INFERRING DEPTH-DEPENDENT RESERVOIR PROPERTIES FROM INTEGRATED ANALYSIS USING DYNAMIC DATA

Transcription

1 INFERRING DEPTH-DEPENDENT RESERVOIR PROPERTIES FROM INTEGRATED ANALYSIS USING DYNAMIC DATA A REPORT SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE By Vinh Qung Phn June 1998

2 I certify tht I hve red this report nd tht in my opinion it is fully dequte, in scope nd in qulity, s prtil fulfillment of the degree of Mster of Science in Petroleum Engineering. Dr. Rolnd Horne (Principl dvisor) ii

3 Astrct To e le to predict reservoir performnce or to optimize reservoir production, the determintion of reservoir properties is required. The reservoir properties re sptilly dependent nd deterministic ut re smpled t only very smll numer of points. It is impossile to determine most of them y direct mesurement. The mition of modern reservoir modeling is to mke integrted use of dynmic dt from multiple sources to infer the reservoir properties. The process of inferring the reservoir properties from indirect mesurement is n inverse or prmeter estimtion prolem. The prmeters of interest in this work re porosity nd solute permeility. These prmeters hve importnt influence in determining the performnce of the reservoir nd in reservoir optimiztion. This work represents wy of estimting such prmeters from vriety of indirect mesurements such s well test dt, long-term pressure nd wter-oil rtio history, nd 4-D seismic informtion nd lso considers the effect of the dt on the uncertinty nd resolution of reservoir prmeters. In prticulr, since erlier work (Lnd, 1997) hs ddressed two-dimensionl prolems, this study focuses on the estimtion of prmeters in three dimensions where properties vry s function of depth The ojective is to find sets of distriutions of permeility nd/or porosity such tht the model response closely mtches the reservoir response. In ddition, esides physicl constrints, the sets of permeility nd porosity must lso stisfy constrints given y other informtion known out the reservoir. iii

4 Acknowledgements I would like to express my sincere grtitude to Dr. Rolnd N. Horne, chirmn of Petroleum Engineering Deprtment, for his vlule guidnce nd counsel s principl dvisor throughout the entire course of this study. This work could not e done without his profound knowledge nd experience. The finncil support provided y SUPRI-D, Consortium on Innovtion in Well Testing, is grtefully cknowledged. iv

5 This report is dedicted to my mother, Nguyen Thi Thi, nd to the memory of my fther, Phn Thon. v

6 Contents Astrct Acknowledgements Tle of Contents List of Tles List of Figures iii iv vi ix x 1 Introduction 1 2 Previous Work 4 3 The Inverse Prolem Principle Forwrd Model Equtions Ojective Function Minimizing the Ojective Function Guss-Newton Method Line Serch Penlty Function nd Step-Size Controller Scling, Mrqurdt Modifiction, nd Cholesky Fctoriztion Pixel Modeling Resolution of Prmeters vi

7 3.5.1 Nonliner Prmeter Estimtes Permeility nd Log-Permeility Spce Sensitivity Coefficients Sustitution Method Computtion of Full Sensitivity Mtrix Computtion of Jcoin Mtrix Computtion of R ỹ (n) Computtion of R α Computtion of Sensitivity Coefficients Derivtives of Wellore Pressure Derivtives of Wter Cut Computtionl Results Appliction of the Method Exmple 1: Uniform Properties within Ech Lyer Exmple 2: Chnnel in Ech Lyer Exmple 3: Verticl Fult Resolution of Permeility nd Porosity Summry Optiml Strtegy for Dt Collection The Mening of Prmeter Estimtes Optiml Strtegy for Dt Collection Conclusion Summry Mjor Results Computtionl Procedures Ares tht Need Further Reserch Nomenclture 120 vii

8 Biliogrphy 123 A Lists of Progrms 126 A.1 Generl Instructions A.2 Dt File Structure A.3 Dt File Contents A.4 Ancillry Progrms A.5 Input Dt Files A.6 Output Dt Files A.7 Exmple Dt Files A.7.1 Input Dt File A.7.2 Flow Rte Dt File A.7.3 Oservtion Dt Files A.7.4 Reservoir Property Dt Files viii

9 List of Tles 4.1 CPU time in seconds Averge certinties nd certinties of the estimtes in gridlocks (1,1,1), (4,1,1), nd (4,4,1) ix

10 List of Figures 3.1 Forwrd nd inverse prolems Generl well completion Three-lyer reservoir model for sensitivity study Injection rte of Well # Long term pressure nd wter cut t Well # Chnge in wter sturtion etween 50 nd 150 dys Sensitivity of pressure nd wter cut with respect to the permeilities in NE-SW digonl t 150 dys Sensitivity of pressure nd wter cut t Well #2 with respect to the permeility in gridlock-(1,20,1) Sensitivity of pressure nd wter cut t Well #2 with respect to the permeility in gridlock (20,1,1) Sensitivity of chnge in wter sturtion etween 50 nd 150 dys Wter sturtion distriution in the ottom lyer t 150 dys Sensitivity of wter sturtion distriution in the ottom lyer Individul lyer well completion Multilyered well completion Nine wells in multilyered reservoir with individul lyer well completion (LP) Nine wells in multilyered reservoir with multilyered well completion (CP) Time-dependent rte history of nine wells x

11 5.6 Long-term pressure nd wter cut dt D seismic dt Wter sturtion t 15 dys: Lyer Production (LP) Mtch of long term pressure nd wter cut dt Mtch of 4-D seismic dt Comprison etween true nd clculted permeility, mtching Lyer Production nd Lyer y Lyer Seismic (LP-LS) Certinty of permeility estimtes, mtching Lyer Production nd Lyer y Lyer Seismic (LP-LS) Comprison etween true nd clculted permeility, mtching Lyer Production nd Depth-Averged Seismic (LP-AS) Uncertinty of permeility estimtes, mtching Lyer Production nd Depth-Averged Seismic (LP-AS) Comprison etween true nd clculted permeility, mtching Commingled Production nd Lyer y Lyer Seismic (CP-LS) Uncertinty of permeility estimtes, mtching Commingled Production nd Lyer y Lyer Seismic (CP-LS) Comprison etween true nd clculted permeility, mtching Commingled Production nd Depth-Averged Seismic (CP-AS) Uncertinty of permeility estimtes, mtching Commingled Production nd Depth-Averged Seismic (CP-AS) Comprison of permeility estimtes etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth- Averged Seismic (LP-AS) exmples Comprison of permeility estimtes etween Commingled Production nd Lyer y Lyer Seismic (CP-LS) nd Commingled Production nd Depth-Averged Seismic (CP-AS) exmples Comprison of the resolution mtrices etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth- Averged Seismic (LP-AS) dttypes xi

12 5.22 Comprison of permeility estimtes etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth- Averged Seismic (LP-AS) dttypes Comprison of permeility estimtes etween Commingled Production nd Lyer y Lyer Seismic (CP-LS) nd Commingled Production nd Depth-Averged Seismic (CP-AS) dttypes Comprison of certinty for four dt types: chnnel cse Resolution mtrices y mtching four dt types: chnnel cse Comprison of permeility estimtes etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth- Averged Seismic (LP-AS) dttypes Comprison of permeility estimtes etween Commingled Production nd Lyer y Lyer Seismic (CP-LS) nd Commingled Production nd Depth-Averge Seismic (CP-AS) dttypes Comprison of certinty (with clculted vlues) for four dt types: fult cse Comprison of certinty (with true vlues) for four dt types: fult cse Resolution mtrices y mtching four dt types: fult cse Estimtes of permeility in different situtions Estimtes of porosity in different situtions Resolution mtrices: either permeility or porosity is known Resolution mtrices: permeility nd porosity re treted independently Certinty in estimtes of porosity Mesure of correltion: permeility nd porosity re treted independently Mesure of correltion etween permeility nd porosity xii

13 Section 1 Introduction The predictions of reservoir performnce, coning effects such s gs coning in oil wells nd wter coning in gs wells, the effect of wter influx from nery quifers, the optimiztion of reservoir production, the plcement of infill wells, nd the predictions of rekthrough time nd recovery ll require the vilility of reservoir simultion model in which rock properties such s porosity nd permeility re specified t ll lock loctions. Moreover, the reservoir model geometry nd types of reservoir oundries such s fults, closed, liner, nd constnt pressure must lso e known in dvnce. For some purposes, reltively simple models such s homogeneous, frctured or dul-porosity reservoir my e dequte nd trditionl well test nlysis is useful tool to provide good reservoir description in the vicinity of well in such models. But other cses, for exmple to study the effect of wter influx nd coning on reservoir performnce, to optimize reservoir production or to predict rekthrough time nd recovery, often require detiled, distriuted descriptions of the reservoir prmeters. Trditionl well test nlysis encounters difficulties in such cses. The min cuse of these difficulties is tht trditionl well test nlysis dels only with reltive simple models such s homogeneity or t most dul-porosity. Another cuse is tht trditionl well test nlysis hndles trnsient pressure dt collected t single well over reltively smll time intervl, nlyzing ech set of collected dt nd estimting ech prmeter individully. As result, trditionl well test nlysis ignores the interction etween different regions of dt. This pproch cn only 1

14 SECTION 1. INTRODUCTION 2 cpture the verge properties in the well vicinity. There exist severl multiple-well nlysis methods to cpture the heterogeneity of the entire reservoir, however, the scle t which prmeters re resolved is reltively corse. Reservoir heterogeneities often control most of the reservoir flow phenomen nd thus the determintion of reservoir prmeters t fine scles is necessry. The focus of this work is on the estimtion of the sptil distriution of reservoir properties y mtching dt of different types t multiple loctions nd time. In prticulr, the study considered the integrtion of well test dt, long term pressure nd production history, nd sptil sturtion chnges s indicted y 4-D seismic surveys. Since erlier uthors hve considered minly two-dimensionl prolems, this work ddressed the determintion of properties s function of depth in fully three-dimensionl spce. This report consists of seven sections. Section 2 of this report outlines list of relted work tht hs een conducted previously. Section 3 descries the principle of reservoir prmeter estimtion in generl nd discusses in detil the solution technique for estimting these prmeters. The method of computing the resolution of prmeters is lso presented. Section 4 demonstrtes n efficient method of computing sensitivity coefficients (derivtives of the response of reservoir with respect to gridlock permeilities nd porosities) for lyered systems, prticulrly where wells intersect severl lyers. The sustitution method is lso presented for the purpose of ensuring the correctness of the computtionl results. Section 5 demonstrtes the viility of the method developed in this work for lyered reservoirs nd exmines severl study cses to nswer fundmentl issues ssocited with the resolution of depth-dependent properties in reservoir chrcteriztion prolems. Section 6 nlyzes the mening of the prmeters estimted from the nonliner prmeter estimtion prolem nd lso presents some importnt issues to e ddressed for the purpose of designing n optiml strtegy for dt collection. Section 7 summrizes the pproch developed in this study nd explores the

15 SECTION 1. INTRODUCTION 3 res tht need further reserch. Severl remrks re lso mde in pplying this pproch to the reservoir chrcteriztion prolem.

16 Section 2 Previous Work Severl previous works hve ddressed the prolem of reservoir chrcteriztion using integrtion of dt from vrious sources. This section discusses some of these relted works in which prmeters were estimted y mking use of grdient method. Chu, Reynolds, nd Oliver (1995) explored the ppliction of grdient methods to the prolem of reservoir chrcteriztion for two-dimensionl, single-phse flow. The permeility field ws estimted y mtching well test pressure dt nd hrd dt. Chu et l. first generted reliztion of permeility field tht honored ll dt loctions y kriging nd then used this reliztion s prior informtion nd s n initil guess for the Guss-Newton method to condition to the well test pressure dt. Although this method results in considerle svings in computtionl time y computing only sensitivities of welllock pressure, the sensitivity coefficients were only pproximte. Chu, Reynolds, nd Oliver (1995) generted reservoir rock property fields nd well skin fctors conditioned to multiwell pressure dt, hrd dt nd prior informtion which included the vriogrms nd the correltion coefficient etween porosity nd permeility. The Guss-Newton lgorithm ws used with sensitivity coefficients computed y the Generl Pulse Spectrum Technique (GPST). The uthors reported tht their lgorithm is very efficient. The convergence ws chieved in five to eight itertions nd did not get trpped t locl minim. Reynolds, He, Chu, nd Oliver (1995) recognized tht the inverse of the Hessin 4

17 SECTION 2. PREVIOUS WORK 5 mtrix t ech itertion in the Guss-Newton lgorithm ecomes very expensive s the numer of permeility nd porosity vlues need to e estimted is lrge (e.g. thousnd of gridlocks). They suggested two methods to improve the computtionl efficiency. The first method used spectrl (eigenvlue/eigenvector) decomposition of the prior model. The second method used the suspce vector to reduce the size of the mtrix tht must e solved t ech Guss-Newton itertion. The uthors showed tht if the prmeters re properly reprmeterized, the computtionl time required to generte the reservoir model decreses significntly. Lnd, Kml, Jenkins, nd Horne (1996) presented method to otin twodimensionl reservoir description for the Pgerungn Field, offshore Indonesi y integrting well test, production, shut-in pressure, log, core, nd geologicl dt. In this work, sensitivity coefficients were computed in very efficient wy using modifiction of GPST method descried y Chu nd Reynolds (1995) nd Tng nd Chen (1985) nd (1989) He, Reynolds, nd Oliver (1996) extended their own previous work from twodimensionl single-phse to three-dimensionl single-phse flow prolems. They used n djoint method for computing sensitivity coefficients in wy tht requires only one dditionl simultion run per well for ech itertion of the inverse prolem. The method is reltively efficient for the cses in which the numer of wells nd the numer of gridlocks re not lrge. For this single-phse prolem, He et l. ssumed no pressure grdient long the well ore. As we discuss in detil lter, these ssumptions significntly ese the computtion of the sensitivity of wellore pressure nd wter cut nd my not e vlid for most of the cses in which the WOR nd GOR will chnge with depth nd gs my e lierted in the wellore t n elevtion ove the lowermost perfortion. In this work, we investigted the use of the grdient-sed method for the threedimensionl two-phse reservoir chrcteriztion prolem in which more generl nd ccurte pproch for computing sensitivity coefficients ws used. These sensitivity coefficients cn then e incorported into the inverse prolem procedures for estimting the reservoir prmeters. Due to the existence of uncertinty oth in mesurements nd in models, these prmeter estimtes re not intended to e used s

18 SECTION 2. PREVIOUS WORK 6 finl nswers for the reservoir prmeters ut rther s reliztion for the reservoir chrcteriztion prolem.

19 Section 3 The Inverse Prolem Reservoir chrcteriztion is prolem of descriing reservoir properties indirectly from remote mesurements. Becuse the mesurements re imprecise we cn never hope to determine the true vlues of reservoir properties with solute certinty. Insted, we cn only chrcterize these uncertinties within confidence intervl. This section first explins the principle of the inverse prolem in reservoir chrcteriztion nd then discusses some methods tht cn e used to estimte the reservoir properties. We will focus in detil on the Guss-Newton method tht we used in this work. 3.1 Principle A physicl sitution cn e descried y n engineer in eqution form to show the reltionship mong certin quntities. These quntities re clssified into vriles nd prmeters. Vriles refer to mesurle quntities nd prmeters refer to the inherent properties of nture. If these prmeters re known within n cceptle ccurcy, computing the ehvior of certin physicl sitution (ehvior of some vriles) in response to n externl perturtion (some other vriles) is referred to s the forwrd prolem. In the reservoir chrcteriztion prolem, vriles cn e pressure nd sturtion nd prmeters cn e solute permeility, reltive permeility, porosity, well 7

20 SECTION 3. THE INVERSE PROBLEM 8 skin fctors etc. These prmeters re deterministic ut in some situtions their vlues re unknown nd need to e estimted. This sitution is referred to s the inverse prolem nd the process of inferring these prmeters is lso clled prmeter estimtion. These concepts re summrized in Figure 3.1. Forwrd prolem prmeters externl perturtion physicl sitution predicted ehvior Inverse prolem true ehvior externl perturtion physicl sitution prmeters Figure 3.1: Forwrd nd inverse prolems. In this work, we delt with the inverse prolem in which the prmeters re estimted given the response of reservoir t series of injector nd producer wells. The generl procedure for the solution of the inverse prolem cn e divided into three mjor steps s follows: 1. Estlishing forwrd model equtions. 2. Defining n ojective function. 3. Minimizing the defined ojective function. Hving estlished the forwrd model equtions nd defined n ojective function, common lgorithm for minimizing the defined ojective function in the inverse prolem is s follows: 1. Compute or resonly guess vlue for n unknown set of prmeters. 2. Compute the response of the mthemticl model.

21 SECTION 3. THE INVERSE PROBLEM 9 3. Compute the vlue of the defined ojective function which is defined s the squre of the discrepncy etween the reservoir response nd the model response. If the vlue is less thn predetermined tolernce then the lgorithm is terminted. 4. Updting y computing chnge in the set of prmeters. If the prmeters re not updted significntly (the chnge is less thn certin predetermined tolernce) then the lgorithm is terminted. 5. Go to step Forwrd Model Equtions The forwrd mthemticl model equtions used in this work were derived from mteril conservtion nd Drcy s lw. In petroleum engineering, the mss conservtion for ny component is normlly converted to the volume conservtion evluted t stndrd conditions or surfce conditions. Consider n ritrry, fixed volume V emedded within permele medium through which is flowing n ritrry numer of component. The conceptul volume conservtion eqution for component c in volume V is: Rte of Rte of Net rte of c ccumultion = production trnsported of c in V of c in V from V (3.1) All terms in Eqution 3.1 re volumetric flow rtes evluted t stndrd conditions. Assuming the mteril trnsported in the porous medium is only y convection: Net rte of c trnsported = np Ũ p.ñ R cp ds (3.2) p=1 S B p from V The surfce integrl in Eqution 3.2 cn e converted to volume integrl through the divergence theorem.

22 SECTION 3. THE INVERSE PROBLEM 10 np p=1 S R cp Ũ p.ñ B p ds = np p=1 V ( ) Ũ p.ñ. R cp dv (3.3) B p Rte of ccumultion = d np dt p=1 V of c in V R cp S p φ B p dv = np p=1 V ( ) S p φ R cp dv (3.4) t B p Rte of production = q c dv (3.5) V of c in V where: q c is volume metric production rte of component c per unit ulk volume evluted t stndrd conditions. Ũ p is the Drcy velocity of phse p. R cp is soluility of component c in phse p. B p is formtion volume fctor of phse p. φ is the porosity of the porous medium. np is the numer of phses. Comining Equtions. 3.2 through 3.5 into Eqution 3.1 gives the following sclr eqution for the component c: np ( ) S p φ np ( ) Ũ p.ñ R cp dv = q c dv. R cp dv (3.6) p=1 V t B p V p=1 V B p Let the ritrry volume V pproch zero: np ( ) np ( ) S p φ Ũ p.ñ R cp = q c. R cp (3.7) p=1 t B p p=1 B p Eqution 3.7 is the differentil form for the component c. The forwrd flow model equtions were constructed in this work with the following fetures: Three-dimensionl flow in Crtesin coordintes. Slightly-compressile fluids (wter nd oil).

23 SECTION 3. THE INVERSE PROBLEM 11 Flow is only y convection. Heterogeneous nd isotropic medium. No cpillry pressure. Applying these fetures to the generl Eqution 3.7 gives the following equtions tht were used in this work for the forwrd mthemticl model: ( ) Ũw. + ( ) φ0 ( x)f(p)s w + q w =0 (3.8) t. B w ( ) Ũo B o B w + ( ) φ0 ( x)f(p)s o + q o =0 (3.9) t B o Ũ w = k rw(s w )k( x) µ w.(φ w ) (3.10) Ũ o = k ro(s o )k( x).(φ o ) µ o (3.11) Φ w = p γ w z (3.12) Φ o = p γ o z (3.13) γ w = ρ wg (3.14) γ o = ρ og (3.15) B w = B w (p) (3.16) B o = B o (p); (3.17) 3.3 Ojective Function The ojective is to estimte the reservoir prmeters y mtching the model response to the ctul reservoir response. To mtch the model response to the reservoir response, we minimize the ojective function which is the discrepncy etween the oservtion dt nd the response computed from the mthemticl model. There

24 SECTION 3. THE INVERSE PROBLEM 12 re severl wys to define n ojective function provided tht it is mesure of discrepncy. One of the most common forms of the ojective function is the Weighted Lest Squre descried y Eqution Ojective Function = E =ẽ T Wẽ (3.18) ẽ is the discrepncy etween the oservtion dt nd model response: ẽ =( d os d cl ) (3.19) d os is the set of dt mesurements nd d cl is the response computed from the mthemticl model. The oservtion dt considered in this work includes long term well pressure, wter cut from production dt, nd the chnge in wter sturtion inferred from 4-D seismic. d = p wf ( α) w ct ( α) S w ( α) (3.20) α =( k, ϕ) (3.21) α Set of unknown prmeters k Permeility distriution ϕ Porosity distriution p wf w ct S w Long term well pressure Wter cut Chnge in wter sturtion W is digonl weighting fctor mtrix which is included into the ojective function to ccount for the different scles of the different mesurements. For exmple, mesurement of wter cut is etween 0 nd 1 while the mesurement of pressure my e from 1000 psi to psi. Comining Eqution 3.18 nd Eqution 3.19 gives the Weighted Lest Squre form of the ojective function tht ws used in this work: E =( d os d cl ) T W( d os d cl ) (3.22)

25 SECTION 3. THE INVERSE PROBLEM Minimizing the Ojective Function Hving defined the ojective function, E = E( α), the next step is to construct n optiml set of prmeters α such tht this function is minimized. E( α )=min α E( α) (3.23) α D D is the domin in which the prmeters re defined. This domin is determined y set of prmeter constrints s will e discussed lter in this section. One of the chrcteristics of the reservoir prmeter estimtion prolem is tht the ojective function to minimize is nonliner with respect to the prmeters. Therefore, finding the optiml point in prmeter spce is n itertive serch process in which the succession of chnges of prmeters re computed to stisfy these conditions: 1. The ojective function must reduce in ech itertion. 2. The prmeters re confined inside the fesile domin. Since the forwrd model is complex nd expensive to compute, the numer of function evlutions should e s smll s possile. As shown in the literture, there re lrge numer of methods for minimizing multivrite function. In devising or choosing n optimiztion method one ttempts to minimize the totl computtion time required for convergence to the minimum. This time is composed primrily of the following two fctors: 1. Function nd derivtive evlutions. 2. Algeric mnipultions such s mtrix inversions or eigenvlue determintions. It is usully possile to trde off these fctors ginst ech other. A method employing more lorious lgeric procedures my require fewer itertions, nd hence fewer function evlutions. This is likely to py off if the ojective function is complicted one. In prmeter estimtion prolems, the ojective function is synthesized from the model equtions nd from the dt otined in mny experiments nd its computtion

26 SECTION 3. THE INVERSE PROBLEM 14 is usully time consuming. We do not hesitte therefore to recommend methods which re sophisticted lgericlly, s long s they re efficient in terms of the numer of required function nd derivtive evlutions. For the reservoir prmeter estimtion prolem, grdient-sed methods hve often een found to e the most effective. The ckwrd-solution technique used in this work to otin the solution of the inverse prolem mde use of the following fetures: 1. Grdient-sed Guss-Newton method to compute direction of descent. 2. Line serch to serch for etter point in the direction of descent. 3. Penlty functions nd step-length controller to constrin prmeters within the fesile domin. 4. Scling, modified Mrqurdt, nd Cholesky mtrix solution techniques for stiliztion. These fetures re discussed in the following sections Guss-Newton Method The Guss-Newton lgorithm ws used in this work to compute direction of descent in the prmeter estimtion prolem. δ α is sid to e direction of descent if there exists smll positive numer ρ such tht E( α + ρδ α) <E( α) (3.24) Theojectiveistofind α defined s α =min α E( α), α D, D R npr (3.25) where npr is the numer of prmeters. α is locl minimum of E in D if E( α + δ α) >E( α ); 0 < δ α ɛ; α + δ α D (3.26)

27 SECTION 3. THE INVERSE PROBLEM 15 α is glol minimum of E in D if E( α + δ α) >E( α ); δ α > 0; α + δ α D (3.27) α is referred to s the optiml point. Since the min purpose of the inverse prolem is to find the minimum point, it is importnt to review some optimlity conditions. The necessry conditions for n interior point α to e locl minimum point of smooth ojective function re: E( α )= E = 0 (3.28) α α α is sttionry point; nd the Hessin mtrix H of E evluted t α is semipositive definite. α T H α 0; α 0 (3.29) The Hessin mtrix H is defined s the second derivtive of E s follows: H = E (3.30) α or, H ij = 2 E = 2 E = H ji (3.31) α i α j α j α i From Eqution 3.31, the Hessin mtrix is symmetric mtrix. The sufficient conditions is the sme s the necessry conditions except tht the Hessin mtrix H is positive-definite. Eqution 3.29 ecomes strict inequlity. Hence t the optiml point the solution of the inverse prolem must stisfy these two conditions: 1. The function E must e t sttionry point. 2. The Hessin mtrix evluted t the sttionry point is positive-definite. The sttionry point cn e found y solving Eqution 3.28 which is nonliner with respect to α. The eqution cn e linerized y pplying Tylor series expnsion for E in the neighorhood of α 0 : E( α 0 + δ α) = E( α 0 )+ E δ α + O( δ α 2 )= E( α 0 )+H 0 δ α + O( δ α 2 ) α α0 (3.32)

28 SECTION 3. THE INVERSE PROBLEM 16 If α = α 0 + δ α is sttionry point, it must e true tht E( α 0 + δ α) =0nd comining with Eqution 3.32 gives: E( α 0 )+H 0 ( α α 0 )+O( α α 0 2 ) = 0 (3.33) Truncting the series fter the first order gives liner system of equtions tht cn e solved for n pproximtion to α : E( α 0 )+H 0 δ α = 0 (3.34) or, H 0 δ α = E( α 0 ) (3.35) If the Hessin mtrix in Eqution 3.35 is computed exctly, the method is known s Newton s method. Although the rte of convergence of this method is fst (qudrtic convergence), this method does not work in mny cses for the following two resons: 1. The Hessin mtrix is not positive-definite. Therefore, δ α is not gurnteed to e direction of descent nd the sttionry point my not e minimum point. 2. The method requires the evlution of the second derivtives. This plces hevy computtionl urden nd my e difficult where the ojective functions re complicted. As ws discussed erlier, the sufficient condition for sttionry point to e minimum point is tht the Hessin mtrix evluted t the sttionry point must e positive-definite. We will show tht if the Hessin mtrix is positive-definite, δ α otined from Eqution 3.35 is direction of descent. Premultiplying Eqution 3.35 y δ α T gives δ α T H 0 δ α = δ α T E( α 0 ) (3.36) Since H 0 is positive definite, the left hnd side of Eqution 3.36 is positive nd hence δ α T E( α 0 ) < 0 (3.37)

29 SECTION 3. THE INVERSE PROBLEM 17 E cn e pproximted in the neighorhood of α 0 using Tylor series expnsion. There exists vlue of ρ positive nd smll enough such tht the following eqution holds. E( α 0 + ρδ α) =E( α 0 )+ρδ α T E( α 0 ) (3.38) Comining Eqution 3.38 with Eqution 3.37 gives E( α 0 + ρδ α) <E( α 0 ) (3.39) As we hve shown, we cn not gurntee δ α to e direction of descent unless the Hessin mtrix is positive-definite. In mny cses it my e necessry to mke chnge in the Hessin mtrix to chive two following properties. 1. The modified mtrix is positive-definite. 2. The modified mtrix is close to the Newton Hessin mtrix. The second property is desirle for the qudrtic convergence. Gill, Murry, nd Wright (1981) descrie severl well-known methods to mke the Hessin mtrix positive-definite while retining the dvntges of the Newton method. Newton- Greenstdt nd Mrqurdt s method re designed to overcome the prolem of indefiniteness, wheres the Guss-Newton nd Singulr Vlue Decomposition (SVD) methods eliminte the need for computing second derivtives. Since this work mde use of the Guss-Newton method, we will descrie this method in detil, in prticulr, how to compute oth sides of Eqution We strt with the definition of the ojective function: E =ẽ T Wẽ =( d os d cl ) T W( d os d cl ) (3.40) E = E α = ( ) T ẽ E α ẽ (3.41) By ssuming the mtrix W is symmetric nd constnt: E =2Wẽ (3.42) ẽ nd, ẽ α = ( d os d cl ) = d cl = G (3.43) α α

30 SECTION 3. THE INVERSE PROBLEM 18 Comining Eqution 3.41 with Eqution 3.43 gives the formul to compute the grdient of the ojective function s follows: E = 2G T Wẽ (3.44) The Hessin mtrix is computed s: H = 2 E = E α (3.45) Considering column i of mtrix H: ut, thus, or, H i = E α i (G T Wẽ) α i = 2 (GT Wẽ) α i (3.46) = GT α i Wẽ + G T W ẽ α i (3.47) H i = 2 GT α i Wẽ 2G T W ẽ α i = 2 GT α i Wẽ +2G T W d cl α i (3.48) [ ] G T H = 2 Wẽ α i H =[H i ]= 2 [ G T α i Wẽ +2G T W d cl α ] = 2 +2 [ [ G T G T W d cl α i α i Wẽ ] ] (3.49) +2G T WG (3.50) In the Guss-Newton method, we neglect the first term in Eqution 3.50, nd the Hessin mtrix H is replced y the Guss-Newton Hessin mtrix: H gn =2G T WG (3.51) Eqution 3.35 ecomes H gn δ α = E( α) (3.52) If the weighting fctor mtrix W is positive-definite, we cn prove tht the Guss- Newton mtrix defined y Eqution 3.51 is t lest semipositive-definite.

31 SECTION 3. THE INVERSE PROBLEM 19 For ll δ α R npr nd δ α 0 δ α T H gn δ α =2δ α T G T WGδ α =2(Gδ α) T W(Gδ α) (3.53) W is positive-definite nd Gδ α 0. Therefore, from Eqution 3.53, we hve δ α T H gn δ α 0 (3.54) Thus, the Guss-Newton mtrix is semipositive-definite. The equlity occurs only when Gδ α = 0. Comining Gδ α = 0 to Eqution 3.51 nd Eqution 3.52 gives E( α) = 0 (3.55) Thus the equlity in Eqution 3.54 occurs only t the minimum point. This rgument shows tht lthough the Guss-Newton mtrix my not e positive-definite t some itertions during the solution process, δ α computed from Eqution 3.52 is still direction of descent provided the Guss-Newton mtrixisnonsingulr Line Serch The purpose of computing the direction of descent is to provide direction (δ α) long which we cn find nother point t which the vlue of the ojective function is lower. Line serch is n lgorithm for serching long the direction of descent for such points. The sic ide ehind line serch is tht we first pick point long the direction of descent ( α + ρ 0 δ α). If the point we hve picked is worse (E( α + ρ 0 δ α) E( α)), we next try smller vlue of ρ 0, nd keep repeting the process until etter point is found. As ws shown in the previous section, if δ α is direction of descent, such ρ 0 lwys exists. Becuse line serch involves only plin function evlution while computing the direction of descent requires oth function nd derivtive evlution which is t much higher cost, it lwys pys to try t lest one other vlue of ρ to see whether we cn do even etter. The optiml vlue of ρ is determined y minimizing qudrtic pproximtion to the ojective function in the neighorhood of α. For ll ρ>0, α+ρδ α is point long the direction of descent (δ α) nde( α+ρδ α) is n univrite function depending only on ρ. E(ρ) =E( α + ρδ α) (3.56)

32 SECTION 3. THE INVERSE PROBLEM 20 The qudrtic pproximtion to the ojective function is of the form: E (ρ) =ρ 2 + ρ + c (3.57) We hve computed: E(0) = E( α) (3.58) E(ρ 0 )=E( α + ρ 0 δ α) (3.59) E (0) = E = δ α T E( α) (3.60) ρ ρ=0 Since the qudrtic pproximtion E (ρ) in Eqution 3.57 must stisfy the conditions descried from Eqution 3.58 to Eqution 3.60, prmeters, nd c re determined s: c = E( α) (3.61) = δ α T E( α) (3.62) = E( α + ρ 0δ α) ρ 0 c (3.63) ρ 2 0 Then the optiml step size ρ is otined y minimizing Eqution 3.57 s: ρ = (3.64) 2 This line serch lgorithm ws descried y Brd(1970). As we will show lter, line serch lone does not work in some cses, prticulrly where constrints re imposed on prmeters nd the ojective function is concve downwrd close to the oundry. In these situtions, the lgorithm my serch for point outside the fesile domin. For this reson it is often necessry to use penlty functions nd step-size controller in the line serch lgorithm Penlty Function nd Step-Size Controller The fesile region in which the prmeter estimte is to e found is limited nd ny serch lgorithm should e confined only to this region. This work mde use of penlty functions nd step-size controller to confine the serch. One of the resons for the serch to step out of the fesile domin is tht the ojective function

33 SECTION 3. THE INVERSE PROBLEM 21 is concve downwrd close to the oundry. Therefore the penlty functions must e designed to ecome very lrge when the prmeters pproch the oundry (the ojective function then is concve upwrd) nd to ecome negligile elsewhere inside the fesile region. Since the penlty function is defined sed on the constrints of prmeters, it is worth to first discuss the types of constrints tht we used in this work. For the reservoir prmeter estimtion prolem tht uses the pixel modeling method (this method will e explined lter) to descrie the permeility nd porosity distriutions, the prmeters re the unknown properties in ech grid cell nd the constrints imposed on those prmeters re rnges of resonle vlues. The generl form of the constrints on the prmeters is expressed s follows: c( α) 0 (3.65) where c R ncons nd ncons is the numer of constrints. These constrints cn e liner or nonliner with respect to the prmeters α. In the cse where the prmeters re permeilities or porosities it is useful to set lower nd upper ounds: k min <α i <k mx ; i (3.66) or, φ min <α i <φ mx ; i (3.67) k min nd φ min cn e set to zero or to resonle lower ounds. Ech constrint in Eqution 3.65 is then expressed s: c j ( α) =k mx α i ; i (3.68) nd, c j+1 ( α) =α i k min ; i (3.69) where i is prmeter index (i =1 npr) ndj is constrint index (j =1 ncons). The new ojective function including the penlty functions is defined s: Ê = E + ncons j=1 ɛ j c j ( α) (3.70)

34 SECTION 3. THE INVERSE PROBLEM 22 Thechoiceofɛ j should e positive nd smll enough tht the ojective function remins lmost unchnged in the interior of the fesile region nd should pproch zero s the serch pproches the minimum point. ɛ j 0 s E 0 (3.71) The initil choice of ɛ j should lso e dictted y the rnge of vlues tht c j ( α) cn tke in the fesile region. One pproprite definition of ɛ j stisfying these conditions is s follows: ɛ j =10 3 (k mx k min )E (3.72) Using penlty functions still does not lwys confine the line serch to remin within the fesile region. Sometimes the serch procedure my step too fr nd cross the fesile region into n infesile one where the vlue of the ojective function my e smller. To gurntee tht the serch lwys stys within the fesile domin, it is necessry to hve step-length controller. The sic ide ehind the step-length controller is tht the step length ρ i t ech itertion is controlled y n upper ound ρ i,mx which is the smllest positive vlue of ρ for which α + ρδ α lies within the oundry of the fesile region. For simple constrints in Eqution 3.66 nd Eqution 3.67, ρ i,mx cn e clculted s follows. Since α nd α + ρδ α re interior points: α min <α j <α mx ; j =1 npr (3.73) nd, α min <α j + ρδα j <α mx ; j =1 npr (3.74) Comining Eqution 3.73 nd Eqution 3.74 we hve or, thus ρ i,mx is expressed s: α min α j ρ i,mx =min j δα j ρ< α mx α j δα j ; if δα j > 0 (3.75) ρ< α min α j δα j ; if δα j < 0 (3.76) δαj <0, α mx α j δα j ; j =1 npr (3.77) δαj >0

35 SECTION 3. THE INVERSE PROBLEM 23 It is lso resonle to hve lower ound for ρ. This lower ound ρ i,min is the positive smllest step size such tht further itertions fil to chnge the prmeter vlues significntly. The lgorithm will e terminted if the step length is forced to e less thn ρ i,min. The smllest llowle chnge of prmeters etween two itertions recommended y Brd (1970) is s follows: Tht mens, we ccept α (i+1) s the solution α provided α (i+1) j ɛ j =10 4 (α (i) j ) (3.78) α (i) j ɛ j ; j =1 npr (3.79) where superscripts denote itertion index nd suscripts denote prmeter index. Since, α (i+1) j comining with Eqution 3.79 gives = α (i) j + ρδα (i) j (3.80) ρ δα (i) j ɛ j ; j =1 npr (3.81) or, ρ ɛ j δα (i) j hence the minimum dmissile ρ for the ith itertion is ρ i,min =min j ɛ j δα (i) j (3.82) (3.83) Summry of Bsic Equtions Hving introduced the penlty function into the definition of the ojective function, it is necessry to recompute the grdient nd the Hessin mtrix. A new set of sic equtions is s follows: ẽ =( d os d cl ) (3.84) G = d cl α (3.85)

36 SECTION 3. THE INVERSE PROBLEM 24 E =ẽ T Wẽ (3.86) Ê = E + ncons j=1 ɛ j c j ( α) (3.87) E = 2G T Wẽ (3.88) ncons Ê = E ɛ j c j (3.89) H = 2 ncons Ĥ = H +2 j=1 [ G T α i Wẽ ɛ j c 3 j j=1 ] c 2 j +2G T WG (3.90) ncons c j c T j j=1 ɛ j c 2 j c j α (3.91) H gn =2G T WG (3.92) ncons ɛ j Ĥ gn = H gn +2 c j=1 c 3 j c T j (3.93) j Ĥ gn δ α = E (3.94) From Eqution 3.91, the new Guss-Newton mtrix Ĥ gn could lso e written s follows: Ĥ gn = H gn +2 ɛ ( ) ncons j c c 3 j c T j c c j j (3.95) j j=1 α If α is fr wy from the jth constrint, the contriution of ɛ j /c j ( α) nd its derivtives is very smll. Close to the jth constrint c j is nerly zero, nd the second term under the summtion of Eqution 3.95 my e neglected reltive to the first term. In either cse, it is sfe to replce Eqution 3.95 y Eqution It should lso e noted tht since c j c T j is semipositive-definite, the new Guss-Newton mtrix Ĥ gn is gurnteed to e positive-definite s is required. The ddition of the penlty functions does not spoil the definiteness of the originl Guss-Newton mtrix. In the cse of liner constrints, prticulrly those specifying only the physicl limits of the prmeters, the second derivtives vnish nywy nd Eqution 3.93 is then exct.

37 SECTION 3. THE INVERSE PROBLEM Scling, Mrqurdt Modifiction, nd Cholesky Fctoriztion As ws shown efore, t ech itertion the ckwrd solution technique using Guss- Newton lgorithm requires the solution of set of simultneous liner equtions: H gn δ α = E (3.96) in which, from Eqution 3.51, H gn is given y: H gn =2G T WG (3.97) where, G = d cl (3.98) α The Hessin mtrix H gn, s ws proved in Section 3.4.1, is semipositive-definite. The lck of strictly positive-definiteness in H gn rises directly from the structure of the sensitivity mtrix G. In reservoir prmeter estimtion, the reservoir normlly does not respond to ll prmeters t the sme order of mgnitude. There re some prmeters tht cuse strong effect on the reservoir ehvior ut others tht show lmost no influence. As result, some columns of the sensitivity mtrix my e zero or very smll compred to the others nd consequently the Hessin mtrix H gn ecomes singulr or very ill-conditioned. If the mtrix is singulr, Eqution 3.84 hs no solution nd is impossile to solve. If the mtrix is ill-conditioned, round-off error cn cuse prolems of ccurcy nd the solution process my not e numericlly stle. To overcome these difficulties, it is importnt to first recll t this point tht our min interest is in finding direction of descent rther thn computing δ α precisely. Therefore, we cn introduce chnge to the mtrix H gn in such wy to prevent the lck of strict positive-definiteness. Oviously the chnge should e slight to retin the qudrtic convergence property of the originl mtrix close to the optimum point. This pproch ws implemented in this work y mens of Mrqurdt nd Cholesky Fctoriztion methods tht will e discussed next. Finlly the stiliztion process is lso enhnced y prescling the mtrix H gn to mke ll the digonl elements unity. This ws ccomplished y first constructing

38 SECTION 3. THE INVERSE PROBLEM 26 digonl scling mtrix F whose elements re the inverse of the squre root of the digonl elements of H gn : F ii =(H gnii ) 1 2 (3.99) nd then pre- nd postmultiplying mtrix H gn y F. Eqution 3.84 ecomes (FH gn F)F 1 δ α = F E (3.100) Solving Eqution 3.88 for (F 1 δ α) is more stle thn solving Eqution δ α then cn e determined s: δ α = F(F 1 δ α) (3.101) Mrqurdt Method This method converts the semipositive-definite Guss-Newton Hessin mtrix into positive-definite one y dding sufficiently lrge numer to its digonl: H gn = H gn + µi (3.102) where I is the identity mtrix nd µ is positive numer. The vlue of µ is chosen sufficiently lrge to void ill-conditioning ut smll enough to retin the closeness etween the modified mtrix nd the originl one. Modified Cholesky Fctoriztion Method This method is modifiction of Cholesky Fctoriztion method to hndle the sitution in which the fctored mtrix is not gurnteed to e positive-definite. The Cholesky fctors of mtrix exist only when the mtrix is positive-definite. If the Cholesky fctoriztion fils then the mtrix is not positive-definite nd the method introduces n incrementl chnge in the digonl elements of the originl mtrix. H gn = H gn + E (3.103) where E is nonnegtive digonl mtrix. This method is descried in detil y Gill, Murry, nd Wright (1981). Since the method converts the mtrix into positivedefinite one nd simultneously stilizes it, this method is most desirle for solving either Eqution 3.35 or Eqution 3.88 to otin direction of descent. The method ws implemented in this work.

39 SECTION 3. THE INVERSE PROBLEM Pixel Modeling This pproch ws used in this work to descrie the permeility nd porosity distriution of the reservoir t the finest level of the simultion grid. Since ech unknown reservoir property in every cell of the simultion grid is considered s one prmeter, this method is ttrctive in terms of the lrge mount of reservoir informtion eing chieved nd my result in sets of permeility nd porosity distriutions tht reproduce the oservtion dt. When the pixel modeling pproch is used long with grdient-sed method, it is necessry to compute the sensitivity of the model response to the permeility nd porosity t every cell of the simultion grid. The method used to compute these sensitivity coefficients in n efficient mnner will e discussed in detil in Chpter Resolution of Prmeters The resolution of prmeters cn e determined y nswering the three questions: 1. How close to the true vlue cn ech prmeter e resolved? 2. How well does the computed response gree with the oserved dt? 3. Wht is the certinty in ech prmeter estimte? For the nonliner prmeter estimtion prolem, nswering these questions quntittively is still left unresolved. However, the resolution of prmeters cn e well understood for the liner cses, in which the response of the model is liner with respect to prmeters, nd is descried in literture y Jckson (1972) nd Menke (1989). In this work, this theory ws used for the nonliner cse in mnner similr to tht of Lnd (1997) nd Dtt-Gupt, Vsco, nd Long (1995).

40 SECTION 3. THE INVERSE PROBLEM Nonliner Prmeter Estimtes The ide is to linerize the ehvior of the system with respect to the prmeters. Tht is, if d denotes the ehvior of the system, then: d α = G = const (3.104) or, d = G α + const (3.105) As ws shown erlier, the prmeters cn e estimted y minimizing the lest squre ojective function: E =( d os d cl ) T ( d os d cl )= d os d cl 2 (3.106) Comining Eqution nd Eqution we hve, E = d os G α 2 (3.107) The prmeters otined y minimizing Eqution represent the unique solution of liner system of equtions descried s: G T G α = G T dos (3.108) or, α =(G T G) 1 G T dos = G g dos (3.109) where G g = (G T G) 1 G T is the generlized inverse of mtrix G which cn e computed sed on the Singulr Vlue Decomposition theory. For ny nonsqure nos x npr mtrix G cn e decomposed s: G = UΛV T (3.110) where U is n orthogonl nos x nos mtrix, V is n orthogonl npr x npr mtrix, nd Λ is digonl nos x npr mtrix. U T U = UU T = I nos (3.111)

41 SECTION 3. THE INVERSE PROBLEM 29 V T V = VV T = I npr (3.112) Λ = Λ p (3.113) where I is the identity mtrix, Λ p is squre digonl p x p mtrix, nd p is the numer of nonzero elements on the digonl of Λ. Mtrices U nd V cn e expressed s: U = [ U p ] U 0 (3.114) V = [ V p ] V 0 (3.115) where U p nd V p re the columns of U nd V respectively corresponding to the nonzero elements in Λ. U 0 nd V 0 re the columns of U nd V respectively corresponding to the zero elements in Λ. U T p U p = I p (3.116) V T p V p = I p (3.117) Comining Eqution with Eqution 3.115, mtrix G cn e given s follows: G = UΛV T = [ ] U p U Λ p 0 VT p 0 = U p Λ p V T 0 0 V0 T p (3.118) Comining Eqution with Eqution to the definition of the generlized inverse of mtrix G in Eqution gives: G g =(G T G) 1 G T = ( (U p Λ p Vp T )T (U p Λ p Vp T )) 1 (Up Λ p Vp T )T = V p Λ 1 p UT p (3.119) Eqution cn e used to compute the generlized inverse of mtrix G nd the est lest squre estimte of the prmeters is: α = G g dos (3.120) However, d os = G α t (3.121)

42 SECTION 3. THE INVERSE PROBLEM 30 where α t is the true vlues of prmeters. Thus, α = G g G α t = R α t (3.122) nd, d cl = G α = GG g dos = S inf dos (3.123) where R = G g G is the resolution mtrix determining the reltionship etween the estimted prmeters nd the true prmeters. If R is close to identity, the estimted prmeters hve good resolution. S inf = GG g is the informtion density mtrix determining the reltionship etween the clculted response nd the true response. If S inf is close to identity, the true response is mtched well. The resolution nd informtion density mtrices cn e determined sed on the singulr vlue decomposition s: R = V p Vp T (3.124) S inf = U p U T p (3.125) According to Eqution 3.119, to compute the generlized inverse requires singulr vlue decomposition s descried in Eqution An lterntive method which ws used in this work to compute the component mtrices in Eqution mkes use of the eigenvlue decomposition. From Eqution G is given y: Then we cn construct mtrix M defined s: G = U p Λ p V T p (3.126) M = G T G = V p Λ 2 pv T p (3.127) The mtrix M is symmetric nd thus cn e decomposed y eigenvlue decomposition to find V p nd Λ 2 p. From Eqution 3.126, U p is then determined y: From Eqution 3.120, the estimted prmeters re: U p = GV p Λ 1 p (3.128) α = G g dos (3.129)

43 SECTION 3. THE INVERSE PROBLEM 31 The covrince mtrix of the prmeter estimtes cn e clculted s: C{ α } = G g C{ d os }G gt = V p Λ 1 p UT p C{ d os }U p Λ 1 p VT p (3.130) where C{ d os } is the covrince of the oserved dt. If the mesurement errors re independent, C{ d os } is digonl mtrix. Comining Eqution 3.127, Eqution 3.128, nd Eqution gives C{ α } = V p Λ 1 p Λ 1 p V T p G T C{ d os }GV p Λ 1 p Λ 1 p V T p (3.131) = V p Λ 2 p VT p GT C{ d os }GV p Λ 2 p VT p (3.132) = M 1 CM 1 (3.133) where the mtrices M nd C re (npr x npr) nd re given s follows: M 1 = V p Λ 2 p VT p (3.134) C = G T C{ d os }G (3.135) The vrince of prmeters cn e otined directly from the digonl of the covrince mtrix of prmeters C{ α } s: σ 2 α i = npr k=1 where σ 2 α i is the vrince of prmeter i. npr j=1 H 1 i,j C j,k H 1 k,i (3.136) Permeility nd Log-Permeility Spce In mny cses, it is convenient to consider the logrithm of permeility (permeility is log-norml distriution or logrithm of permeility is linerly correlted to porosity etc.) It is useful to perform vrince nd resolution nlysis in log-permeility spce. We now show how to trnsform sensitivity coefficients from permeility spce to log-permeility spce. In review of Eqution 3.104, the sensitivity coefficients in permeility spce cn e expressed s: G k = d k (3.137)

44 SECTION 3. THE INVERSE PROBLEM 32 Where G k is the sensitivity coefficients with respect to permeility. The sensitivity coefficients with respect to log permeility re determined s: G lnk = d ln k (3.138) nd the chin rule gives: G lnk = d ln d k = k k ln k = G kdig{ K} (3.139) where dig{ K} is the digonl mtrix whose digonl elements re permeilities. The generlized inverse of the sensitivity coefficients in log-permeility spce cn e computed s: G lnk g = ( G k dig{ K} ) g = dig{ K} 1 G k g = dig{ K 1 }G k g (3.140) where dig{ K 1 } is the digonl mtrix whose digonl elements re the inverse of permeilities. G g k is the generlized inverse of the sensitivity mtrix in permeility spce computed y the SVD lgorithm s shown erlier in Section

45 Section 4 Sensitivity Coefficients To minimize the ojective function using grdient sed method, we need to evlute the derivtives of the ojective function with respect to ll unknown prmeters. In mny prmeter estimtion prolems, these unknown prmeters pper only implicitly in the ojective function. The ojective function depends explicitly on the model response, which in turn depends on the prmeters through the forwrd mthemticl model equtions. To compute derivtives of the ojective function, we first differentite it with respect to the model response, nd then differentite the model response with respect to the prmeters. The derivtives of the model response with respect to the prmeters re clled the sensitivity coefficients. It should e noted tht the response of the model should not e confused with the solution of the forwrd flow equtions. The response of the model refers to the response corresponding to the oserved dt while the solution of the forwrd flow equtions refers to the pressure, sturtion, nd reference pressures in ll the wells. If d m denotes the response of the model nd ỹ denotes the solution of the forwrd flow equtions, then d m nd ỹ re expressed s follows: p wf d m = w ct (4.1) S w 33

46 SECTION 4. SENSITIVITY COEFFICIENTS 34 p 1 S w1 p 2 S w2.. p. ỹ = ỹ = S ỹ w = p n w p ref S wn p ref1 p ref2... p refnw (4.2) where n = nx ny nz is the totl numer of gridlocks nd nx, ny, ndnz re the numer of locks respectively in x, y, ndz directions. nw is the numer of wells. p nd S w re respectively the pressures nd wter sturtions t gridnode loctions. p ref re the well pressures t reference lyers. Since the chnge in wter sturtion nd wter cut cn e computed from the vector solution, the response of the model is indeed function of the forwrd solution nd prmeters: d m = d(ỹ, α) (4.3) Discretizing the forwrd equtions s descried in Eqution 3.8 to Eqution 3.11 in spce nd time, we otin set of residul flow equtions which re used to determine the solution of the forwrd flow prolem: R( k( x), φ( x), ỹ (n), ỹ (n+1), x) = 0 (4.4) As ws discussed erlier in Chpter 3, in the pixel modeling pproch, since ech

47 SECTION 4. SENSITIVITY COEFFICIENTS 35 unknown gridnode permeility nd porosity is considered s one prmeter, Eqution 4.4, in term of prmeters, ecomes: R( α, ỹ (n), ỹ (n+1), x) = 0 (4.5) where R is set of residuls in ll gridlocks, including well constrints. x is set of discrete loctions. α = α( x) R npr is set of discrete prmeters. npr is the numer of discrete prmeters. ỹ = ỹ( α, x, t) is vector solution t discrete loctions x nd time t. ỹ (n) nd ỹ (n+1) re respectively vector solutions t time step n nd (n + 1). In the forwrd flow prolem, the prmeters α in Eqution 4.5 re known nd since this eqution is nonliner with respect to ỹ (n) nd ỹ (n+1), given the solution t time step n, the solution t time step (n + 1) cn only e solved y itertion such s y the Newton-Rphson method. R ỹ (n+1) J ỹ (n+1) = R ỹ (n+1) δỹ(n+1) = R( α, ỹ (n), ỹ (n+1), x) (4.6) where J = is the Jcoin mtrix of R determined t ỹ (n+1). The Newton- Rphson lgorithm cn e summrized s: 1. Set ỹ (n+1) =ỹ (n) s n initil guess for the new time step. 2. Compute the Jcoin mtrix J t ỹ (n+1). 3. Solve Eqution 4.6 for δỹ (n+1) 4. If converged, then replce n +1yn nd go to step Updte the new ỹ (n+1) y: ỹ (n+1) =ỹ (n+1) + δỹ (n+1) 6. Go to step 2. By iterting forwrd in time, the entire vector solution ỹ( α, x, t) cn e otined nd hence the response of the model d = d(ỹ) is lso determined. The new set of prmeters is otined y perturing the next prmeter nd Eqution 4.5 is

48 SECTION 4. SENSITIVITY COEFFICIENTS 36 solved for the new forwrd solution. If this process is repeted until ll prmeters re pertured, the sensitivity coefficients cn e pproximted y finite difference method. This is the sic ide of the sustitution method tht will e discussed in detil next. 4.1 Sustitution Method The purpose is to compute the sensitivity mtrix G defined s: G = d α (4.7) The first order pproximtion to d y Tylor series is: d( α + δ α, x, t) d( α, x, t)+ d δ α (4.8) α Thus, the mtrix G cn e pproximted s: G = d α d( α + δ α, x, t) d( α, x, t) δ α = d α (4.9) or, G i,j = d i (4.10) α j The lgorithm of the sustitution methodimplementedinthisworkissfollows: 1. Set α = α 0 t which the sensitivity is computed. 2. Solve Eqution 4.5 for ỹ =ỹ( α 0, x, t) nd compute d 0 = d(ỹ). 3. For j =1 npr: Pertur α j = α j + δα j. Where δα j is frction of α j ; δα j = F α j. F is positive numer prespecified sed on types of dt nd prmeters (typiclly 10 7 ). Solve Eqution 4.5 for ỹ =ỹ( α, x, t) nd compute d = d(ỹ). Compute G i,j = d i d 0i δα j

49 SECTION 4. SENSITIVITY COEFFICIENTS End. Although this method is strightforwrd nd very esy to implement, it requires npr + 1 simultion runs which is very expensive in terms of computtionl work when the numer of prmeters npr is lrge. A fr more efficient method to compute sensitivity coefficients is y emedding the lgorithm to compute the full sensitivity mtrix inside the forwrd solution procedure s will e shown in the next section. 4.2 Computtion of Full Sensitivity Mtrix The purpose is to compute the full sensitivity mtrix S defined s: S = ỹ (4.11) α In this work, this mtrix ws computed in mnner similr to tht of Anterion, Eymrd, nd Krcher (1989). The size of the vector solution ỹ is the sme s the numer of unknown vriles in the forwrd flow prolem, tht is the totl of pressures nd sturtions in ll gridlocks nd the numer of wells: size =2n + nw (4.12) where n nd nw re the numer of gridlocks nd wells respectively. Therefore, the size of the full sensitivity mtrix S is (size x npr). Approximting the residul flow equtions descried in Eqution 4.5 y first-order Tylor series expnsions with respect to α, ỹ (n),ndỹ (n+1) gives: R( α + δ α, ỹ (n) + δỹ (n), ỹ (n+1) + δỹ (n+1), x) = R( α, ỹ (n), ỹ (n+1), x)+ However, R R δ α + α ỹ (n) δỹ(n) + R ỹ (n+1) δỹ(n+1) (4.13) R( α + δ α, ỹ (n) + δỹ (n), ỹ (n+1) + δỹ (n+1), x) = R( α, ỹ (n), ỹ (n+1), x) = 0 (4.14) thus, R R δ α + α ỹ (n) δỹ(n) + R ỹ (n+1) δỹ(n+1) = 0 (4.15)

50 SECTION 4. SENSITIVITY COEFFICIENTS 38 Dividing oth sides of Eqution 4.15 y δ α nd comining with Eqution 4.11 gives: R α + R ỹ (n) S(n) + R ỹ (n+1) S(n+1) = 0 (4.16) which ecomes, y rerrnging: R ỹ (n+1) S(n+1) = R ỹ (n) S(n) R α (4.17) where S (n) nd S (n+1) re full sensitivity mtrices t time step n nd (n +1)ndre given y: S (n) = ỹ(n) (4.18) α S (n+1) = ỹ(n+1) (4.19) α The function of Eqution 4.17 is to compute the full sensitivity mtrix S. Fromthis eqution, given S (n), to compute S (n+1) we need to compute three mtrices R ỹ (n+1), R,nd R. The first one is the Jcoin mtrix nd is the sme one computed in ỹ (n) α the forwrd solution procedure (Eqution 4.6). The three mtrices re otined y differentiting the set of residul flow equtions with respect to the unknowns t the current time step, the unknowns t old time step, nd the prmeters respectively. The computtion is discussed in detil lter in this section. Overll, S (n+1) is computed y columns. If S i ; i =1 npr denotes column i of mtrix S, Eqution 4.17 is equivlent to npr systems of liner equtions: R S(n+1) ỹ (n+1) i = R S(n) ỹ (n) i R ; i =1 npr (4.20) α i To compute the full sensitivity mtrix S, it is necessry to solve Eqution 4.20 npr times Computtion of Jcoin Mtrix The residul flow equtions in ech gridlock cn e expressed s flow term F, well term Q, nd n ccumultion term à s follows: R = F Q à (4.21)

51 SECTION 4. SENSITIVITY COEFFICIENTS 39 or, t lock index l s: R l = F l Q l A l (4.22) where 6 F l = T lt Φ lt (4.23) t=1 where the lt index denotes the interfce etween lock l nd lock t. Φ lt =( p lt γ lt D lt ) (4.24) Q l = T w (p l p wfl ) (4.25) A l = V ( ) (n+1) ( ) (n) l φs φs (4.26) t n+1 B B l l In this work, the set of finite-difference residul flow equtions re descried y fully implicit scheme. All coefficients re evluted t the new time step. Pressuredependent terms re evluted using mid-point verging. Sturtion-dependent terms re evluted using single-point upstrem pproximtion. V l is the ulk volume of lock l V l =( x y z) l (4.27) T lt = A ltk lt Kr(S) L lt (µb) lt (4.28) (µb) lt = x l(µb) t + x t (µb) l x l + x t (4.29) K lt = k lk t ( x l + x t ) k l x t + k t x l (4.30) L lt =0.5( x l + x t ) (4.31) r 0 = T w = WIΛ (4.32) 2πkh WI = ln ( ) (4.33) r 0 r w + s 0.14 x 2 + y 2 verticl well; 0.14 x 2 + z 2 horizontl well in y direction; 0.14 y 2 + z 2 horizontl well in x direction. (4.34)

52 SECTION 4. SENSITIVITY COEFFICIENTS 40 Λ= Kr(S) µb The residuls for well constrints re: i=1 (4.35) nc R w = Q sp w Q i ; w =1 nw (4.36) where Q sp w is the specified flow rte nd Q i is the flow rte t connecting well lock i. The Jcoin mtrix is computed y differentiting Eqution 4.21 nd Eqution 4.36 with respect to ll unknown vriles which re pressures nd sturtions in ll gridlocks nd reference pressures in ll wells. The ordering of the set of flow equtions descried in Eqution 4.21 nd Eqution 4.36 is wter efore oil eqution in ech lock, then lock y lock in the x direction, then row y row in the y direction, nd finlly lyer y lyer downwrd in the z direction. With this order, the residul vector R is expressed s: R w 1 R o 1 R w 2 R o 2... R = R w (4.37) l R o l... R w n R o n

53 SECTION 4. SENSITIVITY COEFFICIENTS 41 R w = R w1 R w2... R wl... R wnw (4.38) R = R R w (4.39) If ỹ denotes the pressures nd sturtions in gridlocks nd ỹ w denotes the reference pressures in ll wells, tht is: ỹ = p 1 S w1 p 2 S w2... p n S wn (4.40) ỹ w = p ref1 p ref2... p refnw (4.41)

54 SECTION 4. SENSITIVITY COEFFICIENTS 42 then the Jcoin mtrix J is given y: J = R R R R R ỹ = w (ỹ, ỹ w ) = ỹ ỹ w R w R (4.42) w ỹ ỹ w Due to the similrity etween the residul equtions of wter nd those of oil, we will derive only the derivtives of the wter equtions with respect to unknowns. R ỹ is (2x2) lock mtrix nd its elements re (2x2) mtrices given y: Rw i R i R o R w R w i i = i y j (p j,s wj ) = p j S wj R o R o (4.43) i i p j S wj where R w i nd R o i re respectively wter nd oil residul equtions t gridlock i. p j nd S wj re respectively pressure nd wter sturtion t gridlock j. Differentiting Eqution 4.22 gives: R w i = F i w Qw i Aw i (4.44) p j p j p j p j R w i = F i w Qw i Aw i (4.45) S wj S wj S wj S wj Differentiting Eqution 4.23 gives: F w i p j = ( 6 T ) lt ( Φ lt ) Φ lt T lt p j p j t=1 Fi w = S wj Differentiting Eqution 4.28 gives: T lt p j ( 6 t=1 T lt S wj Φ lt ) (4.46) (4.47) = A ltk lt Kr(S) (µb) lt (4.48) L lt (µb) 2 lt p j T lt = A ltk lt Kr(S) (4.49) S wj L lt (µb) lt S wj Differentiting Eqution 4.29 gives: (µb) lt p j = x (µb) t (µb) l p j + x l t p j (4.50) x l + x t

55 SECTION 4. SENSITIVITY COEFFICIENTS 43 Differentiting Eqution 4.24 gives: ( Φ lt ) p j =( ( p lt) p j Differentiting Eqution 4.26 gives: ( ) A w i = V is φ B p j t n+1 p j = V is t n+1 γ lt p j D lt ) (4.51) φ p j B φ B p j B 2 (4.52) A w i = V iφ (4.53) S wj t n+1 B Differentiting Eqution 4.25, Eqution 4.32, nd Eqution 4.35, gives: Q w ( i pi = T w p ) wf i + T w (p i p wfi ) (4.54) p j p j p j p j ( ) Q w i S wj = T w p wf i S wj T w p j T w S wj Λ = Kr(S) p j (µb) 2 + T w S wj (p i p wfi ) (4.55) = WI Λ p j (4.56) = WI Λ (4.57) S wj ( µ B + B µ ) (4.58) p j p j Λ = 1 Kr(S) (4.59) S wj (µb) S wj The prtil derivtives given y Eqution 4.43 to Eqution 4.59 only exist when gridlock j is neighor connection of gridlock i. R ỹ w is (2x1) lock mtrix nd its elements re (2x1) mtrices given y: R i y wj = Rw i R o i = y wj where p refj is the reference pressure t well j. R w i p refj R o i p refj (4.60) R w i p refj = Qw i p refj = T w p wfi p refj (4.61)

56 SECTION 4. SENSITIVITY COEFFICIENTS 44 The prtil derivtives given y Eqution 4.60 nd Eqution 4.61 only exist when gridlock i is connecting welllock of well j. R w ỹ is (1x2) lock mtrix nd its elements re (1x2) mtrices given y: R wi = R [ w i y j (p j,s wj ) = Rwi p j where R wi is the well constrint eqution of well i. Differentiting Eqution 4.36 gives: ] R wi S wj (4.62) R wi p j nc = k=1 Q k p j (4.63) R wi S wj nc = k=1 Q k S wj (4.64) The prtil derivtives given y Eqution 4.63 nd Eqution 4.64 exist only when gridlock j is connecting welllock of well i. R w ỹ w is (1x1) mtrix nd its elements re given y: R wi y wj = R w i p refj (4.65) Differentiting Eqution 4.36 gives: R wi p refj nc = k=1 Q k p refj (4.66) The prtil derivtives given y Eqution 4.66 exist only when well j isthesmes well i (e.g. i = j). As ws shown from Eqution 4.43 to Eqution 4.66, we hve relted the elements of the Jcoin mtrix to terms ll of which, except for the pressure grdient long the wellore nd its derivtives with respect to the solution vector, re known. It is importnt to remrk tht to ese the computtion of the derivtives of wellore pressure with respect to the solution vector in lyered systems, in the forwrd flow simultion one normlly uses semi-implicit scheme for the pressure grdient long the wellore. This mens tht the wellore pressure t ny lyer t the new time step is relted to the pressure t the reference lyer through the pressure grdient t the old time step. In this cse, the derivtives of wellore pressure with

57 SECTION 4. SENSITIVITY COEFFICIENTS 45 respect to the solution vector re simply either one or zero. This semi-implicit scheme is no longer vlid for the prmeter estimtion prolem. Since the downhole pressure nd wter cut depend on the wellore pressure grdient t the new time step, using the pressure grdient t old time step to explicitly evlute the pressure nd wter cut t new time step gives n inccurte response of the model nd consequently the prmeters estimted y mtching downhole pressure nd wter cut dt re distorted. Computing pressure grdient long the wellore nd its derivtives implicitly in lyered systems is reltively complex nd is discussed seprtely s follows. Let us define the following terms (see Figure 4.1): l: index for perforted lyers k: index for the reference lyer Our interest is to relte the pressures in ll lyers to the pressure t the reference depth nd compute the derivtives of these pressures with respect to the solution vector. Considering two djcent lyers l nd l + 1, nd neglecting the pressure loss due to friction long wellore, the difference in pressure etween the two lyers with depth cn e expressed s: p = p l+1 p l = γ(d l+1 D l )=0.5(γ l+1 + γ l ) D (4.67) where γ l nd γ l+1 re the specific weights of fluid present in lyer l nd lyer l +1 nd re given y: γ l = γw l f w l + γl o f o l (4.68) γ l+1 = γw l+1 f w l+1 + γl+1 o fo l+1 (4.69) γ w nd γ o re the specific weights of wter nd oil phses. γ w = γ ref w (1 + c w(p p ref )) (4.70) γ o = γ ref o (1 + c o (p p ref )) (4.71) f w nd f o re lyer frctionl flow rtes of wter nd oil phses respectively. f w = λ w λ w + λ o = 1 (4.72) 1+ kro(sw)µw k rw(s w)µ o

58 SECTION 4. SENSITIVITY COEFFICIENTS 46 Well l=1 2 Reference lyer (k) 3 D Perforted lyer (l) l=l Figure 4.1: Generl well completion Comining Eqution 4.68 with Eqution 4.72 gives: f o =1 f w (4.73) γ l + γ l+1 = γw l f w l + γl+1 w f w l+1 + γl o f o l + γl+1 o fo l+1 = γw ref (1 + c w (p l p ref ))fw l + γw ref (1 + c w (p l+1 p ref ))fw l+1 + γo ref (1 + c o (p l p ref ))fo l + γref o (1 + c o (p l+1 p ref ))fo l+1 = γw ref (fw l + fw l+1 γ ref )+γ ref o (fo l + f o l+1 w c w ((p l p ref )fw l +(p l+1 p ref )fw l+1 )+ )+γ ref o c o ((p l p ref )fo l +(p l+1 p ref )fo l+1 ) (4.74) This eqution is then comined with Eqution 4.67 to give the pressure in lyer l +1

59 SECTION 4. SENSITIVITY COEFFICIENTS 47 s function of the pressure in lyer l nd the wter sturtion in lyer l nd lyer l +1. p l+1 = p l(γw ref c wfw l + γref o c o fo l)+γref w (f w l + f w l+1)(1 p refc w )+γo ref (fo l + f o l+1 )(1 p ref c o ) 2( D) 1 (γw ref c w fw l+1 + γo ref c o fo l+1 ) (4.75) The importnt ppliction of Eqution 4.75 is tht it cn e used in fully implicit scheme to compute pressure grdient long the wellore with depth. The derivtives of wellore pressure with respect to the solution vector cn e computed y the method descried in the following prgrph. Differentiting Eqution 4.67 with respect to S wl, S wl+1,ndp l gives: p l+1 S wl Rerrnging gives: p l S wl p l+1 S wl+1 p l+1 p l p l+1 S wl = =0.5 D =0.5 D ( γl S wl 1=0.5 D + γ l p l + γ ) l+1 p l+1 p l S wl p l+1 S wl ( γl+1 p l+1 + γ ) l+1 p l+1 S wl+1 S wl+1 ( γl + γ ) l+1 p l+1 p l p l+1 p l (4.76) (4.77) (4.78) ( ) p l S wl D γ l p l +0.5 D γ l S wl D γ (4.79) l+1 p l+1 p l+1 S wl+1 = 0.5 D γ l+1 S wl D γ (4.80) l+1 p l+1 γl p l D p = l p l D γ (4.81) l+1 p l+1 where the derivtives of the γ functions t lyer l with respect to wellore pressure nd wter sturtion t the sme lyer re otined y differentiting Eqution 4.68: γ l p l = γl w p l f l w + γl o p l f l o (4.82) γ l S wl = γ l w fw l S wl f l o + γo l (4.83) S wl

60 SECTION 4. SENSITIVITY COEFFICIENTS 48 Differentiting Eqution 4.70 to Eqution 4.73 gives: γ l w p l γ l o p l = γ ref w c w (4.84) = γ ref o c o (4.85) where, f l w S wl = λ w l S wl λ o l λ w l λ o l S wl (λ w l + λ o l )2 (4.86) λ w l S wl = 1 µ w k rw S wl (4.87) λ o l S wl = 1 µ o k ro S wl (4.88) f l o = fl w (4.89) S wl S wl Eqution 4.79 through Eqution 4.81 cn e used recursively to compute the derivtives of the wellore pressure in ech lyer with respect to the solution vector. Since the prtil derivtives exist only t few loctions, the Jcoin mtrix is very sprse mening tht it contins only reltively smll numer of nonzero elements. There re severl prcticl schemes to optimize the computtionl work in solving sprse system of liner equtions, such s the Yle solver (1977) Computtion of R ỹ (n) This mtrix is otined y differentiting the set of residul flow equtions defined t new time step t (n+1) with respect to the unknowns t old time step t (n). R ỹ (n) = R ỹ (n) R w ỹ (n) R ỹ w (n) R w ỹ w (n) (4.90) Since only the ccumultion terms depend on the solution t the old time step, this mtrix is expressed s: R R ỹ = 0 ỹ (n) (n) (4.91) 0 0

61 SECTION 4. SENSITIVITY COEFFICIENTS 49 where R ỹ (n) y: R w i p (n) i is (2x2) lock digonl mtrix whose digonl elements re determined = Aw i p (n) i R i y (n) i = = V i t n+1 Rw i R o i (p (n) i,s w (n) i ) = S ( φ B R w i S (n) w i p (n) i Computtion of R α ) (n) = V i t n+1 R w i p (n) i R o i p (n) i = Aw i = V i S w (n) i t n+1 R w i S w (n) i R o i S w (n) i ( φ S p (n) i (4.92) B φ B p (n) i B 2 ) (n) (4.93) ( ) (n) φ (4.94) B As descried in Section 3.4.5, in the pixel modeling pproch the prmeters eing estimted re the permeility nd porosity t ech lock of the simultion grid. The mtrix R α is therefore function of the unknown permeility nd porosity vectors: R ( α = R k, R ) φ (4.95) 0 R R R α = w = α R i α j = R w i α j Rw i R o i = α j = F i w α j F w i α j = T lt α j Qw i α j ( 6 t=1 R α R w α R w i α j R o i α j T lt α j Φ lt (4.96) (4.97) Aw i (4.98) α j ) (4.99) = A ltkr(s) L lt (µb) lt K lt α j (4.100)

62 SECTION 4. SENSITIVITY COEFFICIENTS 50 K lt α j = ( x l + x t x (k l x t + k t x l ) 2 t kl 2 ) k t + x l k 2 k l t α j α j (4.101) Q w i α j = T w i α j (p i p wfi ) (4.102) T wi α j =Λ i WI i α j (4.103) WI i α j = WI i k i k i α j (4.104) WI i 2πh = k i ln ( ) r 0 r w + s (4.105) A w i = V ( ) (n+1) ( ) (n) i S φ S φ α j t n+1 B α j B α i j i (4.106) φ = f(p) φ 0 α j α j (4.107) R wi α j nc = k=1 Q k α j (4.108) All of the terms in Eqution 4.95 to Eqution depend on k i α j nd φ 0 α j which re the derivtives of permeility nd porosity t gridlock i with respect to tht t gridlock j. These derivtives re either zero or one: k i 1 i = j; = (4.109) α j 0 i j. φ 0i = α j 1 i = j; 0 i j. (4.110) R R Since the three mtrices,,nd R re ll very sprse, to optimize the computtionl work nd sve spce, only their nonzero elements re computed nd stored. ỹ (n+1) ỹ (n) α Hving computed these mtrices, the full sensitivity mtrix S cn e determined y solving Eqution 4.20 npr times. The key simplifiction is tht the left-hnd side of this eqution is held constnt, while only the right-hnd side is chnged. This tsk ws effectively implemented using the Yle solver (1977). As ws mentioned in Section 4.1, our finl interest is to compute the sensitivity of the responses corresponding

63 SECTION 4. SENSITIVITY COEFFICIENTS 51 to the oserved dt with respect to the prmeters of the inverse prolem. These sensitivities cn e computed from the full sensitivity mtrix S s will e explined next. 4.3 Computtion of Sensitivity Coefficients By knowing the vector solution of the forwrd prolem ỹ, we cn completely determine the response of the model d m = d(ỹ, α). Therefore, the sensitivity coefficients cn e computed s: G = d m α = d ỹ ỹ α + d α = d ỹ S + d (4.111) α From Eqution 4.111, hving determined S, to compute G we need to compute d nd ỹ d. Since these two mtrices depend on the type of dt in the prmeter estimtion α prolem, it is importnt to define this set of dt first. The set of dt for the prmeter estimtion prolem in this work consists of: 1. Downhole pressures of wells (denoted y p wf ). 2. Wter cut of wells (denoted y w ct ). 3. Chnge in wter sturtion distriution over the reservoir etween two given instnts (denoted y S w ). Therefore, p wf d m = w ct S w p ỹ = S w G = d m α = p ref p wf α w ct α S w α (4.112) (4.113) (4.114)

64 SECTION 4. SENSITIVITY COEFFICIENTS 52 S = ỹ α = p α S w α p ref α (4.115) From Eqution nd Eqution 4.115, to compute the sensitivity mtrix G, we need to compute the three mtrices p wf, wct,nd S w. The lst one is otined α α α directly from the full sensitivity mtrix S: S w α = S w α S w (4.116) t2 α t1 Where t1 ndt2 re the two given instnts in time t which 3-D seismic informtion is collected to infer the time-lpse (4-D) seismic signl. The first two mtrices cn not esily e otined from the full sensitivity mtrix nd the computtion of these mtrices emphsizes the difference in the level of difficulty etween single-lyer nd multilyered reservoir prolems. In multilyered reservoirs, prticulrly when wells re completed over severl lyers, the derivtives of wellore pressure nd wter cut with respect to the prmeters ecomes extremely complicted. The lyer flow rte is prmeter dependent. Moreover, the pressure grdient vries long the wellore nd is function of pressure nd sturtion in ll connecting welllocks nd thus the derivtives of the wellore pressure with respect to the prmeters cn no longer e derived directly from the derivtives of the welllock pressure in the well constrint equtions. Similrly, the wter cut is no longer just simple function of moility rtio s is the cse in single lyer prolems ut is complex function of moility rtios, pressure, nd sturtion in ll lyers. Thus, the derivtive of wter cut with respect to the prmeters requires the derivtives of moility, pressure, nd sturtion in ech lyer penetrted y the wells. One wy of computing the derivtives of wellore pressure nd wter cut with respect to prmeters from the full sensitivity mtrix in multilyered systems is proposed in the following section Derivtives of Wellore Pressure Wellore pressure nd its derivtives with respect to the prmeters depend upon the loction of the downhole tool. Since the loction of the downhole tool my chnge

65 SECTION 4. SENSITIVITY COEFFICIENTS 53 during the test, the reference lyer my not e the sme s the lyer where the mesurement tkes plce. Therefore, the pressure t the tool depth is not simply the reference pressure ut is relted to the reference pressure y Eqution Applying Eqution for the pressure t the tool depth we hve. p wf α = p wf ỹ ỹ α + p wf α α ỹ = p wf ỹ S + p wf α α ỹ (4.117) From Eqution 4.75, it is esy to see tht wellore pressure does not depend explicitly on the prmeters. Therefore, the lst term in Eqution vnishes. Hence, p wf α p wf α ỹ = p wf ỹ = 0 (4.118) S (4.119) α where the terms p wf re the derivtives of wellore pressure t the depth of the ỹ α downhole tool with respect to the solution vector nd cn e computed y using Eqution 4.79 to Eqution 4.81 recursively. In this work, the reference lyer ws ssumed to coincide with the lyer where the mesurement tkes plce ( p wf = p ref ). p wf ỹ = p ref ỹ α = α Derivtives of Wter Cut 1 ỹ = p ref ; 0 otherwise. (4.120) The wter cut t ech well is defined s the rtio etween the wter nd the totl liquid volume metric flowrtes mesured t surfce conditions. w ct = q w q w + q o (4.121) where q w nd q o re respectively wter nd oil volume metric flow rtes mesured t surfce conditions. For the injectors, since only wter is injected, we hve: w ct = 1 (4.122)

66 SECTION 4. SENSITIVITY COEFFICIENTS 54 nd, w ct α = 0 (4.123) For the producers, differentiting Eqution with respect to prmeters gives: w ct α = w ct q w q w α + w ( ) ct q o q o α = 1 q w q (q w + q o ) 2 o α q q o w (4.124) α Since the inner oundry conditions cn e specified in term of wter, oil, or totl liquid rtes, the derivtives of wter cut with respect to prmeters ccording to Eqution differs with these conditions. If wter flow rte is specified: w ct q w α = q o (q w + q o ) 2 α (4.125) If oil flow rte is specified: w ct α = q o q w (q w + q o ) 2 α (4.126) If totl liquid rte is specified: w ct α = 1 q w q w + q o α (4.127) From Eqution to Eqution 4.127, to compute the derivtives of wter cut, we need to compute the derivtives of wter nd oil flow rtes t the producers with respect to the prmeters. The flow of wter nd oil components from ech well is the sum of the flow of those components from ll the perforted lyers. For wter, nc q w = Q wl (4.128) l=1 For oil, nc q o = Q ol (4.129) l=1 Differentiting Eqution nd Eqution with respect to prmeters gives: nc q w α = l=1 Q wl α (4.130)

67 SECTION 4. SENSITIVITY COEFFICIENTS 55 nc q o α = l=1 Q ol α (4.131) Reclling Eqution 4.25, the lyer flow rte of wter nd oil components is given y: Q wl = T w (p l p wfl ) (4.132) Q ol = T o (p l p wfl ) (4.133) Since the lyer flow rte is function of pressure (p l ) nd sturtion (S l )inthe welllock, the pressure in the wellore (p wfl ) nd the permeility in the welllock (k l ), differentiting Eqution nd Eqution with respect to prmeters gives: For wter, Q wl α = Q w l p l p l α + Q w l S l S l α + Q w l p wfl p wfl α + Q w l k l k l α (4.134) For oil, where, Q wl p l Q wl k l nd Qo l k l α p l nd S l α α Q ol α, Qw l S l, Qo l p l = Q o l p l p l α + Q o l S l S l α + Q o l p wfl p wfl α + Q o l k l k l α (4.135) nd Qo l S l, cn e computed from Eqution 4.54 nd Eqution k l cn e computed from Eqution cn e computed from Eqution cn e otined from the full sensitivity mtrix S. Q wl p wfl = T w (4.136) nd, Q ol = T o (4.137) p wfl where T w nd T o re the trnsmissiilities of wter nd oil components respectively etween the well nd the formtion nd re given y Eqution re the derivtives of wellore pressure t ech lyer penetrted y the well with respect to the prmeters nd re computed y the following steps. p wfl α

68 SECTION 4. SENSITIVITY COEFFICIENTS 56 As ws discussed erlier, since the pressure in lyer l + 1 depends on the pressure in lyer l nd the sturtion in lyer l nd l + 1, differentiting the pressure in lyer l + 1 with respect to prmeters gives: p wfl+1 α = p wf l+1 p wfl p wfl α + p wf l+1 S l S l α + p wf l+1 S l+1 S l+1 α (4.138) If lyer l is the reference lyer then the pressure in lyer l is the reference pressure nd its derivtive with respect to prmeters is otined from the full sensitivity mtrix. Knowing the derivtives of wellore pressure in lyer l, the derivtives of wellore pressure in lyer l + 1 with respect to prmeters cn e computed from Eqution in which p wf l+1 p wfl, p wf l+1 S l,nd p wf l+1 S l+1 re computed from Eqution 4.79 to Eqution 4.81, nd S l nd S l+1 re otined from the full sensitivity mtrix. α α Eqution cn e used recursively to compute the derivtives of wellore pressure in ll lyers penetrted y the well with respect to the prmeters. Up to this point, we hve shown n ccurte wy of computing the sensitivity coefficients for multilyered reservoirs. These sensitivity coefficients hve some fetures tht cn e summrized s follows: The sensitivity coefficients cn e computed from the full sensitivity mtrix. This computtion occupies most of the work nd is very difficult to implement in terms of computer coding. The computtion is much more complex for three-dimensionl multiphse prolems thn for two-dimensionl or single-phse prolems Computtionl Results The correctness nd efficiency of the method of computing the sensitivity coefficients ws tested nd ensured y compring computtionl results to the results otined from the numericl sustitution method. To serve this purpose, model of rectngulr prllelepiped reservoir of uniform thickness ws prepred s depicted in Figure 4.2. The reservoir model consists of three lyers in verticl communiction. Three-dimensionl, two-phse flow of oil nd wter ws considered. The dimensions of the reservoir were chosen to e sufficiently smll to cpture ll possile scenrios

69 SECTION 4. SENSITIVITY COEFFICIENTS 57 for the sensitivity study in short time period. A 600 ft y 600 ft y 150 ft reservoir with no-flow oundries ws modeled using uniform grid with 30 ft y 30 ft y 50 ft cuic gridlocks. The discretized reservoir ws therefore represented s 20x20x3 grid (1200 gridlocks). The reservoir contins four wells. Three producers (well #1, #2, nd #3) penetrte fully through the reservoir thickness. Wells #1 nd #3 produce t constnt totl liquid rte of 750 STB/d. Well #2 produces t constnt oil rte of 750 STB/d. The injector (Well #4) is completed only in the ottom lyer nd injects wter t series of step rtes s depicted in Figure 4.3. The porosity ws set constnt throughout the reservoir (φ 0 =0.2). The permeility is uniformly distriuted within ech lyer, 400md in the top nd ottom lyer nd 600md in the middle lyer. The long term pressure nd wter cut of Well #2 re shown in Figure 4.4. The rupt chnge in slope of pressure t 200 nd 250 dys is due to the step chnge in injection rte. Wter rrives t this well in pproximtely 40 dys. The chnge in wter sturtion etween 50 nd 150 dys in ll three lyers is depicted in Figure 4.5. The sensitivity of pressure nd wter cut t Well #2 with respect to the permeilities in ll cells long the NE-SW digonl (see Figure 4.2) were computed using oth nlyticl nd numericl methods. The comprison is shown in Figure 4.6. Note tht the sensitivity coefficients re lrgest t the gridlock contining the well. The downhole pressure is less sensitive to the permeilities t gridlocks fr from the well. The wter cut is lso less sensitive to the permeilities t gridlocks fr from the well ut regins sensitivity to the gridlocks close to the injector. The sensitivity of wter cut reches its minimum t the point etween the producer nd the injector. Figure 4.7 nd Figure 4.8 show the sensitivity of pressure nd wter cut t Well #2 with respect to the permeility in gridlocks contining the injector (1,20,3) nd the producer (20,1,1) s function of time. These sensitivity coefficients re zero efore wter rrivl t Well #2 (40 dys) nd rpidly increse fter tht. The sensitivities of the chnge in wter sturtion in ll cells etween 50 nd 150 dys with respect to the permeility in cell (10,10,3) were lso computed with oth methods nd re shown in Figure 4.9. These figures indicte tht the results computed using the nlyticl method proposed in this work nd the sustitution method re lmost identicl. In

70 SECTION 4. SENSITIVITY COEFFICIENTS 58 fct, they mtch to six deciml plces. The wter sturtion distriution in the ottom lyer is shown in Figure The effect of permeilities on the wter front nd wter sturtion distriution ws lso studied y computing the sensitivity of the ltter with respect to the permeilities in the upstrem gridlock contining the injector (1,20,3), the downstrem gridlock contining the producer (20,1,3), nd the gridlock tht lies t the wter front etween the injector nd the producer (10,10,3) (see Figure 4.10). This result is shown in Figure It is interesting to oserve tht the most sensitive region my not lwys contin the lock whose permeility is eing studied. This is ecuse the upstrem wter sturtion t lte times ecomes lmost constnt.

71 SECTION 4. SENSITIVITY COEFFICIENTS 59 3 Well Well 2 Verticl Well Well Northing Esting Figure 4.2: Three-lyer reservoir model for sensitivity study.

72 SECTION 4. SENSITIVITY COEFFICIENTS Flow Rte (STB/d) time(dys) Figure 4.3: Injection rte of Well # Pressure: psi Wter Cut: frction time(dys) Figure 4.4: Long term pressure nd wter cut t Well #2.

73 SECTION 4. SENSITIVITY COEFFICIENTS Lyer #1 Lyer #2 Lyer #3 Figure 4.5: Chnge in wter sturtion etween 50 nd 150 dys. Sensitivity of Pressure (psi/md) fg h h f g Anlyticl Method-Sensitivity of Pressure Sustitution Method-Sensitivity of Pressure Anlyticl Method-Sensitivity of Wter Cut Sustitution Method-Sensitivity of Wter Cut gf gf g f g f gf gf gf gf gf gf gf g f gf g f gf gf gf h h h h h h h h h h f h h h h h f h h g g h h Sensitivity of Wter Cut (frction/md) Distnce(ft) Figure 4.6: Sensitivity of pressure nd wter cut with respect to the permeilities in NE-SW digonl t 150 dys.

74 SECTION 4. SENSITIVITY COEFFICIENTS 62 Sensitivity of Wter Cut (frction/md) 1e e-13-1e-7-2e-7-3e-7-4e-7-5e-7 hhhhhhhh h h wct w2 / k (1,20,1) Anlyticl Method-Sensitivity of Wter Cut Sustitution Method-Sensitivity of Wter Cut h h h h hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh time (dys) Sensitivity of Pressure (psi/md) p w2 / k (1,20,1) Anlyticl Method-Sensitivity of Pressure h Sustitution Method-Sensitivity of Pressure hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh time (dys) Figure 4.7: Sensitivity of pressure nd wter cut t Well #2 with respect to the permeility in gridlock-(1,20,1).

75 SECTION 4. SENSITIVITY COEFFICIENTS e-12 hhhhhhhh wct w2 / k (20,1,1) Sensitivity of Wter Cut (frction/md) -5e-6-1.0e-5-1.5e-5-2.0e-5-2.5e-5 h Anlyticl Method-Sensitivity of Wter Cut h Sustitution Method-Sensitivity of Wter Cut h h h hhhhhhhhhhh h h hhh hh hhhh h hh h hh hhhhhhhhhhh hh hh hhhhh time (dys) Sensitivity of Pressure (psi/md) p w2 / k (20,1,1) Anlyticl Method-Sensitivity of Pressure h Sustitution Method-Sensitivity of Pressure hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh h time (dys) Figure 4.8: Sensitivity of pressure nd wter cut t Well #2 with respect to the permeility in gridlock (20,1,1).

76 SECTION 4. SENSITIVITY COEFFICIENTS By Anlyticl Method By Sustitution Method Figure 4.9: Sensitivity of chnge in wter sturtion etween 50 nd 150 dys Figure 4.10: Wter sturtion distriution in the ottom lyer t 150 dys.

77 SECTION 4. SENSITIVITY COEFFICIENTS With Respect To k (10,10,3) With Respect To k (1,20,3) With Respect To k (20,1,3) Figure 4.11: Sensitivity of wter sturtion distriution in the ottom lyer.

78 SECTION 4. SENSITIVITY COEFFICIENTS 66 The oservtions drwn from the computtionl results re summrized s follows: 1. In most cses, the response in given gridlock is more sensitive to the permeility of tht gridlock thn to the permeility of more distnt gridlocks. 2. Downhole pressure nd wter cut show low sensitivity to the permeility efore wter rrivl nd show strong sensitivity fter tht. 3. The downstrem sturtions re more sensitive to the permeilities in the gridlocks t the wter front thn re upstrem sturtions. 4. The chnges in downstrem sturtions re more sensitive to the permeilities in the gridlocks t the wter front thn those in upstrem sturtions. 5. The permeilities in locks ehind wter front show stronger effect on the distriution of wter sturtion in the reservoir thn do those hed of the wter front. 6. Wter sturtion in downstrem gridlock is most sensitive to the permeility in tht gridlock. 7. Wter sturtion in n upstrem gridlock is not necessrily most sensitive to the permeility in tht gridlock. Although these oservtions re drwn from the prticulr reservoir model, mny of them re still vlid in generl nd cn e used to determine principles for dt collection strtegy s will e discussed in the next section. Since the dt with high sensitivity contins more informtion out the prmeters eing estimted, the sensitivity coefficients cn e interpreted in terms of dt informtion s follows: 1. Distriuted dt such s the chnge in wter sturtion re richer in informtion, eing physiclly close to the locks whose permeilities re eing estimted. 2. Single point dt such s long-term downhole pressure or wter cut history collected t ech well re richer in informtion fter wter rrivl.

79 SECTION 4. SENSITIVITY COEFFICIENTS The chnges in downstrem wter sturtions re richer in informtion out the permeilities in the region contining the wter front. The wider the trnsitionl re of this wter front, the more vlues of gridlock permeilities cn e resolved. This is true whether the trnsition re is extended y physicl dispersion, or y the effects of heterogeneity in reservoir properties. 4. Wter sturtion in downstrem gridlock is richest in informtion out the permeility in tht gridlock. 5. Wter sturtion in n upstrem gridlock my not e richest in informtion out the permeility in tht gridlock. The efficiency of the nlyticl method of computing sensitivity coefficients for lyer system proposed in this work cn e nlyzed y considering the computtion time. Let us define the following terms: t s : CPU time required to solve Eqution 4.6. t α : CPU time required to solve Eqution 4.20 with fixed left-hnd side. n p : The numer of prmeters of interest to which the sensitivities re computed. n s : The numer of time steps in one simultion run. t s nd t α depend on the size of the forwrd prolem, the solver lgorithm, nd the computer hrdwre on which the code is run. Note tht the numer of prmeters of interest n p is not necessrily the sme s the numer of prmeters for the inverse prolem npr. For the purpose of the sensitivity study, the sensitivities with respect to only some prmeters whose impct on the reservoir response need to e studied re computed. The CPU time required to compute the sensitivity coefficients with respect to n p prmeters of interest cn e expressed s: By the nlyticl method: t = n s (t s + n p t α ) (4.139) By the sustitution method: t n = n s (n p +1)t s (4.140)

80 SECTION 4. SENSITIVITY COEFFICIENTS 68 Since computer efficiency is defined s the inverse of the CPU time, the comprison of the efficiencies of the two methods is given y rtio. 1 efficiency of the nlyticl method efficiency of sustitution method = t 1 t n Eqution cn e rerrnged s: = t n t = n s(n p +1)t s n s (t s + n p t α ) (4.141) efficiency of the nlyticl method efficiency of sustitution method = n p +1 1+n p t α ts = t t s s t α 1 t t α 1+n α (4.142) p ts Since t α t s, the middle frction in Eqution is greter thn unity nd the method proposed in this work is more efficient thn the sustitution method. The dominnt efficiency of the nlyticl method is emphsized s the numer of prmeters of interest increses. The gretest efficiency t s /t α is rchived s n p pproches infinity. The ctul efficiencies of the two methods were evluted y running the sme code on the sme mchine ( UNIX worksttion running t 400 MHz). The results re shown in Tle 4.1. The nlyticl method is 1.33 times fster thn the Tle 4.1: CPU time in seconds n p Simultion Anlyticl Sustitution Rtio of Two Efficiencies X X X sustitution method for computing the sensitivity coefficients with respect to 1 prmeter. For computing the sensitivity coefficients with respect to 21 prmeters, the nlyticl method is 4.07 times fster thn the sustitution method nd 65 times fster for computing the sensitivity coefficients with respect to 1200 prmeters. To estimte 1200 permeilities for this reservoir model, the nlyticl method tkes out 92 minutes to compute ll sensitivity coefficients tht re required for the inverse prolem while the sustitution method tkes out 4 dys to perform the equivlent work.

81 Section 5 Appliction of the Method This section demonstrtes the ppliction of the prmeter estimtion procedure discried in the previous sections for severl study cses. By exmining these cses, we ddressed some fundmentl issues ssocited with the resolution of depth-dependent properties. These issues re s follows: Wht type of dt is necessry to resolve depth-dependent reservoir properties? Wht type of dt revels most depth-dependent informtion? How much does ech type of dt contriute to reducing the uncertinty of the prmeter estimtes? How well cn the reservoir properties e determined in the depth dimension s compred to the rel resolution? How is the certinty of the estimtes distriuted in the depth dimension? For two-dimensionl prolems, wells cn only e produced in single lyer nd the chnge in wter sturtion ville from 4-D seismic interprettion cn e considered s the chnge in ech cell. For multilyered reservoirs, wells cn intersect severl lyers nd the 4-D seismic dt cn e interpreted s the informtion either within ech lyer or the verge over severl lyers. To ddress the fundmentl issues mentioned ove, the reservoir prmeters were estimted y mtching different types of dt. The types of these dt re descried s follows: LP: Lyer production dt (wells produced in seprte lyers (see Figure 5.1)). 69

82 SECTION 5. APPLICATION OF THE METHOD 70 CP: Commingled production dt (wells produced t severl lyers (see Figure 5.2)). LS: Lyer-y-lyer seismic dt (the chnge of wter sturtion is ville t every gridlock in ech lyer of the reservoir). This type of dt is only fesile for thick lyers. AS: Depth-verged seismic dt (only the verge chnge of wter sturtion in the depth dimension is ville). This type of dt is for thin reservoirs. The prmeters eing estimted re the permeilities in ll cells of lyered reservoir. All individul vlues of gridlock permeility of this exmple reservoir re lredy known nd re considered s the true solution for the prmeter estimtion prolem. The dvntge of hving known solution is tht the solution otined from the procedure developed in this work cn then e compred to the known solution nd the fundmentl issues stted erlier cn then e ddressed. If the prmeters estimted y mtching set of dt re close to the true prmeters, then such dt re sid to revel the reservoir prmeter informtion. The field dt used in these exmples were generted y the numericl simultor using the true permeility distriution. The dt generted therefore contin no noise. The true reservoir ws discretized with seven gridlocks in oth x nd y dimensions nd three gridlocks in the z dimension. The reservoir is produced y five wells nd wter is injected t four other wells. The three-lyer reservoir model with lyer production (LP) isshown in Figure 5.3 nd with commingled production (CP) in Figure 5.4. The flow rte histories of ll nine wells re shown in Figure 5.5. We investigted three exmple reservoirs: one with uniform distriution of permeility nd porosity in ech lyer, one with chnnel, nd one with fult.

83 SECTION 5. APPLICATION OF THE METHOD 71 Well #1 Depth Well #2 Well #3 Well #4 Figure 5.1: Individul lyer well completion. Well #1 Well #3 Well #2 Depth Figure 5.2: Multilyered well completion.

84 SECTION 5. APPLICATION OF THE METHOD Well# 1 Well# 4 Verticl Well# 9 Well# 8 Well# 5 Well# 3 Well# 6 Well# 2 Well# Northing Esting Figure 5.3: Nine wells in multilyered reservoir with individul lyer well completion (LP).

85 SECTION 5. APPLICATION OF THE METHOD 73 Well# Well# 5 Well# 1 Well# 3 2 Well# 2 Verticl Well# 9 Well# Well# 6 Well# Northing Esting Figure 5.4: Nine wells in multilyered reservoir with multilyered well completion (CP).

86 SECTION 5. APPLICATION OF THE METHOD 74 qw#9(stb/d) qw#7(stb/d) qw#5(stb/d) qw#3(stb/d) qw#1(stb/d) time(dys) qw#8(stb/d) qw#6(stb/d) qw#4(stb/d) qw#2(stb/d) time(dys) Figure 5.5: Time-dependent rte history of nine wells.

87 SECTION 5. APPLICATION OF THE METHOD Exmple 1: Uniform Properties within Ech Lyer The purpose of this exmple ws to test if it ws possile to recover the true depthdependent permeilities y mtching seprtely the four different types of dt descried erlier. There were only three independent permeility prmeters, one for ech lyer. However we ssumed tht the uniformity of properties is unknown to us nd hence we must descrie the reservoir model t finer scle to pproch this prolem. The finest scle is t the scle of the reservoir simultor which is 7x7x3. This pproch is known s pixel modeling s descried in Chpter 3. The first type of dt we considered were Lyer Production nd Lyer y Lyer Seismic (LP-LS). The true permeilities vry only in the depth dimension nd re 400md in the top nd ottom lyer nd 800md in the middle lyer. The oserved dt were generted from these true permeilities using the numericl simultor nd re shown in Figure 5.6 nd Figure 5.7. Figure 5.7 shows the chnge in wter sturtion etween 5 nd 15 dys s might e determined from 4-D seismic dt. Figure 5.8 shows the wter sturtion t 15 dys. Wter rrives very erly t the five producers. Thewiggleinoserved pressure efore 20 dys in Figure 5.6 is due to the rpid chnge of specified flow rtes. Figure 5.7 shows no chnge in wter sturtion in the first lyer ecuse wter does not rrive t this lyer efore 15 dys. All nine wells were prtilly penetrted. Wells #1 to #5 re produced t gridlocks (4,4,1), (1,1,2), (7,7,2), (7,1,3), nd (1,7,3) respectively with specified oil rtes. Wells #6 to #9 inject wter t gridlocks (4,1,3), (7,4,3), (4,7,3), nd (1,4,3) respectively with specified wter rtes. The time-dependent flow rtes of ll nine wells were shown erlier in Figure 5.5. Figure 5.9 shows the mtch of long-term pressure nd wter cut t the nine wells. The continuous lines re oserved dt nd the points re the computed vlues. Figure 5.10 shows the mtch of seismic dt t ll three lyers. The dt re mtched perfectly. Figure 5.11 compres the computed permeilities to the true permeilities in ll three lyers. The true permeility vlues re recovered lmost exctly in the third lyer nd firly well in the first nd the second lyers except t few locks.

88 SECTION 5. APPLICATION OF THE METHOD 76 These poorly recovered locks re green in the first lyer where the true color is lue nd red in the second lyer where the true color is green. This cn e explined s follows: The green locks in the first lyer re locted hed of the wter front nd from the sensitivity nlysis shown in Chpter 4, the permeilities in these locks hve wek or no effect on the wter sturtion. In the second lyer, the wter front is the X-shped region contining the two digonls (not including the lue lock). The true color is recovered in this region. This is consistent with the oservtion tht the wter sturtion is most sensitive to the permeilities in the wter front re. The second est re is in the region of the red locks locted just ehind the wter front nd fr from the wells. The worst re is the single lue lock. It is importnt to note here tht lthough this lue lock is locted next to well, the distnce from the well is 220 ft which, ccording to Figure 4.6 is still not close enough to show ny effect on the pressure nd wter cut t this well. Moreover, this lock is still hed of wter front. Figure 5.12 shows the certinty of the permeility estimtes in ech cell nd the verge certinty in ech lyer. The locks in wrmer color represent higher certinty nd the permeilities in these locks cn e resolved with more confidence. The certinty in ech lyer hs common oservtions tht cn e interpreted s follows: The est determined res re t the well loctions. Pressure nd wter cut dt re collected t these wells. The worst determined res re the ones locted hed of the wter front nd fr from the wells. Wter sturtion is only wekly sensitive to the permeilities in downstrem res. The modertely well determined res re the ones locted either next to the wells or ehind the wter front. Wter sturtion is strongly sensitive to the permeilities in upstrem res. The rel verge of the certinty lso increses from the first to the third lyer. This comes from the fct tht there were more wells in the second nd third lyer thn there were in the first. Also the 4-D lyer seismic dt re richer in informtion out the third lyer which is locted upstrem of the wter front.

89 SECTION 5. APPLICATION OF THE METHOD 77 The next type of dt we mtched ws Lyer Production nd Depth-Averged Seismic (LP-AS). The finl results re shown in Figure 5.13 nd Figure The true permeility vlues in the first nd the third lyers were only recovered t the well loctions. One interesting result oserved here in the second lyer is tht the true permeility vlues in most cells re very well recovered s compred to those in the first nd third lyer nd re s well recovered s in the cse of Lyer Production nd Lyer y Lyer Seismic (LP-LS). This is ecuse (see Figure 5.8) the second lyer contins the wter front region nd ccording to the sensitivity nlysis, the permeilities in this region show the strongest effect on wter sturtion. Moreover, lso from Figure 5.8, t 15 dys wter rrives t lmost every gridlock in the second lyer. The first lyer is fr from the wter front in the second lyer nd thus contins no moile wter. The first lyer shows no chnge in wter sturtion nd is therefore insensitive with respect to the permeilities in the second lyer. In fct, the sensitivity of the chnge in wter sturtion in ll three lyers with respect to the permeilities in the second lyer were computed nd the computtionl results show tht the first lyer is insensitive with order of mgnitude while the second lyer is of order 10 6 nd the third lyer is of order Therefore, in terms of resolving the permeilities in the second lyer, mtching depth-verged seismic over the three lyers is essentilly the sme s mtching depth-verged seismic over only two lyers nd mtching seismic dt lyer y lyer is essentilly the sme s mtching seismic dt in the second nd third lyer. The next two types of exmple investigted hve different well completions nd thus the oserved dt sets differ from those of the first two types. The producers re fully penetrted. The results otined y mtching Commingled Production nd Lyer y Lyer Seismic (CP-LS) dt re illustrted in Figure 5.15 nd Figure 5.16 nd the results otined y mtching Commingled Production nd Depth-Averged Seismic (CP-AS) dt re illustrted in Figure 5.17 nd Figure By mtching lyer seismic (LS), permeility vlues re well determined with highest certinty t the injectors ut re poorly resolved t the producers nd everywhere else except in the second lyer. The true permeilities in the second lyer

90 SECTION 5. APPLICATION OF THE METHOD 78 re recovered well if mtching lyer seismic ut poorly recovered if mtching depthverged seismic dt (AS). Mtching Commingled Production (CP) dt does not help in resolving the welllock permeility vlues for multilyered completion ut y dding Depth-Averge Seismic (AS) ndthenlyerylyerseismic(ls) dtthe resolution is significntly improved. We cn see this improvement y, for instnce, looking t the centrl well in Figure 5.15 nd Figure The true permeility vlue in Lyer #3 is only recovered in the center welllock if mtching Depth-Averge Seismic (AS) ut if mtching Lyer y Lyer Seismic (LS) dt ll the true vlues of connecting welllock permeilities re recovered. The three fundmentl issues mentioned t the eginning of this section cn e nswered quntittively y computing the verge of k/σ vlues otined y seprtely mtching the four dt types nd determining the proilities tht the true permeility vlues fll in given cceptle intervl (10% is typicl cceptle rnge for permeilities Horne 1995). To compute these proilities, we ssumed permeilities re log-norml distriutions with men eing the estimtes nd vrince eing σ 2. The results of this nlysis re given in Tle 5.1. The verge vlues of k/σ nd the corresponding verge proilities re otined y verging over ll gridlocks. The lst three columns show the individul proilities in gridlocks (1,1,1), (4,1,1), nd (4,4,1). From these proilities, we cn know how much ech type of dt contriutes to the certinty of the estimtes of permeilities in these locks. Dt type LP-LS gives the highest certinty (71%) on verge nd the closest recovery of the true vlues of permeilities, hence revels the most depth-dependent informtion. Dt type CP-AS gives the lowest certinty (37%) on verge with the poorest recovery of the true vlues of permeilities nd is considered to revel the lest depth-dependent informtion. The permeility vlue in gridlock (1,1,1) is resolved with only 3.2% certinty for type LP-LS nd 2.7% for type LP-AS nd cn not e tken seriously. This is ecuse this lock is locted fr from ny wells nd hed of the wter front. While types CP-LS nd CP-AS cn resolve this gridlock permeility vlue t much higher certinty (26% nd 24%), this comes from the fct tht this lock is now one of the connecting well locks. Type CP-LS gives higher certinty (6.6%) thn LP-LS (5.5%) in determining the permeility vlue in gridlock

91 SECTION 5. APPLICATION OF THE METHOD 79 (4,1,1) ecuse there re more wells intersecting the first lyer. LP-LS nd LP-AS resolve the well lock permeilities with solute certinty (100%) while CP-LS nd CP-AS cn resolve only with 15% certinty. Dt type LP itself cn resolve the vlue of welllock permeility ccurtely with lmost solute certinty while LS type cn only resolve with roughly 20% certinty ehind the wter front nd 3% hed of wter front. This is ecuse the pressure mesured in welllock contins much more informtion out the welllock permeility thn does the chnge in sturtion in the sme lock nd the sturtion is more sensitive to conditions upstrem thn to conditions downstrem of the wter front. However, the verge certinty y mtching the LP type dt increses with the numer of wells. In this prticulr prolem, LP gives higher certinty thn LS dttypeonvergeecusewehvereltively lrge numer of wells compred to the totl numer of gridlocks. Lyer Seismic (LS) dt is richer in informtion thn Averged-Seismic (AS) s is to e expected. This result is emphsized if the time intervl of the 4-D seismic interprettion is lrge enough to show sufficient chnge in wter sturtion t every lyer. Finlly, dt type CP is the worst. Tle 5.1: Averge certinties nd certinties of the estimtes in gridlocks (1,1,1), (4,1,1), nd (4,4,1). Dt Type Averge of k/σ Averge Certinty k (1,1,1) k (4,1,1) k (4,4,1) LP-LS % 3.2% 5.5% 100% LP-AS % 2.7% 3.5% 100% CP-LS % 26% 6.6% 15% CP-AS % 24% 4.9% 14.7% Figure 5.19 nd Figure 5.20 summrize the results of mtching the four different types of dt nd Figure 5.21 compres their resolution mtrices. As ws stted in Chpter 3, the resolution mtrix presents the reltionship etween the estimted prmeters nd the true prmeters. The closer this mtrix to identity or the more dominnt the digonl over the off-digonl elements then the closer the estimted vlues lie to the true vlues. As indicted in Figure 5.21, the resolution mtrix

92 SECTION 5. APPLICATION OF THE METHOD 80 y mtching Lyer y Lyer Seismic (LS) contins only the digonl while the resolution mtrix y mtching Depth-Averged Seismic (AS) does contin significnt off-digonl elements.

93 SECTION 5. APPLICATION OF THE METHOD Well #7 Injection History Well #8 Injection History Well #9 Injection History Pressure Wter Cut Pressure Wter Cut Pressure Wter Cut Well #4 Production History Well #5 Production History Well #6 Injection History Pressure Wter Cut Pressure Wter Cut Pressure Wter Cut Well #1 Production History Well #2 Production History Well #3 Production History Pressure Wter Cut Pressure Wter Cut Pressure Wter Cut Figure 5.6: Long-term pressure nd wter cut dt Lyer #1 Lyer #2 Lyer #3 Figure 5.7: 4-D seismic dt

94 SECTION 5. APPLICATION OF THE METHOD Lyer #1 Lyer #2 Lyer #3 Figure 5.8: Wter sturtion t 15 dys: Lyer Production (LP) Well #7: Injector Well #8: Injector Well #9: Injector Well #4:Producer Well #5:Producer Well #6: Injector Well #1: Producer Pressure Dt Wter Cut Dt Clculted Vlues Well #2: Producer Well #3: Producer Figure 5.9: Mtch of long term pressure nd wter cut dt. 0.8

95 SECTION 5. APPLICATION OF THE METHOD 83 True S w Clculted S w Lyer #1 Lyer #2 Lyer #3 Figure 5.10: Mtch of 4-D seismic dt.

96 SECTION 5. APPLICATION OF THE METHOD 84 True Permeility Clculted Permeility md Lyer #1 Lyer #2 Lyer #3 Figure 5.11: Comprison etween true nd clculted permeility, mtching Lyer Production nd Lyer y Lyer Seismic (LP-LS) k/ σ 10-1 Lyer #1: µ=3.08 Lyer #2: µ=7.60 Lyer #3: µ= Figure 5.12: Certinty of permeility estimtes, mtching Lyer Production nd Lyer y Lyer Seismic (LP-LS).

97 SECTION 5. APPLICATION OF THE METHOD 85 True Permeility Clculted Permeility md Lyer #1 Lyer #2 Lyer #3 Figure 5.13: Comprison etween true nd clculted permeility, mtching Lyer Production nd Depth-Averged Seismic (LP-AS) k/ σ 10-1 Lyer #1: µ=3.05 Lyer #2: µ=4.416 Lyer #3: µ= Figure 5.14: Uncertinty of permeility estimtes, mtching Lyer Production nd Depth-Averged Seismic (LP-AS).

98 SECTION 5. APPLICATION OF THE METHOD 86 True Permeility Clculted Permeility md Lyer #1 Lyer #2 Lyer #3 Figure 5.15: Comprison etween true nd clculted permeility, mtching Commingled Production nd Lyer y Lyer Seismic (CP-LS) k/ σ 1 Lyer #1: µ=0.75 Lyer #2: µ=1.45 Lyer #3: µ= Figure 5.16: Uncertinty of permeility estimtes, mtching Commingled Production nd Lyer y Lyer Seismic (CP-LS).

99 SECTION 5. APPLICATION OF THE METHOD 87 True Permeility Clculted Permeility md Lyer #1 Lyer #2 Lyer #3 Figure 5.17: Comprison etween true nd clculted permeility, mtching Commingled Production nd Depth-Averged Seismic (CP-AS) k/ σ 1 Lyer #1: µ=0.55 Lyer #2: µ=1.31 Lyer #3: µ= Figure 5.18: Uncertinty of permeility estimtes, mtching Commingled Production nd Depth-Averged Seismic (CP-AS).

100 SECTION 5. APPLICATION OF THE METHOD 88 Lyer # Lyer # Lyer #3 Depth-Averged Seismic Lyer y Lyer Seismic True Permeility Figure 5.19: Comprison of permeility estimtes etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth-Averged Seismic (LP-AS) exmples.

101 SECTION 5. APPLICATION OF THE METHOD 89 Lyer # Lyer # Lyer #3 Depth-Averge Seismic Lyer y Lyer Seismic True Permeility Figure 5.20: Comprison of permeility estimtes etween Commingled Production nd Lyer y Lyer Seismic (CP-LS) nd Commingled Production nd Depth- Averged Seismic (CP-AS) exmples.

102 SECTION 5. APPLICATION OF THE METHOD Depth-Averged Seismic Lyer y Lyer Seismic Figure 5.21: Comprison of the resolution mtrices etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth-Averged Seismic (LP-AS) dttypes.

103 SECTION 5. APPLICATION OF THE METHOD Exmple 2: Chnnel in Ech Lyer In Exmple 1, we descried the investigtion of the cse in which the reservoir properties depends only on the depth dimension. In relity, most reservoir properties cn vry in ny or ll dimensions nd cn hve complex distriution chrcteristics. A common type of reservoir considered here is chnnel reservoir. In our exmple, the reservoir ws descried y three different chnnels, one in ech lyer. The true permeility inside the chnnel is 1000md nd outside is 300md. The sme four dt types s in Section 5.1 were considered. The sets of oservtion dt were mtched perfectly in ll four cses. The comprison etween the estimted nd the true permeility vlues otined y mtching these four dt types re shown in Figure 5.22 nd Figure The LS dt type resolved the chnnel geometry firly well, s well s the permeility vlues vlues inside nd outside the chnnel. The true vlues t some locks were not ccurtely recovered. The true vlues could e recovered more ccurtely if we mtched two 4-D seismic intervls insted of one. The AS dt type did not resolve ny fetures of the reservoir except t the well loctions. These results cn e summrized y looking t the resolution mtrices shown in Figure The LS dt type results in stronger digonl dominnce thn does the AS type. Figure 5.24 shows the certinty mps otined y mtching the four dt types. The permeility vlues inside the chnnel re determined with higher certinty thn those outside the chnnel nd the LS dt type gives much higher certinty thn the AS type.

104 SECTION 5. APPLICATION OF THE METHOD 92 Lyer #1 Lyer # Lyer #3 Depth-Averged Seismic Lyer y Lyer Seismic True Permeility Figure 5.22: Comprison of permeility estimtes etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth-Averged Seismic (LP-AS) dttypes.

105 SECTION 5. APPLICATION OF THE METHOD 93 Lyer # Lyer # Lyer #3 Depth-Averged Seismic Lyer y Lyer Seismic True Permeility Figure 5.23: Comprison of permeility estimtes etween Commingled Production nd Lyer y Lyer Seismic (CP-LS) nd Commingled Production nd Depth- Averged Seismic (CP-AS) dt types.

106 SECTION 5. APPLICATION OF THE METHOD 94 Lyer #1 CP_AS Lyer #2 Lyer #3 CP_LS k/ σ LP_AS LP_LS Figure 5.24: Comprison of certinty for four dt types: chnnel cse.

107 SECTION 5. APPLICATION OF THE METHOD CP_LS CP_AS LP_LS LP_AS Figure 5.25: Resolution mtrices y mtching four dt types: chnnel cse.

108 SECTION 5. APPLICATION OF THE METHOD Exmple 3: Verticl Fult In this section, the true reservoir ws descried with single verticl fult. There re only two vlues of true permeility. The one inside the fult is 50md nd the other outside is 500md. We mtched the sme four types of dt s in Sections 5.1 nd 5.2. The oservtion histories were mtched perfectly nd the finl results re shown in Figure 5.26 nd Figure The permeility vlue in the fult nd its shpe in ll three lyers re recovered very well for LP-LS nd CP-LS dt types, while type LP-AS nd CP-AS cn only detect the fult loction nd its true permeility vlue t Lyer #2. It is importnt to note here tht the fult intersects the wter front region in the second lyer. Type LP-AS lso recognized the fult region firly well in the third lyer. This is ecuse there re up to eight LP wells in this lyer. Figure 5.28 nd Figure 5.29 show mps of certinty distriution computed using the clculted nd true vlues of permeilities. The reson of showing oth mps is tht type CP-AS dt re very poor in informtion in terms of the permeilities in the first lyer (most of this lyer is fr hed of the wter front) nd my result in permeility estimtes tht re too high or too low. These too low or too high vlues cn then misled the interprettion of these certinty mps. The oservtions seen from these mps re: The permeility vlues inside the fult re determined with lower certinty thn those outside the fult. The LS dt type resolves fult geometry well nd the fult permeility vlue while the AS type cn not resolve either. However, for the fult segment inside the wter front region, oth geometry nd permeility vlues re recovered well.

109 SECTION 5. APPLICATION OF THE METHOD 97 Lyer # Lyer # Lyer #3 Depth-Averged Seismic Lyer y Lyer Seismic True Permeility Figure 5.26: Comprison of permeility estimtes etween Lyer Production nd Lyer y Lyer Seismic (LP-LS) nd Lyer Production nd Depth-Averged Seismic (LP-AS) dttypes.

110 SECTION 5. APPLICATION OF THE METHOD 98 Lyer # Lyer # Lyer #3 Depth-Averged Seismic Lyer y Lyer Seismic True Permeility Figure 5.27: Comprison of permeility estimtes etween Commingled Production nd Lyer y Lyer Seismic (CP-LS) nd Commingled Production nd Depth-Averge Seismic (CP-AS) dt types.

111 SECTION 5. APPLICATION OF THE METHOD 99 Lyer #1 CP_AS Lyer #2 Lyer #3 CP_LS k Clculted /σ LP_AS LP_LS Figure 5.28: Comprison of certinty (with clculted vlues) for four dt types: fult cse.

112 SECTION 5. APPLICATION OF THE METHOD 100 Lyer #1 CP_AS Lyer #2 Lyer #3 CP_LS k True /σ LP_AS LP_LS Figure 5.29: Comprison of certinty (with true vlues) for four dt types: fult cse.

113 SECTION 5. APPLICATION OF THE METHOD CP_LS CP_AS LP_LS LP_AS Figure 5.30: Resolution mtrices y mtching four dt types: fult cse.

114 SECTION 5. APPLICATION OF THE METHOD Resolution of Permeility nd Porosity From the previous sections, we hve shown in three exmples tht to determine lyer properties it is necessry to collect lyer informtion nd the type of dt tht est provided this kind of informtion is Lyer Production nd Lyer y Lyer Seismic (LP-LS). In this section we used this dt type to study the resolution of reservoir ttriutes (permeility nd porosity) in different situtions tht re descried s follows: 1. Permeility is unknown. 2. Porosity is unknown. 3. Permeility nd porosity re oth unknown ut with known correltion. 4. Permeility nd porosity re oth unknown nd with unknown correltion. 5. Permeility nd porosity re independent. We reused the verticl-fult reservoir model s defined in Section 5.3 s our synthetic exmple. A set of oservtion dt ws generted from this model using the numericl simultor. The true permeilities nd porosities re correlted y Eqution 5.1. log k =6.0φ +1.5 (5.1) Figure 5.31 nd Figure 5.32 show the comprison etween the computed nd the true vlues of permeility nd porosity respectively in ll five different situtions. By exmining the two figures we cn see the permeility mp revels the fult firly well ut not the porosity mp. Inside the fult, permeility is etter determined thn porosity. Most of the true porosity vlues re recovered for the cse in which permeility is ssumed to e known (the ottom three mps in Figure 5.32) ut the true permeility vlues re not recovered for the cse in which porosity is ssumed to e known (the ottom three mps in Figure 5.31). This cn lso e seen y compring the two resolution mtrices in Figure However t nd ner the well loctions, this oservtion is reversed. Permeility is resolved etter thn porosity. This my

115 SECTION 5. APPLICATION OF THE METHOD 103 e ecuse the reltive sensitivity of the seismic dt with respect to porosity is higher thn tht with respect to permeility (the reltive sensitivity is ( S w /S w )/( ϕ/ϕ)). Also the pressure mesured t the wells determines the welllock permeility vlues ut is not strong function of the porosity t the sme lock. Since seismic dt contins informtion out porosity nd is poor in informtion out permeility, hving oth seismic dt nd porosity informtion my e redundnt while hving oth seismic dt nd permeility vlues is not. For the cse in which porosity nd permeility re oth treted s unknown (the third nd fourth rows in oth Figure 5.31 nd Figure 5.32), the true vlues re lmost s well recovered with unknown permeility-porosity correltion s with known correltion. This is proly ecuse the inverse prolem with unknown correltion hs only two more prmeters (the two unknown coefficients in the correltion) nd this increment is very smll compred to the totl of 363 independent prmeters. If permeility nd porosity re treted s independent vriles, the results show tht in the region fr the well (the first nd second lyer for exmple) porosity vlues re firly well determined ut not the permeility wheres t or ner the well loctions (the third lyer) the permeility vlues re recovered etter thn porosity. Figure 5.34 shows tht the resolution mtrix for porosity is closer to identity thn tht for permeility. This mens the true vlues of porosity re recovered etter thn those of permeility if the two re treted independently. The certinty of the estimtes ws lso computed nd is shown in Figure The permeility-porosity correltion gives highest certinty in the estimtes. Moreover, n unknown correltion etween permeility nd porosity gives s much certinty s fixed correltion. The second highest certinty elongs to the cse in which either permeility or porosity is known nd the worst is the independent permeility nd porosity cse. Another interesting oservtion is drwn from the cses in which we ssumed no correltion etween permeility nd porosity. These cses re: Permeility is treted s unknown with known porosity (the ottom three mps in Figure 5.31). Porosity is treted s unknown with known permeility (the ottom three mps in Figure 5.32).

116 SECTION 5. APPLICATION OF THE METHOD 104 Porosity nd permeility re treted independently (the second rows in oth Figure 5.31 nd Figure 5.32). Looking in more detil t these mps, in some res the computed permeility is oserved t high vlues where vlues of porosity re low nd vice vers. This oservtion is not consistent with the nture of typicl reservoir rock where high vlues of permeility re ssocited with high vlues of porosity. This my due to vrious resons. First, the mount of dt is not sufficient to recognize ny correltion etween permeility nd porosity. Second, the numer of unknown prmeters to e estimted in the inverse prolem is lrge s compred to the mount of dt, nd is douled in the independent permeility nd porosity cse. Third, nd most importntly, the sensitivity coefficients with respect to permeility nd porosity show opposite sign in some regions. This mens tht n increment in permeility hs the sme effect s decrement in porosity, or in other words, n increse in permeility cn e compensted y decrese in porosity. The inversion process cn either increse permeility or decrese porosity to otin the sme grdient of dt nd this results in regions with high permeility vlues nd low porosity vlues nd vice vers. If we oserve t some cells tht k>k true nd ϕ = ϕ true (permeility is treted s unknown with known porosity) we lso my oserve t the sme cells tht k = k true nd ϕ<ϕ true (porosity is treted s unknown with known permeility). This effect is seen in regions tht indicte negtive correltion etween permeility nd porosity (s shown in Figure 5.36). Figure 5.37 shows the plots of computed porosity versus computed permeility vlues in ll cells for the cses in which permeility nd porosity re oth unknown. The line shows the true correltion. The squre points show the computed vlues with unknown correltion. The true correltion is recovered very well with c 1 equl to 5.88 compred to the true vlue of 6.0 nd c 2 equl to 1.56 compred to the true vlue of 1.5. The cloud of tringles presents the correltion etween the computed permeility nd porosity vlues for the cse in which they re oth treted s independent vriles. The true permeility-porosity correltion in this cse is recovered very poorly with correltion coefficient vlue of which is fr less thn unity ( correltion coefficient of one represents perfect correltion while vlue of zero

117 SECTION 5. APPLICATION OF THE METHOD 105 represents no correltion). Figure 5.36 shows how the permeility-porosity correltion cn e resolved s function of depth. The correltion coefficients increse from the top (Lyer #1) to the ottom lyer (lyer #3). The first lyer shows no correltion (ρ = 0.014). Some of the res in this lyer lso indicted negtive correltion etween log-permeility nd porosity. This is the reson why (s ws remrked erlier) in some regions the computed permeility vlues re high with low vlues of porosity nd in vice vers. The second lyer shows stronger correltion thn the first lyer ut the vlue of correltion coefficient is still very smll (ρ = 0.097). Only few locks in this lyer show negtive correltion. The third lyer with ρ = indictes the strongest correltion of the three lyers (most of the wells re locted in this lyer) nd t some locks the true correltion etween log-permeility nd porosity is perfectly recovered (ρ = 1.0). However, since the correltion coefficient is still fr from unity, we cn not clim ny correltion to ny resonle certinty.

118 SECTION 5. APPLICATION OF THE METHOD 106 true k k nd φ independent k nd φ unknown with unknown correltion k nd φ unknown with known correltion φ known k unknown Lyer #1 Lyer #2 Lyer #3 Figure 5.31: Estimtes of permeility in different situtions.

119 SECTION 5. APPLICATION OF THE METHOD 107 true φ k nd φ independent k nd φ unknown with unknown correltion k nd φ unknown with known correltion k known φ unknown Lyer #1 Lyer #2 Lyer #3 Figure 5.32: Estimtes of porosity in different situtions.

120 SECTION 5. APPLICATION OF THE METHOD 108 k unknown φ unknown Figure 5.33: Resolution mtrices: either permeility or porosity is known. Permeility Porosity Figure 5.34: Resolution mtrices: permeility nd porosity re treted independently.