WATER RESOURCES RESEARCH,
VOL. 34, NO. 12, PAGES 3335-3347, DECEMBER
1998
Inverse hydrologic modeling using stochasticgrowth algorithms Kevin Hestir,1 StephenJ. Martel, 2 StacyVail, 3 Jane Long,4 Pete D'Onfro, s and William D. Rizer 6
Abstract. We presenta methodfor inversemodelingin hydrologythat incorporatesa physicalunderstanding of the geologicalprocesses that form a hydrologicsystem.The method is basedon constructinga stochasticmodel that is approximatelyrepresentativeof thesegeologicprocesses. This model providesa prior probabilitydistributionfor possible solutionsto the inverseproblem,.The uncertaintyin the inversesolutionis characterized by a conditional(posterior)probabilitydistribution.A new stochastic simulationmethod, calledconditionalcoding,approximatelysamplesfrom this posteriordistributionand allowsstudyof solutionuncertaintythroughMonte Carlo techniques.We examinea fracture-dominatedflow system,but the method can potentiallybe appliedto any system where formationprocesses are modeledwith a stochasticsimulationalgorithm. 1.
Introduction
Determiningthe hydrologicpropertiesof a flow systemfrom well testmeasurements is a fundamentalproblemin hydrology. This problem is an inverseproblem, and a processfor determining a solutionis commonlycalled inversemodeling.One fundamental
issue
is that
well
test measurements
do not
basedon geologictheory.We solvethe inversemodelingproblem by usingthis stochasticmodel to define a broad suite of possiblespatial structures,and we then select samplesfrom this suite that are consistentwith (i.e., conditionedupon) the availablehydraulicresponsedata. The resultingsolutionis a model that is consistentwith flow data and the geologictheory representedin the stochasticmodel. Our inversemodelingmethodis Bayesian,sowe give a brief descriptionof Bayesiantheoryand introducesomenotation.In the Bayesianapproach one first assumesthat the unknown hydraulicgeometryis randomlychosenfrom a givenprobability distributionof possiblehydraulicgeometries.This distribution is called the prior distributionor just the prior. Probabilities given by the prior can be representedwith the notation P(X = X), where X representsthe random hydraulicgeometry andX is a particulargeometrythat X could equal. Before data are collected, the prior contains the only information about possiblevaluesfor X, but after the data are taken some Xs will be more compatiblewith the data and sobecomemore probablewhile otherXs are lesscompatibleand lessprobable. The new probabilitiesfor differentXs are representedthrough a conditionaldistributioncalledthe posteriordistribution.This posteriordistributionis the conditionaldistributionof X given M, where M representsthe flow data or measurements.Prob-
uniquelydeterminethe hydrologicpropertiesof a flow system, so solutionsto the inverseproblem are not unique. To deal with this nonuniqueness, assumptionsare usuallymade that restrict possiblesolutions.These assumptionsfirst impose a spatial structure on the hydrologicproperties [Yeh, 1986]. Once this spatial structureis defined, a number of unknown parameters,suchas permeabilityvalues,are needed to complete the model and to enableit to givepredictionsof well test responses. A functioncan be constructedto measurehowwell the parameterizedmodel predictsthe observedresponses. By minimizingthis functionone obtainsa set of parametersthat give a "best fit." We call this function the distancefunction; others have referred to it as the output error criterion or objectivefunction.Definition of the distancefunctiondepends on assumptionsabout measurement errors and previous knowledgeabout likely valuesof the parameters.Framed in this way, the inversemodelingprocedureinvolvestwo steps: (1) definingthe geometryof hydrogeologic structuresand (2) abilities givenbytheposterior aredenoted P(X = XIM = M). determiningparametervaluesfor the structuresthat minimize The notationM = M is usedbecausethe flow data dependon a distance function. X and are therefore valuesof a stochasticprocess;so M repOne weakness of the above method is that the choice of resentsthe stochasticprocessfor flow data, andM is a possible structurein step 1 is somewhatarbitrary and can lack justifi- realization of M. cationon the basisof flow observations or geologictheory.The In the Bayesianapproach,the posterior distributionis the inversemodelingmethodpresentedhere addressesthisweak- solutionto the inverseproblem. This solutionrepresentsnonnessby definingthe structurewith a stochasticmodel that is uniquenessthrougha probabilitydistributionthat reflectsthe informationabout X containedin M. Using the posteriordis•Department of Mathematics andStatistics, Utah StateUniversity, tribution, it is also possibleto obtain unique answersto the
Logan.
2Department of GeologyandGeophysics, Universityof Hawaiiat Manoa.
3Department of Mathematics, University of Wyoming, Laramie. 4MackaySchoolof Mines,University of Nevada,Reno. SConocoInc., Houston,Texas. 6HoustonAdvancedResearchCenter,Woodlands,Texas. ,
Copyright1998by the AmericanGeophysicalUnion. Paper number 98WR01549. 0043-1397/98/98 WR-01549509.00
inverseproblem,suchastakingthe modeof P(X = XlM = M), but obtaininguniquesolutionsis usuallynot as crucialas understandingthe full posteriorthat quantifiesthe uncertainty and nonuniqueness in the problem.For a generaldiscussion of this Bayesianview of inversemodeling,seePresset al. [1986, pp. 809-817] or Ripley[1988]. Our Bayesianapproachhastwo main components.First we define X usinga physicallybasedstochasticmodel that representsthe geologicprocessesthat form a hydraulicgeometry;
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this is analogousto the processmodels discussedby Koltermann and Gorelick [1996]. Secondwe use a new conditional simulationmethod that allowsuse of this prior in the inverse problem. The physicallybased stochasticmodel differs from standardstochastic models(suchas Gaussianprocesses) becauseit is definedby physicallybasedrules. These rules implicitly determineX without an explicitanalyticalrepresenta-
MODELING
stochastic rules.Theseare appliedrecursivelyto eachfracture in the network
for a fixed number
of iterations.
The rules are
basedon fracture mechanicsconceptsas appliedto a homogeneous,isotropic,elasticmaterialunderplane strainloading [Lawn and Wilshaw,1975]. 3.1.
Basic Model
tion.The Bayesian solution P(X = xIu = M) is impossible The growthrulesfor a singlesetof parallelmodeI (opening to calculateanalyticallybecauseof the rule-basedalgorithmic mode) fractures(i.e., joints) are basedon the followingas(1) The fracturesbehaveasthoughtheywere medefinitionof X andbecauseof the nonlineardependenceof the sumptions: flow measurementsM on X. To make suchposteriordistribu- chanicallyindependent,(2) The stressfield near the tip of a tions useful, we introduce a new conditional simulation fracturedetermineshow the fracturegrows,and (3) All fracare most method called conditionalcoding[Hestir,1995, 1998], which turesgrowparallelto eachother. Theseassumptions generates flowstructures thataresamples fromP(X -- XIM = appropriatein caseswhere the fracturesgrow primarily in M). In this way a geologicmodel for the formationof a flow responseto the far-field (i.e., regional)stressstate and pore systemis incorporatedinto an inverse modeling procedure. pressurein the rock rather than to fractureinteractions[Olson The resultingsamples fromP(X -- XIM - M) aremodelsof and Pollard, 1989]. A commonlyusedcriterionfor fracturepropagationis that the hydrogeologic systemthat are basedon flow measurements and a geologictheory for how the systemformed. The tech- fracturesgrow in a way that maximizesthe fracture energy nique is quite general and has potential applicationto any release rate G, systemwhere a physicallybased prior is defined through a S = cl(g•2q-g•2•r) q-½2Ki2ii (1) simulationalgorithm. In this paper we first describea model X for a fracture flow The termsK•, K•, andK• are the respectivestressintensity systemthat usesstochasticrules to form a fracture network. factorscorresponding to opening,sliding,andtearingdisplaceNext we considerthe inverseproblem usingflow data M and mentsof the fracturewalls, and c • and c2 are elasticconstants the rule-baseddefinitionof X as a prior distribution.We then [Lawnand Wilshaw,1975].Stressintensityfactorsdescribethe describethe conditionalcodingprocedurefor samplingfrom strengthof the theoreticalstresssingularityat the fracturetip. P(X = xIu = M) andconclude withtwoexamples. They reflectthe geometryof the fractureand the loadson the fracture. For an isolated mode I fracture, K• and K• are
2.
Physically Based StochasticModels
Solutionsof inversemodelingproblemsare commonlyused for prediction.For example,a solutionthat matchestest measurementsfrom one region couldbe usedto predict measurementselsewhere,or it couldbe usedto predicta responsefrom a different type of test. An inversesolutioncan be improved owing to an increasein the quantity, quality, or variety of information on which it is based.For example,even an improvementin the qualitativegeologicconstraintscan yield a more useful hydrologicmodel. A physicalunderstandingof geologicprocesses is particularlyimportantif a model is to be extendedto a region where direct geologicobservationsare scarceor absent.For a discussionof the philosophyof using physicallybasedstochastic models,seeFreedman[1985]. Another benefit of usingphysicallybasedstochasticmodels in inversemodelingis that they can enhancegeologicunderstandingof flow systemformation.Theoreticaldevelopmentof stochasticmodels requires representingthe important processeswith stochasticrules. Implementing these rules and comparingthe resultsto geologicdata can further the understandingof formation processes. 3.
Stochastic
Models
for Rock Fractures
zero, and
=
(2)
whereA is the half-lengthof the fractureand tr is the driving pressureinsidethe fracture, that is, the fluid pressureminus the componentof the remote compressive stressthat actsnormal to the fracture.Assuminga constantdrivingpressure (1) and (2) lead to (3) To castthis fracture mechanicsexpressioninto a stochastic growthrule we set the probabilityof fracturegrowthproportional to G. This assumption, coupledwith (3), leadsto
Prob(fracture growth)= min (-•,1)
(4)
whereL is a cutoffhalf-lengthrelatedto the materialstrength and driving stresses.During a given iteration of the fracture growth algorithm,a fracture of half-lengthlessthan L may grow(probabilitylessthan one) and a fractureof half-length greaterthanL mustgrow(probabilityequalto one). Fracture growthusesone of two rules exclusively. The first rule is called random incremental extension; a fracture of
length2A growsto a lengthof (1 + B U)2A, whereB is the In this study all fracture network models are two-dimen- maximum growth increment and U is a number chosenunisional with fracturesrepresentedas coplanar line segments. formly at random from the interval [0, 1]. The secondlength These line segmentsmark the intersectionof vertical,planar, increaserule is a fixed incremental extension,so that a fracture strata-bound fractures witha horizontal planeparallel to the of length 2A growsto a length of 2A + •, where • is a fixed bedding.The matrix is assumedto be impermeable.In the positiveconstant.Field observationsof natural fracturessugexample presentedlater the two-dimensionalassumptionis gestthat the parameterB in the randomincrementalextension geologicallyreasonable,but there is likely somematrix flow rule is generallymuch lessthan 1 [Engelder,1987];we arbithat is not representedin our model. trarily set B = 0.02 in our simulations. The formationof a fracturenetworkis simulatedby a set of To beginthe process,a set of parallel "starter"fracturesof
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1 st iteration
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350 iterations
_
_
--_
i
15
i
20
i
25
i
30
i
35
4O
15
20
meters
15
i
20
i
25
30
35
40
meters
440 iterations
i
25
[
30
fracture map
i
35
i
40
15
meters
i
20
!
i
25
30
i
35
[
40
meters
Figure l. Stagesof growth in the basic fracture growth model as describedin text and a fracture map modifiedfrom Segalland Pollard [1983]. Shadedarea is the boundaryof the fracture map.
equal length is defined accordingto a Poissonprocess.The parametersof the stochasticgrowthprocessare as follows: l0 starter fracture length; • rate of Poissonprocessof starterfractures(i.e., spatial densityof fracturenuclei); B the maximalrelative growthincrement,or •i equalsthe fixed growth increment; L cutoff half-lengthfor fracturegrowthas givenin (4); n number of iterationsof growthprocess.
3.2.
Two-Set
Fracture
Growth
Model
In manyplacesmore than a singlesetof fracturesexists(e.g., Figure 2), and the presenceof multiple setsof fracturescan dramaticallyaffect the mechanicaland hydrologicbehaviorof a rock mass. The two-set fracture
model that we introduce
here
Figure 1 showsa fracture systemgrown through different numbersof iterationsalongwith a map of fracturetracesfrom a granite outcrop that is visuallysimilar to the result of the growthprocess.The parametersusedin Figure 1 are l0 = 0.25 m, • = 0.38 fracturesper squaremeter, B = 0.02 usingthe
representsa commontype of fracture pattern, two orthogonal setsof fracturesof different age. We subsequentlyapply this model to a fracturedlimestoneat the ConocoTest Facility in Kay County,Oklahoma [Queenand Rizer, 1990;Datta-Guptaet al., 1994]. The two-set fracture growth model growsfractures in two stages.First set 1 is grown usingthe basicgrowth model describedin section3.1. Set 2 is then grown using a modified versionof the basicgrowthmodel; the modifiedrules account
random extension increment rule, L = 2.5 m, and n = 1,
for first-order
mechanical
interactions
of the set 2 fractures
350, and 440. The simulatedand actual fracture trace maps with the preexistingfracturesof set 1. These first-orderinterare similar but not identical. One might ask how to best de- action rules are as follows: 1. If a tip of a set 2 fractureintersectsa set 1 fracture,then termine the parameters, •, B, etc., for a given geological setting.For the purposesof this work we choseparameters that tip will no longerpropagate.If both tips of a set 2 fracture throughtrial and error that givefracturepatternsthat visually intersecta set 1 fracture,then the set 2 fracture stopsgrowing. match, and we have approximatelythe samespatialdensityof This rule is consistentwith the fracture trace pattern in Figure fracturesshownin the fracture mapsof Figures 1 and 2. 2 and with the mechanicaltendencyof an openingmode frac-
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MODELING
(A) o o
-
o
_
o
-
meters
-150
-100
-50
0
50
100
150
meters
Figure 2. Fracture pattern mapped on pavement surface, modifiedfrom Queenand Rizer [1990].
ture to terminateagainsta preexistingopenfracture [Keerand Chen, 1981]. 2. If exactlyone tip of a set 2 fracture intersectsa set 1 fracture,then the effectivehalf-lengthA of the set2 fractureis doubled.Recall that the half-lengthA is used in determining both the probabilityand amountof incrementalgrowth.When one end of an openingmode fracture intersectsan extensive free surface,the fracture can open more, increasingthe mechanicalleverageon the "free" tip of the fracture.Doublingof the effectivehalf-lengthof the set 2 fracture reflectsthe approximatedoublingof the stressintensityfactor at the "free" tip of the fracture that accompaniesthe intersectionprocess [Tada et al., 1973, p. 8.1]. Calculationof flow in the model can be made computationally efficientby restrictingfracturesto lie on a fixedlattice.The two-setfracture growthmodel with realizationsrestrictedto a lattice will be called the lattice-restrictedfracture growth model. Figure 3 showssuch a model in a map view, with fracturesof one setstrikingeast-westandthoseof the other set strikingnorth-south,north being to the top of the page. A modificationof growthrules is required to translatethe model to a lattice. The minimum fracture length in a latticerestrictedmodelis the latticespacing(i.e., the elementlength). Starter fracturesare chosenas singleelements.Set 1 starter fracturesare chosenfirst by pickingat random(with replacement) rn• of the elementsin the lattice. If an element thus chosenis east-west,it is kept as a set 1 starter fracture. If elementswith a commonendpoint are chosenand kept, then they are coalescedto form a singlestarterfracture.The resulting random number of set 1 starter fracturesare all east-west and have random lengthsdependingon the number that are coalesced.Set 2 starterfracturesare chosenin the sameway usingm2, rather than m•, choicesfrom the elementsin the lattice,keepingthe choicesthat are north-southand coalescing elementswith commonendpoints.The methodof generatinga random number of starterfracturesfor eachset approximates the spatial Poissonprocessfor starter fracturesused in the
(B)
-150
-1 O0
-50
0
50
1 O0
150
meters
(c) o
_
o
_
i
i
i
i
i
i
i
-150
-100
-50
0
50
100
150
meters
Figure 3. (a and b) A simulatedfracturepattern usingthe lattice-restrictedfracture growth model and (c) the fracture pattern from Figure 2. In Figures 3b and 3c only the area exposedin the fracture map is shown.
HESTIR
non-lattice-restricted
ET AL.: INVERSE
model. The fixed incremental
HYDROLOGIC
extension
Table
1.
MODELING
Parameter
Values
3339
for the Lattice-Restricted
rule for lengthincreaseis usedwith 6 equalto a single-element Fracture Growth Model Used to Generate the length. Parametersused to generatethe network in Figure 3 Figures3a, 3b, and 10 are givenin Table 1; theseparametervalueswere chosento Parameter Description give a fracturepattern that resemblesthe map in Figure 2. l0 lattice element length A major conceptualdifferencebetweencontinuumand lattotal number of lattice elements tice-restrictedgrowth is that lattice-restrictedgrowth intronumber of random choices for horizontal ducesthe possibilityof two collinear fracturesgrowinginto starter cracks each other. If this occurs, the fractures are coalesced into a
singlefracture,a mechanismthat is not presentin the continuum model but is nonethelessgeologicallyreasonablefor nearly coplanarfractures. Once the lattice-restricted network is formed, the elements
are assigned hydrologicpropertiesof transmissivity T and storativityS. Hydraulic propertiesof a natural fracture are determined by how it forms and by subsequent dissolutionor precipitationof mineralsalongit. Thus T andS mightbe modeled on an element-by-elementbasisby a stochasticprocess.We have not done this here. Instead, we make the simplifying assumptionthat T and S are constantacrossthe fracture network [Longe! al., 1989].This has the effect of accountingfor flow responseby network geometryrather than by individual fracturecharacteristics, an approachfundamentallysimilarto that of Mauldon et al. [1993]. The assumptionof constantvaluesof T and S for all elementshassomecomputationaladvantages for the flow datawe are using.These flow data are drawdownsmeasuredat a number of locationsduring constantflux pumpingin a particular well. In a modelfor suchdata, drawdowncurvesfor In (head) versusIn (time) will shiftverticallyor horizontallywith changes in T or S [Fair e! al., 1966]. Once drawdowncurvesare calculated for specificvaluesof T and S, drawdowncurvesfor other valuesof T and S canbe obtainedthroughshiftingrather than by recalculatingflow in the whole network.The sameshifting principalis usedto determinetransmissivity and storativityin a uniformlylayeredmediumby shiftingobservedIn (head) versusIn (time) curvesto matcha Theis type curve.A similar shiftingprocedurewith inversemodelingis alsodescribedby Doughtyet al. [1994, p. 1728]. A simpleway to make T and S part of the stochasticmodel is to let the horizontalandverticalshiftseachbe independently chosenuniformlyat randomovergivenranges;or equivalently, let In (T) andIn (T/S) haveindependentuniformdistributions overgivenranges.The upperandlowerboundsof theseranges are model parameterswhich are chosento bracket the horizontal and vertical shifts that allow numerical
drawdown curves
to approximatelymatchthe data. The assumptionof independentuniformdistributions reflectsa lackof preferencefor any specificvalues of the horizontal and vertical shifts.Further, some elementary probability calculationsshow that this inducesa distributionon T and S with a positivecorrelation,so if T is large,S tendsto be large aswell.
number of random starter cracks
Networks
choices for vertical
length of growth incrementequalsl 0 probabilityof growthby extensionfor set 1 probabilityof growthby extensionfor set 2 cutoff length for fracture growth for
p• p2 L
in
Value 2.69" 44700 1500 5100
2.69" 1.0 1.0 75"
sets 1 and 2
(ao, bo)
intervalfor uniform distributionof
(0.8, 3.5)
In (T/S) in unitsof In (m2/s) (at, Or)
intervalfor uniformdistributionof
(-39.4, 20.6)
In (T) in unitsof In (m2/s) Subscripts refer to the fracturesetnumber.Intervalsfor the uniform distributionof In (T) and In (T/S) are,alsogiven. "Units
are meters.
1998], sowe refer the reader to thosereferencesfor technical details omitted
in the discussion below.
In conditionalcoding,pseudorandomnumbersare used in the computersimulationalgorithmfor X; thesenumbersare viewedas parametersthat can be optimizedto match a hydraulic response.If the optimizationis done with simulated annealing[Bertsiinasand Tsitsiklis,1993], then a probability argument showsthat the optimized pseudorandomnumbers generatea hydraulicgeometryfrom the posteriordistribution
P(X = XIM = M). This indirectmethodof samplingis desirablebecauseit usesonly the algorithmsfor generatingX (thelattice-restricted two-setfracturemodelalgorithm)andM (the simulationof flow in X). Hence samplesfrom posterior distributionsthat are defined through simulationalgorithms can be formed.
To seehow pseudorandomnumberscan be thoughtof as a vector of parameters,we need a precise descriptionof the relationshipbetweenX, the pseudorandomnumbers,and M. Recall that M is a vector of data values that are measurements
on X with randommeasurementerrors added.Letting E denote the measurement errors, we have
M = in(X) + E = In[#(U)] + E
(5)
where U is the vector whoseentries are the pseudorandom numbersused to generateX; !7 representsthe fracture simulation algorithm,whichcanbe thoughtof as a functionacting on U, so a simulationof X is X = 17(U);andIn(X) represents the numericalmodel of flow responsein X without measurement error. Equation (5) showsthat U can be viewed as a parameterthat couldpossiblybe optimizedto createan M that 4. Conditional Coding givesa goodmatchto data (Figure4). With X defined by the lattice-restrictedtwo-set fracture Theoretically,optimizationof U with simulatedannealing growthmodel,we wish to implementa Bayesianinversemodrandomlygeneratesa vectorof numbersUc that matchesdata eling schemeby sampling from the posterior distribution through(5), but if the correctdistancefunctionis used,it can P(X = XIM = M). In thissection we discuss the computabe shown [Hestir, 1998] that Uc is also a sample from the tional method,called conditionalcoding,that can be usedto conditional distribution, approximatelysamplefrom this posterior.The mathematical justificationof conditionalcoding is given by Hestir [1995, e(u = VIM = M) (6)
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U
MODELING
20.2 (11) fvlM(U) =k•exp (-- ½{Mm[g(U)]}) Specifically,let M = (M•, M2, ..., M,), where the Mis representall of the datavaluesandlet m[ #(U)] = {m•[#(U)], m2[#(U)], ..., m,[#(U)]}, wherethemi[#(U)]s represent all of the simulated
data values based on the flow network X
generatedfrom U. Then we can define
d(U) = ½{M- m[g(U)]}= • {M,- m,[g(U)])2 Distance
Fracture
Function
Network
(12)
X
whichis simplythe sumof the squareddifferencesbetweenthe data and the simulation.Using simulatedannealingto optimize d(U) with a temperatureschedulethat has a minimum
d(U)
temperature of 2o'2givesa samplefroma distribution with a
x/
Flow Data M
Figure 4. Schematicillustrationshowingthe relationshipbetweenthe pseudorandom numbers,the stochastic model, and the flow data. The distance function measures the match be-
tween the real flow data and the flow data simulated in X,
whichis generatedusingthe pseudorandom numbersin U.
A propertyof conditionaldistributionsthen impliesthat the networksimulatedwith Uc(X = #(U0) is a samplefrom P(X
=XM=
M).
densitygivenby (7), and soit is a samplefrom the distribution in (6). Formulation of an error distribution is required for our method aswell as for the maximumlikelihood approach[Carrera and Neuman, 1986] and other Bayesianinversemethods [Ripley,1988]. Traditionally,the distributionsused are independentGaussianas in (9) or Gaussianwith nonzerocorrelationsas in Carreraand Neuman [1986].IndependentGaussian errorslead to the distancefunction(12), whichis perhapsthe simplestdistancefunctionthat one could define. Indeed, we have chosenthe Gaussiandistributionin this paper becauseof this resultingsimplifieddistancefunction.Other error distributions can be used that would result in more complicated forms;in fact, conditionalcodingcanbe usedwith very general error densitiesand evenwith nonadditiveerrors [Hestir,1998].
'5. Inverse Modeling Example With Synthetic Data
To see what the correct distance function of U is, we note
In this section and in section 6 we provide examplesof conditionalcoding solutionsto inverseproblemsin fracture hydrology.The solutionsmatch hydraulicdata, and the fracture patternsare consistentwith the lattice-restrictedfracture model. fG(u)= k exp tmin (7) generation The first example uses syntheticdrawdown data derived wherefG is the density,k is a normalizationconstant,and tmin from a known fracture network, shownin Figure 5, that was is the smallest(final) temperaturein the temperatureschedule generatedby the lattice-restrictedfracturegrowthmodel. Pa[Bertsimas and Tsitsiklis,1993]. Now (6) can be written as a rametersusedfor the growthmodel are givenin Table 2. The syntheticflow data for this example,and for all flow simuladensityin this form. To see this,we first note that tionspresentedin this paper,were numericallycalculatedus(8) ing the Trinet program [Karasaki,1987]. Drawdown curves fUIM(U)= fe{M - m[g(U)]} were calculatedat the sixwells of Figure 5, with well 3 as the wherefUIM(U) is theconditional probability density function well. The flow boundaryconditionsare a constant of U givenM andfE is the probabilitydensityfunctionof E. pumping flux at well number 3 and constant head on the outside boundFor example,if we assumethat E is a vectorof independent identicallydistributedGaussianerrors with mean zero and ary of the mesh.All elementswere givenconstantvaluesfor T and S. Errors in the drawdowndata are simulatedby adding standarddeviation0.,we havefor E -- (E•, E2, ..., E,), independentnormallydistributedrandom numbers(mean 0 andstandarddeviation1) to the simulatedIn (head)valuesfor (9) eachof the sixdrawdowncurves.The resultingsimulateddata 20 .2 with errorsaddedare plottedin Figure 6. Thesedata form the with vectorM, the componentsof which are the drawdowndata at
that simulatedannealingoptimizesa distancefunctiond(U) by generatinga samplefrom the Gibbsdistribution,
-a(u))
fE(E)=k•exp(-½(E))
various times for each well.
½(E)= • E,2 i=1
Thus
(10)
The syntheticflow datareflecthowthe pumpingwell (well 3) is connectedto the rest of the network.For example,the flow data showthat well 4, althoughclosestto the pumpingwell, does not respondto the pumping and therefore is not con-
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generateX. Five samplesgeneratedin this way are shownin Figure7, alongwith a plot of the distancefunctiond (U) versus iteration numberfor sample5. Each of the samplesin Figure 7 required -60 hours of computationon a 70 MHz Sparc 20 workstation,with -90% of CPU time being spent in flow calculationand the majorityof the rest in networkgeneration. Valuesfor d(U), givenin Table 3, rangefrom 286.8 to 574.5.
I
Drawdown curves, distance function values, and data for sam-
ple 5, whichhasthe highestvaluefor d (U) andhencetheworst fit, are shownin Figure 8. Note that each samplenetwork in Figure 7 has a pattern definedby the stochasticgrowthmodel, 1 showsno connectionof well 4 to the pumpingwell, and has 2 long flow pathsfrom the pumpingwell to wells 5 and 6. Other featuresin the five samplesvary,illustratingthat the drawdown responsesdo not highlyconstrainthe network. With a larger number of samplenetworks,posteriorprobabilitiesfor somefeaturesthat appearin severalof the samples couldbe estimated.For example,four of the samples,samples 1, 2, 4, and 5, have a singleeast-westfracture connectingwells 1 and 2, sowe can estimatethe probabilityof sucha feature to -0.4 -0.2 0.0 0.2 0.4 be 4/5. Note that samples2, 3, and 5 have a fairly direct [L] connectionbetweenwells5 and 6: this suggests a probabilityof 3/5 for this feature, even though it is absent from the actual Figure 5. Simulated fracture pattern for lattice-restricted stochasticfracture growthmodel. Numbered circlesmark the network(Figure 5). Clearly,a muchlargernumberof sample location of the wells where drawdowns are observed. networkswouldbe requiredfor suchprobabilisticestimatesto be well founded.
Somedetailsof the annealingprocedureare appropriateto nected to the fracture network that intersectsthe pumping discussbefore turning to our secondexample.The simulated well. Also,wells5 and 6 are closeto the pumpingwell but have annealingalgorithmstartswith a randomlygeneratedvalue of a delayedresponsecomparedwith wells1 and2, sowells5 and 6 U. The randomperturbationof U in the Metropolis iterations areconnected to well 3 by flowpathwith a longerdelaythanwells [Bertsimasand Tsitsiklis,1993] is done in one of two ways 1 and 2, with well 6 havinga connectionwith the longestdelay. chosenat random:either the componentsusedfor generating To solvethe inverseproblem,we need two more piecesof In (T/S) and In (T) are replacedby new uniform random information in addition to the simulated drawdown data. First numbersor the componentsfor generatingthe fracture netwe need boundsa D, bD, a r, and br for the uniform distribu- work are perturbed.The perturbationof the componentsfor tionsofln (T/S) andIn (T). The valuesofar•, bz>,at, andbr generatingthe fracture network is done by randomlychoosing are determinedby bracketinga rangeof verticalandhorizontal entries in U and replacing those entries with new uniform shiftsthat allow the simulateddrawdowncurvesto approxi- random numbers. The vector U has -15,000 entries, and the matelymatchthe drawdowncurves.Secondwe need parame- number of perturbed entriesis randomlychosenat each iterters for the syntheticgrowthmodel. These are assumedto be ation to be between 1 and 300 when componentsfor generatknown and equal to those used to generate the network in ing the fracture network are perturbed. One important issuefor generatingconditionalcodingsamFigure 5. The inverseanalysiscan now proceed.Using the distance plesis the determinationof a temperatureschedule[Bertsimas functiond (U) from (12), simulatedannealingis usedto sam- and Tsitsiklis,1993].This is a sequenceof decreasingnumber ple Ucfrom the distributionin (6). A samplefrom the posterior ("temperatures") t•, ..., tmwithtm = tmin = 2rr2 (tmin -- 2 distribution P(X = XIM = M) is formedby usingUc to in thisexample)and a specification of the numberof iterations
Table
2.
Parameter
Values for the Lattice-Restricted
Fracture
Growth Model Used to
Generate the Networks in Figures5 and 7 Parameter
Description
l0
lattice element length
N
total number
m• m2 & p• P2 L
number of random choicesfor horizontal starter cracks number of random choicesfor vertical starter cracks lengthof growthincrementequalsl 0 probabilityof growthby extensionfor set 1 probabilityof growthby extensionfor set 2 cutoff length for fracture growthfor sets1 and 2 intervalfor uniformdistribution of In (T/S) in unitsof In [(L) 2/time] intervalfor uniformdistribution of In (r) in unitsof In [(L) 2/time]
(ar•, br•) (at, br)
of lattice
Value 0.02 [L]
elements
4900
150 220 0.02 [L] 1.0 1.0 1.25 [L] (7.55, 10.91) (-41.71, 18.29)
Subscripts refer to fracturesetnumber.Intervalsfor the uniformdistributionof In (T) andIn (T/S) are also given.
3342
HESTIR
ET AL.'
INVERSE
HYDROLOGIC
well 1
e ee
ee
eee
MODELING
well 2
ee e
ß
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well 3
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ee
ee
ßß eeeeeeeeeeeee ß
ee
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-14
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-10
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-8
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In(time)
well 4
In(time)
well 5
well 6
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0
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i
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-8
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-
ß ß
i
i
!
i
i
i
-16
-14
-12
-10
-8
-6
In(time)
In(time)
Figure 6. Simulatedflow data from Figure 5 with Gaussianerrorsadded.
at each temperatureti. Finding an appropriatetemperature scheduleis somewhatproblematicbut can be done usingnumerical studies.The annealingalgorithmusesa trial temperature scheduleto get the vectorUc. The trial temperature scheduleis then adjusted,by addingmore intermediatetemperaturesand increasingthe numberof iterationsat eachtemperature,until valuesof d(Uc), the distancefunction,are reachedthat are belowa giventolerance.This tolerancecanbe defined through numerical studiesto establishlower bounds for d(Uc) that (1) can be obtainedin a reasonableamountof computertime and that (2) lead to drawdowncurvesthat visuallymatchthe input data. Choiceof thistoleranceis similar to settinga tolerancein latticeannealing[Mauldonet al., 1993]. A roughlowerboundfor d(U•), and hencea lowerboundfor the tolerance,can be made by assumingthat U is such that #(U), the predictednetwork(e.g.,Figure7a), exactlymatches the networkthat yieldedmeasurements M (e.g., Figure5). In thiscased(U) reflectsonlymeasurement errorsandis thusthe sum of the squaresof n independentN(0, 1) randomvariables.For our first example,n = 120, so d(U) will be ap-
proximately anN(120, 2X/•) random variable. Thusa reasonable lowerboundford(U) is 120+ 2X/•-6,or ---135.In practice,this value generallycan not be reached,but it does
givea roughestimatefor a tolerance.
A generalproblemwith simulatedannealingis the lack of a widely applicablecriteria for determiningthe number of iterationsrequiredfor convergence. Hence a temperatureschedule must be devisedusing numerical trial and error as describedin thissection.This stepwouldbe greatlyfacilitatedby the theoreticaldevelopmentof rigorousconvergence criteria.
6. Inverse Modeling Example With Data From the Conoco Test Facility Our secondexampleis basedon a hydrologictestconducted in the Fort Riley limestoneat the ConocoBoreholeTest Fa-
Table 3. Values of h (U) for SamplesFrom Posterior
Distribution P(X = XIM = M) for Synthetic Data M in Figure 6 and the Networksin Figure 7 Sample
d (Uc)
1
573.1
2
475.0
3 4 5
363.3 286.8 574.5
Sample#1
Sample#2
II,
--I "--'
o
I
-0.4
-0.2
0.0
0.2
0.4
-0.2
I
0.4
0.2
0.4
Sample #4
iiL i -0.4
I
0.2
Sample #3
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[L]
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Sample #5
Sample #5 Distance Function
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-0.2
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0.0
[L]
0.2
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0
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I
2'1 (Y'6
Iteration
Figure7. Fivesamples fromtheposteriordistribution P(X = X M = M) withM fromthe simulated data in Figure 6. The lastplot showsthe distancefunctiond(U) versusiterationfor sample5.
well 1
well 4
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el
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lee I
ß I
I
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-8
In(time)
In(time)
well 2
well 5
ß
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i
I
I
-16
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In(time)
In(time)
well 3
well 6
ß
ß
I
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I
I
I
i
i
i
I
-16
-14
-12
-10
-8
-6
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-8
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In(time)
In(time)
Figure 8. Numericallycalculateddrawdowncurves(solidlines)from the networkin samplein Figure7 and simulateddata from Figure 6.
HESTIR
ET AL.'
INVERSE
cility (CBTF) in Kay County,Oklahoma [Datta-Guptaet al., 1994].The Fort Riley limestone,a nearlyhorizontalunit, lies at a depthof 45 m at the CBTF but iswell exposed5 km to the east-northeast, at Vap's Pass.Queenand Rizer [1990]mapped two setsof orthogonal,nearlyvertical,strataboundfracturesin the limestoneat Vap's Pass(Figure 2). The dominantset at Vap's Passstrikeseast-northeast, and the secondaryset strikes north-northwest.These fracturesappear to be part of a regionalsetof fracturesmappedin easternOklahomaby Melton [1929]. Five shallow(45 m) wells, designatedGW-1 to GW-5 in Figure 9a, were drilled and completedat the CBTF. During the hydrologictest,calledPump58 [Datta-Guptaet al., 1995], water was producedat a constantrate from well GW-5, and pressureresponseswere observedin GW-1 through GW-5. Backgrounddata collectedbefore the beginningof the test indicatedthat the wellswere recoveringfrom a rainfall event when
the test was started.
Hence
the drawdown
MODELING
3345
(A)
2[2 ......
½1
,i ,1•
'
[i-
,,,I i
-200
_•,,•ß-
100
i
200
meters
data were
corrected for rainfall effects before further analysis[DattaGuptaet al., 1995].The drawdowncurvesare shownin Figure 9b. One interesting feature of these drawdown curves, not unusualfor a fracture network, is that GW-2 respondsmore quicklythan the other wells even though it is furthest from
O
HYDROLOGIC
(B)
ø1
_
(A)
o GW-1
0 O
-
O Od
-
GW-5
GW-2
o GW-3
- 100
!
meters
GW-4 o r,l:) !
-50
o
_
Figure 10. Sample from the posterior distributionP(X =
i
i
-60
i
i
-20
i
0
20
i
i
40
60
X M = M) for the Pump 58 test including(a) the whole networkand (b) a closeupof the networknear the wells.
meters
GW-5
(B)
GW-2 GW-3
•' •
o
GW-1
'•
GW-4
i
i
i
i
8
10
12
14
In(timein seconds)
Figure 9. (a) Well locationsand (b) drawdowncurvesfor the Pump 58 test.
GW-5. This is indicativeof a direct fractureflow path between GW-5 and GW-2; thus it is interestingto see how the conditional codingsamplerepresentsthis fracture connection. Requirementsfor implementingthe inversemodelingprocedurehere are (1) growthparametersfor the lattice-restricted growth model, (2) parametersao, bo, a r, and br for the uniform distributionsof In (T/S) and In (T), (3) a function d(U) as in (12), determinedby the error distributionin In (head) values,and (4) a temperatureschedulefor simulated annealing.We now examineeach of theserequirements. The growthparametersusedin the inversionare thoseused to generatethe networkin Figures3a and 3b and are givenin Table 1. Theseparameterswere found throughtrial and error with the goalof approximatelymatchingthe fracturespacingin the map of Figure 2. Implicationsfor usingparametersdetermined in this way are discussedin section7. Values for ao, bo, a r and br are determinedby first generatingan examplefracturenetworkand numericallycalculating drawdown curves.The fracture networks are generated over a large enougharea so that the drawdowncurvessimulated
in the mesh do not show substantial
effects
from
the
constanthead boundaryconditionon the outer boundary.By
3346
HESTIR
ET AL.:
INVERSE
GW-5
HYDROLOGIC
MODELING
GW-2
GW-1
o''" !
i
i
i
!
I
10
12
14
8
10
12
log(timein sec.)
14
I
[
8
10
GW-4
12
log(timein sec.)
log(timein sec,)
GW-3
eß
ß
i
i
i
i
I
i
i
i
8
10
12
14
8
10
12
14
log(timein sec.)
log(timein sec.)
Figure 11. Numericallycalculateddrawdowncurves(solidlines) from the networkin Figure 10 and data from the Pump 58 test.
determininga range of horizontal and vertical shiftsneededto allow simulateddrawdowncurvesto approximatelymatch the data, we get valuesfor ao, bo, a r, and b r. These valuesare given in Table 1. Determinationof a distancefunctiond (U) requiresa probabilitydistributionfor measurement errorsasshownin (9). For this inverse model we assumeindependent Gaussianerrors, which leadsto the simplifieddistancefunction(12) discussed in section4. This choiceis somewhatarbitrary, and we did not attempt a detailed error analysis,which could establishthat
ing the primary feature of the well test data as describedin this section.With enoughcomputertime manysuchsamplescould be generatedgivingestimatesof posteriorprobabilitiesfor the existence,extent, and location of this short flow path. The fracture models so generatedwill have features that are reasonablefrom a geologicpoint of view, will match field measurements,and naturally quantify the information about the
errors are in fact correlated.
7.
The value of tr = 0.69 is reached
throughnumericalexperimentswith the temperatureschedule
fracture
network
contained
in the flow data.
Summary and Conclusions
We have presenteda general method for inversemodeling described below. that incorporatesa geologictheory for systemformation. The A temperature scheduleis derived in the sameway as was method is not limited in application to fracture networks, done for the syntheticexampledescribedabove.We numeri- rather it providesa general meansfor incorporatinggeologically experimentwith a seriesof temperature schedulesand cally reasonablemodelsinto inverseproblems. establishpracticallowerboundsfor valuesof d(Uc). The minThe method requirestwo main steps.The first is devisinga imal value of t in the temperature schedulerequired to reach geologicallyrealistic stochasticmodel and a meansfor simuthislowerboundis thensetequalto 202. In thisexample the lating field response.The secondstep is implementinga new minimal value reachedafter many experimentswas d(Uc) = simulationtechnique,called conditionalcoding,for sampling 15.34at tmin= 0.94, givinga value of tr = 0.69. Somecare must from the conditional (posterior) distributionof the model be used in this overall approach.If the errors are large and given field measurements.The conditionalcoding technique highlycorrelated,then overfittingcan occur,i.e., the matching requires onlythe algorithmsfor simulatingthe model and field of noise rather than signal. measurements, so it is very general and can be used on a wide A fracture network obtained with conditional coding is shownin Figure 10, and the flow responseof the network is variety of inverseproblems. The strengthof the method is that it leads to inversesoluplotted in the Figure 11. Computationof thisnetworkrequired tions that should be more realistic because they are con2 weeks(on a 70 MHz Sparc20 workstation). A major feature of the conditionalcodingnetwork is a rel- strainedby an understandingof the geologicsystem.The posativelyshortflow path betweenwellsGW-5 and GW-2, reflect- terior distributionformulationis alsoa naturalway to quantify
HESTIR
ET AL.:
INVERSE
HYDROLOGIC
the uncertaintyin the solutionthrough a range of predictions givenby Monte Carlo sampling. Drawbacks of the method are that it is highly computer intensiveand lacksa theoreticalconvergencecriteria. Both of theseproblemswouldbe alleviatedby theoreticaldevelopment of applicable convergencecriteria and annealing schedules basedon statisticalanalysisof the Markov chain. One further problem,broughtout in our example,is that the parametersof the stochasticmodel are estimatedthrough trial and error. In fact, it is a difficult problem, in general, to relate mapped patternsto the parametersof formation processes.It is theoretically possibleto use conditionalcoding on thig problem, and we are currentlyworking on this approach. The stochastic modelwe havepresentedis a rather simplistic representation of the complex processof fracture system growth.It could clearlybe improvedby accountingmore fully for fracture interactionand, in general,by incorporatingmore of the physicsof fracture growth.We would argue, however, that a simplified stochasticmodel of some kind will almost alwaysbe requiredfor modelingcomplexmechanicalsystems. Reasonsfor this are that detailed deterministicmodelsare very computationallyintensiveand require unattainableinformation on the precisegeologicboundaryconditionsand material properties. Others have usedgeologicprocessmodelsand related ideas to generatereservoirstructure[Koltermannand Gorelick,1996; Wang, 1996; Bogdan and Lerche, 1985]. Among these approachesare stochasticmodels that duplicate structurebut , that are not explicitlyrelated to formation processes,suchas the "structure-imitating"processesdescribedby Koltermann and Gorelick[1996]. Structure-imitatingmodelsstochastically generategeologicallyreasonablestructuresbut are not based on geologicprocesses.It is clear that a structure-imitating processcouldlikely be devisedthat producesfracturenetworks comparableto the networksgeneratedby our physicallybased model. However, usinga structure-imitatingmodel meansthat lessphysicsis usedin formingthe inversesolution.Indeed, the main weaknessof our exampleapplicationis that it doesnot contain enoughphysicsto be clearly better than a structureimitating model. This is not a weaknessof the general inverse modeling procedurewe have outlined. Further, if one insists on using a structure-imitatingmodel, the conditional coding methodis, of course,still applicable.So this classof stochastic models can also be conditioned
on field data for inverse solu-
tions.
MODELING
3347
Doughty, C., J. C. S. Long, K. Hestir, and S. M. Benson,Hydrologic characterizationof heterogeneousgeologicmedia with an inverse method based on iterated function systems,Water Resour. Res., 30(6), 1721-1745, 1994. Engelder,T., Jointsand shearfracturesin rock, in FractureMechanics of Rock, editedby B. K. Atkinson,pp. 27-69, Academic,San Diego, Calif., 1987.
Fair, G. M., J. C. Geyer, and D. A. Okun, Water and Wastewater Engineering,Water Supply and WastewaterRemoval, vol. 1, John Wiley, New York, 1966. Freedman,D. A., Statisticsand theScientificMethod,CohortAnalysisin SocialResearch:Beyondthe IdentificationProblem,edited by W. M. Mason and S. E. Fienberg, Springer-Verlag,New York, 1985. Hestir, K., Conditional Coding, paper presentedat Winter Conference, Am. Stat. Assoc.,Raleigh, N. C., 1995. Hestir, K., Conditionalcoding:A generalmethod for simulatingconditionaldistributions, J. Comput.GraphicalStat.,7(1), 77-91, 1998. Karasaki, K., A new advection-dispersion code for calculatingtransport in fracturenetworks,Earth Sci.Div. Annu. Rep.LBL-22090, pp. 55-58, LawrenceBerkeleyLab., Univ. of Calif., Berkeley,June 1987. Keer, L. M., and S. H. Chen, The interactionof a pressurizedcrack with a joint, J. Geophys.Res., 86, 1032-1038, 1981. Koltermann,C. E., and S. M. Gorelick, Heterogeneityin sedimentary deposits:A reviewof structure-imitating,processimitating,and descriptiveapproaches,WaterResour.Res.,32(9), 2617-2658,1996. Lawn, B. R., and T. R. Wilshaw,Fractureof BrittleSolids,Cambridge Univ. Press, New York, 1975.
Long, J. C. S., K. Hestir, K. Karasaki, A. Davey, J. Peterson,J. Kemeny, and M. Landsfeld,Fluid flow in fractured rock: Theory and application,in TransportProcesses in PorousMedia, editedby J. Bear and M. Y. Corapcioglu,Kluwer Acad., Norwell, Mass., 1989. Mauldon, A.D., K. Karasaki,S. J. Martel, J. C. S. Long, M. Landsfeld, and A. Mensch, An inverse techniquefor developingmodels for fluid flow in fracture systemsusingsimulatedannealing,WaterResour.Res.,29(11), 3775-3789, 1993. Melton, R. A., A reconnaissance of the joint systemsin the Ouachita Mountains and Central Plains of Oklahoma, J. Geol., 37, 729-746, 1929.
Olson, J., and D. D. Pollard, Inferring paleostresses from natural fracture patterns:A new method, Geology,17, 345-348, 1989. Press,W. H., B. P. Flannery, S. A. Teukolsky,and W. T. Vetterling, NumericalRecipes,CambridgeUniv. Press,New York, 1986. Queen, J. H., and W. D. Rizer, An integratedstudyof seismicanisotropy and the natural fracture systemat the Conocoborehole test facility, Kay County, Oklahoma,J. Geophys.Res.,95, 11,255-11,273, 1990.
Ripley, B. D., StatisticalInferencefor Spatial Processes,Cambridge Univ. Press, New York, 1988.
Segall,P., and D. D. Pollard, Joint formation in the graniticrock of the Sierra Nevada, Geol. Soc. Am. Bull., 94, 563-575, 1983.
Tada, H., P. C. Paris, and G. R. Irwin, The StressAnalysisof Cracks Handbook,Del Res. Corp., Hellertown, Pa., 1973. Wang, L., Modelingcomplexreservoirgeometrieswith multiple-point statistics, Math. Geol.,28(7), 895-907, 1996. Yeh, W. W.-G., Review of parameter identification proceduresin groundwaterhydrology:The inverseproblem, WaterResour.Res., 22(2), 95-108, 1986.
Acknowledgment. This work was supportedby DOE/BES grant DE-FG03-95ER14526.
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K. Hestir, Department of Mathematics and Statistics,Utah State University, Logan, UT 84322-3900. (e-mail:
[email protected]. usu.edu) J. Long, Mackay School of Mines, MS 172, University of Nevada, Reno, NV 89557-0138.
S. J. Martel, Department of Geologyand Geophysics,Universityof Hawaii at Manoa, 2525 Correa Road, Honolulu, HI 96822. W. D. Rizer, Houston Advanced Research Center, 4800 Research Forest Drive, Woodlands, TX 77381.
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(ReceivedMay 13, 1997;revisedMarch 5, 1998; acceptedMay 7, 1998.)
