Advances in Water Resources 33 (2010) 241–256

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Advances in Water Resources

journal homepage: www.elsevier .com/ locate/advwatres

Bayesian uncertainty quantification for flows in heterogeneous porous mediausing reversible jump Markov chain Monte Carlo methods

A. Mondal a, Y. Efendiev b,*, B. Mallick a, A. Datta-Gupta c

a Department of Statistics, Texas A&M University, College Station, TX 77843, United Statesb Department of Mathematics, Texas A&M University, College Station, TX 77843, United Statesc Petroleum Engineering Department, Texas A&M University, College Station, TX 77843, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 June 2009Received in revised form 17 October 2009Accepted 24 October 2009Available online 15 January 2010

Keywords:Reversible jumpMCMCPorous mediaUpscalingMultiscaleFaciesLevel sets

0309-1708/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.advwatres.2009.10.010

* Corresponding author. Tel.: +1 979 845 1972.E-mail addresses: anirban@stat.tamu.edu (A. Mond

(Y. Efendiev), bmallick@stat.tamu.edu (B. Mallick),Datta-Gupta).

In this paper, we study the uncertainty quantification in inverse problems for flows in heterogeneous por-ous media. Reversible jump Markov chain Monte Carlo algorithms (MCMC) are used for hierarchical mod-eling of channelized permeability fields. Within each channel, the permeability is assumed to have a log-normal distribution. Uncertainty quantification in history matching is carried out hierarchically by con-structing geologic facies boundaries as well as permeability fields within each facies using dynamic datasuch as production data. The search with Metropolis–Hastings algorithm results in very low acceptancerate, and consequently, the computations are CPU demanding. To speed-up the computations, we use atwo-stage MCMC that utilizes upscaled models to screen the proposals. In our numerical results, weassume that the channels intersect the wells and the intersection locations are known. Our results showthat the proposed algorithms are capable of capturing the channel boundaries and describe the perme-ability variations within the channels using dynamic production history at the wells.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Subsurfaces are complex geological formations encompassing awide range of physical and chemical heterogeneities. The goal ofstochastic models is to characterize its different attributes suchas permeability, porosity, fluid saturation, and etc. Flow in the sub-surface is controlled by the connectivity of the extreme permeabil-ities (high and low) which are generally associated with geologicalpatterns that create preferential flow paths/barriers.

In many geologic environments, the distribution of subsurfaceproperties is primarily controlled by the location and distributionof distinct geologic facies with sharp contrasts in properties acrossfacies boundaries [42]. For example, in a fluvial setting, high per-meability channel sands are often embedded in a nearly imperme-able background causing the dominant fluid movement to berestricted within these channels. Under such conditions, the orien-tation of the channels and channel geometry determine the flowbehavior in the subsurface rather than the detailed variations inproperties within the channels. Traditional geostatistical tech-niques for subsurface characterization have typically relied on

ll rights reserved.

al), efendiev@math.tamu.edudatta.gupta@pe.tamu.edu (A.

variograms that are unable to reproduce the channel geometryand the facies architecture [2,6,13,26,28,29,31,36,37]. Variousother approaches have been applied for modeling facies, e.g., dis-crete Boolean or object-based models [21]. The success of these ob-ject-based models is heavily dependent on the parameters tospecify the object size, shapes, proportion and orientation.

Several authors have used the adjustment of paleochannelparameters as a mechanism to match the production data and up-date the facies models. This approach allows us to take advantageof the gradient-based inverse methods but is limited with respectto channel shapes and geometry. For example, Landa and Horne[30] used trigonometric functions to model the channel bound-aries. The channel boundaries were moved to match the dynamicresponse but were always kept parallel. This was generalized byBi et al. [5] to accommodate more flexible channel geometry. Thechannel shapes and orientations were specified using principaldirection, horizontal and vertical sinuosity of the channel and thewidth and aspect ratio of the channel. However, the use of geologicobjects restricted the ability to generate multiple facies architec-ture. The introduction of truncated plugaussian models allowedfor considerable flexibility in terms of facies textures and shapes[22]. The approach requires specification of at least two covariancemodels and truncation thresholds but allows for multiple faciesand a variety of facies association. The conditioning of these mod-els to dynamic data is again complicated by the discrete represen-

242 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

tation of the facies that makes the application of gradient-basedmethods difficult and often inefficient [32]. Recently geostatisticalmodels based on multipoint statistics have been proposed forreproduction of complex channel architectures. These methodsrely on training images that can be difficult to obtain. Also, currentmultipoint methods are well suited for subsurface characterizationusing static data only and do not allow for efficient integration ofdynamic data. A rigorous formalism for uncertainty quantificationis largely missing in all of the methods discussed above.

In this paper, our goal is twofold: (1) hierarchical modeling ofpermeability fields with channelized architecture; (2) efficientsampling of the posterior probability distribution with hierarchicalpriors. In hierarchical modeling, the permeability field is repre-sented by facies boundaries and variogram-based permeabilityfields within each facies. Typically, the parameters representing fa-cies boundaries are highly uncertain, particularly in the earlystages of subsurface characterization [7,13]. In a channel type envi-ronment, the channel sands may be observed at a few well loca-tions. The observations at the well locations can be used inconjunction with reversible jump Markov chain Monte Carlo meth-ods to construct the parameterization of the facies in a computa-tionally efficient manner. This is one of our objectives in thispaper. There are many plausible channel geometries that will sat-isfy the channel sand distribution, orientation and well intersec-tions. Thus, the stochastic models for channels will requirespecification of random variables that govern the channel principaldirection, its horizontal and vertical sinuosity, channel width tothickness ratio, etc. All these parameters have considerable uncer-tainty associated with them but will profoundly impact fluid flowin the subsurface. In this paper, the channel boundaries are repre-sented using piecewise linear functions – an approach capable ofreproducing a wide variety of channel geometry. The shape ofthe channel boundaries is updated with dynamic data usingreversible jump MCMC where the number of points representingthe channel boundaries is assumed to be unknown. In the revers-ible jump MCMC method the dimension of the parameter spaceis also taken to be random. Note that in a conventional MCMCmethod the dimension of the parameter space is fixed. This flexi-bility allow us to have a Birth or a Death Step at each iteration ofthe reversible jump MCMC method. In a Birth Step we add onemore point on the channel boundary and thus increase the dimen-sion by one where as in a Death Step we delete one point on theboundary and thus reduce the dimension by one. We can also havea jump step like in conventional MCMC methods at each iterationof the reversible jump MCMC method. Thus in reversible jumpMCMC method we can move the points along the horizontal direc-tions by having a Birth or a Death Step and we can move the pointsin vertical directions by having a jump step. In standard MCMCmethod we only have the jump step, so the points on the channelboundaries can only move in vertical directions. So the reversiblejump MCMC method is more flexible in spanning all possible chan-nel shapes and thus allows an efficient search in the uncertaintyspace.

Within each facies, a variogram-based permeability field isused. To represent variogram-based permeability fields, Karhun-en–Loéve expansion [34] is used. Karhunen–Loéve expansion al-lows significant reduction in the number of parameters forcorrelated permeability fields. This is very advantageous in historymatching because it allows to perform the search in a smallerparameter space. Because the permeability fields are independentwithin different channels, the uncertainty space is still quite large.The permeability field is further conditioned at the well locations.The conditioning can be performed within Karhunen–Loéveexpansion.

The sampling of the posterior is done using reversible jumpMetropolis–Hastings MCMC. Each proposal is screened by running

detailed fine-scale models. It turns out that the acceptance rate ofthis algorithm is very small. To speed-up the algorithm, we employtwo-stage MCMC [9,17–19,35,33] methods, where coarse-scalesimulations are used to screen the proposals. In this paper, we con-sider simple flow-based upscaling techniques by averaging thepermeability field within each channel. This gives very coarsedescription of the media. Our numerical results show that such up-scaled models can improve the acceptance rate by several folds. Inthis paper, we present the formulation of two-stage reversiblejump MCMC which differs from two-stage MCMC proposed earlierbecause of the associated birth and death processes. The accep-tance rate of two-stage MCMC is further improved by using mixedmultiscale finite element methods (MsFEM) for preconditioning ofreversible jump MCMC methods.

Numerical results are presented for two dimensional perme-ability fields. The channel boundaries are modeled with reversiblejump MCMC where the number of points is assumed to be un-known. Within each channel, the permeability field is character-ized by two-point correlation functions. We assume that thevalues of the permeabilities are known at the wells. In the numer-ical results, we use priors for the number of points at the channelinterfaces, the locations of these points, death and birth processes,and the permeability fields within each channel. The initial loca-tions of the interfaces are taken un-informative. For example, wetake the initial channel boundaries to be flat line segment, whilethe reference channel has substantial lateral variations. As for thepermeability within each channel, we take the initial permeabilityto be homogeneous permeabilities, while the reference permeabil-ities are chosen to be heterogeneous. Our numerical results showthat the proposed algorithm can adequately predict the boundariesof the channels. Our algorithm produces some small oscillationsalong the boundaries; however, the main features of the bound-aries are correctly predicted. The acceptance rate of reversiblejump MCMC is improved by screening the proposal with upscaledmodels and mixed multiscale finite element methods. In particular,an error model (cf. [24]) is constructed based on offline computa-tions of fine- and coarse-scale models to allow for the bias-correc-tion from coarse-scale models.

The paper is organized as follows: In next section, we state theproblem and present some preliminaries. In Section 3, the param-eterization of the permeability field and the sampling methods arediscussed. Finally, in Section 4, we present numerical results.

2. Problem statement and preliminaries

2.1. Fine model

In this section we briefly introduce the fine-scale model used inthe simulations. We consider two-phase flow in a subsurface for-mation (denoted by X) under the assumption that the displace-ment is dominated by viscous effects. For clarity of exposition,we neglect the effects of gravity, compressibility, and capillarypressure, although our proposed approach is independent of thechoice of physical mechanisms. Also, porosity will be consideredto be constant. The two phases will be referred to as water andoil (or a non-aqueous phase liquid), designated by subscripts wand o, respectively. We write Darcy’s law for each phase as follows:

v j ¼ �krjðSÞlj

krp; ð1Þ

where v j is the phase velocity, k is the permeability tensor, krj is therelative permeability to phase j ðj ¼ o;wÞ; S is the water saturation(volume fraction) and p is the pressure. In this work, a single setof relative permeability curves is used and k is taken to be a diago-nal tensor. Combining Darcy’s law with a statement of conservation

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 243

of mass allows us to express the governing equations in terms ofpressure and saturation equations:

r � ðkðSÞkrpÞ ¼ Q s; ð2Þ@S@tþ v � rf ðSÞ ¼ 0; ð3Þ

where k is the total mobility, Qs is a source term, f is the fractionalflux of water, and v is the total velocity, which are respectivelygiven by:

kðSÞ ¼ krwðSÞlw

þ kroðSÞlo

; ð4Þ

f ðSÞ ¼ krwðSÞ=lw

krwðSÞ=lw þ kroðSÞ=lo; ð5Þ

v ¼ vw þ vo ¼ �kðSÞk � rp: ð6Þ

The above descriptions are referred to as the fine-scale model of thetwo-phase flow problem.

2.2. Coarse models

Next, we briefly describe the coarse models used in the paper.

2.2.1. Single-phase flow upscaling of multi-phase flow and transportNext, we will briefly describe single-phase flow upscaling pro-

cedure for two-phase flow in heterogeneous porous media. Thistype of approaches for upscaling are discussed by many authors;see e.g., [4,10,14]. The main idea of this approach is to upscalethe absolute permeability field k on the coarse-grid (see Fig. 1),then solve the original system on the coarse-grid with upscaledpermeability field. Below, we will discuss briefly the upscaling ofabsolute permeability used in our simulations.

Consider the fine-scale permeability that is defined in the do-main with underlying fine-grid as shown in Fig. 1. On the samegraph we illustrate a coarse-scale partition of the domain. To calcu-late the upscaled permeability field at the coarse-level, we use thesolutions of local pressure equations. The main idea of the calcula-tion of a coarse-scale permeability is that it delivers the same aver-age response as that of the underlying fine-scale problem locally.For each coarse domain K, we solve the local problems

div ðkðxÞr/jÞ ¼ 0; ð7Þ

with some coarse-scale boundary conditions. Here kðxÞ denotes thefine-scale permeability field. A typical boundary condition is givenby /j ¼ 1 and /j ¼ 0 on the opposite sides along the direction ej

and no flow boundary conditions on all other sides. For these

Coarse−grid

K

Fig. 1. Schematic description of fine- and coarse-grids. Bold lines illustrate a coarse-scale

boundary conditions, the coarse-scale permeability tensor is givenby

ðk�ðxÞej; elÞ ¼1jKj

ZKðkðxÞr/jðxÞ; elÞdx; ð8Þ

where /j is the solution of (7) with prescribed boundary condi-tions. Various boundary condition can have some influence onthe accuracy of the calculations, including periodic, Dirichlet,etc. These issues have been discussed in [44]. In particular, fordetermining the coarse-scale permeability field one can choosethe local domains that are larger than the target coarse-block,K, for (7).

Once the upscaled absolute permeability is computed, the origi-nal equations are solved on the coarse-grid, without changing theform of relative permeability curves. This is an inexpensive calcu-lation, since the pressure update involves only solving the pressureequation on the coarse-grid, and one can take larger time step forsolving the transport equation. For example if the fine-grid iscoarsened 10 times in each direction, then this provides 100 timesspeed-up for each pressure update. As a result, the upscaling oftwo-phase flow based on absolute permeability upscaling providesmore than 100 times speed-up. For solving the saturation equation,we employ upwind finite volume method. Note that the upscalingof the saturation equation does not take into account subgrid ef-fects. These kinds of upscaling techniques in conjunction withthe upscaling of absolute permeability has been used in groundwa-ter applications (see e.g. [14–16]).

2.2.2. Mixed multiscale finite element methods (MsFEM) forcoarsening flow equations

In this section, we present multiscale finite element method forsolving the flow equation on the coarse-grid. This technique is sim-ilar to upscaling introduced earlier, except that instead of comput-ing effective properties, multiscale basis functions are calculated.These basis functions are coupled through a variational formula-tion of the problem. For multi-phase flow and transport simula-tions, the conservative fine-scale velocity is often needed. Forthis reason, mixed MsFEM is used [1,3,8].

First, we re-write the elliptic equation in the form

ðkkÞ�1v þrp ¼ 0;

divðvÞ ¼ Q s:ð9Þ

In mixed multiscale finite element methods, the basis functions forthe velocity field, v ¼ �krp, are needed. The basis functions for thevelocity in each coarse-block K are given by

Fine−grid

partitioning, while thin lines show a fine-scale partitioning within coarse-grid cells.

coarse grid block

n =1/|e|

ψ

ψψ. n =1/|e|

.

n =0ψ .

n =0ψ .

div( )=1/|K|

ψ n =0

=k grad( )φ

Fig. 2. Schematic description of a velocity basis function construction in a coarse-grid block.

244 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

div kr/Ki

� �¼ 1jKj in K;

kr/Ki � n ¼

gKi on eK

i

0 else;

( ð10Þ

where gKi ¼ 1

jeKij and eK

i are the edges of K (see Fig. 2). Note that these

basis functions are defined for each edge by imposing constant fluxalong an edge (constant Neumann boundary condition) and zeroflux over all other edges of the coarse-grid block. In order to pre-serve the total mass and have well-posed system, some source termis needed. The source term is taken to be constant.

We define the finite dimensional space for the velocity by

Vh ¼ spanfwKi g;

where wKi ¼ kr/K

i . For each edge ei, one can combine the basis func-tions in adjacent coarse-grid blocks and obtain the basis functionfor the edge ei denoted by wi (or wei

). More precisely, if we denoteby K1 and K2 adjacent coarse-grid blocks, then wi solves (10) in K1

and solves div ðwiÞ ¼ � 1jK2 j

in K2, and gK2i ¼ � 1

jei jK2on eK2

i and 0 other-

wise. In other words, wi ¼ wK1i in K1 and wi ¼ �wK2

i in K2, where wKi is

defined via the solution of (10).The basis functions for the pressure are piecewise constant

functions over each K. We denote the span of these basis functionsby Ph. The multiscale basis functions attempt to capture the smallscale information of the media. The functions wK

i are the basis func-tions for the velocity field and conservative both on the fine andcoarse-grids provided the local problems are solved using a conser-vative scheme. An approximation of the fine-scale velocity fieldcan be obtained if average velocities along the coarse edges areknown, i.e., if ve is the average normal flux along the edge e andwe is the corresponding basis functions, then v �

Pevewe is an

approximation of the fine-scale velocity field. The mixed finite ele-ment framework, presented next, couples the velocity and pressurebasis functions and provides an approximation of the global solu-tion (both p and v).

To formulate the mixed MsFEM, we use the numerical approx-imation associated with the lowest order Raviart–Thomas mixedfinite element to find fvh; phg 2Vh � Q h such that vh � n ¼ ghon@X, where gh ¼ g0;h � n on @X and g0;h ¼

Pe2 @K

T@X;K2Thf g

Re gds

� �we;we 2Vh is the corresponding basis function corresponding toedge e,Z

XðkkÞ�1vh �wh dx�

ZX

div ðwhÞph dx ¼ 0; 8wh 2V0hZ

div ðvhÞqh dx ¼Z

fqh dx; 8qh 2 Ph; ð11Þ

X X

where V0h is a subspace of Vh with elements that satisfy a homoge-

neous Neumann boundary conditions. The above formulation wasthe mixed MsFEM introduced in [8].

In our simulations, the multiscale basis functions are computedfor the velocity once with k ¼ 1. These basis functions are used la-ter without any update for solving two-phase flow equations. As aresult, we obtain coarse-scale velocity field that is used for solvingthe transport equation on the coarse-grid. We note that mixedMsFEM can be implemented on unstructured grids [20].

2.3. The sampling problem

Our goal in this paper, is to sample fine-scale permeability fieldbased on fractional flow, specifically, the fraction of water pro-duced in relation to the total production rate. For two-phasewater–oil flow, the fractional flow or water-cut FðtÞ (denoted sim-ply by F in further discussion) is defined as the fraction of water inthe produced fluid and is given by qw=qt , where qt ¼ qo þ qw, withqo and qw the flow rates of oil and water at the production edge ofthe model,

FðtÞ ¼R@Xout vnf ðSÞdlFR

@Xout vndlF ; ð12Þ

where @Xout is outflow boundary and vn is normal velocity field. Ourobjective is to sample from the conditional probability distributionPðkjFÞ that will be discussed in a later section. Typically, the coarse-and fine-scale water-cut curves can be different. However, withinthe sampling approach, the strong correlation between fine- andcoarse-scale water-cut curves plays an important role.

3. Sampling with hierarchical models

3.1. Permeability parameterization

3.1.1. Parameterization of interfaces with level setsThe permeability field is decomposed into subregions where

each region represent a facies. Within each facies, the permeabilityfield will be populated using log-Gaussian fields which are de-scribed with the covariance matrix (see Fig. 3 for illustration). Thistype of a hierarchical representation of the permeability field al-lows us to write the following expression

kðxÞ ¼X

i

IDiðxÞkiðxÞ; ð13Þ

where ID is an indicator function of the region D (i.e., IðxÞ ¼ 1 if x 2 Dand IðxÞ ¼ 0 otherwise).

In this paper, we seek the boundaries of the facies using adap-tive representation. More precisely, level set functions s represent-

Fig. 3. Illustration of the permeability field with facies.

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 245

ing the facies boundaries are defined such that s ¼ si for differentinterfaces. For the update of the facies, the level set equations (e.g.,[38,40]) will be used. More precisely, we assume

@s@sþw � rs ¼ 0; ð14Þ

where w is a vector field and s is a pseudo-time. Eq. (14) is used forthe update of the interface. This equation is a linear transport equa-tion where one needs to specify the velocity field w. We take w to bea random divergence-free field with a deterministic flow direction.One can use random forcing instead of random velocity. Because theflow direction is deterministic, Eq. (14) will be solved using stream-line approaches (e.g., [12]). Streamline approaches reduce (14) intoODE along the characteristic of flow directions. In particular, thestreamlines are defined by

dls

dsx¼ w;

where sx is the spatial coordinates along the streamlines. In our sim-ulations, vertical streamlines are used. If we denote by s the pointsof the interface, then the update of these points will be given by

snþ1ðlsðxÞÞ � snðlsðxÞÞ ¼ Rdsx;

where R is a random variable. The equation above is a physicalinterpretation of the instrumental proposal distribution for s usedon subsequent algorithms.

In one dimensional case, the interface is a point and there is noneed to define level set surfaces. In this case, the motion of theinterface is described by the ODE

@x@s¼ const:;

where x describes the location of the interface.Next, we discuss the parameterization of kiðxÞ within each

facies.

3.1.2. Parameterization of the permeability within faciesOne of the commonly used stochastic descriptions of spatial

fields is based on a two-point correlation function of log perme-ability. To describe it, we denote by Yðx;xÞ ¼ log½kðx;xÞ�. For per-meability fields described with a two-point correlation function, itis assumed that Rðx; yÞ ¼ E½Yðx;xÞYðy;xÞ� is known, where E½�� re-fers to the expectation (i.e., average over all realizations) and x; yare points in the spatial domain. In applications, the permeabilityfields are considered to be defined on a discrete grid. In this case,Rðx; yÞ is a square matrix with Ndof rows and Ndof columns, whereNdof is the number of grid blocks in the domain. For permeabilityfields described by a two-point correlation function, one can usethe Karhunen–Loève expansion (KLE) [43] to obtain permeabilityfield description with possibly fewer degrees of freedom. This isdone by representing the permeability field in terms of an optimalL2 basis. By truncating the expansion, we can represent the perme-ability matrix by a small number of random parameters.

We briefly recall some properties of the KLE. For simplicity, weassume that E½Yðx;xÞ� ¼ 0. Suppose Yðx;xÞ is a second order sto-chastic process with E

RX Y2ðx;xÞdx <1. Given an orthonormal

basis fUig in L2ðXÞ, we can expand Yðx;xÞ as a general Fourierseries

Yðx;xÞ ¼X

i

Y iðxÞUiðxÞ; YiðxÞ ¼Z

XYðx;xÞUiðxÞdx:

We are interested in the special L2 basis fUig which makes the ran-dom variables Yi uncorrelated. That is, EðYiYjÞ ¼ 0 for all i – j. Thebasis functions fUig satisfy

E½YiYj� ¼Z

XUiðxÞdx

ZX

Rðx; yÞUjðyÞdy ¼ 0; i – j:

Since fUig is a complete basis in L2ðXÞ, it follows that UiðxÞ areeigenfunctions of Rðx; yÞ:Z

XRðx; yÞUiðyÞdy ¼ kiUiðxÞ; i ¼ 1;2; . . . ; ð15Þ

where ki ¼ E½Y2i � > 0. Furthermore, we have

Rðx; yÞ ¼X

i

kiUiðxÞUiðyÞ: ð16Þ

Denote hi ¼ Yi=ffiffiffiffikip

, then hi satisfy EðhiÞ ¼ 0 and EðhihjÞ ¼ dij. It fol-lows that

Yðx;xÞ ¼X

i

ffiffiffiffiki

phiðxÞUiðxÞ; ð17Þ

where Ui and ki satisfy (15). We assume that the eigenvalues ki areordered as k1 P k2 P � � �. The expansion (17) is called the Karhun-en–Loève expansion. In the KLE (17), the L2 basis functions UiðxÞare deterministic and resolve the spatial dependence of the perme-ability field. The randomness is represented by the scalar randomvariables hi. After we discretize the domain X by a rectangularmesh, the continuous KLE (17) is reduced to finite terms and UiðxÞare discrete fields. Generally, we only need to keep the leading or-der terms (quantified by the magnitude of ki) and still capture mostof the energy of the stochastic process Yðx;xÞ. For an NKL-term KLEapproximation YNKL ¼

PNKLi¼1

ffiffiffiffikip

hiUi, define the energy ratio of theapproximation as

eðNKLÞ :¼ EkYNKLk2

EkYk2 ¼PNKL

i¼1kiP1i¼1ki

:

If ki; i ¼ 1;2; . . . ; decay very fast, then the truncated KLE would be agood approximation of the stochastic process in the L2 sense.

In our numerical examples, we will use log-normal fields,though the method is not restricted to this particular covariancestructure. Rðx; yÞ in this case is defined as

Rðx; yÞ ¼ r2 exp � jx1 � y1j2

2l21

� jx2 � y2j2

2l22

!; ð18Þ

l1 and l2 are the correlation lengths in each dimension, andr2 ¼ EðY2Þ is the variance.

In the numerical experiments, we first generate a reference per-meability field using all eigenvectors and compute the correspond-ing fractional flows. To propose permeability fields from the prior(unconditioned) distribution, we keep M terms in the KLE. Supposethe permeability field is known at MH distinct points. This condi-tion is imposed by setting

XMH

k¼1

ffiffiffiffiffikk

phk/kðxjÞ ¼ aj; ð19Þ

where ajðj ¼ 1; . . . ;MHÞ are prescribed constants. In this system, weidentify MH unknowns for which the system will be solved bychoosing the rest of M �MHh’s normally distributed. These un-knowns are found by searching all MH �MH minors of M �MH

matrices with the best condition number. Here M must be chosensuch that MH is less than M.

3.2. Reversible jump MCMC

Our main objective is to sample the permeability field givenfractional flow measurements. We also incorporate the informa-tion that the permeability field is known at some spatial locationscorresponding to wells. The fractional flow is an integrated re-

246 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

sponse and the map from permeability field to the fractional flowis not one-to-one. So there may exist many different permeabilityrealizations for a given production data. The measured fractionalflow or water-cut data F contains measurement errors. For a givenpermeability field k, we denote the fractional flow as Fk. Fk can becomputed by solving the model Eqs. (1)–(3) on the fine-grid. Thecomputed Fk will contain both modeling error and measurementerror. Assuming the combined error as a random error � we canwrite the model as

F ¼ Fk þ �; ð20Þ

where � is distributed as Nð0;r2f IÞ. i.e., PðFjkÞ is assumed to be

NðFk;r2f IÞ. From the parameterization of interfaces with level sets

we can say that the permeability field k is completely known givenh and s. Our goal is to generate the permeability field given thewater-cut data Fobs. Using Bayes’ theorem the posterior distributioncan be written as

pðkÞ ¼ PðkjFobsÞ / PðFobsjkÞPðkÞ ¼ PðFobsjkÞPððh; sÞ0Þ¼ PðFobsjkÞPðhÞPðsÞ½since h and s are independent�: ð21Þ

We note that the hierarchical structure of the model is due to faciesand a permeability distribution with each facies. One can show thatthe posterior measure is continuous with respect to the data in thetotal variation distance (following [11,27]). We assume that thepermeability field is given on a finite grid. This assumption is prac-tical because permeability field is not defined on very small scales(e.g., pore scale). To show the continuity of the posterior with re-spect to data, we define

pFobsðkÞ ¼ 1

Zexp

�kFobs � Fkk2

2r2f

!p0ðkÞ; ð22Þ

where kFobs � Fkk2 ¼Pn

i¼1ðFobsðtiÞ � FkðtiÞÞ2. Here ti are time instantswhen water-cut is observed and FkðtÞ is the water-cut curve (see(12)) for permeability k. One can also write this expression in termsof k ¼ ðh; mÞ. Here,

Z ¼Z

exp�kFobs � Fkk2

2r2f

!p0ðkÞdk

¼ZZ

exp�kFobs � Fkk2

2r2f

!p0ðhÞp0ðmÞdhdm: ð23Þ

Then, one can show that for all r > 0, there exists C ¼ CðrÞ such thatthe posterior measures p1 and p2 for two different data sets F1

obs andF2

obs with kF1obs _ F1

obskl2 6 r satisfy

kp1�p2kTV ¼Z

Z�11 exp

�kF1obs� Fkk2

2r2f

!

� Z�12 exp

�kF2obs� Fkk2

2r2f

!!p0ðkÞdk6 CkF1

obs� F2obskl2

;

ð24Þ

where Z1 and Z2 are defined by (23) for F1obs and F2

obs, respectively.The proof is given in the Appendix B.

If the dimension of the parameters s and h is fixed then we canuse standard Metropolis–Hastings algorithm to sample from theposterior distribution PðkjFÞ.

The algorithm is as follows.

Algorithm. Metropolis–Hastings MCMC [39]Suppose at the nth step we are at sn; hn and permeability field kn.

� Step 1. Generate s from a distribution qsðsjsnÞ and h from a dis-tribution qhðhjhnÞ. Then the generated permeability field withineach facies is given by logðkðxÞÞ ¼

PMi¼1

ffiffiffiffikip

hiUiðxÞ with the con-

straints as given in (19), where the covariance function is ofthe same type for each facies as given in (18) (but the correlationlengths can be different for different facies). The entire perme-ability field is proposed using (13).

� Step 2. Accept k as a sample with probability

aðkn; kÞ ¼min 1;pðkÞqðknjkÞpðknÞqðkjknÞ

� �

¼min 1;PðFobsjkÞPðFobsjknÞ|fflfflfflfflfflffl{zfflfflfflfflfflffl}

likelihood ratio

� PðsÞPðhÞPðsnÞPðhnÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

prior ratio

� qsðsnjsÞqhðhnjhÞqsðsjsnÞqhðhjhnÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

proposal ratio

8>>><>>>:9>>>=>>>;ð25Þ

i.e., take knþ1 ¼ k with probability aðkn; kÞ, and knþ1 ¼ kn with prob-ability 1� aðkn; kÞ.

Starting with an initial permeability sample k0, the MCMC algo-rithm generates a Markov chain fkng with the transition kernel as

Krðkn; kÞ ¼ aðkn; kÞqðkjknÞ þ 1�Z

aðkn; kÞqðkjknÞdk�

dkn ðkÞ

¼ aðkn; kÞqsðsjsnÞqhðhjhnÞ

þ 1�Z

aðkn; kÞqsðsjsnÞqhðhjhnÞdsdh

� dsn ðsÞdhn ðhÞ:

The target distribution pðkÞ is the stationary distribution of theMarkov chain kn, so kn represent the samples generated from pðkÞafter the chain converges and reaches a steady state.

As an example we can use standard random walk Metropolis–Hastings algorithm to generate samples from the posterior distri-bution. Then at the nth step, we propose s ¼ sn þ hsus, where us

is generated from a Nð0; IÞ distribution. Similarly, we proposeh ¼ hn þ hhuh, where uh is also generated from a Nð0; IÞ distribution.Here hs and hh represent the step size of the jump in each step ofthe Metropolis–Hastings algorithm. The values of hs and hh controlthe convergence of the MCMC algorithm. The prior distribution of scan be taken to be Nðso;r2

sIÞ. Similarly, the prior distribution of hcan be taken to be Nðho;r2

h IÞ. Then our acceptance probability isgiven by

aðkn;kÞ

¼min 1;exp �kFobs�Fkk2

2r2f

� exp �kFobs�Fkn k

2

2r2f

� |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

likelihood ratio

�exp �kh�hok2

2r2h

þ�ks�sok2

2r2s

�exp �khn�hok2

2r2h

þ�ksn�sok2

2r2s

�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

prior ratio

� 1|{z}proposal ratio

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;:

ð26Þ

One of the main disadvantages of using direct Metropolis–Hastingsalgorithm is the dimension of s is fixed, so the number of the pointswhich determines the interfaces and their positions in the horizon-tal direction remains fixed. The points only jump at fixed positionsgiving different s’s and hence the resulting interfaces may not cap-ture the actual boundaries of the facies very well. If the number ofthe points and their positions are allowed to vary along with thejumps in each step then the resulting algorithm would offer moreflexibility in terms of channel shapes and smoothness of the chan-nel boundaries. If we vary the number of points that determines theinterfaces then the dimension of s will also change in each step.This jumping between different dimensions in the parameter spacecan be achieved through reversible jump Markov chain Monte Carlo

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 247

methods as proposed in [25]. First, we briefly discuss a general ap-proach of the reversible jump MCMC [41].

3.2.1. Brief introduction to reversible jump MCMCSuppose we want to generate from a target distribution pðzÞ,

where z ¼ ðy;mÞ. Here m, also a random variable, is the dimensionof the random vector x. Suppose m 2 A;A ¼ f1;2;3 . . . ; Ig. Herewe assume I to be a finite integer. Let m has probability distribu-tion PðmÞ and y has probability distribution PðyjmÞ. Then we canwrite

pðzÞ ¼ PðyjmÞPðmÞ: ð27Þ

We would like to construct a reversible Markov chain fzng whichhas stationary distribution pðzÞ. Suppose at the nth step we are atzn ¼ ðyn;mnÞ. Then, we propose a new m with probability pðmjmnÞ,where

Pm2ApðmjmnÞ ¼ 1. Given m, we generate u as random vector

of dimension dmnm from qmnmðujynÞ and propose y ¼ g1mnmðyn;uÞ.

Here g1mn m: Rmnþdmnm ! Rm is a deterministic mapping. There are

two crucial conditions for such a step from ðyn;mnÞ toðy;mÞ ¼ ðg1mnm

ðyn;uÞ;mÞ and the reverse move from ðy;mÞ toðyn;mnÞ ¼ ðg1mmn

ðy;u0Þ;mnÞ.

� Condition 1.

mn þ dmnm ¼ mþ dmmn ð28Þ

i.e., the dimension of the proposal random variables ðyn;uÞ andðy;u0Þ must be equal.

� Condition 2. There exist functions g2mnm: Rmnþdmnm ! Rdmmn and

g2mmn: Rmþdmmn ! Rdmnm , such that the mapping gmnm given by

ðy; u0Þ ¼ gmnmðyn;uÞ ¼ ðg1mnmðyn;uÞ; g2mnm

ðyn; uÞÞ ð29Þ

is one-to-one with

ðyn;uÞ ¼ g�1mnmðy;u0Þ ¼ gmmn

ðy;u0Þ ¼ ðg1mmnðy; u0Þ; g2mmn

ðy;u0ÞÞð30Þ

and gmnm is differentiable.

We accept z ¼ ðy;mÞ with probability

amnmðyn;yÞ ¼min 1;pðzÞpðmnjmÞqmmn

ðu0jyÞpðznÞpðmjmnÞqmnmðujynÞ

@gmnmðyn;uÞ@yn@u

���� ����� �¼min 1;

PðyjmÞPðmÞpðmnjmÞqmmnðu0jyÞ

PðynjmnÞPðmnÞpðmjmnÞqmnmðujynÞ@gmnmðyn;uÞ

@yn@u

���� ����� �;

ð31Þ

i.e., take ðynþ1;mnþ1Þ ¼ ðy;mÞ with probability amnmðyn; yÞ, andðynþ1;mnþ1Þ ¼ ðyn;mnÞ with probability 1� amnmðyn; yÞ.

In our case using the hierarchical Bayes’ model, the posteriorcan be written as

PðkjFobsÞ / PðFobsjkÞPðhÞPðsÞ ¼ PðFobsjkÞPðhÞPðsjxloc;mÞPðxlocjmÞPðmÞ:ð32Þ

Here m is the number of points considered and xloc denotes the loca-tions of those points in the horizontal direction that determine theinterfaces. We use the reversible jump process as a birth and deathprocess. At each step either we add a new point or delete a point orconsider only jumps at fixed positions. The dimension of s may varyin each step but the dimension of h is always the same.

The algorithm is as follows.

Algorithm. Reversible Jump MCMC as Birth and Death Process[25]Suppose at the nth step we are at sn; hn; xloc

n ;mn and permeabilityfield kn. Here xloc

n is a mn dimensional vector ðxlocn;1; x

locn;2; . . . ; xloc

n;mnÞ0.

We have three options: add a point with probability paddmn

; delete apoint with probability pdel

mn; or just propose a new h and s and thus

have a jump step with the locations remained fixed with proba-bility pj

mn , where paddmnþ pdel

mnþ pj

mn ¼ 1; 8 mn.

� Birth Step. Here we add one point from the remaining pointsand the proposed m is mn þ 1 with probability pðmjmnÞ ¼ padd

mn.

So in this case, dmnm ¼ 1 and dmmn ¼ 0. We generate u fromqðujxloc

n Þ and the proposed locations are given byxloc ¼ ðxloc

1 ; xloc2 ; . . . ; xloc

mnþ1Þ0 ¼ g1mnm

ðxlocn ;uÞ ¼ gmnmðxloc

n ;uÞ, whereg1mnm

ðxlocn ;uÞ is a deterministic function. The proposed h is gener-

ated from qhðhjhnÞ. The acceptance probability is given by

abmnmnþ1ðxloc

n ;xlocÞ

¼min 1;PðFobsjkÞPðFobsjknÞ|fflfflfflfflfflffl{zfflfflfflfflfflffl}

likelihood ratio

�PðhÞPðsjxlocÞPðxlocjmnþ1ÞPðmnþ1ÞPðhnÞPðsnjxloc

n ÞPðxlocn jmnÞPðmnÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

prior ratio

8>>><>>>:

�qhðhnjhÞpdel

mnþ1

qhðhjhnÞpaddmn

qmnmnþ1ðujxlocn Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

proposal ratio

�@gmnmðxloc

n ;uÞ@xloc

n @u

���� ����|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Jacobian

9>>>=>>>;: ð33Þ

� Death Step. Here we delete one point from the existing pointsand the proposed m is mn � 1 with probability pðmjmnÞ ¼ pdel

mn.

So here dmnm ¼ 0 and dmmn ¼ 1. The proposed locations are givenby the function ðxloc;u0Þ ¼ ðxloc

1 ; xloc2 ; . . . ; xloc

mn�1;u0Þ ¼ gmnmðxloc

n Þ ¼ðg1mnm

ðxlocn Þ; g2mnm

ðxlocn Þ and xloc

n ¼ g�1mnmðxloc;u0Þ ¼ g1mnm

ðxloc;u0Þ. Theproposed h is generated from qhðhjhnÞ. The acceptance probabil-ity is given by

admnmn�1ðxloc

n ; xlocÞ

¼min 1;PðFobsjkÞPðFobsjknÞ|fflfflfflfflfflffl{zfflfflfflfflfflffl}

likelihood ratio

� PðhÞPðsjxlocÞPðxlocjmn � 1ÞPðmn � 1ÞPðhnÞPðsnjxloc

n ÞPðmnÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}prior ratio

8>>><>>>:�

qhðhnjhÞpaddmn�1qmn�1mn

ðu0jxlocÞqhðhjhnÞpdel

mn|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}proposal ratio

�@gmnmðxloc

n Þ@xloc

n

���� ����|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}Jacobian

9>>>=>>>;: ð34Þ

� Jumps at Fixed Location. Here the number of points and theirlocations in horizontal direction are fixed, so the algorithm issame as Metropolis–Hastings Algorithm as described before.The acceptance probability is given by

aðkn; kÞ ¼min 1;PðFobsjkÞPðFobsjkÞ|fflfflfflfflffl{zfflfflfflfflffl}

likelihood ratio

� PðsÞPðhÞPðsnÞPðhnÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

prior ratio

� qsðsnjsÞqhðhnjhÞqsðsjsnÞqhðhjhnÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

proposal ratio

8>>><>>>:9>>>=>>>;:ð35Þ

3.2.2. An exampleAs an example, suppose we have a N � N fine-grid perme-

ability field and let the prior distribution for h be Nðho;r2h Þ

and that of s to be Nðso;r2sÞ. The prior distribution of m is taken

to be discrete uniform distribution, i.e., PðmÞ ¼ 1ðmmax�mminþ1Þ ;

m ¼ mminð1Þmmax. Given m, the prior distribution of the locations

xloc is given by PðxlocjmÞ ¼ 1ðN�2ÞCðm�2Þ

, i.e., ðxlocð2Þ; x

locð3Þ; . . . ; xðm�1ÞÞloc0 are

distributed as order statistics of a sample of size ðm� 2Þ drawnwithout replacement from a population of size N � 2. Note that,here we have N � 2 instead of N because the two ends of eachof the interfaces are fixed and known and are assumed to bethe channel intersections at the wells. Given m and xloc; s is a

248 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

vector denoting the heights of the points of the interfaces at gi-ven m locations. Also given m, the points for the different inter-faces are assumed to be independent. At every step, we join them points in each interfaces by linear interpolation which com-pletely defines the boundaries of the facies (See Fig. 4 for a de-tailed explanation of the facies update for a Birth, Death andJump step.). Then the reversible jump algorithm will be asfollows.

� Birth Step. We randomly choose one of the fine-grid interval inðxloc

n;ð1Þ; xlocn;ð2Þ; . . . ; xloc

n;ðmnÞÞ. Without loss of generality, let the interval

be ½xlocn;ð1Þ; x

uplocn;ð2Þ �, suppose there are C known locations in this inter-

val and the ordered locations are lx1; lx2; . . . ; lxC . We assign proba-bility pðiÞ to each of the length li ¼ lxi � xloc

n;ð2Þ; i ¼ 1ð1ÞC. Then wegenerate one of the lengths from the previously defined probabil-ity distribution and add it to xloc

n;2 to get the newly added location.

So the proposed location vector is given by xloc ¼ ðxlocð1Þ; x

locð2Þ; . . . ;

xlocðmnÞþ1Þ

0¼ðxlocn;ð1Þ;x

locn;ð2Þþut ;xloc

n;ð2Þ;...;xlocn;ðmnÞÞ

0¼gmnmnþ1ðxlocn ;uÞ, where u

is drawn from the probability distribution of lðiÞ’s. Hence in thiscase qmnmnþ1ðujxloc

n Þ¼pðiÞ , if u¼ lðiÞ.Here we consider

pðiÞ ¼PN

ðlðiÞþlðiþ1ÞÞ2 �

ðxlocn;ð1Þ�xloc

n;ð2Þ Þ

2rl

!� PN

ðliþli�1 Þ2 �

ðxlocn;ð1Þ�xloc

n;ð2Þ Þ

2rl

!

PN

ðlCþlCþ1Þ2 �

ðxlocn;ð1Þ

�xlocn;ð2Þ

Þ

2rl

!� PN

ðlð1Þþlð0ÞÞ2 �

ðxlocn;ð1Þ

�xlocn;ð2Þ

Þ

2rl

! ; ð36Þ

8 i ¼ 1 ð1Þ C, with l0 ¼ xlocn;ð1Þ � xloc

n;ð2Þ and lCþ1 ¼ 0, where PNð�Þ isthe standard normal cdf function. We generate u from this prob-ability distribution, which can be considered as a discretized ver-sion of normal distribution and which guarantees that the addednew point lies in the middle of the two locations with very highprobability. We propose h ¼ hn þ hhuh, where uh is generatedfrom a Nð0; IÞ distribution. Then the acceptance probability is gi-ven by

abmnmnþ1ðxloc

n ; xlocÞ ¼min 1; likelihood ratio� prior ratiof� proposal ratio� Jacobiang; ð37Þ

where

existing points

new point

updated points

Fig. 4. An illustration of the birth, death and jump pr

likelihood ratio ¼exp �kFobs�Fkk2

2r2f

� exp �kFobs�Fkn k

2

2r2f

� ; ð38Þ

prior ratio ¼exp �kh��hok2

2r2h

�exp �khn�hok2

2r2h

�� exp �ks��sok2

2r2s

�exp �ksn�sok2

2r2s

�� prior-multiplier;

ð39Þ

proposal ratio ¼pdel

mnþ1

paddmn

� proposal-multiplier; ð40Þ

Jacobian ¼ 1: ð41Þ

The prior-multiplier and the proposal-multiplier depend on thenumber of interfaces considered. If there are eI interfacies andwe assume the locations for different interfacies are independentwith each other then, prior-multiplier ¼ ðmn�1Þffiffiffiffi

2pp

rsðN�mnÞ

�eI,

proposal-multiplier ¼ 1qðu1Þqðu2Þ���qðueI Þ.

� Death Step. We randomly choose one of the points in

xlocn;ð2Þ; x

locn;ð3Þ; . . . ; xloc

n;ðmn�1Þ

n o. Without loss of generality, let the

point be xlocn;ð2Þ. Suppose there are C1 þ C2 þ 1 known locations

within the interval ½xlocn;ð1Þ; x

locn;ð3Þ� and let the locations are denoted

by lx11; lx12; . . . ; lx1C1 ; xlocn;ð2Þ; lx21; lx22; . . . ; lx2C2 . We delete the point

xlocn;ð2Þ and hence propose xloc ¼ ðxloc

ð1Þ; xlocð3Þ; . . . ; xloc

ðmnÞÞ0. Here we take

u0 ¼ ðxlocð2Þ � xloc

ð3ÞÞ and hence in this case gmnmn�1ðxlocn Þ¼

ðg1mnmn�1ðxloc

n Þ;g2mnmn�1ðxloc

n ÞÞ¼ðxlocn;ð1Þ;x

locn;ð3Þ; . . . ; xloc

n;ðmnÞ;ðxlocn;ð2Þ �xloc

n;ð3ÞÞÞ0,

and

oce

qmn�1mnðu0jxlocÞ ¼

PN

lx1C1þxloc

n;ð2Þ2 �

ðxlocn;ð1Þþxloc

n;ð3Þ Þ

2rl

!� PN

xlocn;ð2Þþlx21

2 �ðxlocð1Þþxloc

ð3Þ Þ

2rl

!

PN

lx2C22 �

xlocð1Þ2

rl

!� PN

lx112 �

xlocð3Þ2

rl

! ;

ð42Þ

where PNð�Þ is the standard normal cdf function. We proposeh ¼ hn þ hhuh, where uh is generated from a Nð0; IÞ distribution.The acceptance probability is given by

removed interface segments

existing interface

added interface segmentsupdated interface

ss in reversible jump MCMC on an interface.

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 249

admnmn�1ðxloc

n ; xlocÞ ¼min 1; likelihood ratio� prior ratiof� proposal ratio� Jacobiang; ð43Þ

where

likelihood ratio ¼exp �kFobs�Fkk2

2r2f

� exp �kFobs�Fkn k

2

2r2f

� ; ð44Þ

prior ratio ¼exp �kh�hok2

2r2h

�exp �khn�hok2

2r2h

�� exp �ks�sok2

2r2s

�exp �ksn�sok2

2r2s

�� prior-multiplier;

ð45Þ

proposal ratio ¼padd

mn�1

pdelmn

� proposal-multiplier; ð46Þ

Jacobian ¼ 1: ð47Þ

The prior-multiplier and the proposal-multiplier depend on the

number of interfaces considered. If there are eI interfaces andwe assume the locations for different interfaces are independent

from each other then, prior-multiplier ¼ffiffiffiffi2pp

rsðN�mnþ1Þmn�2

�eIand

proposal-multiplier ¼ qðu1Þqðu2Þ � � � qðueI Þ� Jumps at Fixed Locations. Here the number of points and their

locations are fixed so we only have jumps at those fixed loca-tions. We propose s ¼ sn þ hsus, where us is generated from aNð0; IÞ distribution. Similarly propose h ¼ hn þ hhuh, where uh isalso generated from a Nð0; IÞ distribution. This step is same asa simple random walk Metropolis–Hastings step. The accep-tance probability is given by

aðkn;kÞ ¼min 1; likelihood ratio�prior ratio�proposal ratiof g; ð48Þ

likelihood ratio¼exp �kFobs�Fkk2

2r2f

� exp �kFobs�Fkn k

2

2r2f

� ; ð49Þ

prior ratio¼exp �kh�hok2

2r2h

þ�ks�sok2

2r2s

�exp �khn�hok2

2r2h

þ�ksn�sok2

2r2s

� ; ð50Þ

proposal ratio¼ 1: ð51Þ

Here we take paddmn¼pdel

mn¼pj

mn ¼ 13 ; 8mn¼ðmminþ1Þð1Þðmmax�1Þ.

paddmmin¼ 2

3; pdelmmin¼0; pj

mmin¼ 1

3.paddmmax¼0; pdel

mmax¼ 2

3; pjmmax ¼ 1

3.

3.3. Two-stage reversible jump MCMC

The main disadvantage of the above reversible jump MCMCalgorithm is very high computational cost in solving the couplednonlinear PDE system (1)–(3) on the fine-grid to compute Fk inthe target distribution pðkÞ. Typically, in our simulations, revers-ible jump MCMC method converges to the steady state after thou-sands of iterations and the acceptance rate is also very low. A largeamount of CPU time is spent on simulating the rejected samples,making the direct (full) reversible jump MCMC simulations veryexpensive.

The direct reversible jump MCMC method can be improved byadapting the proposal distribution qðkjknÞ to the target distribu-tion using a coarse-scale model. This can be achieved by a two-stage reversible jump MCMC method, where we compare thefractional flow curves on the coarse-grid model, first. If the pro-posal is accepted by the coarse-scale test, then a full fine-scalecomputation will be conducted and the proposal will be furthertested as in the direct reversible jump MCMC method. Otherwise,the proposal will be rejected by the coarse-scale test and a new

proposal will be generated from qðkjknÞ. The coarse-scale test fil-ters the unacceptable proposals and avoids the expensive fine-scale tests for those proposals. The filtering process essentiallymodifies the proposal distribution qðkjknÞ by incorporating thecoarse-scale information of the problem. The algorithm for a gen-eral two-stage MCMC method with upscaling was introduced in[17]. Our hierarchical model can also take an advantage of inex-pensive upscaled simulations to screen the proposals. Here weextend the algorithm to two-stage reversible jump MCMC meth-od. Let F�k be the fractional flow computed by solving the coarse-scale model of (1)–(3) for the given k. This is done either withupscaling methods or mixed MsFEM. The fine-scale target distri-bution pðkÞ is approximated on the coarse-scale by p�ðkÞ. Herewe have

pðkÞ / exp �kFobs � Fkk2

r2f

!� PðkÞ; ð52Þ

p�ðkÞ / exp �ðGðkFobs � F�kkÞÞ2

r2c

!� PðkÞ; ð53Þ

where the function G is estimated based on offline computationsusing independent samples from the prior. More precisely usingindependent samples from the prior distribution, the permeabilityfields are generated. Then both the coarse-scale and fine-scalesimulations are performed and kFobs � Fkk vs kFobs � F�kk are plot-ted. This scatterplot data can be modeled by

kFobs � Fkk ¼ GðkFobs � F�kkÞ þw; ð54Þ

where w is a random component representing the deviations of thetrue fine-scale error from the predicted error. Using the coarse-scaledistribution p�ðkÞ as a filter, the two-stage reversible jump MCMCcan be described as follows.

Algorithm. Two-stage reversible jump MCMC as Birth and DeathProcessSuppose at the nth step we are at sn; hn; xloc

n ;mn and permeabilityfield kn.

� Step 1. This step is the same as the reversible jump MCMCmethod described earlier. The only difference is the frac-tional flow F�k is computed by solving the coarse-scalemodel. At kn, generate a trial proposal ~k from distributionqð~kjknÞ the same way as in the reversible jump MCMCdescribed earlier.

� Step 2. Take the proposal as

k ¼~k with probability apðkn;

~kÞ;kn with probability 1� apðkn;

~kÞ:

(

If we are at Birth Step then the acceptance probability is given by

apðkn;~kÞ

¼min 1;P�ðFobsj~kÞP�ðFobsjknÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}likelihood ratio

� PðhÞPðsjxlocÞPðxlocjmn þ 1ÞPðmn þ 1ÞPðhnÞPðsnjxloc

n ÞPðxlocn jmnÞPðmnÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

prior ratio

8>>><>>>:�

qhðhnjhÞpdelmnþ1

qhðhjhnÞpaddmn

qmnmnþ1ðujxlocn Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

proposal ratio

�@gmnmðxloc

n ; uÞ@xloc

n @u

���� ����|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Jacobian

9>>>=>>>;: ð55Þ

Note that, PðsÞPðhÞPðxlocjmn þ 1ÞPðmn þ 1Þ is the same as the priorprobability Pð~kÞ as defined in (53).If we are at Death Step then the acceptance probability is given by

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ek

E* k

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

Ek

E* k

Fig. 5. Cross-plot between Ek ¼ kFobs � Fkk and E�k ¼ kFobs � F�kk. Left: cross-plot using three-coarse-block case. Right: cross-plot using nine-coarse-block case.

reference initial

realization realization

realization realization

median mean

0123

0123

0123

0123

0123

0123

0123

0123

Fig. 6. Top left: the true log permeability field. Top right: initial log permeabilityfield. Middle four figures: four accepted realizations of log permeability field.Bottom left: the log of the median of the sampled permeability field. Bottom right:the log of the mean of the sampled permeability field from two-stage reversiblejump MCMC in three-coarse-block case.

250 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

apðkn;~kÞ

¼min 1;P�ðFobsj~kÞP�ðFobsjknÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}likelihood ratio

�PðhÞPðsjxlocÞPðxlocjmn�1ÞPðmn�1ÞPðhnÞPðsnjxloc

n ÞPðxlocn jmnÞPðmnÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

prior ratio

8>>><>>>:�

qhðhnjhÞpaddmn�1qmn�1mn

ðu0jxlocÞqhðhjhnÞpdel

mn|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}proposal ratio

�@gmnmðxloc

n Þ@xloc

n

���� ����|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}Jacobian

9>>>=>>>;: ð56Þ

If we are going to have Jumps at Fixed Locations then the accep-tance probability is given by

apðkn;~kÞ ¼min 1;

P�ðFobsj~kÞP�ðFobsjknÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}likelihood ratio

� PðsÞPðhÞPðsnÞPðhnÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

prior ratio

� qsðsnjsÞqhðhnjhÞqsðsjsnÞqhðhjhnÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

proposal ratio

8>>><>>>:9>>>=>>>;:ð57Þ

Therefore, the final proposal k is generated from the effectiveinstrumental distribution

QðkjknÞ ¼ apðkn; kÞqðkjknÞ

þ 1�Z

apðkn; kÞqðkjknÞdk�

dkn ðkÞ: ð58Þ

To show that the reversible jump MCMC sampling generates a Mar-kov chain, whose stationary distribution is the candidate distribu-tion it is sufficient to show that the transition kernel satisfies thedetailed balance condition. The proof is shown in Appendix A.In our paper, we use a simple relation for modeling coarse- and fine-scale errors. In particular, G is taken to be a linear function with thecondition Gð0Þ ¼ 0. Then our p�ðkÞ becomes

p�ðkÞ / exp �kFobs � F�kk2

r2c

!� PðkÞ; ð59Þ

i.e., on the coarse-scale Fobsjk is assumed to follow NðF�k;r2c IÞ distri-

bution, i.e.,

P�ðFobsjkÞ / exp �kFobs � F�kk2

r2c

!; ð60Þ

where rc is the precision associated with the coarse-scale model.The parameter rc plays an important role in improving the accep-tance rate of the preconditioned MCMC method. The optimal value

of rc depends on the correlation between kF � Fkk and kF � F�kk,which can be estimated by offline computations.

� Step 3. Accept k as a sample with probability�

af ðkn; kÞ ¼ min 1;

QðknjkÞpðkÞQðkjknÞpðknÞ

; ð61Þ

i.e., knþ1 ¼ k with probability af ðkn; kÞ, and knþ1 ¼ kn with probabil-ity 1� af ðkn; kÞ.Using the same argument as in [17], the acceptance probability (61)can be simplified as

af ðkn; kÞ ¼ min 1;pðkÞp�ðknÞpðknÞp�ðkÞ

� : ð62Þ

Assuming that on the fine-scale Fobsjk follows a NðFk;r2f IÞ distribu-

tion, i.e.,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Cross−plot of fractional flows

sampled fractional flow

true

fract

iona

l flo

w

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pore volume injected

F

Fractional flows

exact F(t)initial F(t)sampled F(t)s

Fig. 7. Left: cross-plot between the reference fractional flow and sampled fractional flows from two-stage reversible jump MCMC in three-coarse-block case. Right: solidblack line designates the fine-scale reference fractional flow, the dashed blue line designates the initial fractional flow and the dashed red line designate fractional flowcorresponding to sampled permeability fields from two-stage reversible jump MCMC in three-coarse-block case. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

accepted iterations

fract

iona

l flo

w e

rror

Fractional flow error vs iterations

full RJMCMCtwo−stage RJMCMC with three blocks in coarse scaletwo−stage RJMCMC with nine blocks in coarse scale

Fig. 8. Fractional flow errors vs. accepted iterations for two-stage and full reversible jump MCMC.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

Ek

E* k

0 1 2 3 4 5 6 70

1

2

3

4

5

6

Ek

E* k

0 1 2 3 4 50

1

2

3

4

5

6

Ek

E* k

Fig. 9. Cross-plot between Ek ¼ kFobs � Fkk and E�k ¼ kFobs � F�kk when the variance of the log permeability field is 2. Left: cross-plot using three-coarse-block case. Middle:cross-plot using nine-coarse-block cases. Right: cross-plot using mixed MsFEM.

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 251

reference initial

realization realization

realization realization

median mean

−10123

024

−10123

−10123

−10123

−10123

−202

−202

Fig. 10. Top left: the true log permeability field. Top right: initial log permeabilityfield. Middle four: four accepted realizations of log permeability field. Bottom left:the log of the median of the sampled permeability field. Bottom right: the log of themean of the sampled permeability field from two-stage reversible jump MCMCusing mixed MsFEM when the variance of the log permeability field is 2.

252 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

PðFobsjkÞ / exp �kFobs � Fkk2

r2f

!; ð63Þ

the acceptance probability (62) becomes

af ðkn; kÞ ¼min 1;exp �kFobs � Fkk2r2

f

�exp � kFobs�F�kn

k2

r2c

� exp �kFobs�Fkn k

2

r2f

� exp �kFobs�F�kk

2

r2c

�0BB@

1CCA:ð64Þ

To show that the reversible jump MCMC sampling generates aMarkov chain, whose stationary distribution is the candidate dis-tribution it is sufficient to show that the transition kernel satisfiesthe detailed balance condition. The proof is almost same as in [17]and is given in the Appendix A.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

sampled fractional flow

true

fract

iona

l flo

w

Cross−plot of fractional flows

Fig. 11. Left: cross-plot between the reference fractional flow and sampled fractional flowdesignates the fine-scale reference fractional flow, the dashed blue line designates the initsampled permeability fields from two-stage reversible jump MCMC using mixed MsFEMreferred to the web version of this article.)

In the above algorithm, if the trial proposal ~k is rejected by thecoarse-scale test (Step 2), kn will be passed to the fine-scale test asthe proposal. Since af ðkn; knÞ 1, no further (fine-scale) computa-tion is needed. Thus, the expensive fine-scale computations canbe avoided for those proposals which are unlikely to be accepted.In comparison, the regular reversible jump MCMC method requiresa fine-scale simulation for every proposal k, even though most ofthe proposals will be rejected at the end. Since the computationof the coarse-scale solution is very cheap, Step 2 of the precondi-tioned MCMC method can be implemented very fast to decidewhether or not to run the fine-scale simulation. The second stepof the algorithm serves as a filter that avoids unnecessary fine-scale runs for the rejected samples. It is possible that the coarse-scale test may reject an individual sample which will otherwisehave a (small) probability to be accepted in the fine-scale test.

We can use the same illustrating example as presented in Sec-tion 3.2 and the numerical results shows how the two-stagereversible jump becomes more efficient in terms of CPU. Whileusing this example in the two-stage algorithm in Step 1 we add anew location or delete a location or consider jumps as given loca-tions in the same way as we did in reversible jump MCMC method.In Step 2, the acceptance probability for the Birth Step, Death Step,and jumping step remains the same as in (37), (43) and (48),respectively with Fk; Fkn and r2

f in the likelihood ratio replaced byF�k; F

�kn

and r2c , respectively. Later, we describe how Langevin algo-

rithms can be implemented.

4. Numerical results

In our first numerical example, we consider a 50� 50 fine-scalepermeability field on the unit square. We consider the case withonly one high conductivity layer. Thus there are two interfaces,one for the upper interface and one for the lower interface. Thepermeability field is known at 8 locations along x ¼ 0 and x ¼ 1boundaries. The ends of the interface are fixed at 0.4 and 0.6.One injection well at (0, 0.5) and one production well at (1, 0.5)are placed. Two-phase flow model with quadratic relative perme-abilities krw ¼ S2 and kro ¼ ð1� SÞ2 are considered. The log of thepermeability field within the channel (middle facies) is assumedto be Gaussian process with mean 3 and covariance function givenby (18), where l1 ¼ 0:3; l2 ¼ :1 and r2 ¼ :32. The log of the perme-ability field outside the high conductivity is assumed to be Gauss-ian process with mean 0 and the same covariance function, where

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.4

0.5

0.6

0.7

0.8

0.9

1

Pore volume injected

F

Fractional flows

actual F(t)initial F(t)sampled F(t)s

s from two-stage reversible jump MCMC using mixed MsFEM. Right: solid black lineial fractional flow and the dashed red line designate fractional flow corresponding to. (For interpretation of the references to colour in this figure legend, the reader is

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

accepted iterations

fract

iona

l flo

w e

rror

Fractional flow error vs accepted iterationfull RJMCMCtwo−stage RJMCMCusing multiscale model

Fig. 12. Fractional flow errors vs. accepted iterations when the variance of the logpermeability field is 2.

reference initial

realization realization

realization realization

median mean

0123

0123

0123

0123

0123

0123

0123

0123

Fig. 13. Top left: the true log permeability field. Top right: initial log permeabilityfield. Middle four: four accepted realizations of log permeability field. Bottom left:the log of the median of the sampled permeability field. Bottom right: the log of themean of the sampled permeability field from two-stage reversible jump MCMC withfive coarse blocks.

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 253

l1 ¼ :2; l2 ¼ :2 and r2 ¼ :32. We retain the first 20 terms in the KLE.Initially s’s are taken to be equidistant points on the straight linejoining the two ends of the interfaces. We first run the full revers-ible jump MCMC taking r2

h ¼ 0:16 and r2s ¼ 0:04. The acceptance

rate of the full reversible jump MCMC is very low, approximately0.002, using r2

f ¼ :004.Next, we implement two-stage reversible jump MCMC to in-

crease the acceptance rate. Because of mild variations within thefacies, one can take volume average of the permeability and avoidmore costly single-phase upscaling. We consider two cases. In thefirst case, the permeability is upscaled via simple volume averagingto three coarse blocks corresponding to facies. In the second exam-ple, we divide the domain into three equal vertical parts and up-scale the permeability within each of nine blocks. We find suchsimple and very coarse upscaling works well (i.e., improve theacceptance probability substantially) for the cases where the per-meability does not vary too much within facies. The efficiency ofthese simple upscaling techniques deteriorates as we increasethe variance within the facies. One can improve these methodsby taking coarse-grid blocks at the kink points of the interface.We have not implemented these coarsening approaches. We sug-gest the use of mixed MsFEM for the cases with high permeabilityvariations within the facies.

To assess the accuracy of two-stage MCMC, we perform coarse-scale vs. fine-scale simulations for permeability samples from theprior. More precisely, the cross-plot between Ek ¼ kFobs � Fkk andE�k ¼ kFobs � F�kk for both three-coarse-block and nine-coarse-blockcases is shown in Fig. 5. The correlation coefficient between thetwo errors are 0.86 and 0.93 for three-coarse-block and nine-coarse-block cases, respectively. This correlation coefficient de-creases if the variance of the permeability within facies increases.The high correlation coefficient provides a favorable results fortwo-stage MCMC. The acceptance rate for the two-stage reversiblejump MCMC in the cases of three-coarse-block and nine-coarse-block are 0.33 and 0.47, respectively, using r2

f ¼ 0:004 andr2

c ¼ 0:01. In Fig. 6, the reference log permeability field, the initiallog permeability field, some of the sampled log permeability field,the log mean and median of the sample permeability field for thetwo-stage reversible jump MCMC are shown. In Fig. 7 (right plot),we plot the initial fractional flow and the fractional flows corre-sponding to some of the sampled permeability fields. We observesubstantial improvement in fractional flow predictions. On the leftof Fig. 7, we depict the cross-plot of fractional flows correspondingto the right figure. We can see that the sampled permeability fieldsare very close to the reference permeability field.

The convergence of two-stage MCMC is plotted in Fig. 8. It isclear from this figure, that both two-stage and fine-scale reversiblejump MCMC have similar convergence properties, i.e., they reachto the steady state within the same number of iterations. The for-mal convergence diagnosis can be performed using multiple chainsmethod-based convergence diagnosis [23]. In this paper, our goal isto compare two-stage and direct reversible jump MCMC. We re-strict ourselves to only showing errors vs. the number of iterations.We note that the convergence diagnostics has nothing to do withthe rate of convergence, which depends on the second largesteigenvalue of the transition matrix of the Markov chain. For thecomplex chains, the calculation of these eigenvalues is not simple.

In our next numerical example, the same setup is chosen exceptthe variance of the log permeability field is increased to r2

f ¼ 2. Thefull reversible jump MCMC performs as before. However, theacceptance rate becomes very low, approximately 0.001. For two-stage reversible jump MCMC, the correlation betweenEk ¼ kFobs � Fkk and E�k ¼ kFobs � F�kk becomes low, 0.43 and 0.46for three-coarse-block and nine-coarse-block cases, respectively(see Fig. 9). In this case, mixed MsFEM is preferred. The correlationbetween Ek ¼ kFobs � Fkk and E�k ¼ kFobs � F�kk is very high, approxi-

mately 0.99, when mixed MsFEM is used (see Fig. 9). With mixedMsFEM, the acceptance rate of two-stage reversible jump MCMCincreases to 0.31. In Fig. 10, the reference log permeability field,the initial log permeability field, some of the sampled log perme-ability field, the log mean and median of the sample permeabilityfield for the two-stage reversible jump MCMC using mixed MsFEMare shown. In Fig.11 (right plot), we plot the initial fractional flowand the fractional flows for some of the sampled permeabilityfields using two-stage reversible jump MCMC using mixed MsFEM.On the left of Fig. 7, we depict the cross-plot of fractional flows cor-responding to the right figure. We plot the fractional flow errors vs.iteration number for the two-stage reversible jump MCMC usingmixed MsFEM in Fig. 12.

In our next set of numerical examples, we consider two highconductivity facies. Thus, there are four interfaces. We assume per-meabilities are known in the middle of each facies along x ¼ 0 andx ¼ 1. As before, the ends of the facies are assumed to be fixed. Two

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pore volume injected

F

Fractional flows

exact F(t)initial F(t)sampled F(t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pore volume injected

F

Fractional flows

exact F(t)initial F(t)sampled F(t)s

Fig. 14. Left: solid black line designates the fine-scale reference fractional flow, the dashed blue line designates the initial fractional flow and the dashed red line designatesfractional flow corresponding to sampled permeability fields from full reversible jump MCMC. Right: solid black line designates the fine-scale reference fractional flow, thedashed blue line designates the initial fractional flow and the dashed red line designates fractional flow corresponding to sampled permeability fields from two-stagereversible jump MCMC in three-coarse-block case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

accepted iterations

fract

iona

l flo

w e

rror

Fractional flow error vs iterationsfull RJMCMCtwo−stage RJMCMC

Fig. 15. Fractional flow errors vs. accepted iterations for the example with twochannels.

254 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

injection wells at (0, 0.4) and (0, 0.75) and two production wells at(1, 0.4) and (1, 0.75) are placed. Two-phase flow model with qua-dratic relative permeabilities krw ¼ S2 and kro ¼ ð1� SÞ2 are consid-ered. The log of the permeability field inside high conductivityregions is assumed to be Gaussian process with mean 3 and covari-ance function given by (18), where l1 ¼ 0:3; l2 ¼ 0:1 and r2 ¼ :32.The log of the permeability field outside the high conductivity fa-cies is assumed to be Gaussian process with mean 0 and the corre-lation lengths l1 ¼ 0:2; l2 ¼ 0:2 and r2 ¼ 0:32. The rest of the set upis same as the first example. As before, the full reversible jumpMCMC still predicts the interfaces quite accurately. The acceptanceprobability of reversible jump MCMC is very low, nearly 0.001. Thetwo-stage reversible jump MCMC with five spatial coarse-blocks(corresponding to facies) speeds up the process with acceptancerate nearly 0.63 without sacrificing the convergence. In Fig. 13,we plot the permeability fields obtained using two-stage algo-rithm. As we see, the prediction is quite accurate in the two-stagereversible jump MCMC as the sampled permeability fields resem-ble the reference permeability field very closely. The correspondingfractional flows are plotted in Fig. 14. Note that there is a substan-tial improvement in fractional flows when comparing the initialsample and a sample from the posterior. Finally, in Fig. 15, we pres-ent the fractional flow errors vs. the number of iterations to dem-onstrate that two-stage reversible jump MCMC has similarconvergence as fine-scale reversible jump MCMC.

5. Conclusions

In this paper, we study uncertainty quantification in inverseproblems for heterogeneous subsurfaces where the permeabilityfields have channelized structure. Hierarchical models are usedto model the channel boundaries as well as the permeability distri-bution within the channels that are assumed to be independent.We assume that the channel information at the wells are known;however, no other information is assumed to be given about thechannel shape. The channel boundaries are modeled with variablenumber of points resulting to changing dimension in the uncer-tainty space. Reversible jump Markov chain Monte Carlo algo-rithms are used in such modeling. Within each channel, thepermeability is assumed to have a log-normal distribution. Thesearch with Metropolis–Hastings algorithm results to very lowacceptance rate, and consequently, the computations are CPUdemanding. To speed-up the computations, we use coarse-scalemodels to screen the proposals. Our computations show that theproposed algorithms are capable of capturing the channel bound-aries and result to accurate predictions of subsurface properties.

In future, Langevin proposals will be used to improve the algo-rithms. Langevin proposals employ gradient information in makingnew proposals [35]. To use the gradient information, we will splitthe jump process into two parts: (1) adding/deleting new point; (2)perturbing the channel boundaries. Langevin proposal will be com-puted in the second stage that will provide an easy implementa-tion of two-stage MCMC. Moreover, we will use gradientinformation based on coarse-scale models for computing the gradi-ents. More precisely in Step 2 of the two-stage reversible jumpMCMC algorithm we would choose the proposal generator qð~kjknÞas

~k ¼ kn þDðs;hÞ

2r log p�ðknÞ þ

ffiffiffiffiffiffiffiffiffiffiffiDðs;hÞ

q�n; ð65Þ

where �n are independent Gaussian vectors. This will further speed-up the computations.

Acknowledgment

We would like to acknowledge NSF CMG 0724704 This work ispartly supported by Award Number KUS-C1-016-04, made by KingAbdullah University of Science and Technology (KAUST).

A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256 255

Appendix A. Detailed balance

In this Appendix we will show the transition kernel of the two-stage reversible jump MCMC satisfies the detailed balance condi-tion. The transition kernel of the Markov chain kn generated bythe two-stage reversible jump MCMC is given by,

Kðkn; kÞ ¼ af ðkn; kÞQðkjknÞ; for k – kn; ð66Þ

Kðkn; knÞ ¼ 1�Z

k–kn

af ðkn; kÞQðkjknÞdk; ð67Þ

where, QðkjknÞ is defined as in (58).So, the transition kernel is continuous when k – kn ans has po-

sitive probability for the event fk ¼ kng. We have to show that thetransition kernel satisfies the detailed balance condition.

pðknÞKðkn; kÞ ¼ pðkÞKðk:knÞ: ð68Þ

Proof. Equality in (68) is obviously true for k ¼ kn, Fork – kn;pðknÞKðkn; kÞ ¼ pðknÞaf ðkn; kÞQðkjknÞ ¼minðQðkjknÞpðknÞ;QðknjkÞpðkÞÞ ¼min QðkjknÞpðknÞ

Qðkn jkÞpðkÞ ;1 �

QðknjkÞpðkÞ ¼ af ðk;knÞQðknjkÞpðkÞ¼ pðkÞKðk;knÞ.

Hence the proof follows. h

Appendix B. Regularity of the posterior distribution

In this Appendix, we will show that the posterior measure is con-tinuous with respect to the data in the total variation distance Weconsider a layered permeability field, k ¼ kðx2Þ and the flow alongthe layers. More precisely, we assume p ¼ 1 at x1 ¼ 1 and p ¼ 0 atx1 ¼ 0 and no flow on lateral boundaries. In this case, one can easilyshow that the velocity is given by ðkðx2Þ;0Þ. We consider the dataFkðtÞ ¼

Rout v � nSdl. Then, FkðtÞ ¼

R 10 vðx2ÞSð1; x2; tÞdx2. For simplic-

ity, we assume that Sðx1; x2; t ¼ 0Þ ¼ S0ðx1Þ. In this case,FkðtÞ ¼

R 10 kðx2ÞS0ð1� kðx2ÞtÞdx2. To avoid sub-indices in the deriva-

tions, we replace x2 by g, thus, FkðtÞ ¼R 1

0 kðgÞS0ð1� kðgÞtÞdg. As forthe permeability field, we assume that it is log-Gaussian in each sub-interval ðmðiÞ; mðiþ1ÞÞ, where m ¼ ðmð1Þ; mð2Þ; . . . ; mðIÞÞ0. i.e., mð2Þ . . . mðI�1Þ areorder statistics drawn from a truncated distribution truncated atsome fixed numbers ðmð1Þ; mðIÞÞ. Here, mðiÞ’s represent the boundariesof the channels. For example, we have mð1Þ ¼ 0; mðIÞ ¼ 1 andmð2Þ; . . . ; mðI�1Þ are order statistics from truncated normal distributiontruncated at (0,1). In each sub-interval permeability will be de-scribed using Karhunen–Loéve expansion.

We define Wððh; mÞ; FobsÞ ¼Pn

i¼1ðFobsðtiÞ�FkðtiÞÞ2

2 . Without loss of gen-erality we assumed r2

f ¼ 1. Thus, pFobsðh; mÞ ¼ 1Z expð�Wððh; mÞ;

FobsÞÞ. Here, FkðtiÞ ¼PI�1

j¼1

R mðjþ1ÞmðjÞ

expPM

l¼1hjlU

jl

ðgÞÞSo 1� expðPM

l¼1hjlU

jlðgÞ

�tiÞdg. The prior measure is given by dloðh; mÞ

¼ dl0ðhÞdl0ðmÞ ¼ p0ðhÞp0ðmÞdhdm, where, p0ðhÞ is the joint pdf ofM independent standard Gaussian variables. p0ðmÞ is the joint pdfof I � 2 order statistics from a truncated normal distribution, withtruncation at (0, 1).

Lemma B.1. For every r > 0 9 C ¼ CðrÞ > 0 such that if kFkk 6 r,then

Wððh; mÞ; FobsÞ 6 CðrÞXI�1

j¼1

Z mðjþ1Þ

mðjÞexp 2

XM

l¼1

hjlU

jlðgÞ

!dgþ 1

" #:

ð69Þ

Proof. First, we note that 0 6 S0 6 1.

Wððh;mÞ;FobsÞ ¼Xn

i¼1

XI�1

j¼1

Z mðjþ1Þ

mðjÞexp

XM

l¼1

hjlU

jlðgÞ

!So

"

� 1�expXM

l¼1

hjlU

jlðgÞ

!ti

!dg� FobsðtiÞ

#2

6

Xn

i¼1

2CXI�1

j¼1

Z mðjþ1Þ

mðjÞexp 2

XM

l¼1

hjlU

jlðgÞ

!þ2jFobsðtiÞj2

" #

6 CXI�1

j¼1

Z mðjþ1Þ

mðjÞexp 2

XM

l¼1

hjlU

jlðgÞ

!dgþ1

" #: ð70Þ

Here, C ¼maxð2rn;2CnÞ. h

Lemma B.2. For every r > 0 9 C ¼ CðrÞ > 0 such that for everyF1

obs; F2obs with kF1

obs _ F1obsk 6 r, we have

jWððh; mÞ; F1obsÞ �Wððh; mÞ; F2

obsÞj

6 CXI�1

j¼1

Z mðjþ1Þ

mðjÞexp

XM

l¼1

hjlU

jlðgÞ

!dgþ 1

" #kF1

obs � F2obskl2

: ð71Þ

Proof. jWððh;mÞ;F1obsÞ�Wððh;mÞ;F2

obsÞj ¼Pn

i¼1jF1obsðtiÞ� F2

obsðtiÞj� jF1obs

ðtiÞ þ F2obsðtiÞ � 2

PI�1j¼1

R mðjþ1ÞmðjÞ

expPM

l¼1hjlU

jlðgÞ

�So 1 � exp

PMl¼1h

jlU

jl

ðgÞÞtiÞdgj 6 kF1

obs � F2obskl2

2r þ 2PI�1

j¼1

R mðjþ1ÞmðjÞ

expPM

l¼1hjlU

jlðgÞ

�dg

h i6 CkF1

obs � F2obskl2

1 þPI�1

j¼1

R mðjþ1ÞmðjÞ

expPM

l¼1hjlU

jlðgÞ

�dg

h i. h

Theorem B.3. If p1 and p2 are two posterior measures for two differ-ent datasets F1

obs and F2obs with kF1

obs _ F1obsk 6 r, then for every

r > 0 9 C ¼ CðrÞ such that

kp1 � p2kTV 6 CkF1obs � F2

obskl2: ð72Þ

Proof. First we show that Z is bounded above.

Z ¼ZZ

expð�Wððh; mÞ; FobsÞÞdl0ðhÞdl0ðmÞ;

where; Wððh; mÞ; FobsÞ ¼Xn

i¼1

ðFobsðtiÞ � FkðtiÞÞ2 P 0

) Z 6 1:

ð73Þ

Next we show that Z is bounded below by a positive quantity.

Z¼Z Z

expð�Wððh;mÞ;FobsÞÞdl0ðhÞdl0ðmÞ

PZ Z

exp �CðrÞXI�1

j¼1

Z mðjþ1Þ

mðjÞexp 2

XM

l¼1

hjlU

jlðgÞ

!dgþ1

" # !dl0ðhÞdl0ðmÞ

P exp �CðrÞZ Z XI�1

j¼1

Z mðjþ1Þ

mðjÞexp 2

XM

l¼1

hjlU

jlðgÞ

!dg

"

�YMl¼1

1ffiffiffiffiffiffiffi2pp expð�ðhj

lÞ2=2Þdhj

l dl0ðmÞ#

¼ exp �CðrÞXI�1

j¼1

Z mðjþ1Þ

mðjÞ

YMl¼1

Z1ffiffiffiffiffiffiffi2pp exp 2hj

lUjlðgÞ

�"�expð�ðhj

lÞ2=2Þdhj

l dgdl0ðmÞi

¼ exp �CðrÞZ XI�1

j¼1

Z mðjþ1Þ

mðjÞ

YMl¼1

exp 4ðUjlðgÞÞ

2 �

dgdl0ðmÞ" #

P exp �CðrÞZðAMÞdm

�; where; A¼max 4ðUj

lðgÞÞ2

�¼ expð�CðrÞAMÞ>0: ð74Þ

Lemma B.1 is used in the first step of the above calculations. Wehave,

256 A. Mondal et al. / Advances in Water Resources 33 (2010) 241–256

jZ1 � Z2j 6ZZ

exp � Wððh; mÞ; F1obsÞ ^Wððh; mÞ; F2

obsÞ �h i

� Wððh; mÞ; F1obsÞ �Wððh; mÞ; F2

obsÞÞ��� ���dl0ðh; mÞ

6

ZZjWððh; mÞ; F1

obsÞ �Wððh; mÞ; F2obsÞÞjdl0ðhÞdl0ðmÞ

6 CkF1obs � F1

obskl2

ZZ XI�1

j¼1

Z mðjþ1Þ

mðjÞexp

XM

l¼1

hjlU

jlðgÞ

!"� dgþ 1�dl0ðhÞdl0ðmÞ6 CkF1

obs � F1obskl2

: ð75Þ

Lemma B.2 is used in the third step of the above calculations. Thus,

kp1 � p2kTV ¼ZZ

Z�11 exp �Wððh; mÞ; F1

obsÞ �h

� Z�12 exp �Wððh; mÞ; F2

obsÞ �i

dl0ðhÞdl0ðmÞ6 I1 þ I2; ð76Þ

where

I1 ¼1Z1

ZZexp �Wððh; mÞ; F1

obsÞ ����

� exp �Wððh; mÞ; F2obsÞ

����dl0ðhÞdl0ðmÞ; ð77Þ

I2 ¼jZ1 � Z2j

Z1Z2

Zexp �Wððh; mÞ; F2

obsÞ �

dl0ðhÞdl0ðmÞ: ð78Þ

From (74) we obtain that Z is bounded below. From (75) and thefact that Z1 is bounded below (see (74)), it follows thatI1 6 B1ðrÞkF1

obs � F1obskl2

. By the upper bound of Z in (73), and lowerbound of Z in (74) and also the bound as in (75) we haveI2 6 CkF1

obs � F1obskl2

. Thus, combining these results, we have

kp1 � p2kTV 6 BkF1obs � F2

obskl2: ð79Þ

This completes the proof of the theorem. h

The proof of Theorem B.3 can be extended to general permeabil-ity fields. This will be presented elsewhere.

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