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Ž .Journal of Applied Geophysics 45 2000 203–223www.elsevier.nlrlocaterjappgeo

Hydraulic well testing inversion for modeling fluid flow infractured rocks using simulated annealing: a case study at

Raymond field site, California

Shinsuke Nakao a,), Julie Najita b,1, Kenzi Karasaki b,1

a Geological SurÕey of Japan, 1-1-3 Higashi, Tsukuba, Ibaraki, 305-8567, Japanb Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Received 2 August 1999; accepted 23 June 2000

Abstract

Ž . Ž .Cluster variable aperture CVA simulated annealing SA is an inversion technique to construct fluid flow models infractured rocks based on the transient pressure data from hydraulic tests. A two-dimensional fracture network system isrepresented as a filled regular lattice of fracture elements. The algorithm iteratively changes element apertures for a clusterof fracture elements in order to improve the match to observed pressure transients. This inversion technique has been appliedto hydraulic data collected at the Raymond field site, CA to examine the spatial characteristics of the flow properties in afractured rock mass. Two major conductive zones have been detected by various geophysical logs, geophysical imagingtechniques and hydraulic tests; one occurring near a depth of 30 m and the other near a depth of 60 m. Our inversion resultsshow that the practical range of spatial correlation for transmissivity distribution is estimated to be approximately 5 m in theupper zone and less than 2.5 m in the lower zones. From the televiewer and other fracture imaging logs it was surmised thatthe lower conductive zone is associated with an anomalous single open fracture as compared to the upper zone, which is anextensive fracture zone. This would explain the difference in the estimated practical range of the spatial correlation fortransmissivity. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Hydraulic well test; Inversion; Fluid flow model; Fractured rocks; Simulated annealing

1. Introduction

For engineering concerns such as environmentalremediation, it is quite important to characterize

) Corresponding author. Fax: q81-298-61-3702.Ž .E-mail addresses: [email protected] S. Nakao ,

Ž [email protected] J. Najita , [email protected]Ž .K. Karasaki .

1 Fax: q1-510-486-5686.

subsurface fractures that control groundwater flowand transport properties of fractured rocks. Pressureinterference testing due to pumping or injection us-ing multiple wells is a useful and direct method tocollect information on transmissivities of the fracturesystem. Once pressure transient data is obtained, aninverse method can be applied to match the observeddata and hence to construct a hydrological modelŽCarrera and Neuman, 1986; Finsterle, 1993;

.Doughty et al., 1994 .

0926-9851r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0926-9851 00 00029-X

( )S. Nakao et al.rJournal of Applied Geophysics 45 2000 203–223204

Hydrological characterization of fractured rocks isextremely difficult because fracture distributions arehighly heterogeneous and discontinuous. If the rockis extensively fractured and all the fractures arehydrologically active, the medium would behave likean equivalent porous medium. However, cross-holeand tracer tests show that this is not usually the caseŽ .Billaux et al., 1989 . In the case where the rockmatrix can be considered impermeable, major hy-draulic behavior is governed by the geometry of thefracture network. In order to overcome some of theseproblems, after extracting fracture zones by geologi-cal and geophysical methods, fracture network mod-els with a partially filled lattice were used to charac-

Žterize the heterogeneity of the zone Long et al.,.1991 . Our study follows this approach.

Ž .Simulated annealing SA , on the other hand, is astochastic search method and has attracted attentionas a scheme for optimizing the objective function,because the algorithm selects parameters outside theneighborhood of a local minimum. Although in prac-tical applications obtaining the global minimum isnot guaranteed, near-optimal solutions can be oftenfound by representing SA as a Markov chain with

Žadequate parameters Geman and Geman, 1984; van.Laarhoven and Aarts, 1987 . SA has been used in a

Žvariety of optimization problems Kirkpatrick et al.,.1983 , and also been applied to geophysical explo-Žration e.g. Sen and Stoffa, 1991; Vasudevan et al.,

.1991 , and hydrology to develop a groundwater man-Ž .agement strategy Dougherty and Marryott, 1991

Žand stochastic reservoir modeling Deutsch and Jour-.nel, 1994 .

In hydrological applications, Lawrence BerkeleyŽ .National Laboratory LBNL has been developing an

Žinverse method for well test data using SA Mauldon.et al., 1993 . A partially filled lattice represents a

fracture network and an individual fracture elementis changed randomly from conductive to nonconduc-tive or vice versa. The effect of this change isexamined by numerically simulating well tests andcomparing simulations with field test data. JacobsenŽ . Ž .1993 and Najita and Karasaki 1995 developed a

Ž .SA method called cluster variable aperture CVASA that uses a distribution of fracture apertures. Thealgorithm changes element transmissivities by chang-ing element apertures following the cubic law. Inaddition, the algorithm will change the property of a

cluster of elements instead of a single element. Clus-ter size and shape, which represent simplified frac-ture sets, are the parameters specified in the inver-sion. Sensitivity studies, investigating optimal clustersize using synthetic models with spatially correlatedtransmissivity, showed that the optimal cluster sizeseems to be 20–40% of the practical range of spatial

Žcorrelation for transmissivity distribution Nakao et.al., 1999 .

To develop multi-disciplinary field testing tech-niques and analysis methods for characterizing hy-draulic properties of fractured rocks, a dedicatedfield site was established near the town of Raymond,

Ž .CA Karasaki et al., 1994 . Questions that are beingŽ .addressed at this site include: 1 the number of

boreholes, the number and type of the hydrologicand geophysical well tests needed to characterize a

Ž .given volume of rock, 2 the possibility of predict-Ž .ing fluid transport based on fracture geometry, 3

how to scale-up the observations made at a smallerŽ .scale, and 4 how to relate geometric fracture infor-

mation from outcrops and boreholes to hydrology.Various geophysical and hydrologic tests have beenconducted in a cluster of nine wells at this site toimage the hydrologic connections of a fractured rockmass. The results from these tests indicate that flowis mainly confined in the two dominant upper and

Ž .lower conductive zones Cohen, 1993 . The prelimi-nary result of hydrologic modeling for the upperconductive zone using SA was presented in Nakao et

Ž .al. 1999 . In this article, comprehensive results ofhydrologic modeling using SA will be presented toexamine the spatial characteristics of the flow prop-erties in both the upper and lower conductive zonesat the Raymond field site. We will start with a briefdescription of the Raymond field site, then proceedto describe the hydraulic well testing inversion usingSA and the application to the field test data.

2. Raymond field site

The Raymond field site is located in the foothillsof the Sierra Nevada, approximately 3.2 km east ofRaymond, CA. The site lies within the Knowlesgranodiorite which is light-gray, equigranular andnon-foliated, and is widely used as a building mate-

( )S. Nakao et al.rJournal of Applied Geophysics 45 2000 203–223 205

Žrial in California Bateman and Sawka, 1981; Bate-.man, 1992 . Nine boreholes were drilled in an in-

verted V pattern with increasing spacing betweenŽ .boreholes Fig. 1a . Driller’s logs indicate that rela-

tively unweathered granite is located beneath lessthan 8 m of soil and regolith. The wells are cased toapproximately 10 m and vary in depth between 75and 100 m. The water level is normally between 2and 3 m below the top of casing.

Various geophysical logs, geophysical imagingtechniques and hydraulic tests have been conductedto image the hydrologic connection of the fractured

Žrock mass Cohen, 1993; Karasaki et al., 1995;.Cook, 1995; Cohen, 1995; Vasco et al., 1996 . Re-

gional characterization and site specific fracturemeasurements show that there are two sets of subver-tical tectonic fractures: one set strikes at N30W and

Ž .the other strikes at N60E Cohen, 1995 . The current

Ž . Ž .Fig. 1. a Well configuration at the Raymond field site. Large dots and small dots indicate 25 and 15 cm diameter wells, respectively. bConceptual model of hydrogeologic structure.


Ž .conceptual model Fig. 1b consists of two dominantconductive zones: one occurring near a depth of 30m and the other between 54 and 60 m. The presenceof these zones are also imaged by ground penetratingradar reflection and a seismic tomography surveyŽ .Vasco et al., 1996 . It should be noted that thesesurveys respond to different physical properties, i.e.the seismic method responds to a rock stiffnesscontrast, while the radar responds to the electro-mag-netic properties of the media. Furthermore, the re-sults from hydraulic tests indicate that there is a highdegree of heterogeneity in transmissivity distribu-

Ž .tions within the two conductive zones Cohen, 1993 .The main purpose of the application of SA is tocharacterize the heterogeneity of transmissivity dis-tributions within the zones.

3. Analysis method

3.1. SA in general

SA has its origins in thermodynamics and themanner in which liquid metals cool and anneal. Inphysical annealing, a metal is heated and allowed tocool very slowly in order to obtain a regular molecu-lar configuration having the lowest possible energystate. If the temperature T is held constant, thesystem approaches thermal equilibrium and the prob-ability distribution for the configuration with energy

Ž .E approaches the Boltzmann probability Pr E sexpŽ .yErk T ; where k is the Boltzmann constant.b b

Ž .Metropolis et al. 1953 first introduced a simplealgorithm to incorporate these ideas into numericalcalculations. The following criterion known as theMetropolis algorithm is applied in determiningwhether a transition to another configuration occursat the current temperature. For the ny1st configura-tion X and the nth configuration X with ener-ny1 n

gies E and E , respectively, the transition proba-ny1 n

bility at some system temperature T is given by

Pr X ™XŽ .ny1 n

1 if E yE -0n ny1s ½ exp y E yE rT if E yE )0.Ž .n ny1 n ny1

1Ž .

This criterion always allows a transition to a config-uration if system energy is decreased and sometimesallows a transition to a configuration with higherenergy. This stochastic relaxation step allows SA tosearch the space of possible configurations withoutalways converging to the nearest local minimum. InSA, the objective function for an optimization prob-lem is analogous to energy state and the set of free

Ž .parameters configuration is analogous to the ar-rangement of molecules. ATemperatureB is simply acontrol parameter in a given optimization problem.We will refer to the objective function as Aobjective

Ž .function energy B and also refer to the temperatureas Acontrol parameter T B in the following sentences.

In general, the SA algorithm consists of the fol-Ž .lowing tasks: 1 generate or randomly change sys-

Ž .tem configuration, 2 calculate values of the objec-Ž . Ž .tive function energy , 3 perform the Metropolis

algorithm to determine whether a new configurationŽ .is accepted or not, and 4 adjust the current control

Ž .parameter T according to the annealing coolingschedule.

Several choices of annealing schedule are possi-ble. A computationally practical schedule is the

Ž .widely used decrement rule Press et al., 1986 .Given an initial control parameter T , assign0

T sT a k ; ks0,1,2,3 . . . , 2Ž .k 0

where a is between 0 and 1. This general form hasbeen implemented by others with values of a rang-

Žing from 0.5 to 0.99 e.g. van Laarhoven and Aarts,.1987 . In this schedule, the current control parameter

T is kept fixed until a finite number of transitions,L , have been accepted, then the parameter T isk

lowered.

3.2. Implementation to well test inÕersion

In order to use SA for the inversion of well tests,we consider a two-dimensional fracture networkmodel as shown in Fig. 2. Fractures at multipledepths are not considered. Each element represents asimplified fracture that controls the transmissivity.

Ž 2 . Ž y1 .Transmissivity m rs , specific storage m , aper-Ž . Ž .ture m , unit thickness 1 m and length are imposed

on each element. Well locations are specified at nodeŽpoints and pressure transients hydraulic head


Ž .Fig. 2. Fracture network models: each element represents a simplified fracture. Example of an isotropic fracture cluster left and anŽ .anisotropic fracture cluster right .

.changes due to pumping or injection are calculatedat each node using the finite element code TRINETŽ .Karasaki, 1987 . Note that the neither SA algorithmnor TRINET is limited to the two-dimensional model.We use a two-dimensional model because of thecomplexity and large computational requirements fora 3-D representation. Fluid flow along fractures isassumed to be laminar. It is also assumed that thetransmissivity of any fracture element follows the

Ž 3cubic law Tsr gb r12m — r: density of water,g: acceleration due to gravity, m: dynamic viscosity

.of water, b: aperture . Whether the cubic law isapplicable here is not pertinent to the present study,although the subject itself is very important.

In CVA SA, the objective is to find a near-opti-mal fracture network geometry by modifying clusters

Ž .of element apertures hence transmissivities and cal-culating pressure transient curves in order to simulta-neously match observed head data at observationwells. At each step of the algorithm, a cluster ofelements is randomly selected using the Wolff algo-

Ž .rithm Wolff, 1989 and their aperture is changed toan aperture chosen at random from a discretizeddistribution. In this study, we use a finite list ofaperture choices. The number and the values of

replacement apertures are user-specified parameters.In the case of Fig. 2, two discretized apertures, thickŽ . Ž .high transmissivity and thin low transmissivitylines, are specified. The transmissivity of the clusterof elements is also updated according to the cubiclaw. This step is analogous to the perturbation step inthe general SA algorithm.

Cluster elements are dependent on the maximumcluster size and the location of the cluster origin. Thecluster origin is an element chosen at random fromthe fracture network. In the first stage of clusterformation, all conductive elements that are connectedto the cluster origin are determined. Each of theseneighbors is added as a new cluster element. If themaximum cluster size is not reached, the algorithmproceeds to a second stage where conductive ele-ments connected to new cluster elements are identi-fied and accepted as new cluster elements as long asthe element is not already in the cluster. This processcontinues until the maximum cluster size is reachedor no new elements can be added. The maximumcluster size is a parameter specified by a user and isheld constant throughout the iterations. In general, acluster becomes random shape, because the regularlattice is used. We call it isotropic. As the maximum


cluster size increases, however, the cluster shapemore likely becomes circular. Instead of the randomcluster shape, as shown in Fig. 2, we also have anoption in the cluster selection so that the cluster

Ž .shape can be anisotropic elliptical to incorporate apriori information, namely, geological informationabout regional fractures such as strikes and scales.These two parameters, cluster size and shape, play akey role in the inversion, because these are a directrepresentation of simplified fracture sets. As men-tioned before, the optimal cluster size approximatelycorresponds to the spatial correlation length of trans-

Ž .missivity distributions Nakao et al., 1999 .Following the perturbation step, well tests are

simulated on the fracture network and the calculatedpressure transients are compared with observed data.

Ž .The objective function energy to be minimized isthe sum of the squared differences between thecalculated and observed pressure transients due topumping or injection. In this study, all wells in-

Ž .cluded in the objective function energy are un-weighted under the assumption that errors in fieldmeasurements due to noise or the accuracy of pres-sure transducers are approximately equal for all wells.At the ith iteration:

w xntimes jawells2E s o yp , 3Ž . Ž .Ý Ýi jk i jk

js1 ks1

where o refers to the observed pressure transient atjk

the k th time step for jth well. p refers to thei jk

simulated pressure transient at the ith iteration forthe jth well and k th observed time step.

The Metropolis algorithm is applied to determinewhether the current fracture configuration is accept-

Ž .able based on the Eq. 1 . When L acceptances atk

control parameter T have been achieved, the param-kŽ .eter is reduced to T via Eq. 2 with as0.9.kq1

T stays constant until L transitions have beenkq1 kq1

accepted at T . This process continues until thekq1

annealing schedule is exhausted or the number ofiterations has reached a user-specified maximum. Atthis point, the best model found thus far is expectedto be close to the global minimum. In the following,

Ž .we refer to Aminimum objective function energy B;however, this is used to describe the smallest objec-

Ž .tive function energy found at the end of the inver-sion.

4. Hydraulic well test data

Hydraulic well testing was conducted at the Ray-Žmond field site on several occasions e.g. Cohen,

1993; Karasaki et al., 1995; Cook, 1995; Cohen,.1995 . The data used in this study was taken in 1995.

Each of the nine wells was injected systematicallywith fresh water using a straddle packer system. Thedistance between the straddle packers was roughly 6m. A typical injection test was conducted for 10 min.The pressure in the water tank was held constantusing compressed air. Neither the flow rate nor thedownhole pressure was actively controlled; theyspontaneously adjusted themselves accordingly tothe transmissivity of the injection interval. The ad-vantages of this method are the simplicity of theset-up and the ease of test execution. After each test,the packer string was lowered by approximately 6 m.Depth intervals sealed by packers during a particularinjection were kept unobstructed during the next, sothat the entire length of the well was tested. Therewere approximately 15 injection tests per well in allnine wells. While these injections were conducted,the pressures in the remaining 31 intervals weresimultaneously monitored. As a result, a total of4200 interference pressure transients were recorded.A schematic of the packer set up in the site is shownin Fig. 3. To analyze connections between wellsusing a large number of interference data, a binary

Žinversion method was developed Cook, 1995;.Karasaki et al., 1995 . In their method, each set of

Fig. 3. Unfolded view of the wells and typical packer locationsduring the systematic injection tests. Note the length of theinjection interval is much shorter than the packed-off zones innon-injection wells.


pressure transient data was reduced to a binary set: 1Ž .yes if an observation zone responds to an injection,

Ž .and 0 no otherwise, and they successfully visual-ized connections between wells. The result stronglysupports the existence of two separated conductivezones, one at a depth of approximately 30 m andother at 60 m.

From the large number of interference data de-scribed above, three sets of injection data in serieswere selected for each zone: injection into wells 0-0,SE-1 and SE-3 for the upper zone, and injection intowells 0-0, SW-1 and SW-3 for the lower zone. Theseinjection intervals are located approximately betweendepths of 20 to 30 m and between depths of 50 to 60m. Intervals with interference response are also lo-cated in these depths. We believe these intervals areconfined in the upper and lower conductive zonesshown in Fig. 1b. Injection flow rates were, on theaverage, 6.4 lrmin for well 0-0, 5.9 lrmin for wellSE-1 and 6.5 lrmin for well SE-3 in the upper zone.In the lower zone, injection flow rates were 4.5lrmin for well 0-0, 5.3 lrmin for well SW-1 and 1.4lrmin for well SW-3. It should be noted that for apair of wells, injecting at the first and observing atthe second does not always give the same pressure

Žresponse when their roles are reversed i.e. injecting.at the second and observing at the first . For exam-

ple, injection into well 0-0 gives about 1.5 m changeof head in well SE-3; on the other hand, injectioninto well SE-3 gives 0.2 m change of head in well

Ž .0-0 for the upper zone see Fig. 6 . This is becausethe packer configurations were changed before doingthe reciprocal tests.

Ž .The template model of the site consists of a32=32 regular grid with 2.5 m fracture elementswhich covers an 80=80 m area both for the upperand lower zones. The total number of nodes andelements are 1089 and 2112, respectively. WellsSW-4 and SE-4 are not included in the model,because the interference response is not observedduring a 600-s injection. In all the calculations dis-cussed below, we impose an initial head value of 0m for all nodes, so that the relative head changes ofwells are examined. We also assign constant pressure

Ž .conditions 0 m on the outer boundary. As an initialconfiguration, all fracture elements have transmissiv-

y5 Ž 2 . y4ity 10 m rs , aperture 2.30=10 m. This ini-tial transmissivity is the average value of the aquifer

estimated from the drawdown analyses of the threeŽ .wells, 0-0, SW-1 and SE-1 Cohen, 1993 . Note that

the transmissivity of the outer region beyond a radiusof 37.5 m from the origin is held fixed at 10y5

Ž 2 .m rs throughout the inversion, because the pres-sure transients are not so sensitive to the transmissiv-ity distribution of the outer region. In other words, itis difficult to resolve the regions away from thewells and in particular, the regions near the bound-aries.

The equivalent specific storage of the fractureelements are set to be 0.1 and 0.01 my1 for theupper and lower zones, respectively, and are heldconstant throughout the inversion. These values areselected by trial and error. It is generally necessaryto use a higher-than-usual specific storage in discretefracture network models where the conductivity ofan element is coupled to its volume through thecubic law. Because it is impossible to model all thefractures and pore spaces in the rock enumerativelydue to the computational limitations, the specificstorage value in the model is an effective one thatrepresents the storage of the volume of rock that

Ž .surrounds the fracture element. Mauldon et al. 1993used a similar approach for their simulation of thefracture flow experiments conducted at Stripa Minein Sweden. Our focus, too, is to estimate the distribu-tion of the transmissivity contrast. Therefore, we feelit is justified to use the effective value for thespecific storage.

In our forward simulation, we set the boundarycondition at the well as the constant rate using theaverage flow rate over the injection period. Exceptfor the very early time, when the compliance of theinjection plumbing affected the apparent flow rate,the flow did not change greatly. Replacement aper-tures specified in the inversion are 1.07=10y3,4.96=10y4 , 2.30=10y4 and 1.07=10y4 m, whichcorrespond to transmissivities of 10y3, 10y4 , 10y5

y6 Ž 2 .and 10 m rs , respectively. The upper and lowerzones were modeled separately.

5. Results

5.1. The upper zone

Four cluster sizes, 1, 10, 20 and 40 were used forthe inversion to investigate spatial characteristics of


the transmissivity distribution within the upper con-ductive zone. For each cluster size, inversions wererun seven times starting from seven different randomseeds. Inversions using cluster size of 10 withanisotropic clusters, with strikes at N30W and N60E,were also conducted. These major fracture sets wereidentified from the measurements of outcrops and

Ž .the acoustic televiewer logs. Annealing coolingschedules for four cluster sizes are shown in Fig. 4.In these schedules, the total number of the elementchange in an inversion is set to be same for allcluster size cases. Namely, in the case of cluster size40, the control parameter T is lowered after 20configuration changes are accepted, whereas the con-trol parameter T is lowered after 800 changes areaccepted in the case of cluster size 1. We set allinversions will be stopped when the annealing sched-ule is exhausted. This means that the inversionsusing larger cluster sizes need less number of itera-tion. For example, the inversions using cluster sizes1 and 40 requires approximately 22,460 and 980total iterations, respectively.

Fig. 5 summarizes the inversion results. The low-Ž .est minimum objective function energy , 4.7, is

reached in the seed 4 of the cluster size 10, startingŽ .from the initial objective function energy of 3.33=

3 Ž .10 . The minimum objective function energy variesto some extent depending on a random seed for agiven cluster size. The averaged minimum objective

Ž .function energy is smallest for a cluster size of 1.As the cluster size decreases, the averaged minimum

Ž .objective function energy also decreases. Compari-son of pressure transients between observed andcalculated data for the minimizing configuration ob-

Ž .tained using the cluster size 10 seed 4 is illustratedin Fig. 6. A relatively good match is achieved.

In order to represent average properties of theannealing solutions obtained using the same clustersize, seven annealing solutions are combined to pro-duce one configuration by means of median filters,which are commonly used as velocity filters in data

Žprocessing of vertical seismic profiling e.g. Hardage,.1985 . That is, the median transmissivity value

Ž .transmissivity value itself, not its logarithm amongseven values is selected for each fracture element soas to produce one median configuration. Five me-dian-filtered configurations are shown in Figs. 7–9.The first result is found using a cluster size of 1. The

Ž .Fig. 4. Annealing cooling schedules used in the inversion.


Ž .Fig. 5. Inversion results for the upper zone: minimum objective function energy vs. cluster sizes.

second, third and fourth results are found usingcluster sizes of 10, 20 and 40 with isotropic clusters.

The fifth result is found using a cluster size of 10with anisotropic clusters, with strikes at N30W and

Ž .Fig. 6. Pressure match for minimizing configuration obtained using cluster size of 10 seed 4 for the upper zone. Symbols and linesrepresent the observed and calculated data, respectively.


Ž . Ž .Fig. 7. Median-filtered inversion result with cluster size 1 U1 and cluster size 10 U2 for the upper zone. The injection test data into wellŽ .0-0, SE-1 and SE-3 in series filled circles was used in the inversion.


Ž . Ž .Fig. 8. Median-filtered inversion result with cluster size 20 U3 and cluster size 40 U4 for the upper zone.


Ž .Fig. 9. Median-filtered inversion result with cluster size 10 and anisotropic shapes for the upper zone U5 .

N60E. We refer to these as cases U1 through U5. Itis difficult to observe the major feature common toall the results from these figures. However, the an-nealing results consistently show high transmissivi-ties around well 0-0. SW-3 is relatively isolated inU3 and U5 results. This is because SW-3 had a verylow interference response from all of three well

Ž .injections see Fig. 6 . Aside from these two fea-Ž .tures, U1–U4 isotropic clusters do not seem to

show a particular spatial pattern.Since the inversion result is based on limited

information and the hydraulic inversion is inherentlynon-unique, it is difficult to say how ArealB suchresults are. Rather, we believe that assessing thevalidity of the result can best be done by testing how

Ž .the result model predicts hydraulic behavior of thedata, which is not used to build the model. Thus, toevaluate the inversion results, a cross validation testwas conducted. We modeled the SW-2 injection test,which was not included in the inversion as injection

Ž .data. TRINET Karasaki, 1987 was again used tosimulate injection into SW-2. Fig. 10 shows predic-tion errors for injecting at SW-2 for each cluster size

Ž .results U1–U5 . The prediction error, which is thesum of the squared difference between the calculated

Ž .and observed data given by Eq. 3 , is approximately4 for cases U1 and U2, while it is more than 15 for

U3, U4 and U5. This result indicates that cases U1and U2 are the good models among the results offive cluster types. From these results, we observethat cluster sizes of 1 and 10 are suitable to obtaingood results.

ŽThe omnidirectional semi-variogram for U2 clus-.ter size of 10 is given in Fig. 11. The directional

semi-variograms were also calculated to check theexistence of geometric anisotropies. However, nosignificant anisotropies are observed in Fig. 11. Apractical range of the spatial correlation is approxi-

Žmately 4.0 m since both the element length and.spacing are 2.5 m, we refer to this value as 5.0 m .

This practical range is a little smaller than thatobtained from each annealing solution because of thesmoothing effect of the median filters. However, it is

Ž .nearly consistent with the range 5–7.5 m estimatedfrom the optimal cluster size, in that the optimalcluster size corresponds to 20–40% of the number offractures within the practical range of spatial correla-

Ž .tion Nakao et al., 1999 .

5.2. The lower zone

We repeated the same procedure for the lowerconductive zone. Fig. 12 summarizes inversion re-


Fig. 10. Predicted errors of injection into well SW-2 for five annealing solutions for the upper zone.

Žsults. The lowest minimum objective function en-.ergy , 140.0, is reached in the seed 7 of the cluster

size 1, starting from the initial objective functionŽ . 3energy of 4.82=10 . The averaged minimum ob-

Ž .jective function energy is smallest for a cluster size

of 1. As the cluster size decreases, the minimumŽ .objective function energy also decreases. For clus-

ter size of 10, the anisotropic-shapes case reachessmaller energy than that of isotropic-shapes case.Comparison of pressure transients between observed

Ž .Fig. 11. Omnidirectional and directional semi-variograms of transmissivity for the inversion results with cluster sizes10 U2 for the upperzone.


Ž .Fig. 12. Inversion results for the lower zone: minimum objective function energy vs. cluster sizes.

and calculated data for the minimizing configurationŽ .obtained using the cluster size 1 seed 7 is illus-

trated in Fig. 13. Although a good match betweenobserved and calculated data was obtained in the

Ž .Fig. 13. Pressure match for minimizing configuration obtained using cluster size of 1 seed 7 for the lower zone. Symbols and linesrepresent the observed and calculated data, respectively.


Ž . Ž .Fig. 14. Median-filtered inversion result with cluster size 1 L1 and cluster size 10 L2 for the lower zone. The injection test data into wellŽ .0-0, SW-1 and SW-3 in series filled circles was used in the inversion.


Ž . Ž .Fig. 15. Median-filtered inversion result with cluster size 20 L3 and cluster size 40 L4 for the lower zone.


Ž .Fig. 16. Median-filtered inversion result with cluster size 10 and anisotropic shapes for the lower zone L5 .

lower zone, the absolute head value of the injectioninto SW-3 is quite large. This causes the relatively

Ž .high minimum objective functions energies ratherthan those of the upper zone cases.

Five median-filtered configurations are shown inFigs. 14–16. The first result is found using a clustersize of 1. The second, third and fourth inversionsolutions are found using cluster sizes of 10, 20 and

Fig. 17. Predicted errors of injection into well SW-2 and SE-1 for five annealing solutions for the lower zone.


Ž .Fig. 18. Omnidirectional and directional semi-variograms of transmissivity for the inversion results with cluster sizes1 L1 for the lowerzone.

40 with isotropic clusters. The fifth result is foundusing a cluster size of 10 with anisotropic clusters,with strikes at N30W and N60E. We refer to these ascases L1 through L5. A major feature common to allthe results is that well SW-3 is clearly isolated. Thisis because SW-3 had a very high head change while

Ž .injection see Fig. 13 . Annealing results also show arelatively high transmissivity area between wellsSW-2 and SE-2.

In order to evaluate the inversion results, a crossvalidation test was again conducted. We modeled theSW-2 and the SE-1 injection tests, which were notincluded in the inversion as injection data. Fig. 17shows predicted errors for injecting at SW-2 and

Ž .SE-1 for each cluster size results L1–L5 . For clus-Ž .ter size of 10, the anisotropic-shapes case L5 ob-

tains smaller error than that of isotropic-shapes caseŽ .L2 . This finding will be discussed in the nextsection. The prediction errors are smallest for L1 in

Žboth injections. This result indicates that L1 cluster.size of 1 is the most suitable among the results of

five cluster types.The directional and omnidirectional semi-vario-

Ž .grams for L1 cluster size of 1 are given in Fig. 18.A practical range of the spatial correlation is less

Žthan 2.5 m since both of the element length and.spacing are 2.5 m, we refer to this value as 2.5 m .

This practical range is consistent with the rangeŽ .around 2.5 m estimated from the optimal cluster

Ž .size Nakao et al., 1999 . It is impossible to observewhether geometric anisotropies of the correlationstructure exist within the range of 2.5 m, because ofthe resolvable scale used in the configuration. Notethat this is the first attempt to estimate the horizontalcorrelation length of transmissivity for both the up-per and lower conductive zones at Raymond fieldsite.

6. Discussion

In the preliminary analysis of the upper zoneŽ .Nakao et al., 1999 , we conducted the SA inversionjust once for each cluster size, resulting in a littleoverestimation of the practical range of the spatial

Ž .correlation for transmissivity 5–10 m . In this study,Ž .seven realizations annealing solutions starting from

seven different random seeds were obtained and


stacked to produce one average configuration bymeans of the median filters. This is because wereally want to extract the average properties of thetransmissivity distribution, which may be specific forthe results obtained using each cluster size.

For the upper zone, we showed that the configura-tion achieved using cluster size of 1 or 10 with

Ž .isotropic cluster shapes cases U1 or U2 are themost suitable among five cluster types for obtaininggood fluid flow models. Furthermore, the inversionusing cluster size of 10 requires approximately onetenth of the iteration number for the cluster size of 1,which means that the cluster size of 10 is optimal toenhance convergence of the inversion from the view-point of computational effectiveness. From thesemi-variogram analysis of U2, the practical range ofthe correlation structure is estimated to be 5.0 m, andno obvious geometric anisotropies are observed.

For the lower zone, we showed that the configura-Ž .tion achieved using cluster size of 1 case L1 is the

most suitable among five cluster types for obtaininggood fluid flow models. On the other hand, both of

Ž .the minimum objective function energy and predic-tion errors are smaller for the anisotropic cluster

Ž .shape case L5 than those for the isotropic clusterŽ .shapes case L2 . This suggests the possibility of the

lower zone being anisotropic with possible strikes atN30W and N60E, and that the geometric fractureinformation from outcrops and boreholes may becorrelated to hydrology. Even though this is the case,the practical range of the correlation structure isestimated to be less than 2.5 m from the semi-vario-

Ž .gram analysis of the best result L1 .From the televiewer, television and optical scan-

ner logs it was determined that the lower zone isassociated with an anomalous single open fracture ascompared to the upper zone, which is an extensivefracture zone. This would explain the difference inthe estimated practical range of the spatial correla-

Ž .tion for transmissivity. Vasco et al. 1996 analyzedŽ .seismic velocity structures and Q attenuation struc-

tures at this site and reported that the velocity andattenuation anomalies appeared to coincide with theextensively fractured upper zone, although some dis-crepancies between the velocity and attenuationanomalies existed in the lower zone. They suggestthat velocity and attenuation anomalies indicate frac-ture zones rather than individual fractures. This also

supports the difference in fracture types between theupper and lower conductive zones.

We showed that the process of selecting a reason-able cluster size and shape using this algorithm bytrial and error may lead to an estimated range ofspatial correlation. With a possible range of spatialcorrelation parameters, we have much more valuableinformation, such as how to extrapolate model re-sults to a larger region. Note that a cluster size of,say, five can be used to fine-tune the optimal clustersize. However, because we chose to use an orthogo-nal lattice, the resolvable length scale is nonethelesslimited to the multiple of 2.5 m when we estimatethe practical range of spatial correlation for transmis-sivity.

It is important to conduct cross validation tests todetermine the variability in solutions, because themodels we build use very limited data. This differsfrom building models based on a complete knowl-edge of the ArealB fracture system. For future study,in order to reduce the uncertainty in the hydrologicalmodel, inversion of both hydraulic and tracer testdata is planned, because the tracer test data analysisis more effective to estimate fluid flow paths withinthe fracture network. It is also important to conductsensitivity studies to develop a field test design, suchas the relative number of injection or pumping wellsand observation wells, and the location of the wells.These factors are clearly related to the ability todetect heterogeneity of the hydraulic conductivitystructure.

7. Conclusions

The inversion of hydraulic well tests using CVASA is applied as an inverse technique to constructfluid flow models at the Raymond field site. Wemodel two fracture zones, which are extracted fromvarious geophysical and geological methods. Inver-sions are run seven times starting from seven differ-ent random seeds for four cluster sizes of 1, 10, 20and 40 to investigate spatial characteristics of thetransmissivity distribution within the fracture zones.In order to represent average properties of the an-nealing solutions obtained for a same cluster size,


seven annealing solutions are combined to produceone configuration by means of median filters.

Since a cluster size of 1 and 10 gives the mini-Ž .mum objective function energy , the practical range

of the spatial correlation of transmissivity is esti-mated to be 5 m in the upper extensive fracture zone.On the other hand, the practical range of the spatialcorrelation of transmissivity is estimated to be lessthan 2.5 m in the lower fracture zone, because thecluster size of 1 gives minimum objective functionŽ .energy . These results are confirmed by the crossvalidation of the injection test data, which is not usedin the inversion.

Acknowledgements

This work was supported by Science and Tech-Ž .nology Agency of Japan STA . It was also partially

supported by the Japan Nuclear Cycle DevelopmentŽ .Institute JNC , through the U.S. Department of En-

ergy Contract Number DE-AC03-76SF00098. Wewould like to thank two anonymous reviewers fortheir helpful comments.

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