise to errors as high as 70% in estimates ofwater saturation prior to mud-filtrate invasion. To obtain acceptableresults from commercial software, we assumed that volumetric con-centration of shale was known. Unknown parameters were then lim-ited to volumetric concentrations of quartz, calcite, and dolomite,porosity, and water saturation, whereas in other inversion methodsthe volumetric concentration of shale was assumed to be unknown.Results from depth-by-depth nonlinear inversion are also accu-

rate in thick beds (Figure 10). Even though errors correspondingto estimates of porosity, water saturation, and volumetric concen-tration of shale obtained from depth-by-depth inversion are gener-ally lower than 20%, we observe errors higher than 50% inestimates of dolomite and calcite. These significant errors aredue to shoulder-bed effects. In general, in the presence of successivethin beds, method 2 provides the most accurate results among all theinversion methods.After mud-filtrate invasion, conventional linear inversion yields

results with increased errors, as high as 90% and 25% in the esti-mates of water saturation in the gas- and water-bearing zones, re-spectively. Increased errors in estimates of water saturation aftermud-filtrate invasion are due to limitations of commercial softwareto take into account invasion effects in a quantitative manner. Wealso observe error increase in the assessment of volumetric concen-trations of mineral constituents after mud-filtrate invasion.

DISCUSSION

This paper introduced three new methods to estimate porosity,fluid saturation, and volumetric concentrations of mineral constitu-ents from conventional well logs. The main objective of our workwas to introduce inversion methods that can rigorously take intoaccount shoulder-bed and invasion effects on well logs when per-

forming the estimations. Method 1 is the most general, applicable tothe evaluation of invaded and thinly bedded formations. Method 2explicitly accounts for shoulder-bed effects on well logs but doesnot account for mud-filtrate invasion. Method 3 is only recom-mended for evaluation of formations where shoulder-bed andmud-filtrate invasion effects on well logs are negligible. AssociatedCPU times for a 15.24-m (50-ft) formation with 25 beds are fivehours, one hour, and 10 minutes for methods 1, 2, and 3, respec-tively, when implemented on a PC with a 3 GHz processor and4 GB RAM. Compared to conventional petrophysical interpretationmethods, the advantages of the inversion methods introduced in thispaper are the explicit assimilation of shoulder-bed and invasion ef-fects on well logs, and the accurate numerical simulation of welllogs for arbitrary perturbations of petrophysical and mineral/fluidcompositional properties of rock formations. Such advantagesare most clearly evident in cases of thinly bedded formations, com-plex mineral compositions that include shale, presence of invasion,and presence of gas, or else combinations of all of the above.Input data and parameters for the evaluation of invaded forma-

tions include: (1) conventional well logs such as density, neutron

Figure 7. Synthetic case 2: Comparison of final porosity, watersaturation, and volumetric concentrations of shale and mineral con-stituents obtained from nonlinear (center panel) joint inversion(method 1) and conventional linear (right-hand panel) inversionof resistivity, density, neutron, GR, and PEF logs against actualvalues (left-hand panel). The formation is fully saturated with water.Mineral composition and petrophysical properties are equal in allthe four permeable layers. None of the permeable zones is invadedwith mud filtrate.

Table 6. Synthetic case 3: Summary of assumed Archie’sparameters and matrix, mud, fluid, and formationproperties.





Connate water salt concentration 80 kppm NaCl

Bound water resistivity at 230°F 0.018 ohm-m

Mud-filtrate salt concentration 3.5 kppm NaCl





Mud-filtrate density 1.00 g∕cm3

Mud-filtrate viscosity 1.00 cp


Initial formation pressure 25.16 MPa

Mud hydrostatic pressure 26.20 MPa

Mud-cake reference permeability 0.03 mD

Mud-cake reference porosity 0.30 ( )

Mud solid fraction 0.06 ( )

Mud-cake maximum thickness 1.02 cm

Mud-cake compressibility exponent 0.40 ( )

Mud-cake exponent multiplier 0.10 ( )

Absolute permeability 50 mD

Shale porosity 0.10 ( )

Volumetric concentration of clay 0.50 ( )

Wet shale density 2.64 g∕cm3


Formation maximum invasion time 3.00 days

WA92 Heidari et al.


porosity, PEF, GR/GR-spectroscopy, and electrical resistivity;(2) number, type, and chemical formula for mineral and fluid con-stituents; (3) porosity-saturation-resistivity model and associatedparameters (e.g., dual-water model, electrical resistivity of connatewater, tortuosity factor, cementation exponent, saturation exponent,etc.); (4) formation and fluid properties such as density, perme-ability, and saturation-dependent relative permeability and capillarypressure; (5) invasion parameters such as overbalance pressure, timeof invasion, and mud/mud-cake properties; and (6) bed-boundarylocations. Invasion parameters and dynamic petrophysical proper-ties are not needed in the estimation when invasion is negligibleor radially deep. Inversion results consist of porosity, hydrocarbonsaturation, and volumetric/weight concentrations of mineralconstituents.Uncertainty in any of the input parameters affects inversion

results. For instance, uncertainty of four orders of magnitude in per-meability in synthetic case 2 (i.e., permeability of 0.005 mD insteadof 50 mD) causes 37% uncertainty in estimates of water saturationfor the thickest bed (3.05 m [10 ft]). This uncertainty increased to55% in the thinnest bed (0.15 m [0.5 ft]). Estimates of porosity andvolumetric concentrations of mineral constituents, however, exhib-ited less sensitivity to uncertainty in permeability (uncertainty lowerthan 20% in the assessment of volumetric concentrations of mineralconstituents and lower than 5% in the assessment of porosity).In synthetic case 1, however, it was found that the sensitivityof estimated parameters to one order of magnitude uncertainty inpermeability was negligible.Another cause of uncertainty of inversion results is noisy well

logs. We showed that this uncertainty increases across thin beds.The regularization parameter included in the three inversion algo-rithms is intended to stabilize the estimations in the presence of

Table 7. Synthetic case 3: Values assumed for nonshaleporosity ϕs, nonshale water saturation Sws, and volumetricconcentrations of shale Csh, quartz CQ, limestone CL, anddolomite CD for each layer.

Layerthickness (m) ϕs ( ) Sws ( ) Csh ( ) CQ ( ) CL ( ) CD ( )

0.91 (3 ft) 0.00 1.00 1.00 0.00 0.00 0.00

0.91 (3 ft) 0.15 0.20 0.19 0.25 0.30 0.11

0.61 (2 ft) 0.12 0.25 0.03 0.35 0.23 0.27

0.30 (1 ft) 0.14 0.22 0.19 0.12 0.32 0.23

0.30 (1 ft) 0.10 0.26 0.18 0.23 0.15 0.34

0.91 (3 ft) 0.08 0.28 0.18 0.31 0.25 0.18

0.61 (2 ft) 0.04 0.32 0.17 0.28 0.16 0.35

0.91 (3 ft) 0.06 0.30 0.20 0.21 0.24 0.29

0.91 (3 ft) 0.00 1.00 1.00 0.00 0.00 0.00

0.91 (3 ft) 0.15 0.75 0.19 0.25 0.30 0.11

0.61 (2 ft) 0.12 0.85 0.03 0.35 0.23 0.27

0.30 (1 ft) 0.14 0.80 0.19 0.12 0.32 0.23

0.30 (1 ft) 0.10 0.87 0.18 0.23 0.15 0.34

0.91 (3 ft) 0.08 0.90 0.18 0.31 0.25 0.18

0.61 (2 ft) 0.04 0.95 0.17 0.28 0.16 0.35

0.91 (3 ft) 0.06 0.92 0.20 0.21 0.24 0.29

0.91 (3 ft) 0.00 1.00 1.00 0.00 0.00 0.00

Figure 8. Synthetic case 3, consisting of multilayer, successive thin formations in two hydraulically separated gas-bearing and water-bearingzones. Comparison of final simulated well logs (dash-dotted black line) and input well logs (solid line). The left-hand panel shows the spatialdistribution (radial and vertical directions) of water saturation after three days of WBM-filtrate invasion. Results are shown for array-inductionresistivity (second left-hand panel), PEF (third left-hand panel), GR (fourth left-hand panel), density and neutron porosity (water-filled lime-stone porosity units, fifth left-hand panel) logs. The three right-hand panels compare volumetric concentrations of shale and mineral con-stituents, porosity, and water saturation obtained from nonlinear joint (method 1) and conventional linear inversion of resistivity, density,neutron, GR, and PEF logs against actual values.



noisy and/or inadequate data. Large regularization parameter valuesprovide a smooth solution, whereas small values cause oscillatorysolutions. Hansen’s (1994) L-curve method estimates the regular-ization parameter in each iteration based on the difference betweenactual measurements and their numerical simulations. Conse-quently, the regularization parameter takes large values at the initialiterations to guarantee convergence to actual properties; it graduallydecreases to secure stable convergence to the solution, especially inclose proximity to the minimum of the quadratic cost function.When the total number of unknown properties and equality

constraints is equal or less than the number of available welllogs, the inversion is rendered evendetermined or overdetermined.Accurate and reliable estimates of petrophysical/compositional

properties are possible in evendetermined or overdetermined inver-sions when input parameters are reliable. We showed that inversionresults for such cases were stable and independent of the initialguess. In the case of evendetermined or overdetermined inversions,results remained accurate even when assuming mineral componentswhich were not present in the examined rock formations.When the total number of unknown properties and equality con-

straints are greater than the number of available well logs, the in-version is rendered underdetermined. In those cases, nonuniquenessof results is a significant concern and, consequently, the choice ofinitial guess becomes critical to secure accurate and reliable results.Two approaches to overcome this problem are: (1) choosing an in-itial guess close to actual values and (2) performing the inversion

Figure 9. Synthetic case 3, consisting of multilayer, successive thin formations in two hydraulically separated gas-bearing and water-bearingzones. Comparison of final simulated well logs (dash-dotted black line) and input well logs (solid line). The left-hand panel shows the spatialdistribution (radial and vertical directions) of water saturation before WBM-filtrate invasion. Results are shown for array-induction resistivity(second left-hand panel), PEF (third left-hand panel), GR (fourth left-hand panel), density and neutron porosity (water-filled limestone porosityunits, fifth left-hand panel) logs. The three right-hand panels compare volumetric concentrations of shale and mineral constituents, porosity,and water saturation obtained from nonlinear joint (method 2) and conventional linear inversion of resistivity, density, neutron, GR, and PEFlogs against actual values.

Figure 10. Synthetic case 3: Comparison of finalporosity, water saturation, and volumetric concen-trations of shale and mineral constituents obtainedfrom conventional linear (right-hand panel), meth-od 2 (second right-hand panel), and depth-by-depth implementation of method 3 (thirdright-hand panel) against actual values (fourthright-hand panel). The left-hand panel shows thespatial distribution (radial and vertical directions)of water saturation before WBM-filtrate invasion.

WA94 Heidari et al.


with different initial guesses. We suggested different choices forinitial guess close to actual formation properties to avoid trappinginto local minimums as well as to expedite convergence. In the caseof underdetermined inversion, a wrong choice of assumed mineralcomponents caused significant errors in inversion results. Experi-ence shows that the corresponding error is smaller for porosity,water saturation, and volumetric concentration of shale, than forvolumetric concentrations of mineral constituents. The uncertaintyof inversion products depends on the type and properties of as-sumed minerals, petrophysical properties, and on the initial guess.To secure reliable petrophysical and compositional results, we

suggest inputting the maximum number of available well logs tothe inversion. However, it is possible that some well logs are notreliable due to noise (e.g., PEF in the presence of barite inmud). The effect of unreliable well logs on inversion results canbe controlled by the data weighting matrix. Absence of PEF,GR, or density in underdetermined problems can change estimatedproperties by more than 50% but does not affect estimates of por-osity and water saturation significantly (synthetic case 1). Absenceof electrical resistivity measurements, however, has a significant in-fluence on estimates of porosity and water saturation (changes of25% for porosity and 200% for water saturation). Among all miner-al constituents, shale concentration is the most stable outcome ofinversion whenever GR is available.We quantified the accuracy of the introduced inversion methods

when interpreting petrophysical and compositional properties ofthinly bedded formations. It was shown that inversion methodsyielded accurate results only when beds were thicker thantwice the log-sampling interval. Furthermore, uncertainty in bed-boundary locations significantly affected inversion results for thinbeds. For instance, it was found that bed-boundary uncertaintyequal to one log-sampling interval in beds as thick as four timesa log-sampling interval affected inversion results by more than 30%.

CONCLUSIONS

We introduced and successfully tested new nonlinear inversionmethods to estimate porosity, fluid saturation, and volumetricmineral concentrations from combinations of well logs. The estima-tion was made possible by a recently developed algorithm to rapidlysimulate nuclear logs. Synthetic examples indicate that the newnonlinear inversion methods are effective to reduce shoulder-bedand invasion effects on logs in the estimation compared to commer-cial software based on linearized well-log responses. Moreover,we showed that nonlinear inversion is reliable when permeablebeds include gas saturation and complex mineralogy because thesimulation of well logs accurately reproduces the generally non-linear relationship between formation and fluid properties and mea-surements. It was also shown that the integration of the process ofmud-filtrate invasion in the estimation of petrophysical properties isimportant when electrical resistivity logs indicate a relativelysmooth radial invasion profile. Even though commercial softwarebased on linearized inversion is fast and accurate in siliciclastic se-quences with clean and thick beds, it does not perform well in in-vaded and/or thinly bedded formations, which include complexmineral compositions and light hydrocarbons.The number and type of available well logs, location of bed

boundaries, and the number of unknown properties influence theresults obtained with nonlinear inversion. Results are stable andaccurate when noise-free density, neutron porosity, PEF, GR, and

electrical resistivity logs are available for the estimation. Sensitivityanalysis indicated that perturbations of bed-boundary locationsequal to one log-sampling interval could cause relative errors higherthan 30% when estimating unknown properties of beds thinner than0.61 m (2 ft). This error becomes negligible in thick beds. Resultsindicate that proper estimation of bed boundaries and correction ofdepth-matching errors on well logs are important to secure reliableinversion results in thin beds.Combining the nonlinear inversion methods with neutron

spectroscopy measurements is a good alternative to improve thedetection and estimation of mineral volumetric concentrations.We anticipate that nonlinear inversion could be effective in the pet-rophysical interpretation of carbonate formations and unconven-tional plays such as tight-gas sands and gas/oil shales, wheremineral/fluid constituents exhibit significant spatial variability.

NOMENCLATURE

List of Symbols

a = Archie’s factor ( )Ccl = Volumetric concentration of clay ( )CD = Volumetric concentration of dolomite ( )CL = Volumetric concentration of limestone ( )CQ = Volumetric concentration of quartz ( )Csh = Volumetric concentration of shale ( )CðxÞ = Cost function ( )d = Vector of simulated logsdm = Vector of measured or model logseh = Hydrocarbon phase experimental exponent for

Brooks-Corey’s equation ( )ep = Pore-size distribution exponent ( )ew = Water phase experimental exponent for Brooks-

Corey’s equation ( )K = Matrix of FSFs for density, PEF, or GR-

spectroscopy toolskrh = Hydrocarbon phase relative permeability ( )k0rh = krh end point ( )krw = Water relative permeability ( )k0rw = krw end point ( )Lm = Neutron migration length (cm)m = Archie’s porosity exponent ( )n = Archie’s saturation exponent ( )nb = Number of beds ( )nc = Number of mineral constituents ( )nl = Number of well logs ( )nsp = Number of sampling points in each well log ( )nu = Number of unknowns ( )p = Vector of simulated layer-by-layer propertiespm = Vector of model layer-by-layer propertiesp0 = Vector of reference layer-by-layer model

propertiesPc = Capillary pressure (MPa)P0c = Coefficient for Pc equation (MPa:Darcy1∕2)

R = Electrical resistivity measurements (ohm-m)Sh = Total hydrocarbon saturation ( )Shr = Residual hydrocarbon saturation ( )Sw = Total water saturation ( )Swr = Irreducible water saturation ( )Sws = Nonshale water saturation ( )



tinv = Time of mud-filtrate invasion (hours, days)U = Volumetric photoelectric cross section ( )Vcl = Volume of clay (cm3)Vsh = Volume of wet shale (cm3)Vsilt = Volume of wet silt (cm3)Vr = Volume of rock including fluids (cm3)Wd = Data weighting matrix ( )α = Regularization parameter ( )Δt = Sonic interval transient time (ms)ϕD = Density porosity ( )ϕD;sh = Density porosity in a pure shale zone ( )ϕshD = Density porosity corrected for shale ( )

ϕN = Neutron porosity ( )ϕN;sh = Neutron porosity in a pure shale zone ( )ϕshN = Neutron porosity corrected for shale ( )

ϕs = Nonshale porosity ( )ϕsh = Shale-porosity ( )ϕt = Total porosity ( )ρb = Bulk density (g∕cm3)ρcl = Clay density (g∕cm3)ρf = Fluid density (g∕cm3)ρh = Hydrocarbon density (g∕cm3)ρsh = Shale density (g∕cm3)ρsilt = Silt density (g∕cm3)ρw = Water density (g∕cm3)σ = Electrical conductivity (S∕m)

List of Acronyms

AIT® = Array-induction toolCPU = Computer processing unitDLT® = Dual laterolog toolFSF = Flux sensitivity functionGR = Gamma-rayK = Potassiumkppm = Kilo parts per millionL = LinearLS = Long-spacedLWD = Logging-while-drillingMID = Matrix identificationNL = NonlinearOBM = Oil-base mudPEF = Photo electric factorSNUPAR = Schlumberger nuclear parameter codeSS = Short-spacedT = Transpose of a matrixTh = ThoriumUr = UraniumUTAPWeLS = University of Texas at Austin’s

petrophysical and well-log simulatorWBM = Water-base mudXRD = X-ray diffraction1D = One-dimensional2D = Two-dimensional

ACKNOWLEDGMENTS

The work reported in this paper was funded by The University ofTexas at Austin’s Research Consortium on Formation Evaluation,jointly sponsored by Anadarko, Apache Corporation, Aramco,

Baker Hughes, BG, BHP Billiton, BP, Chevron, ConocoPhillips,ENI, ExxonMobil, Halliburton, Hess, Maersk, Marathon OilCorporation, Mexican Institute for Petroleum, Nexen, Pathfinder,Petrobras, Repsol, RWE, Schlumberger, Shell, Statoil, TOTAL,and Weatherford. We thank John Quirein for technical discussionsconcerning linear and quasi-linear estimation of mineral and fluidconcentrations from well logs. A note of special gratitude goes toAlberto Mendoza and Olabode Ijasan for their technical assistancein the numerical simulation of nuclear logs. Our heartfelt thanks goto Chris Skelt, Andre Revil, and one anonymous reviewer for theirconstructive technical and editorial comments that improved thequality of the original manuscript.

APPENDIX A

DEFINITION OF UNKNOWN PARAMETERSESTIMATED WITH JOINT INVERSION

OF WELL LOGS

To enforce the constraints expressed in equations 17 and 18, weinvoke volumes/weights instead of volumetric/weight concentra-tions of minerals and fluids. Additionally, working in logarithmicspace ensures nonnegative estimates of unknown properties.Consequently, we transform the unknown variable vector xi tovector x 0

i , given by

x 0i ¼ ½log V1;i; log V2;i; : : : ; log Vnc ;i; log Vsh;i; log Vps;i; log Sw;i�T ;

(A-1)

where Vj;i is volume of the jth mineral in the ith bed, Vsh;i is vo-lume of shale in the ith bed, Sw;i is total water saturation in the ithbed, and Vps;i is volume of nonshale pore space in the ith bed. Therelationships between volumes and volumetric concentrations ofminerals and fluids are given by

Cj;i ¼Vj;iPnc

j¼1 Vj;i þ Vsh;i þ Vps;i; (A-2)

Csh;i ¼Vsh;iPnc


and

ϕs;i ¼Vsh;iPnc


where Cj;i is volumetric concentration of the jth mineral in the ithbed, ϕs;i is nonshale porosity in the ith bed, and Csh;i is volumetricconcentration of shale in the ith bed.There is no need to enforce an explicit constraint on the summa-

tion of volumes when volumetric concentrations are transformed tovolumes in the minimization process. However, we implicitly satis-fy the constraint in equation 18 by applying equations A-2–A-4 tothe estimated volumes. Normalized volumes are then used as inputsto the forward simulation of well logs at each iteration of the non-linear minimization process.

WA96 Heidari et al.


APPENDIX B

SEPARATE LINEAR INVERSION OF DENSITY,PEF, AND GR-SPECTROSCOPY LOGS TOESTIMATE PHYSICAL PROPERTIES OF

INDIVIDUAL LAYERS

Separate inversions are performed for density, PEF, and GR-spec-troscopy logs to estimate layer-by-layer values of density, PEF, andUr, Th, and K concentrations, respectively. Input data consist ofmeasured well logs. The objective of the separate inversion is to“sharpen” each of the logs by reducing shoulder-bed effects. Itis assumed that bed boundaries have been detected prior to perform-ing the inversion. Bed physical properties are estimated by minimiz-ing the quadratic cost function given by

CðpÞ ¼ kdðpÞ − dmk22 þ α2kp − p0k22; (B-1)

where dðpÞ is the vector of numerically simulated density, PEF, GR-spectroscopy, electrical resistivity, or neutron porosity logs, dm isthe vector of available well logs, α is a regularization (stabilization)parameter, p designates the vector of corresponding layer-by-layerproperties (e.g., density, PEF, Ur, Th, and K concentrations, neutronmigration length, and electrical conductivity), and p0 is a referencemodel vector that can be selected based on (1) center-bed values,(2) the average of well logs in each bed, or (3) the average of welllogs in the entire depth interval considered for inversion. Vectors dand p are given by

d ¼ ½d1; : : : ; dnsp �T; (B-2)

and

p ¼ ½p1; : : : ; pnb �T; (B-3)

respectively, where the superscript T indicates transpose, nsp is thenumber of sampling points in each well log, and nb is the numberof beds.The stabilization parameter in equation B-1 provides a way to

adjust the relative importance of fitting the data and estimating arelatively smooth vertical distribution of properties; it is obtainedwith Hansen’s (1994) L-curve strategy. A data-weighting matrixis not needed here because inversion is applied separately to eachof the well logs. For density, PEF, and GR-spectroscopy measure-ments, we assume a deterministic linear dependence between bedphysical properties (i.e., density, PEF, and Ur, Th, and K concen-trations) and measured logs. This linear dependence is formallyexpressed as

dðpÞ ¼ Kp; (B-4)

whereK designates a matrix of precalculated FSFs for density, PEF,or GR-spectroscopy measurements. For more accurate density in-version results, one can opt to treat short-spaced and long-spaceddensity measurements as separate input measurements. Mendozaet al. (2009) implemented the same linear inversion method for mul-tisector LWD density measurements.In the case of a linear relationship between bed physical proper-

ties and measured logs, equation B-1 is solved via

½KTKþ α2I�p ¼ KTdm þ α2Ip0; (B-5)

where I is the unity matrix.

APPENDIX C

SEPARATE NONLINEAR INVERSION OF NEU-TRON AND APPARENT RESISTIVITY LOGS

The functional relationships between (1) neutron porositymeasurements and layer-by-layer neutron migration length, and(2) electrical resistivity measurements and layer-by-layer resistivityare not linear. Thus, equation B-4 is not valid for electricalresistivity and neutron porosity measurements. To circumvent thatproblem, we minimize the quadratic cost function (equation B-1)using a nonlinear approach. This is accomplished by calculating nu-merically (finite differences) the entries of the Jacobian matrix atevery linear iteration and by implementing the Levenberg-Marquardt minimizationmethod (Marquardt, 1963). In doing so, lin-ear iterative refinement (Mendoza et al., 2010) is used for the rapidnumerical simulation of neutron porosity logs. For the case of inver-sion of AIT resistivity logs, the corresponding entries of the Jacobianmatrix are obtained via finite-difference calculations of apparentresistivity logs. A similar approach could be implemented withraw borehole-corrected electrical conductivity measurements.To improve the convergence rate of the minimization method, we

transform neutron porosity logs into migration-length logs, andperform the inversion to yield layer-by-layer values of migrationlength. Likewise, the inversion of apparent resistivity logs isimplemented with apparent conductivity logs (inverse of apparentresistivity logs) to yield layer-by-layer values of electricalconductivity.

APPENDIX D

NONLINEAR JOINT INVERSION OF INDIVIDUALLAYER PROPERTIES

The objective of joint inversion of layer-by-layer density, migra-tion length, electrical conductivity, PEF, and GR-spectroscopy va-lues is to estimate porosity, water saturation, and volumetric/weightconcentrations of mineral constituents. The inversion is performedfrom layer-by-layer properties obtained from the separate inversionof well logs described in Appendices B and C.Physical properties for each layer are estimated by minimizing

the layer-by-layer quadratic cost function given by

CðxÞ ¼ kWd½pðxÞ − pm�k22 þ α2kxk22; (D-1)

subject to

0 ≤ xi ≤ 1; (D-2)

and

Xnci¼1

Ci þ Csh þ ϕs ¼ 1; (D-3)

where Wd is a data-weighting matrix, pðxÞ is the vector ofphysical properties in each bed (e.g., density, PEF, Ur, Th, and



K concentrations, neutron migration length, and electrical conduc-tivity), pm is the vector of model physical properties estimated fromwell logs in each bed, α is a regularization (stabilization) parameter,nc is the predefined number of mineral constituents, Ci is volu-metric concentration of the assumed mineral constituents, Csh is vo-lumetric concentration of shale, ϕs is nonshale porosity, and x is thevector of unknown formation properties, given by

x ¼ ½C1; C2; : : : ; Cnc; Csh;ϕs; Sw�T; (D-4)

where Sw is total water saturation. The vector of numerically simu-lated layer-by-layer properties is given by

p ¼ ½Lm; ρb; PEF;σ; Ur; Th; K�T; (D-5)

where Lm is neutron migration length, ρb is bulk density, and σdesignates apparent electrical conductivity logs with variable radiallengths of investigation (multiple logs depending on the type of log-ging tool used to acquire resistivity measurements).The data-weighting matrix in equation D-1 controls the impor-

tance of each physical property included in the inversion; it isgiven by

Wd ¼

2664

1pm;1

: : : 0

..

. . .. ..

.

0 : : : 1pm;nl

3775; (D-6)

where nl is the number of available well logs.We minimize the cost function defined in equation D-1 with the

Levenberg-Marquardt method (Marquardt, 1963) and use SNUPARcommercial software (McKeon and Scott, 1989) to estimate neutronmigration length and PEF and for the numerical calculation (finitedifferences) of the corresponding entries of the Jacobian matrix(partial derivatives of the entries of vector p with respect to entriesof vector x). The stabilization parameter α included in equation D-1is selected via Hansen’s (1994) L-curve method.

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http://dx.doi.org/10.1109/23.34634

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Improved estimation of mineral and fluid volumetric ...calculate volumetric mineral and fluid concentrations together with their uncertainty (Busch et al., 1987). Even though constrained