The future of stochastic and upscaling methods in hydrogeology

Beno�t Nœtinger · Vincent Artus · Ghassem Zargar

Abstract Geological formations are complex featuresresulting from geological, mechanical, and physico-chemical processes occurring over a very wide range oflength scales and time scales. Transport phenomenaranging from the molecular scale to several hundreds ofkilometers may influence the overall behavior of fluidflow in these formations. Heterogeneities that cover alarge range of spatial scales play an essential role tochannel fluid-flows, especially when they are coupledwith non-linearities inherent to transport processes inporous media. These issues have considerable practicalimportance in groundwater management, and in the oilindustry, particularly in solving new problems posed byprojects concerned with the trapping of CO2 in the sub-surface. In order to manage this complexity, one must beable to prioritize the respective influences of variousrelevant geological and physico-chemical phenomenaoccurring at several ranges of length and time scales aswell as understand and use the increasingly rich andcomplex geostatistical models to provide realistic simu-lations of subsurface conditions. Multiscale simulation offluid transport in these formations should help engineersto focus on the crucial phenomena that control the flow.This provides a natural framework to integrate data, tosolve inverse problems involving large amounts of data,resulting in a reduction of the uncertainties of the sub-surface description that must be evaluated. This allows inturn the making of more relevant practical decisions. Inthis paper, some perspectives on the development of up-scaling approaches are presented, highlighting some re-cent multiscale concepts, discarding the fractured media

case. Upscaling can be used as a useful framework tosimultaneously manage scale-dependant problems, sto-chastic approaches and inverse problems. Actual andpotential applications of upscaling to the elaboration ofsubsurface models constrained to observed data, and themanagement of uncertainties and sensitivity studies in aglobal multiscale framework is emphasized. In particular,upscaling can help in finding the parameters that controlthe overall behavior of the flow. Finally, upscaling ap-proaches of non-linear transport equations appear as anew frontier in this area of research.

R�sum� Les formations g�ologiques sont des entit�scomplexes r�sultant de processus g�ologiques, m�ca-niques et physico-chimiques, � de larges palettesd’�chelles de temps et d’espace. Les ph�nom�nes detransport, qui sont d�finis de l’�chelle mol�culaire �l’�chelle pluri-kilom�trique, peut influencer le compor-tement global des �coulements dans les formations. Lesh�t�rog�n�it�s qui recouvrent plusieurs �chelles spatialesjouent un r�le essentiel pour les �coulements canaliser, etplus sp�cialement lorsque les �coulements sont coupl�savec des non-lin�arit�s inh�rentes aux processus detransport dans les milieux poreux. Les probl�mes ont uneimportance pratique consid�rable sur la gestion de l’eausouterraine, et dans l’industrie du p�trole, et plus parti-culi�rement pour la r�solution de nouveaux probl�mespos�s par les projets concern�s par le pi�geage du CO2 ensubsurface. De mani�re � g�rer cette complexit�, il fautÞtre capable de donner la priorit� aux influences respec-tives des ph�nom�nes g�ologiques et physico-chimiquesutiles, intervenant � diff�rentes �chelles de temps etd’espace, et comprendre et utiliser les mod�les g�osta-tistiques, riches et complexes, permettant de se munir desimulations r�alistes des conditions de sub-surface. Lessimulations de l’�coulement � plusieurs �chelles, dans cesformations, peuvent aider les ing�nieurs � se focaliser surles ph�nom�nes cruciaux qui contr�lent l’�coulement.Ceci apporte un cadre de travail pour int�grer les donn�es,r�soudre les probl�mes inverses impliquant une grandequantit� de donn�es, r�sultant en une r�duction de l’in-certitude de la description de la subsurface devant Þtre�valu�e. Ceci permet de prendre des d�cisions plus per-tinentes et pratiques. Dans cet article, certaines perspec-tives sur le d�veloppement d’approches «upscaling»(changement d’ �chelle) sont pr�sent�es, mettant en lu-

Received: 6 May 2004 / Accepted: 9 December 2004Published online: 26 February 2005

� Springer-Verlag 2005

B. Nœtinger ()) · V. ArtusInstitut Fran�ais du Petrole,1-4 avenue de Bois-Preau, 92852, Francee-mail: benoit.noetinger@ifp.frTel.: +33-1-47-52-56-80Fax: +33-1-47-52-56-17

G. ZargarNational Iranian South Oil Company,Ahwaz, 61335, Iran

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

mi�re des concepts mutli-�chelles r�cents, en mettant dec�t� le probl�me du milieu fractur�. L’upscalling peut Þtreutilis� comme un cadre utile pour g�rer simultan�ment lesprobl�mes d’�chelle de travail, les approches stochas-tiques et les probl�mes inverses. Les applications effec-tives et potentielles de l’upscalling sont mis en lumi�re,dans l’�laboration de mod�les de subsurface contraintsaux donn�es observ�es, et la gestion des incertitudes etdes �tudes de sensibilit� dans un cadre multi-�chelleglobal. L’upscaling peut, en particulier, aider � trouver lesparam�tres qui contr�lent le comportement global del’�coulement. Finalement, les approches par upscaling des�quations non-lin�aires de transport apparaissent commeune nouvelle fronti�re dans cette aire de recherche.

Resumen Las formaciones geol�gicas son entidadescomplejas que resultan de procesos geol�gicos, mec�ni-cos, y fsico-qumicos que ocurren en un rango muyamplio de escalas espaciales y escalas temporales. Losprocesos de transporte que varan de la escala molecular avarios cientos de kil�metros pueden influenciar el com-portamiento global del flujo de fluido en estas forma-ciones. Las heterogeneidades que cubren un rango ampliode escalas espaciales juegan un papel esencial para ca-nalizar flujos de fluidos, especialmente cuando est�nacoplados con no linealidades inherentes a procesos detransporte en medios porosos. Estos asuntos tienen im-portancia pr�ctica considerable en la gesti�n de aguassubterr�neas, y en la industria petrolera, particularmenteen la soluci�n de problemas nuevos ocasionados porproyectos relacionados con el entrampamiento de CO2 enel subsuelo. Para gestionar esta complejidad uno tiene quepriorizar las influencias respectivas de varios fen�menosrelevantes geol�gicos y fsico-qumicos que ocurren envarios rangos de escalas espaciales y escalas temporalesas como entender y utilizar modelos geoestadsticoscomplejos y cada vez m�s acaudalados para aportar si-mulaciones realisticas de los condiciones del subsuelo. Lasimulaci�n en escalas mfflltiples del transporte de fluidosen estas formaciones debera de ayudar a los ingenieros aenfocarse en fen�menos cruciales que controlan el flujo.Esto aporta un marco natural para integrar datos, resolverproblemas inversos que involucran gran cantidad de da-tos, resultando en una reducci�n de las incertidumbres dela descripci�n del subsuelo que tiene que evaluarse. Estopermite a la vez tomar m�s decisiones practicas relevan-tes. En este artculo se presentan algunas perspectivassobre el desarrollo de los enfoques de escala alta, resal-tando algunos conceptos de escalas mfflltiples recientes,descartando el caso de medio fracturado. La escala altapuede utilizarse como un marco ffltil para manejar si-mult�neamente problemas dependientes de escala, enfo-ques estoc�sticos y problemas inversos. Se enfatiza lasaplicaciones reales y potenciales de escala alta para laelaboraci�n de modelos subsuperficiales confrontadoscon datos observados, y el manejo de incertidumbres yestudios de sensitividad en un marco global de escalasmfflltiples. En particular, la escala alta puede ayudar aencontrar los par�metros que controlan el comporta-

miento global del flujo. Finalmente, los enfoques de es-cala alta de ecuaciones de transporte no lineal aparecencomo una nueva frontera en esta �rea de la investigaci�n.

Nomenclature

Roman symbolsC Unit CPU cost of a typical fluid flow simulation ·Ctot Total CPU cost of a complete study · ct Totalcompressibility value (bar�1) · c(r, t) Local tracerconcentration (mole m�3) · C(r) Log permeabilitycovariance evaluated at lag r · C(q) Fourier transformof C(r) at wave vector q(mD) · CG (r) Filtered logpermeability covariance function · D Flow dimension(D=1, 2 or 3) · d(u) Dispersivity tensor (m) ·dobs Observed data · f(r, t) Local source term (m) ·J(k) Objective function to be minimized ·k(r) Permeability value at point r(m2) · kapp(t) Apparentwell test permeability value at time t(m2) · kv/kh Verticalto horizontal permeability anisotropy ratio · Keff Effectivepermeability value (m2) · Kg Geometric mean ofpermeability (m2) · KgG Renormalized geometric mean ofpermeability (m2) · kri(S) Relative permeability value forfluid i(i=1, 2) · lc Permeability correlation length (m) ·lh/lv Vertical to horizontal correlation length ratio ·L Overall size of the porous medium · Nsens Numberof sensitivity runs · Ninv Number of inversion loops ·NMC Number of Monte Carlo iterations · p(r) pressurevalue at point r. (bar) · q wave vector (m�1) ·R(t) Investigation radius of a well test (m) · S Watersaturation · u(r) Local fluid velocity at point r (m/s) ·Y(r) Logarithm of permeability

Greek symbolsa=(kv/kh)1/2 lh/lv, Global anisotropy ratio · dx Finegrid size (m) · DX Coarse grid size (m) · r(r, t) Localmass density (Kg m�3) · f(S) Fractional flow function ·G Wave vector permeability cut-off. (m�1) ·l(S) Total mobility function · s2=C(r=0) Logpermeability variance · smeas2 Measurements errorsvariance · s2

mod Numerical modeling errors variance ·S(r, t) Diffusion kernel · m Fluid viscosity · mi i thfluid viscosity · w(a) Permeability averaging exponent

Introduction

In this paper, some aspects of the upscaling problem offluid flow in aquifers or oil reservoirs are reviewed, andsome promising research directions are suggested, usingthe practical point of view of an engineer faced with a realcase. In the upstream oil industry the basic goal of fluidflow simulation is to provide tools allowing engineers toperform accurate sensitivity studies, in order to optimizean oil recovery scenario. In practice, this goal is reachedby optimizing the position and design of the wells, andalso by planning a water or gas injection scenario

185

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

(Cosentino 2001). People involved in water resourcesmanagement may have to predict the displacement of apollutant, in order to make decisions to maintain waterquality and to remediate the aquifer while minimizing thecost of the prediction. This is the basic goal of reservoir oraquifer simulations. Let C be the typical ComputationalCost (CC) of a single fluid flow simulation on a givengeological formation under specified boundary condi-tions. In general, engineers want to perform sensitivitystudies to test for example the influence of the numberand position of the wells. Sampling the influence of thesecontrol parameters implies doing Nsens simulations whosecost scales is therefore Nsens�C.

A considerable improvement in the accuracy and ro-bustness of the resulting solution can be obtained bytaking into account all the available data (Alabert 1989;Galli et al. 1990; Gu�rillot et al. 1991; Deutsch andJournel 1992; Blanc et al. 1996; Gorell and Basset 2001).This is the so called History Matching problem of oilreservoir engineers. These input data can be either ofgeological expertise and seismic origin, or be derivedfrom dynamic data related to fluid flow such as a head (orpressure) measurements, the watercut (ratio of water andoil rates) or even repeated seismic acquisition indicatingsubsurface fluid displacements (Pianelo et al. 2000a;Kretz et al. 2002). So, one of the major concerns of en-gineers is to be able to integrate different data comingfrom different measurement processes at different scales(Pianelo et al. 2000b; Schaaf et al. 2002). These huge datasets are intrinsically defined and measured over ratherdifferent support sizes, such as core plug permeabilitymeasurements over some centimeters or a well test in-terpretation at a 100 m scale, or seismic acquisitions overlonger distances.

Ideally, the engineer wants to forecast the fluid trans-port modifying only the geological model, allowing nu-merical aspects to be managed by the simulator itself. Theinverse problems to be solved are based on large gridswhere the number of grid blocks may vary from 104 to106. The measured data can also be quite numerous, es-pecially in the case of repeated seismic acquisition.Generally, the solution of these inverse problems useoptimization procedures that minimize a cost functioncharacterizing the discrepancy between the actual simu-lation results and the observed data (Tarantola 1987). Thisrequires numerous repeated calls to the fluid flow for--ward simulator, resulting in a typical cost equal toNsens�Ninv�C.

Here, the number of forward simulations Ninv is anincreasing function of the original number of grid blocks,N, and of the number of measured data Ndata. In addition,the strength of the non-linearities of the fluid flowequations have a global influence on the number of iter-ations.

Even with careful use of all the available data, theengineer still wants to quantify and to manage the re-maining uncertainties inherent in any geoscience model-ing. An important debate is thus to choose between sto-chastic and deterministic descriptions (Matheron 1967a;

Dagan 1989). However, a more popular approach is totake into account the fact that the knowledge of thesubsurface is intrinsically incomplete. Any reservoir mapis considered as being a single realization of a randomprocess. In addition to the intrinsic uncertainty of themeasurements, uncertainties related to lack of knowledgeof the geological description are accounted for. Due toincreasing geological knowledge, traditional “two point”geostatistical models defined by their mean and theirvariogram are currently being replaced by “processbased” or “multi point” models (Strebelle and Journel2000).

In order to test the influence of randomness, the en-gineer is thus led to perform extensive Monte Carlosimulations involving many NMC fluid flow simulationsover random (although perhaps constrained) realizations(in practice NMC equal to 100, even though in theoryseveral thousands are required to get a statistically validanswer). So, the total computational cost Ctot to perform acomplete study involving sensitivity tests, historymatching with actual data and a Monte Carlo uncertaintyevaluation, becomes Ctot ffi Nsens � NMC � Ninv � C. Thisnumber can be considerably greater than the original cost,C.

To reach these goals, in the usual practice, engineershave first transformed the original 3D geological map intoa numerical model that is well suited for solving thediscrete equations arising from any numerical treatmentof single or multiphase fluid flow equations. It will beassumed that this numerical modeling process is basicallycorrect: which means that if the engineer should be able torun an indefinitely refined model, he will get an exactsolution of the underlying fluid flow equations. So thebasic physics of fluid flows, like choices concerning rel-ative permeabilities, capillary pressures description andphenomenological laws are assumed to be properly de-scribed.

A basic issue is simply to be able to perform thesecalculations in an acceptable computational cost andwith accuracy. The objective is thus to reduceNtot ffi Nsens � NMC � Ninv � C. This can be first achievedby reducing C. The simplest approach is thus to try toreduce the number of grid blocks and of unknowns to geta tractable problem. This classical issue of upscaling,sometimes called pseudoization for multiphase fluid flowproblems, is the subject of a huge amount of literature(see Ahmadi et al. 1993; Ahmadi and Quintard 1996;Artus et al. 2004; Renard and de Marsily 1997 for anextensive review in the single phase case). It will beshown that upscaling or a multiscale approach can alsohelp to reduce NMC, Ninv as well as C, resulting in im-proved advantages of upscaling.

The basic question is thus to be able to account for thesmall scale disorder of the medium (or “subgrid” effectsin the language of people involved in ComputationalFluid Dynamics, CFD Lesieur and M�tais 1996) at thiscoarser scale. Depending on the academic background ofthe authors, the focus is on mathematical asymptotictheories of homogenization and more recently on sto-

186

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

chastic Partial Differential Equations (PDE) (Jikov et al.1994; Badea and Bourgeat 1995, 1996; Bourgeat 1996).People having a physics and fluid mechanics backgroundwill adopt methods arising from mechanics or physicssuch as volume averaging (Quintard and Whitaker 1994;Cherblanc 1999; Wood et al. 2003) effective medium(Dagan 1989), and sometimes percolation theory in theextreme cases of highly heterogeneous media such asfractured rocks (Stauffer 1985; Heiba et al. 1992; Hunt1998, 2001 and references therein). Some sophisticatedperturbation methods, originally devised to treat questionsarising from quantum mechanics, next adapted to studyturbulent transport and including renormalization groupapproaches, are also employed in the stochastic context(King 1989; Glimm et al. 1992; Zhang 1992; Christakoset al. 1993, 1995; Nœtinger 1994; Avellaneda 1996;Jaekel and Vereecken 1997; Stepanayants and Teodor-ovitch 2003). More recently, very interesting purely nu-merical techniques based on new finite element ap-proaches proposing a fully multiscale treatment wereproposed (see e.g. Gu�rillot and Verdi�re 1995; Hou andWu 1997, 1999; Gautier et al. 1999; Arbogast and Bryant2001). Analogous treatments can be found in the vastdomain of CFD, where the Large Eddy Simulation (LES)technique of turbulent flows is very close in spirit topresent considerations (Germano 1992). In addition, thereexists a considerable amount of applied oil reservoir en-gineering or hydrology literature proposing practical so-lutions that can be justified a posteriori using rigorousapproaches.

Concerning the uncertainty management, it can beremarked that fluid flow simulation becomes intrinsicallyprobabilistic: the goal is not really to obtain a very ac-curate result about a single subsurface image that is likelyto be wrong, but mainly to predict a mean and variance ofthe quantities under study. A good model must be unbi-ased, i.e. the average properties must be recovered, andideally the variance should also be correctly estimated.These considerations can help to justify upscaling inpractice: if the errors due to upscaling are smaller thatthose due to uncertainties; one can safely use up-scaledmodels (Glimm et al. 1992; Gorell and Basset 2001;Schaaf et al. 2002). One must remain careful with sys-tematic errors corresponding to using a biased model.

An important general feature of upscaling is that it isclosely related to ergodicity and averaging thanks to aninteresting feature of many diffusive transport phenome-na, called the self-averaging property, whose direct con-sequence is that the overall behavior of a complex het-erogeneous medium can often be described at large scaleby a rather small number of parameters (Goldenfeld 1992;Koslov 1993). Among those parameters, some are clearlyrelated to small scale data, and others appear as the resultof a complex combination of coupled effects of smallscale physics with large scale heterogeneities. A typicalexample is the coupling of viscous instabilities of Saff-man-Taylor type with preferential permeability streaks(Saffman and Taylor 1958). This coupling can give rise tonon-trivial large scale behavior that may be described by

equations that are qualitatively different from the smallscale Darcy equations (Langlo and Espedal 1994;Lenormand and Thiele 1996; Artus et al. 2004). One ofthe more important and difficult tasks is to predict andidentify the occurrence of such phenomena that may bedominant when dealing with uncertainties. Within thispoint of view, a multiscale description defines implicitly atheory of data measurement: what is really measured inthe laboratory when measuring a single phase perme-ability measurement or a two phase displacement: a localproperty or an averaged one ? This question is essentialfor well tests interpretation and most reservoir scalemeasurements. This major conceptual and practical issuewas first recognized by Cushman 1984 and is very closein spirit to the modern renormalization group point ofview (Goldenfeld 1992). It will be seen that this kind ofconsideration has deep consequences even on the nu-merical simulations.

The goal of this paper is to review some exciting andpromising flow related problems in upscaling of hetero-geneous and stochastically characterized subsurfaceaquifers or oil reservoirs. It is organized as follows : In thefirst section, the problem of pressure diffusion in a ran-dom heterogeneous medium is discussed. It is shown thatthe average head is driven by an effective equation whoseform reflects the existence of spatial correlations. Theemergence of a large-scale effective permeability tensorcontrolling the asymptotic long-time long-distance be-havior of the solution of this equation is thus described.This allows the introduction of the useful concept of self-averaging equations, that helps to understand the practicalutility of the stochastic approach of subsurface flow: onesingle large realization can contain the overall statistics.

Next, an alternative point of view of upscaling, termedthe filtering approach is developed. This point of view is apractical version of the renormalization approach widelyused in statistical physics (Goldenfeld 1992). It consists inupscaling directly of the probability distributions of theparameters such as the permeability rather than the indi-vidual realizations. This allows one to imbed the wholeupscaling process in a probabilistic framework, wellsuited for solving inverse problems and to quantify un-certainties. In addition, this approach could help shed newinsights on fundamental theoretical issues regarding thefamous Landau-Lifschitz-Matheron (LLM) conjecturerelating the large-scale effective permeability to localstochastic parameters (Landau and Lifschitz 1960;Matheron 1967b).

Next, several applications of the proposed approachare proposed, regarding a multiscale solution of inverseproblems, where the corresponding stochastic point ofview, termed the Bayesian approach, is introduced. Themain idea is to minimize the functional arising from in-verse problem theory directly at the most relevant lengthscale corresponding to the support size of the data to behistory-matched.

Next, after a short discussion about the large scalebehavior of passive tracer transport equations, some re-cent results concerning the upscaling of non-linear

187

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

transport processes are introduced. Feedback effectswhere there is a retroaction between the flow pattern andthe local transport properties can change drastically thelarge scale physics, especially when instable behaviorsare encountered. This appears for example in two phasewater oil flows in oil reservoirs, or in unsaturated flows,or in acidification processes near wells. The couplingbetween heterogeneities and transport becomes essential.The large scale equations describing transport can thus geta form that is qualitatively different from the local scaleequations. In the opinion of the authors,the most exitingchallenges belong to this class of problems.

Pressure diffusion in heterogeneous media

In order to manage the stochastic nature of subsurfacemodeling, a natural mathematical framework is to con-sider stochastic Partial Differential Equations (PDE).Generally, these PDE are characterized by coefficientsdepending randomly on position, and in some cases ran-dom source terms and boundary conditions can be added.The goal of this section is to check if the average of thesolution of fluid flow equations over the infinite set ofpossible realizations, is itself the solution of a similarPDE having a similar form and renormalized effectivecoefficients. The working example in this paper will bethe diffusivity equation driving the head variations, whichis a good candidate to highlight the self-averaging prop-erty: in what sense a single “large realization” can beequivalent to averaging a huge number of independentreplicas of the subsurface model of this paper.

Equations and notationsTo have a practical basis of discussion, consider first awell test in a 2D heterogeneous reservoir. Here, no flowboundaries are supposed to be at infinity. A drawdownwell test is considered, where the flow rate is equal to Qafter t=0. The well is supposed to be located at the originof coordinates. The pressure derivative @p0 r;tð Þ

@t ¼ p0 r; tð Þ isshown to obey the following equation (de Marsily 1981):

fct@p0 r; tð Þ@t

¼ r � k rð Þmrp0 r; tð Þ

� �þ Qd rð Þd tð Þ ð1Þ

In other words, p0 (r, t) is the Green’s function of thediffusion problem. In the well test interpretation, (Daviau1986), it is common to compute the logarithmic timederivative of the pressure measured at the well that isgiven by:

tp0 r ¼ 0; tð Þ ¼ tdp tð Þ

dt¼ mQ

4pkð2Þ

The last equality is valid in the homogeneous case, andit is the basis of the well test interpretation providing anestimation of the reservoir permeability. In the hetero-geneous case, Eq. 2 can be used to define an “instanta-neous apparent permeability” k(t) that will depend ontime (Feitosa et al. 1993).

In order to fix the ideas, the heterogeneous reservoirwill be supposed to be a single realization of a log normalprocess of correlation length lc. The permeabilityk(r)=exp Y(r) is supposed to be a stationary log normal(i.e. its logarithm has a multigaussian distribution) ran-dom field having the following properties:

Kg ¼ exp Ln k rð Þh i; C rð Þ ¼ Lnk rð ÞKg

Lnk r ¼ 0ð Þ

Kg

� �

ð3ÞThe symbol <...> represents the average of the quantity

in brackets with respect to all the possible permeabilitymaps conveniently weighted by the retained probabilitydensity. The geometric average Kg, and C(r) is the logpermeability correlation function.

An effective equation for the averaged headConsider the ensemble average of the local pressure de-noted by hPi(r, t) that could be obtained in practice afteraveraging Monte Carlo numerical simulations. Is it pos-sible to find an equation driving hPi(r, t), and allowingone to avoid the costly Monte Carlo step? Using pertur-bation methods arising from statistical physics, it is pos-sible to show that the following equation governs theevolution of the average pressure hp0i (r, t):

fct@ p0h i r; tð Þ

@t¼ r kh i

mr p0h i r; tð Þ

� �

þZ t

0dt0r

Zdr0X

r0; t0ð Þr p0h i r� r0; t � t0ð Þ þ Qd rð Þd tð Þ

ð4ÞIf one were able to estimate the memory kernel S(r0, t0)

(a second order symmetric tensor), a single numericalsolution of this equation will thus replace expansiveMonte Carlo simulations. This linear equation has thestructure of an integro differential equation, and was de-rived independently by Indelman 1996, and Nœtinger andGautier 1998 who used a Feynman graph approach. Thememory kernel S(r0, t0) depends explicitly only on thecorrelation functions of the permeability of arbitrary or-der. It has a somewhat local structure: a typical spatialrange equal to lc, and a time range of l2

c=D0 which is thetypical diffusion time over a typical correlation lengthlc D0 ¼ kh i=fmctð Þ. A study of an expansion of S(r, t) in apower series of the permeability variance s can be per-formed using Feynman graph techniques that provide asystematic method to compute any order term of pertur-bation theory, and to reorder the resulting series expan-sion in a physically appealing way (Nœtinger and Gautier1998). Starting from an explicit expression of the averagepressure hp(r, t)i, these techniques allow one to find ex-plicitly the form of the equation (4) that governs theevolution of the average pressure hp(r, t)i. In particular,the kernel S(r, t) that is called the “mass operator” inother areas of physics can be computed at any desiredorder as being the sum of all the corresponding “irre-ducible graphs”. It shows that the average head is driven

188

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

by an equation that has essentially a Darcy structure. Itmay be shown that up to second order (the “one loop”approximation):X

r0; t0ð Þ ¼ s2

m kh i rrP0r0; t0ð Þ½ �C r0ð Þ ð5Þ

Here P0(r0, t0) denotes the Green’s function of thediffusion operator of diffusivity D0 with an instantaneouspoint source at the origin. It shows that this kernel has atypical spatial range equal to lc, and a time range of l2

c=Dwhich is the typical diffusion time over lc D ¼ kh i=fmctð Þ.Equation (4) can be written in a conservation equationform when introducing a local velocity V(r, t) as:

fct@ p0h i r; tð Þ

@tþr:V r; tð Þ ¼ Qd rð Þd tð Þ ð6Þ

V r; tð Þ ¼ � k

m

� �r p0h iðr; tÞ

�Z t

0dt0Z

dr0X

r0; t0ð Þ:r p0h i r� r0; t � t0ð Þ

ð7ÞIndelman 1996 showed that this velocity V(r, t) is

exactly the average of the local Darcy flux defined byV r; tð Þ ¼ �kðrÞ=mrp r; tð Þh i. This means that the averagelocal flow rate at point r is a weighted average of thewhole pressure gradients close to r, in a region of typicalsize lc. The influence of statistical correlations entering inthe averaging process is to transform the originally localDarcy’s law in a non-local formula.

It would be very interesting to be able to test thepresent approach by solving Eq. 4 directly using e.g.Fourier transform methods, and comparing the solutionwith Monte Carlo results. As the estimation (5) of thediffusion kernel is obtained by means of a small pertur-bation expansion, a method allowing us to treat highlyheterogeneous media is to be set up.

Another important problem should be to be able toprovide a robust estimation of S(r, t) valid for large logpermeability variance. The present estimation (5) wasobtained for a small variance assumption : is it possible tofind a more robust formulation, using for example a large-scale effective permeability evaluation ?

A generalized effective permeability tensorUsing the Fourier-Laplace transform, defined by:f q; sð Þ ¼

R10 dt exp �stð Þ

Rdr eiq:rf ðr; tÞ, it is possible to

generalize the concept of effective permeability by in-troducing a wave-vector dependant effective permeabilitytensor Keff (q, s). Using the (q, s) variables, relation (4)takes a simpler form: a simple product involvinga wave-vector dependant Darcy’s law, withKeffðq; sÞ ¼ kh i1þ m

Pq; sð Þ, where 1 denotes the unit

tensor. The limit of Keff (q, s) for a low wave vector, along time (q, s)!(0, 0) can be identified to Keff, thesteady state effective permeability tensor. Here, this canbe justified heuristically, because for a long time, one can

expect that hp0i (r0, t0) varies quite smoothly in both r andt domains. So it can be approximated by hp0i (r, t) underthe integral sign of (4) or (7). This gives:

Vðr; tÞ � � k

m

� �r p0h iðr; tÞ

�Z t

0dt0Z

dr0X

r0; t0ð Þ�:r p0h i r; tð Þ�

ð8Þ

This results in an effective Darcy’s law, with

Keff � kh i1þ mZ t

0dt0Z

dr0X

r0; t0ð Þ !t!1

kh i1

þ mZ 1

0dt0Z

dr0X

r0; t0ð Þ ð9Þ

In the Fourier Laplace domain, this last result can beidentified with Keff (q=0, s=0).

Emergence of the self-averaging propertySo far, these results are valid after averaging the solutionsof the driving PDE over a theoretically infinite number ofsimulations. In practice, in the present case, we are evi-dently considering a well test that is performed on a singlerealization : the actual geological formation. So what isthe practical significance of these results? Considering awell test in one realization, for a short time the well testwill provide an estimation of the permeability at the welllocation, for a long time the effective permeability Keff ofthe reservoir (Nœtinger and Gautier 1998). In between, itcan be shown that at first order in the permeability fluc-tuation, the apparent permeability is a weighted averageof the local permeabilities with a kernel whose supportsize R(t) scales as R(t)ffi(2D0t)1/2, that corresponds to theso-called investigation radius (Blanc et al. 1996) of thewell test, is continuously increasing with time (Oliver1989; Feitosa et al. 1993; Gautier and Nœtinger 1998). Atlong times, one is thus led naturally to measure the av-erage of a great number of independent events. Centrallimit theorems can be invoked and one can expect that theapparent permeability will converge “almost surely” tothe steady state value (Goldenfeld 1992; Jikov et al.1994). This is exactly what happens in practice.

Nœtinger and Gautier 1998 showed that the ap-parent permeability kapp(t) defined by inverting Eq. 2,which is a random function depending on time, hasthe following statistical properties : Lim

t!1kapp tð Þ� �

¼Keff ; kapp tð Þ � Keff

� 2D E

/ 1t

This means that for a long time t, the apparent per-meability is “almost surely” equal to Keff. In practice, awell test simulation on a sufficiently large single real-ization is quite equivalent to applying a Monte Carloaveraging, at least for large scale phenomena. This iswhat is called the self-averaging property. This result,also discussed by Hunt (1998), may be qualitatively ex-plained as follows. Consider that the apparent perme-ability kapp(t) is a weighed average of the small-scalepermeabilities within the disk of radius R(t)ffi(2D0 t)1/2

around the well (Oliver 1989). For large t, there are a

189

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

number NðtÞ � pRðtÞ2=l2c ¼ 2D0t=l2

c of independent per-meability units in this disk. Thus the central limit theoremshows that the variance of kapp(t) will vary like 1/N(t)�1/t: this corresponds to the prediction of the detailed brutalforce calculation of Nœtinger and Gautier 1998. It allowsone to understand how the measurement support size mayinfluence the determination of permeability distributions.

Feitosa et al. (1993), Gautier and Nœtinger (1998), andHaas and Nœtinger (1995, 1996), used these results toelaborate a practical method allowing one to constructreservoir images constrained by the observed well test.The idea was to replace the costly well test simulation bycomputation of quasi-analytical simple averages, yieldinga more easy to handle inverse problem.

These results are illustrated in Fig. 1, where the resultsof well test simulations over five independent realizationsare plotted. After a transient period, all the well testsprovide an equivalent permeability whose value is closeto the so called effective permeability. If one were con-sidering an infinite medium, this will be a well definedquantity depending only on the geostatistical structure ofpermeability, which in the present case is its mean andcovariance structure. Some equivalent permeability fluc-tuations that reflect the finite size of the considered me-dium can be observed. In the 2D isotropic case, it can beshown that there is an excellent correlation between thiseffective permeability and the empirical geometric mean

Kg � exp 1N

Pi¼1;N

Ln ki

" #of the realization (Desbarats 1992;

Neuman and Orr 1993;Gautier and Nœtinger 1997).

Another point of view: direct upscalingof geostatistics, the filtering approachand the missing scale problemIn this section, the goal is to analyze if it is possible toestimate directly the probability distributions of the so-lutions of fluid flows in random porous media. In thepreceding approach, the costly Monte Carlo step was re-placed by the solution of a single more complex equationwhose numerical solution should be cheaper. It providesan estimation of the average property. At this stage, thevariability of the results cannot be estimated. It is alsorestricted to a single class of heterogeneities, mainly log-normally distributed permeabilities, small variance, and tosimple boundary conditions.

In practice, one must be able to consider more generalmedia, such as those generated using “multiple point”geostatistics or object models, like, e.g. fractured media,and more general boundary conditions, Mallet (1997).

A possible solution is the filtering approach that is veryclose in spirit to the large eddy simulation (LES) nu-merical approach used to model very large Reynoldsnumber turbulent fluid dynamics (Germano 1992; Lesieurand M�tais 1996) or renormalization group theory instatistical physics (King 1989; Goldenfeld 1992; Jaekeland Vereecken 1997). It must be acknowledged that flowin natural porous media generally implies laminar flow atlow Reynolds number. The small scale disorder due toturbulence must be replaced by the disorder due to het-erogeneities. Ideas and techniques to incorporate subgridphysics coming from LES can thus be useful.

The basic idea is to consider that it would be powerfulto upscale the geostatistics itself, rather than individualrealizations. Consider a stationary log normal perme-ability distribution of geometric mean Kg0, and of cova-riance function C(r). For the sake of simplicity, consideran isotropic situation where both permeability tensors andgeometric properties are statistically isotropic. An effi-cient practical way of generating realizations Y(r)=Lnk(r) is as follows: consider the Fourier transformof Y(r); denoted by Y(q) which is defined byY qð Þ ¼

Rdr e�iq:rYðrÞ, where dr denotes the natural in-

tegration measure and the summation runs over thewhole space. One has the Fourier inversion formulaY rð Þ ¼ 1

2pð ÞDR

dre�q:rYðqÞ. It can be shown that for astationary multigaussian process Y(r), Y(q) is also amultigaussian process which has the following properties:

Y qð Þh i ¼ 0; Y qð ÞY q0ð Þh i ¼ 2pð ÞDC qð Þd q0 � qð Þ: ð10ÞHere d(q0-q) is a D dimensional Dirac delta function,

C(q) is the Fourier transform of the covariance functionC(r) and Y is the complex conjugate of Y. In other words,this means that the Fourier transform renders the covari-ance matrix diagonal. This property remains exact whenconsidering a discretized representation of Y on a regularCartesian grid. So the vector Y(q)=C(q)1/2 z(q) where z(q)is a multigaussian vector whose covariance matrix is theidentity matrix has the required covariance matrix.

Fig. 1 Five “instantaneous apparent permeability” curves corre-sponding to five well test simulations performed over five inde-pendent realizations of a 3D log normal permeability Kg=10 mD,s2=1. The curves are interpreted using Eq. 2. The stabilization ofthe well test corresponding to the emergence of an homogeneousequivalent medium can be observed after t=100,000s. The ratherlow ratio between the size of the reservoir and the correlationlength explains the fact that stabilization of the well test providesrealization dependent equivalent permeabilities

190

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

Coming back to real space by means of an inverse Fouriertransform, one can get

Y rð Þ ¼ 1

2pð ÞDZ

dq eiq:rC1=2 qð Þz qð Þ ð11Þ

This is the basis of the Fast Fourier Transform MovingAverage (FFT MA) method that implies the followingcalculations :

– Generate a vector of uncorrelated deviates z(r)– then by FFT compute z(q) and thus Y(q)=C(q)1/2 z(q)– come back to real space by an inverse FFT to evaluate

Y(r).

This method has the advantages of combining the gen-eration of z(r) in real space that allows one to localize theheterogeneity and of Fourier transform methods acceler-ated by Fast Fourier Transform techniques (le Ravalec etal. 2000). Gradual deformation techniques can be easilyset up using this technique to generate subsurface real-izations constrained by both geostatistical and fluid flowdata, Hu et al. (2001). It is a good basis that can also serveas a basis to solve directly the Darcy flow equations usingFFT algorithms.

Filtering methodIt is now possible to introduce a cut off scale denoted G(m�1). A new random vector denoted YG(q) can be com-puted by means of the following definition:

YG rð Þ ¼ 1

2pð ÞDZ

q<Gdqeiq:rC1=2 qð Þz qð Þ ð12Þ

where the integration is restricted to wave vectors whosemodulus are smaller than G.

So, once z(q) is given, a complete family of YG(q) canbe obtained depending on the value of the cut off G. Thiscut off plays the role of a filtering parameter in real space,the proposed operation will correspond to a convolutionby a Bessel function whose typical support size would beLffi2p/G. In other words, permeability fluctuations whosefrequencies are larger than G are filtered out. If G goes toinfinity, the original Y is recovered Fig. 2.

Figure 3 displays the permeability maps correspondingto this process for different values of the cut off G. Thesmoothing effect is clear. Physically, these new perme-ability maps correspond to maps of upscaled permeabilityat scale G.

Potential applicationsThis filtering approach has a broad range of practicalapplications.

The first application is upscaling and grid coarseningitself. Notice that when high frequency components arefiltered out, the resulting map is still described by thesame number of grid blocks. But, as the filtered realiza-tion is smoother than the original one, it is appealing tocoarsen the grid using a typical grid size DX<1/G. As thefiltered permeability map is now smooth, the upscalingprocess can be simplified using e.g. the value at the coarsegrid block center that became representative of the wholeblock permeability. This can be observed in Fig. 4 wherean hypothetical coarse grid, refined close to a well wassupposed to be used.

In particular, if one wants to set a permeability at-tribute to an irregular grid, one could attribute to a gridblock of characteristic size 1/G, the permeability providedby kG(r) evaluated at the grid block center.

Once the small scale z(r) map of gaussian numbers isgenerated, kG(r) can be evaluated for any G.

Returning to the permeability map by introducingkG(r)=KgG expYG(r), notice that a renormalized geometricmean KgG(r) depending on the cut off parameter G wasintroduced, because it is impossible to assert that this

Fig. 2 Fourier transform of the covariance function of the G filteredrealizations. Each Fourier mode of wavevector > G is smoothed out

Fig. 3 Filtered realizations: Due to the basic principle of theFFTMA algorithm, the same set of random numbers is used togenerate these different images. These four images differ as Fourier

modes of wavevector > G have been cut off. The first image iswithout filtering, and then G2>G3 >G4

191

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

average should not be changed. The practical issue is thusto find a method to determine KgG.

A simple idea can be implemented as follows: considerthat one is interested in the equivalent permeability of agiven large realization of Y (so G=1). Using for examplea numerical simulation, an equivalent permeability Keff,called KeffðG ¼ 1Þ ¼ Keff1 can be obtained. Now, thecorresponding filtered realization with cut off G, can beused to compute its equivalent permeability of the filteredrealization, which may not coincide with Keff1. It willevidently be proportional to KgG, so one can adjust thegeometric mean KgG in order to get hydraulically equiv-alent realizations providing the same large-scale effectivepermeability Keff1. In particular, considering the limit ofG tending to zero, as most of the heterogeneities aresmoothed out, KgG)0 ¼ Keff1. In other words, the geo-metric mean of these completely filtered realisations willbe equal to the effective permeability of the medium, andas most of the permeability fluctuations were smoothedout, the variance will be equal to zero. In other words, thiswill provide the equivalent homogeneous medium Fig. 5.

To summarize, using kG(r) is equivalent to consideringa log-normal random map such that Ln kGðrÞh i ¼ Ln KgG;

CG rð Þ ¼ LnkG rð ÞKgG

LnkG r ¼ 0ð Þ

KgG

� �¼ 1

2pð ÞDZ

q<Gdqeiq:rC qð Þ

ð13ÞThis provides a useful alternative interpretation of the

filtering approach: it can be viewed as a direct upscalingof geostatistics itself.

Depending on the support scale, the associated geo-statistical properties can be directly computed. Whenfiltering most frequencies (G<<1/lc), we have by a directevaluation of (13):

s2 Ln kGð Þ �VD

2pð ÞDC q ¼ 0ð ÞGD;

where VD is the volume of the sphere of unit radius in Ddimensions. This result is another manifestation of thepreviously discussed self-averaging property. To illustratewhat happens, consider the case of a Gaussian covariancefunction C(r)=C(r=0)exp �(r/lc)

2. After an explicitcalculation, s2 Ln kGð Þ � C r ¼ 0ð Þ G lcð ÞD ¼ C r ¼ 0ð Þ=n.So, when G goes to zero, the variance tends to zero. Thisexpression has a simple interpretation: the quantity n=(Glc)

�D corresponds to the number of “statistical units”, i.e.the number of really independent degrees of freedom thatare present in a volume of typical size L=1/G. It is veryappealing from the theoretical point of view, as thisquantity is intrinsic, involving only physical parameters.In particular, it does not depend at all on the chosendiscretization. This result is closely related to the centrallimit theorem and it highlights the homogenization orsmoothing effect.

A classical result of Matheron (1967b) shows that thisrenormalized average permeability, KgG is constant in 2Dand remains equal to the input geometric mean Kg of thedistribution.

Following an approach that was proposed by Jaekeland Vereecken 1997 to model scale effects on the up-scaling of the dispersion coefficient, and using againperturbation theory, and a mean-field approximation, it ispossible to derive the following differential equation(Nœtinger 2000) :

d Log KgG�

d G¼ 1

2pð ÞD12� 1

D

� �SDGD�1C Gð Þ ð14Þ

with SD equal to the surface of the unit sphere in D di-mensions. The mean-field approximation considers thatfluctuations of permeability of wavevector between G andG+dG interact via an effective homogeneous medium ofpermeability equal to KgG, rather than an homogenousmedium at Kg. Integrating this ordinary differentialequation yields:

Fig. 4 Using the filtering ap-proach to simultaneously usefine and coarse grid data. Theupscaling and downscalingaspects are managed simulta-neously

Fig. 5 Equivalence criterion between filtered and non-filteredgeostatistics. The retained criterion is that when computing theaverage large-scale effective permeability Keff, one should recoverequal values. A possible solution is renormalizing the mean per-meability Kg, KgG

192

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

Log KgG�

¼ Log Kg þZ 1

Gd G

1

2pð ÞD12� 1

D

� �SDGD�1C Gð Þ

In particular, when G=0 (i.e. all permeability fluctua-tions are averaged out), one gets:

Log Kg G¼0�

¼ Log Kg

þZ 1

0dG

1

2pð ÞD12� 1

D

� �SD GD�1C Gð Þ

so Kg G¼0 ¼ Kg exp12� 1

D

� �Cðr ¼ 0Þ

ð15ÞThis is the Landau and Lifschitz (1960), Matheron’s

(1967b) (LLM) conjecture, which appears as a mean fieldapproximation. It may be rewritten in an alternative form:

Keff ¼ kð1�2=DÞD E1=ð1�2=DÞ

¼ kwh i1=w ð16Þ

where w=�1 in 1D (harmonic mean), w=0 in 2D (geo-metric mean), w=1/3 in 3D. This LLM conjecture will bediscussed later.Another very interesting point of view isto interpret these renormalized geostatistics as corre-sponding to measurements performed on a support whosetypical size is 1/G. For example, starting from a high-resolution permeability map, consider the observedstatistical properties of the permeability of plugs of size1/G�dx, studying the geostatistical properties of thesedata measured on these plugs. Due to the homogenizationeffect, a reduced variance is expected. Further tests areneeded to check the evolution of geostatistical distribu-tions with the measurement scale. Preliminary tests showthat a log normal law remains log normal (not a trivialresult). Generalization to anisotropic cases are currentlyunder investigation. The generalization to categoricalmodels with a finite number of different geological faciesat small scales presents a major difficulty.

Theoretical issuesIt should be stressed that the direct upscaling, togetherwith the filtering approaches are very close to the renor-malization-group theory of quantum and statistical phys-ics (Goldenfeld 1992), where it is explicitly recognizedthat it is more important to quantify the evolutions withthe measurement scale of the apparent parameters thatdescribe elementary particle (like electrons) and fields(like the electromagnetic field) fundamental interactionsrather than the infinite resolution value that is completelyinaccessible. This can help one to give a precise meaningto the notion of an “uncorrelated” medium, a mediumwhose correlation length tends to zero, the so called“nugget effect” of geostatistics. A simple dimensionalreasoning shows that the large-scale effective perme-ability Keff should not depend on lc, because the dimen-sionless ratio Keff/Kg cannot depend on a single length-scale. Another reference lengthscale is needed, and thatcan only be the measurement scale lmes. Considering that

lc is smaller than lmes means only that the measuredpermeabilities appear as being uncorrelated at lmes.

These considerations have deep practical conse-quences regarding the fluid-flow simulations in randomaquifers, when the issue is to “refine” the grid repre-senting an uncorrelated medium. The simplest method isto refine the grid blocks by attributing for example thesame permeability to the refined grid blocks. This choiceis arbitrary, and probably any refinement procedure suchthat the overall equivalent permeability of the consideredgrid block is preserved should be equally acceptable(Romeu and Nœtinger 1995). One can note that this cri-terion is exactly the equivalence criterion that was pro-posed above. The quantity that is fixed is the apparentLn(k) variance, as measured at a length scale lmes greaterthan the lc whose precise value is unknown and notmeasurable with the available data. Using the asymptoticexpression (13), one has s2 Ln (kG) � C(r=0) (G lc

)D with(G=1/lmes). This means that in order to let lc tend to zero,another parameter must be changed to ensure that theobserved variance at the reference scale is kept fixed. Thesimplest solution is to assume that the product C(r=0)(lc)

D must remain constant, so C(r=0) varies as 1=lDc . This

result means simply that due to homogenization effects,an uncorrelated medium should exhibit heterogeneousbehavior at a finite scale only if the variance is infinite.This is simply a description of the so-called geostatistical“nugget” effect (Dagan 1989). In theoretical physics, thisstrategy corresponds to the asymptotic renormalizationprocedure of the coupling constants describing the parti-tion function of liquids at the critical point (Goldenfeld1992). The basic idea is that completely local values arenot measurable in practice (Cushman 1984). Any mea-surement is in essence a partial upscaling process, so whatis important is to recover the observed parameters, byadjusting the local values in order to match the observedones. It is probably in that sense that the famous LLMconjecture becomes exact.

This conjecture has a quite long history. Its simplicityand elegance inspired many theoretical studies. From amore practical point of view, Neuman and Orr (1993)obtained very good agreement with numerical resultseven for relatively large log permeability variance (up toseven), Nœtinger and Jacquin (1991) tested the formulawith real data and they got also an excellent agreement.

This conjecture is evidently exact for D=1, and Aba-bou (1994) showed that it was exact for D=1 at all ordersusing perturbation theory. In the 2D case, Matheron(1967b) proved its validity. In the general case, it is exactup to second order in a series expansion in powers ofC(r=0) (see Neuman and Orr 1993 and references there-in). Using the general perturbation series, Nœtinger(1994) showed that a partial summation of a whole per-turbation subseries provides the LLM result, and claimedthat the formula could be valid for a “vanishing correla-tion length” case, without giving at this time a precisemeaning to this term.

On the other hand, de Witt (1995) and later indepen-dently Indelman and Abramovich (1994, 1995), have

193

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

shown by explicit calculations that are both mathematicaltours de force that LLM conjecture is correct up to thenext term involving C(r=0)2, but that it is incorrect at thenext order. More precisely, these authors explicitlycomputed the third order term in C(r=0)3 involving in-teractions between 6 permeability fluctuations. They alsoshow that this term depends explicitly on the whole shapeof the covariance function C(r), and not only on the localvalue C(r=0). At first sight, this fact has a more severeconsequence: it implies that a single purely local aver-aging formula like LLM cannot exist at all, because non-local correlation effects must be accounted for. Stepa-nayants and Teodorovitch 2003 came to similar conclu-sions.

A major conceptual progress could be attained if it waspossible to show that LLM conjecture is asymptoticallycorrect using the procedure proposed above to let thecorrelation length lc tends to zero. The filtering approachis a first step towards this goal. This will allow one toreconcile the apparently contradictory results presentedpreviously. In addition, such results would be a goodstarting point to set-up more powerful perturbationmethods. It can be mentioned that a rather analogousphilosophy was followed by Koslov (1993) using ho-mogenization theory. He found a result similar to LLMfor multiscaled permeability maps. This is an additionalindication that the LLM formula is important, and reflectssome hidden properties.

This approach is being generalized in two cases, wherethe local permeability is an anisotropic tensor, and in thecase of geometrical anisotropies. In that case, kG(r) willbecome a full tensor kG(r). A first result was to obtain anexpression of the averaging exponent w as a function ofthe global anisotropy ratio a=(kv/kh)1/2 lh/lv, w(a). Usingsecond order perturbation theory, the present authors wereable to get an analytical determination of w(a), in veryclose agreement with results of numerical simulations(Duquerroix et al. 1993), and close to estimations pro-vided by other authors using analogous approaches (seee.g. Dagan 1989) Fig. 6:

wðaÞ ¼ arc tgap� arc tga

ð17Þ

Transposing the filtering approach to anisotropic me-dia leads to solving a set of integro differential equationsinvolving the whole permeability tensor having a formclose to (14). This work is still in progress.

This approach can be applied to categorical models.Suppose that the model is characterized by two differentrock types having two permeabilities k1 and k2. Geosta-tistical methods can be used to generate such maps. Thesemaps are not log normal at all, but it must remain possibleto upscale geostatistics directly. Consider now theequivalent permeability probability distribution at a pre-scribed scale L=1/G, which could be obtained by means ofnumerical simulations. If L is very small when comparedto lc, the original bimodal distribution is recovered. WhenL�lc, one can expect to get intermediate values lying inthe interval defined by the two permeabilities. Theirprecise distribution will depend on the details of theheterogeneity pattern of the considered medium. For verylarge L, once again an homogenization process shouldoccur, and the equivalent permeability probability densitydistribution should become continuous with a meantending to Keff, and a vanishing variance. Such a studycould be interesting, mainly to know if a limiting (lognormal?) distribution exists, and at which scale the dis-crete character of the original distribution is lost. Thiswould provide some more quantitative insights about thecharacteristic size of the so-called “representative ele-mentary volume” which is important to predict fluctua-tions. In addition, the interesting case of a large contrastk1/k2!0 would display interesting and non-trivial uni-versal critical effects at a percolation threshold dependingon the correlation structure of the medium (Stauffer1985).

Upscaling and inverse problems

Most modern approaches of computer-aided historymatching uses the Bayesian paradigm (Gavalas and Shah1976; Tarantola 1987; Hu et al. 1999a, b; le Ravalec et al.1999, 2001). Consider some observed data denoted col-lectively by the vector dobs(k). The vector k represents theunknowns (these can be either permeabilities, log per-meabilities, or porosities etc...). An approximation d(k)can be computed by means of a numerical simulator.

History matching a model is equivalent to searchingfor k*=arg Min J(k), where the objective function J(k) isFig. 7:

J kð Þ ¼ 1

2sd2 d kð Þ � dobs kð Þð Þ2

þ 12

Xi;j

ki � kprior;i�

C�1ij kj � kprior;j�

ð18Þ

In the Bayesian framework s2d is the variance of

measurements € the variance of physical and numericalmodeling errors, s2

d ¼ s2meas þ s2

mod. The matrix C�1ij is

Fig. 6 Variations of the averaging exponent w(a) as a function ofthe global anisotropy ratio a. A good agreement is observed be-tween different sets of curves and the theoretical prediction

194

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

the inverse of the covariance matrix of k, whose elementsare given by Cij ¼ Cðrj � riÞ ¼ ki � kih ið Þ kj � kj

� �� � �where ri is the position vector of the center of grid-blocki. In the gaussian case, and assuming that d(k) is linear,the vector kprior can be the a priori average hki, in thatcase the resulting k* will be the mean of the posteriordistribution, or it can be any kprior sampling the priordistribution: in that case k* will sample the posteriordistribution (Oliver et al. 1996).

In practice it is difficult to minimize J(k) because k isgenerally of high dimensions. In addition, the model d(k)represents the reservoir simulator with all the associatedcomplexities. Due to non-linearities of d(k), J(k) candisplay many local minima. Finally, when dealing with amultimodal variable, the minimization process becomesmore severe.

The filtering approach could help to improve theminimization of J(k). Consider now the minimization of Jusing filtering. A renormalized functional JG(k) can bedefined by:

JG kð Þ ¼ 1

2sd2 d kð Þ � dobs kð Þð Þ2

þ 12

Xi;j

ki � kprior;i Gð Þ�

C�1Gij kj � kprior;j Gð Þ�

ð19ÞHere, CGij=CG(rj-ri) is the covariance matrix of the

filtered permeabilities introduced in the last section. Theidea is to filter out the high frequencies of k, and tominimize J(k) from coarse scales to smaller and smallerscales. Here, kprior(G) represents the renormalized averagepermeability depending on the filtering parameter G. Notethat working directly with the Fourier components of k,minimizing JG(k) with respect to k is in fact a mini-mization with respect to the low frequency components.This means that the dimensionality of the problem islowered in the Fourier domain: here the intrinsic numberof unknowns is not the number of grid blocks, but thenumber of uncorrelated regions that are contained in thereservoir volume: a ratio (L/lc)

D, where L is a typicallengthscale of the reservoir, proportional to the number of

wave vectors q for which the correlation matrix C(q) isnot equal to zero.

Thus far, no explicit upscaling of the numerical modelwas performed, only the problem parameterization hasbeen simplified. The numerical model remains the same,and now what is needed is a drastic reduction in thenumber of minimization parameters. As stated before,since the model is now smoother, grid coarsening be-comes easier. It can be implemented to get the followingproblem: minimize with respect to k, J0G (k) defined by:

J0G kð Þ ¼ 1

2s2D Gð Þ dup kð Þ � dobs kð Þ

� 2

þ 12

Xi;j

ki � kprior;i Gð Þ�

C�1G ij kj � kprior;j Gð Þ�

ð20ÞHere, dup(k) represents the simulator working with a

coarsened grid, the variance of the numerical model isincluded in s2

D Gð Þ ¼ smeas2þ smod2 Gð Þ, where smod2 Gð Þincludes now upscaling errors. The full process is com-pleted: a restricted parameterization in the practical realdomain, as well as an upscaled reservoir model. Noticethat both processes remain quite uncoupled.

Some improvements could be added to this method:e.g. starting with a very low cut off G, and once theminimization has been performed, increasing G in order toadd new degrees of freedom representing a more detailedreservoir description and so on.

So far, this analysis has been restricted to Gaussianquantities: how could these ideas be adapted to treat bi-modal or multimodal parameter distributions?

This approach may be followed to examine uncer-tainties, by carefully studying the posterior distributions.

Another interesting option is to work with dup(k),keeping the fine description for the geological part of theobjective function.

Upscaling of transport equationsand non-linear problems

Transport equationsConsider the convection-dispersion equation governingthe time evolution of the concentration c(r, t) of a spike ofpassive tracer (see Marle 1981; de Marsily 1981):

f@c r; tð Þ@t

þr � c r; tð Þu r; tð Þð Þ ¼ r � d uð Þ � rc r; tð Þð Þ

ð21ÞSmall scale disorder is usually modelled by accountingfor fluctuations of the local velocity, u r; tð Þ ¼uh i r; tð Þ þ du r; tð Þ. The local dispersion tensor d(u) is

assumed to be known explicitly. The P�clet numberwhich compares the relative importance of transport byconvection and by dispersion is defined by Pe=Ulc/d(u).

These velocity fluctuations may in turn be due to theunderlying permeability disorder. The goal is to find the

Fig. 7 Schematic representation of the multiscale history matching.Each curve displays typical variations of the objective functionwith any degree of freedom. Strong filtering results in a verysmooth curve that can be easily minimized. Details can be added ifdesired by varying the filtering parameter G

195

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

form of the equation governing large-scale volume-aver-aged concentration C(r, t) or the ensemble averagedconcentration hc(r, t)i. It is possible to derive an equation,corresponding to the purely convective limit at infiniteP�clet number, even if the local dispersion tensor d(u) isequal to zero.

f@ ch i r; tð Þ

@tþr � ch i r; tð Þ uh i r; tð Þð Þ

¼ r � D uh ið Þ � r ch i r; tð Þð Þ ð22ÞA fundamental problem is to relate the large-scale

macrodispersivity tensor, D(hui), to the statistical prop-erties of the underlying disorder, and to know at whichscale this representation is valid. More detailed studies,using a power series expansion of the disorder magnitude,and Feynman graph reorganization of the series show thatD(hui) may be replaced by a memory kernel. Many au-thors, among which major contributions come from Gel-har (1993), Dagan (1989) and Adler (1992) computedD(hui) from the covariance structure of permeability.Each obtained a formula of the form

D uh ið Þ~a lcs2 u uh i uh i þ b lcs2u 1� uh i uh ið Þ;where uh i uh i denotes the dyadic tensor build on the av-erage normalized velocity uh iand a and b are constant oforder unity depending on the precise shape of the cova-riance function. Using a renormalization technique simi-lar to the one presented while discussing the filteringapproach, Jaekel and Vereecken (1997) proposed a nu-merical method allowing accounting for heterogeneitiesof large variance covering multiple lengthscales. A dis-persivity tensor D(hui L) can be computed which dependsexplicitly on the averaging scale L. Their method quan-tifies the observed variations of the apparent dispersivitywith the lengthscale. More complex heterogeneousreservoirs including multifractal cases were treated byGlimm et al. (1992) and Zhang (1992). In that case,anomalous diffusion may arise: the large-scale drivingequation can have a different form than (22). Dual po-rosity descriptions accounting for the rather contrastingdiffusion times can be advantageously used, Coats andSmith (1964), Cherblanc (1999), Wood et al. (2003) in thevery heterogeneous case.

For miscible flows, where the viscosity of the mixturedoes vary with concentration, the interaction between thetransport equation and the velocity field may induceviscous fingering phenomena. These fascinating phe-nomena are still the subject of considerable experimentalnumerical modeling and theoretical efforts. They arestrongly related to the upscaling of non-linear transportproblems to be discussed in the next section.

Non-linear problems

Background and basic equationsTwo phase flow problems have been intensively studiedin the oil industry, where the goal is to model water or gasinjection schemes to enhance oil recovery. In this cir-cumstance, upscaling is an ensemble of practical methodsallowing a coarser computational grid in order to decreasethe simulation time (Barker and Thibeau 1996). Theemergence of stochastic reservoir modeling techniquesrequires one to reconsider these previous upscalingmethods (Zhang et al. 1999). For example, when dealingwith an infinite number of equally likely reservoir mod-els, should the upscaling process lead to the same simu-lation results for each realization of the permeability field,or should it only estimate the uncertainty on the recoverycurve with respect to the uncertainty on the field and onthe production scenario?

In the simplest case, capillarity and gravity effects willbe neglected. The equations are (Marle 1981):

ui ¼K rð Þkri Sð Þ

mi� rp r; tð Þ ð23Þ

r � u ¼ �r � K rð Þl S r; tð Þð Þrp r; tð Þð Þ ¼ 0 ð24Þ

@S r; tð Þ@ t

þr � u r; tð Þf

j S r; tð Þð Þ�

¼ 0 ð25Þ

l Sð Þ ¼ kr1 Sð Þm1þ kr2 Sð Þ

m2ð26Þ

For 1D displacements, where water is displacing oilfrom left to right; the hyperbolic character of the satura-tion transport Eq. 25 yields sometimes to the formation of

Fig. 8 (a) Relative permeabilities; (b) fractional flow f(S); (c) saturation profiles

196

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

a shock front, i.e. a saturation discontinuity that propa-gates at the mean flow velocity, the value of the saturationjump Sf being determined by a Rankine-Hugoniot crite-rion, Marle (1981). This front is followed by a rarefactionwave (Fig. 8). Here, the object of the investigation will beto characterize the shape of the front when the same twophase flow occurs in a heterogeneous medium (Fig. 9).

Water-oil front structure in heterogeneous media

In the heterogeneous case, simulations of 2D water-oildisplacement show the existence of a well defined sharpfront described by its position x(y, t), corresponding to theshock position: S(x, y, t)=Sf (Fig. 10).

Following Saffman and Taylor (1958) who studied thestability of fronts, depending on the total mobility l(S)ratio jump at the front, two regimes are likely to appear:

– In a stable regime, where the front is slightly distortedby heterogeneities and appears to be statistically welldefined, its apparent thickness does not grow withtime.

– In an unstable regime, characterized by the existenceof viscous fingering, the fingers’ typical length in-

creases continuously, providing a thickness that varieslinearly with time.

This phenomena appears to be a consequence of thecoupling between Eqs. (24) and (25). This phenomenawas neglected by most authors attempting to generalizestochastic first order expansions of this problem. It wasthe case of Langlo and Espedal (1994), Zhang et al.(1999), Cvetkovic and Dagan (1996) and Dagan andCvetkovic (1996).

Alternative approaches avoiding this approximationwere proposed by Artus et al. (2004) and Nœtinger et al.(2004). Let the deviation of the position of the front bedenoted by dx(y, t), so dx(y, t)=x(y, t)-Ut. The goal is tofind a relation between dx(y, t) and the underlying het-erogeneities. Using analytical methods proposed by King(1989), and series perturbation expansions in powers ofsLog k, it is possible to derive the following equationdriving the Fourier normal modes dxqðtÞ ¼Rdxðy; tÞexp iq � y dy of the front:

@dxqðtÞ@t

¼ � uh iq Mf � 1Mf þ 1

dxqðtÞ þ hqðtÞ ð27Þ

The mobility ratio Mf is defined at the shock:

Mf ¼lð0ÞlðSf Þ

¼ m1kr2ðS ¼ 0Þm2kr1ðSf Þ

� 1þ m1kr2ðSf Þm2kr1ðSf Þ

� �1

ð28Þ

The first term in the right hand side (rhs) correspondsto the relaxation (or amplification) due to the viscouscoupling, and the second one, denoted by hq(t) is a ran-dom excitation source term describing the random effectof heterogeneities.

Notice that the variable developed in small perturba-tions is the quantity dx(y, t), related to the noise hq(t) bymeans of a first order relation, and not directly the satu-ration dS(r, t). Expanding this last variable in terms ofpermeability fluctuations is not possible, due to thepresence of the saturation shock that corresponds to adiscontinuity. This is the technical difficulty encounteredby many authors. Equation 27 is exact up to first order insLog k.

Coming back to Eq. 27, once the statistical properties(mean, covariance function) of the noise hq(t) are known,Dagan (1989), its explicit solution can be found analyti-cally. In the absence of the relaxation term, the equationdescribes the dynamics of a passive tracer leading to themacrodispersion Eq. 22 presented in the last section.

In the stable case (Mf>1), the solution of this equationis the result of a competition between the viscous relax-ation and the random excitation. The marginal case Mf=1corresponds to the passive tracer case. If Mf<1, the flow isinstable, and our developments are irrelevant, as strongnon-linearities must be accounted for to describe fullydeveloped viscous-fingering patterns. Koval like de-scriptions can be applied to some special regimes such asvertical equilibrium (Yortsos 1992).

Fig. 9 One example of a permeability map, and a water saturationmap

Fig. 10 Shape of the front at three different times in the isotropiccase

197

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

Statistical properties of the frontThe formal solution of Eq. 27 provides an estimation ofthe statistical properties of the front as a function of thepermeability covariance function C(r). Having hdx(y,t)i=0, and defining the long time front covariance as thelimit Cdx dx hð Þ ¼ Limt!1 dx y; tð Þ dx yþ h; tð Þh i, one canfind the relation between Cdxdx(h) and C(r).

Introducing its Fourier transform defined byCdx dx qð Þ ¼

RCdx dx hð Þ exp iq � h dh, gives:

Cdx dx qð Þ ¼Z

dqxhqhq

� �qx; qð Þ

Uh i2q2x þ A2q2

ð29Þ

A ¼ uh iMf � 1Mf þ 1

ð30Þ

The knowledge of the covariance of hq(t) providesCdx dx(h). Notice that the presence of the 1/q2 kernelimplies divergences. In particular, the variance Cdx dx(0)which allows us to estimate the front thickness is given byCdx dx 0ð Þ ¼ Limt!1 dx y; tð Þ2

D E:

Cdx dx 0ð Þ ¼ 14p2

Zdq dqx

hqhq

� �qx; qð Þ

Uh i2q2x þ A2q2

ð31Þ

In 2D, this integral is infra-red (q=0) divergent.In order to avoid this singularity, it is more convenientto compute the variogram gxðhÞ ¼ 1=2ð ÞLimt!1

dx yþ h; tð Þ � dx y; tð Þð Þ2D E

. The singularity behavior isrecovered studying the behavior when h!+1 of gx(h). Inthe stratified case :

gx hð Þ / Cy q ¼ 0ð Þ2A2

h; h! þ1 ð32Þ

In the 2D isotropic case:

gx hð Þ / Cy q ¼ 0ð Þ2A2

Log h; h! þ1 ð33Þ

In 3D, the integral (31) is convergent, so the varianceis well defined.

Monte Carlo testsA systematic Monte Carlo study was run for the stratifiedcase (strata parallel to mean flow direction); and forthe heterogeneous isotropic permeability case. In eachcases, one begins by considering a very stable situation(m1/m2=10). The variogram estimation is obtained through100 simulations of two phase flows in heterogeneous maps.

The stratified case shows the occurrence of a stationaryfront Fig. 11. This is compatible with the solutions ofEq. 27 in stratified media. Such stationary fronts wereobserved in laboratory experiments using miscible fluidsby Loggia et al. (1996). The associated variograms areplotted in Fig. 12.

In the stable stationary isotropic case, a limiting sta-tionary variogram emerges, meaning that a somewhatstationary large-scale effective description can be at-tained. An effective dispersion equation like 22, coupledwith a suitable renormalization of the convective term viaa modification of fractional flow function could be pro-

Fig. 11 Front displacement over time in the stratified case. Thefront becomes stable with time

Fig. 12 Variograms gx (h)with time in the stratified case,emergence of a limiting behav-ior

198

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

posed (Langlo and Espedal 1994; Lenormand and Thiele1996; Artus et al. 2004). In addition, the singularity (33)is qualitatively recovered. It should be interesting to ex-amine 3D cases, to test if the variance remains finite, butactual simulation capabilities remain strongly unsuffi-cient. Further theoretical and numerical studies areneeded to improve our description of these interestingcoupled problems Fig. 13.

Conclusions and future prospects

In this paper, an upscaling framework, which is similar tothe concepts of the renormalization group of statisticalphysics, can be at the heart of a global approach to fluidflow problems in stochastic heterogeneous media in-volving many time scales and length scales. Managingexisting data, and predicting uncertainties is essential forapplications, and practical concepts and tools must bedeveloped for operational engineers. In that sense, up-scaling techniques can be very helpful to set up aframework to reach these goals.

It helps to identify dominant data and phenomena thatwill control the large scale behavior of the flow. Multiplescale approaches, mixing stochastic and deterministicapproaches suggest promising research issues. In partic-ular, finding the optimal gridding scale to make the bal-ance between the resolved heterogeneities and the unre-solved ones is an open problem.

It seems essential to be able to manage the entire scalespectrum as a whole, and to be able to recover fine scaleinformation, by suitable down scaling procedures thatmust be consistent with the upscaling approach. Thisimplies developing numerical tools, such as multiscalefinite elements or double mesh methods allowing one tomanage that. The stochastic framework provides an ele-gant way to embed both averaging, upscaling and inverseproblems, and will probably become the dominant para-digm in a near future.

The non-linear case exhibits very exiting non-trivialfeatures such as the emergence of different flow regimes(stable or unstable) that are described at large scale byqualitatively different equations that can be used for thesmall scale ones. Finding if an adequate Eulerian for-malism can provide a correct description of stronglycoupled flows and heterogeneities is a very open andstimulating challenge.

References

Ababou R (1994) Solutions of stochastic groundwater flow by in-finite series, and convergence of the one-dimensional expan-sion. Stoch Hydrol Hydr 8:139–155

Adler PM (1992) Porous media: geometry and transports. Butter-worth/Heinemann

Ahmadi A, Labastie A, Quintard M (1993) Large scale propertiesfor flow through a stratified medium various approaches.SPERE pp 214–220

Ahmadi A, Quintard M (1996) Large scale properties for two phaseflow in random porous media. J Hydrol 183:69–99

Alabert FG (1989) Constraining description of randomly hetero-geneous reservoirs to pressure test data: A Monte-Carlo study,SPE 19600 SPE 64th Ann. Tech. Conf. and Exh. San Antonio,207–221

Arbogast T, Bryant SL (2001) Numerical subgrid upscaling forwaterflood simulations, SPE No. 66375, SPE Reservoir Simu-lation Symposium, Houston, 11—14 Febr 2001

Artus V, Nœtinger B, Ricard L (2004) Dynamics of the water oilfront for two-phase, immiscible flow in heterogeneous porousmedia. 1 Stratified Media. Trans Por Med 56:283–303

Avellaneda M (1996) Homogenization and renormalization, themathematics of multiple scale random media and turbulentdiffusion. Lect Appl Math 31:251–268

Badea A, Bourgeat A (1995) Homogenization of two phase flowthrough randomly heterogeneous porous media. Proceedings ofthe Conference on “Mathematical Modelling of Flow throughPorous Media”, St Etienne, Bourgeat, A

Badea A, Bourgeat A (1996). Homogenization of two-phase flowsin randomly heterogeneous porous media. Zeitschrift f rAngewandte Mathematik und Mechanik. ZAMM 76:(4)81–84

Barker JW, Thibeau SA (1996) A critical review of the use ofpseudo-relative permeabilities for upscaling SPE paper 35491,EUROPEC, Stavanger Norway

Fig. 13 Variograms gx (h) withtime in the statistically isotropiccase : emergence of a limitingbehavior

199

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

Blanc G, Gu�rillot D, Rahon D, Roggero F (1996) Building geo-statistical models constrained by dynamic data-a posterioriconstraint. spe 35478, European 3D Reservoir Modeling Con-ference, Stavanger, Norway, pp 19–33

Bourgeat A, Quintard M, Whitaker S (1988) El�ments de Com-paraison entre la m�thode d’homog�neisation et la Prise deMoyenne avec Fermeture. CR Acad Sci Paris 306(2):463–466

Bourgeat A (1996) Homogenization of two-phase flow throughporous media. Chapter 5 In: Hornung U (ed) Homogenizationand Porous Media. Interdisciplinary Applied Mathematics Se-ries, Springer, Berlin Heidelberg New York

Cherblanc F (1999) Etude du Transport Miscible en Milieu PoreuxH�t�rog�nes: Prise en Compte du Non Equilibre. [Study ofmiscible transport in heterogeneous porous media accounting ofnon equilibrium] Th�se de l’Universit� Bordeaux 1

Christakos G, Miller CT, Oliver DS (1993) Stochastic perturbationanalysis of groundwater flow. spatially variable soils, semiinfinite domains and large fluctuations. Stoch Hydrol Hydrauli7(3):213–239

Christakos G, Hristopulos, Miller CT (1995) Stochastic diagram-matic analysis of groundwater flow in heterogeneous porousmedia. Water Resour Res 31(7):1687–1703

Coats KH, Smith BD (1964) Dead-end pore volume and dispersionin porous media. SPE J 4:73–84

Cosentino L (2001) Integrated Reservoir Studies, Editions. Tech-nip, Paris

Cushman JH (1984) On unifying the concepts of scale, Instru-mentation, and stochastics in the development of multiphasetheory. Water Resour Res 20(11)1668–1676

Cushman JH, Ginn TR (1993) Non local dispersion in porous mediawith continuously evolving scale of heterogeneity. Trans Po-rous Med 13(1):123–138

Cushman JH, Bennethum LS, Hu BH (2002) A primer on upscalingtools for porous media. Adv Water Resour 25:1043–1067

Cvetkovic V, Dagan G (1996) Reactive transport and immiscibleflow in geological media. II-Applications. Proc Royal Soc Lon452:303–328

Dagan G, Cvetkovic V (1996) Reactive transport and immiscibleflow in geological media. I-General Theory. Proc Royal Societyof London 452:285–301

Dagan G (1989) Flow and transport in porous formations, Springer,Berlin Heidelberg New York

Daviau F (1986) Interpr�tation des essais de puits [Well test in-terpretation] Editions Technip, Paris

de Marsily G (1981) Hydrogeologie quantitative [QuantitativeHydrology] Masson

Desbarats AJ (1992) Spatial averaging of hydraulic conductivity in 3-dimensional heterogenous porous-media’. Math Geol 24:249–267

Deutsch CV, Journel AG (1992) Geostatistical software library anduser’s guide. Oxford University Press

de Witt A (1995) Correlation structure dependence of the effectivepermeability of heterogeneous porous media. Physics of Fluids7(11):2553–2562

Duquerroix JP, Lemouzy P, Nœtinger B, Kruel Romeu R (1993)Influence of the permeability anisotropy ratio on the large scaleproperties of heterogeneous reservoirs. SPE 26648 68 th An-nual Tech. Conf. and Exhb. of the SPE Houston 29–40

Durlovsky LJ (1991) Numerical calculation of equivalent gridblock permeability tensors for heterogeneous porous media.Water Resour Res 27:699–708

Eschard R, Doligez BD, Rahon D, Ravenne C, Leloch G (1991) Anew approach for reservoirs description and simulation usinggeostatistical methods, Advances in Reservoir Technology,Characterization Modeling and Management

Feitosa GS, Lifu Chu, Thompson LG, Reynolds AC (1993) De-termination of permeability distribution from well test pressuredata. SPE 26:407

Galli A, Gu�rillot D, Ravenne C, Heresim-Group (1990) Combin-ing geology, geostatistics and multiphase fluid flow for 3Dreservoir studies. In: Gu�rillot D, Guillon O (eds) ECMOR II -2nd European Conference on the Mathematics of Oil Recovery,Arles, Sep. 11–14, Proceedings, pp 11–19

Gautier Y, Nœtinger B (1997) Preferential flow-paths detection forheterogeneous reservoirs using a new renormalization tech-nique. Trans Por Med 26:1–23

Gautier Y, Nœtinger B (1998) Determination of geostatistical pa-rameters using well-test data, SPE 73rd ATCE, 27–30, New-Orleans, USA

Gautier Y, Blunt MJ, Christie MA (1999) Nested gridding andstreamline-based simulation for fast reservoir performanceprediction. Comput Geosci 3:295–320

Gavalas GR, Shah PC (1976) Reservoir history matching bybayesian estimation. Soc Pet Eng J 16(6):337–350

Gelhar LW (1993) Stochastic subsurface hydrology. Prentice HallGermano M (1992) Turbulence: the Filtering Approach. J Fluid

Mech 238:325–336Glimm J, Lindquist B, Pereira F, Peirls R (1992) The multifractal

hypothesis and anomalous diffusion. Math App Comput11(2):189–207

Goldenfeld N (1992) Lectures on Phase Transitions and theRenormalization Group, Frontiers in physics, Addison Wesley

Gomez-Hernandez J (2005) A Stochastic Approach to the Simu-lation of Block Conductivity Fields Conditioned upon DataMeasured at a Smaller Scale, PhD at Stanford University

Gorell S, Basset R (2001) Trends in reservoir simulation: bigmodels, scalable models? Will you please make up your mind?SPE 71 596, SPE ATCE New Orleans

Granjeon D, Joseph P, Doligez B (1998) Using a 3D stratigraphicmodel to optimise reservoir description. Hart’s Petro Eng Int,Nov, pp 51–58

Gu�rillot DR, Rudkiewicz JL, Ravenne C, Renard G, Galli A(1989) An integrated model for computer aided reservoir,description: from outcrop study to fluid flow simulations. Im-proved oil recovery. 5th European symposium, Budapest, April25–27, Proceedings, pp 651–660

Gu�rillot DR, Lemouzy P, Ravenne C (1991) How reservoircharacterization can help to improve production forecasts.Improved oil recovery. 6th european symposium, Stavanger,May 21–23 1991, Proceedings, pp 3–12

Gu�rillot DR, Verdi�re S (1995) Different pressure grids for res-ervoir simulation in heterogeneous reservoirs. SPE 29148, 1995SPE Symposium on Reservoir Simulation San Antonio

Haas A, Nœtinger B (1996) Stochastic reservoir modelling con-strained by well test permeabilities, Fifth International Geo-statistics Congress 22–27 Sept., Wollongong, Australia

Haas A, Nœtinger B (1995) 3D permeability averaging for sto-chastic reservoir modelling constrained by well tests, Reservoirdescription forum—The Heriot-Watt and Stanford Universi-ty—10–14 Sept. 1995, Puebles Hydro, UK

Heiba AA, Sahimi M, Scriven LH (1992) Percolation theory of twophase relative permeability SPE. Reservoir Evaluation 123:132

Hou TY, Wu XH (1997) A multiscale finite element method forelliptic problems in composite materials and porous media.J Comput Phys 134:169–189

Hou TY, Wu XH, Cai X (1999) Convergence of a multiscale finiteelement method for elliptic problems with rapidly oscillatingcoefficients. Math Comput 227:913–943

Hu LY, le Ravalec M, Roggero F, Blanc G, Nœtinger B, Haas A(1999a) An overview of the gradual deformation approach forconstraining Gaussian-related stochastic models to dynamicdata. EAGE/SPE International Symposium on PetroleumGeostatistics, 20–23 April 1999, Toulouse

Hu LY, le Ravalec M, Blanc G, Corre B, Haas A, Nœtinger B,Roggero F (1999b) Reducing uncertainties in production fore-casts by constraining geological modeling using dynamic data,74th Annual Technical Conference and Exhibition, 3–6 Oct.1999, Houston, USA, SPE 56703

Hunt AG (1998) Upscaling in subsurface transport using clusterstatistics of percolation. Trans Porous Med 30:177–198

Hunt AG (2001) Application of percolation theory to porous mediawith distributed local conductances. Adv Water Res 24:279–307

Indelman P, Abramovich B (1994) A higher order approximation toeffective conductivity in media of anisotropic random structure.Water Resour Res 30(6):1857–1864

200

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0

Indelman P, Abramovich B (1995) Effective permittivity of lognormal isotropic random media. J Phys A: Math Gen 28:693–700

Indelman P, (1996) Averaging of unsteady flows in heterogeneousmedia of stationary conductivity. J Fluid Mech 310:39–61

Jaekel U, Vereecken H (1997) Renormalization group analysis ofmacrodispersion in a directed random flow. Water Resour Res33(10):2287–2299

Jikov VV, Kozlov SM, Oleinik OA (1994) Homogenization ofDifferential Operators, Springer, Berlin Heidelberg New York

King P (1987) The use of field theoretic methods for the study offlow in heterogeneous porous medium. J Phys A.: Math Gen20:3935–3947

King P (1989) The use of renormalisation for calculating effectivepermeability. Trans Por Med 4:37–58

Koslov SM (1993) Central limit theorem for multiscaled perme-ability. Porous Media Meeting, Birkhauser, pp 19–33

Kretz VM, leRavalec M, Roggero F (2002) An integrated reservoircharacterization study matching production data and 4D seis-mic. SPE—Annual Technical Conference and Exhibition of theSociety of Petroleum Engineers, San Antonio, 29 Sep.–2 Oct.

Landau LD, Lifschitz EM (1960) Electrodynamics of continuousmedia. Pergamon, Oxford

Langlo P, Espedal (1994) Macrodispersion for two-phase, immis-cible flow in porous media. Adv Water Resour 17:217–316

Lenormand R, Thiele M (1996) Determining flow equations fromstochastic properties of a permeability field: the MHD model.SPE J pp 179–190

le Ravalec M, Hu LY, Nœtinger B (2001) Stochastic reservoirmodeling constrained to dynamic data: Local calibration andinference of structural parameters, SPE 68883, SPE J

le Ravalec M, Hu LY, Nœtinger B (1999) Stochastic reservoirmodel constrained to dynamic data: local calibration and in-ference of the structural parameters, 74th Annual TechnicalConference and Exhibition, 3-6 Oct. 1999, Houston, USA, SPE56556

le Ravalec M, Hu LY, Nœtinger B (2000) The FFT moving average(FFT-MA) method: an efficient tool for generating and condi-tioning Gaussian simulations. Math Geol 32(6)

Lesieur M, M�tais O (1996) New Trends in Large Eddy Simula-tions of Turbulence. Ann Rev Fluid Mech 28:45–82

Loggia D, Rakotomalala N, Salin D, Yortsos Y (1996) PhaseDiagram of Stable Miscible Displacements in Layered PorousMedia. Europhys Lett 36(2):105–110

Mallet JL (1997) Discrete Modeling for Natural Objects. J MathGeol 29(2):199–219

Marle CM (1981) Multiphase flow in porous media, TECHNIPMarle CM, Simandoux P, Pacsirszky J, Gaulier C (1967) Etude du

d�placement de fluides miscibles en milieu poreux stratifi�,’[Study of the displacement of miscible fluids in stratified po-rous media]. Revue de L’Institut Fran�ais du P�trole 22:272–294

Matheron G (1967a) El�ments pour une th�orie des milieux poreux[Elements of a theory of porous media] Masson, Paris

Matheron G (1967b) Composition des Perm�abilit�s en MilieuPoreux H�t�rog�ne: M�thode de Schwydler et R�gles dePond�ration, [Composing permeabilities in heterogeneous po-rous media. Swindler method and averaging rules]. Revue del’Institut Fran�ais du P�trole 22(3):443–466

Neuman SP, Orr S (1993) Prediction of steady state flow innonuniform geologic media by conditional moments: exact nonlocal formalism, effective conductivities and weak approxi-mation. Water Resour Res 29(2):341–364

Nœtinger B, Artus V, Ricard L (2004) Dynamics of the water oilfront for two-phase, immiscible flow in heterogeneous porousmedia. 2 Isotropic Media. Trans Por Med 56:305–328

Nœtinger B (2000) Computing the effective permeability of log-normal permeability fields using renormalization methods. C.R.Acad Des Sciences, Sciences de la Terre et des Plan�tes331:353–357

Nœtinger B, Gautier Y (1998) Use of the Fourier-Laplace trans-formation and of diagrammatical methods to interpret pumping

tests in heterogeneous reservoirs. Adv Water Resour 21:581–590

Nœtinger B (1994) The effective permeability of a heterogeneousporous medium. Transport in Porous Media 15:99–127

Nœtinger B, Haas A (1996) Permeability averaging for well-tests in3D stochastic reservoir models, SPE Annual Technical Con-ference and Exhibition, 6–9 Oct. 1996, Denver, USA, SPE36653

Nœtinger B, Jacquin C (1991) Experimental tests of a simplepermeability composition formula, 66th Annual TechnicalConference and Exhibition of the Society of Petroleum Engi-neers, 6–9 Oct. 1991, Dallas, USA, SPE 22841

Oliver DS (1989) The averaging process in permeability estimationfrom well test data, SPE 19845 SPE 64th Ann. Tech. Conf. andExh.

Oliver DS, He N, Reynolds AC (1996) Conditioning permeabilityto well test data. 5th ECMOR conf. proc. Leoben, pp 259–268

Pianelo L, Gu�rillot D, Gallou�t T (2000a) Inversion simultan�e desdonn�es sismiques et des donn�es de production. [Simulata-neous inversion of seismic data and production data] Oil andGas Science and Technology—Revue de l’IFP 55(3):235–248

Pianelo L, Gu�rillot D, Gallou�t T (2000b) Coupled inversion ofpermeability and acoustical impedance an outstanding dataintegration. ECMOR VII—7th European Conference on theMathematics of Oil Recovery, Baveno, Sept. 5–8 2000, Pro-ceedings, Paper M-16, 8p

Quintard M, Whitaker S (1994) Transport in ordered an disorderedporous media ii: generalized volume averaging. Trans PorousMed 14:179–206

Ram� M, Killough JE (1991) A new approach to the simulation offlows in highly heterogeneous porous media SPE 21247. SPESymposium on Reservoir Simulation, Anaheim

Renard Ph, de Marsily G (1997) Calculating equivalent perme-ability: a review. Adv Water Resour 20(5–6):253–378

Romeu RK, Nœtinger B (1995) Calculation of internodal trans-missibilities in finite-difference models of flow in heterege-neous media. Water Resour Res 31(4):943–959

Romeu RK, Lara AQ, Nœtinger B, Renard G (1996) Well testing inheterogeneous media: a general method to calculate the per-meability weighting function. Four Latin American and Car-ibbean Petroleum Engineering Conference, 23–26 April 1996,Port-of-Spain, Trinidad and Tobago

Saffman PG, Taylor G (1958) The penetration of a fluid into aporous medium or hele shaw cell Containing a more viscousfluid. Proc Royal Soc Lond A245:312–329

Schaaf TM, Mezghani M, Chavent G (2002) Direct conditioning offine-scale facies models to dynamic data by combining thegradual deformation and numerical upscaling techniques. Proc.8th European Conference on the Mathematics of Oil Recovery,Freiberg, Germany

Stauffer D (1985) Introduction to percolation theory, Taylor andFrancis, London, Philadelphia

Strebelle S, Journel A (2000) Sequential simulation drawingdtructures from training images. In: Kleingeld, Krige (eds)Geostatistics 2000 Cape Town, vol. 1

Stepanayants YA, Teodorovitch EV (2003) Effective hydraulicconductivity of a randomly heterogeneous porous medium.Water Resour Res 39(3):1065

Tarantola A (1987) Inverse problem theory. ElsevierWood BD, Cherblanc F, Quintard M, Whitaker S (2003) Volume

averaging for determining the effective dispersion tensor:Closure using periodic unit cells and comparison with ensembleaveraging. Water Resour Res 39(8)

Yortsos YC (1992) Analytical studies for processes at verticalequilibrium. SPE paper 26022

Zhang D, Li L, Tchelepi HA (1999) A stochastic formulation foruncertainty assessment of two phase flow in heterogeneousreservoirs, SPE paper 51930, SPE Reservoir Simulation Sym-posion, Houston, TX

Zhang Q (1992) A multiple scale theory of the anomalous mixinglength growth for tracer flow in heterogeneous porous media.J Stat Phys 505:485–501

201

Hydrogeol J (2005) 13:184–201 DOI 10.1007/s10040-004-0427-0