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Bayesian Uncertainty Quantification for SubsurfaceInversion Using a Multiscale Hierarchical ModelAnirban Mondala, Bani Mallicka, Yalchin Efendievb & Akhil Datta-Guptac

a Department of Statistics, Texas A&M University, College Station, TX 77843 (; )b Department of Mathematics, Texas A&M University, College Station, TX 77843 ()c Petroleum Engineering Department, Texas A&M University, College Station, TX 77843 ()Accepted author version posted online: 06 Sep 2013.Published online: 24 Jul 2014.

To cite this article: Anirban Mondal, Bani Mallick, Yalchin Efendiev & Akhil Datta-Gupta (2014) Bayesian UncertaintyQuantification for Subsurface Inversion Using a Multiscale Hierarchical Model, Technometrics, 56:3, 381-392, DOI:10.1080/00401706.2013.838190

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Bayesian Uncertainty Quantificationfor Subsurface Inversion Using a MultiscaleHierarchical Model

Anirban MONDAL and Bani MALLICK

Department of StatisticsTexas A&M University

College Station, TX 77843(anirban@stat.tamu.edu; bmallick@stat.tamu.edu)

Yalchin EFENDIEV

Department of MathematicsTexas A&M University

College Station, TX 77843(efendiev@math.tamu.edu)

Akhil DATTA-GUPTA

Petroleum Engineering DepartmentTexas A&M University

College Station, TX 77843(datta.gupta@pe.tamu.edu)

We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is arandom field (spatial or temporal). The Bayesian approach contains a natural mechanism for regularizationin the form of prior information, can incorporate information from heterogeneous sources and providea quantitative assessment of uncertainty in the inverse solution. The Bayesian setting casts the inversesolution as a posterior probability distribution over the model parameters. The Karhunen-Loeve expansionis used for dimension reduction of the random field. Furthermore, we use a hierarchical Bayes’ modelto inject multiscale data in the modeling framework. In this Bayesian framework, we show that thisinverse problem is well-posed by proving that the posterior measure is Lipschitz continuous with respectto the data in total variation norm. Computational challenges in this construction arise from the need forrepeated evaluations of the forward model (e.g., in the context of MCMC) and are compounded by highdimensionality of the posterior. We develop two-stage reversible jump MCMC that has the ability to screenthe bad proposals in the first inexpensive stage. Numerical results are presented by analyzing simulated aswell as real data from hydrocarbon reservoir. This article has supplementary material available online.

KEY WORDS: Bayesian hierarchical model; Bayesian inverse problems; Karhunen-Loeve expansion;Two-stage reversible jump MCMC.

1. INTRODUCTION

Mathematical models are studied using computer simulationsin almost all areas of applied and computational mathematics.The indirect estimation of model parameters or inputs fromobservations constitutes an inverse problem. Such problemsarise frequently in science and engineering with applicationsin weather forecasting, climate prediction, chemical kinetics,and oil reservoir forecasting. In practical settings, observationsare inevitably noisy and may be limited in number or resolu-tion. Quantifying the uncertainty in inputs or parameters is thenessential for predictive modeling and simulation-based decisionmaking. For definiteness, in the following, the focus of this arti-cle is on a petroleum reservoir problem. However, we hope thatthe developed theory, methodology, and the computational toolswill be of general interest and value.

Reservoir simulation models are widely used by oil and gascompanies for production forecasts and for making investmentdecisions. If it were possible for geoscientists and engineersto know the physical properties like locations of oil and gas,the permeability, the porosity, and the multiphase flow prop-erties at all locations in a reservoir, it would be conceptually

possible to develop a mathematical model that could be usedto predict the outcome of any action. This model is usually aset of partial differential equations. These physical propertiesare also known as model variables or the input variables. If themodel variables are known, outcomes (output variables) can bepredicted, usually running a numerical reservoir simulator thatsolves a discretized approximation to those partial differentialequations. This is known as the forward problem.

Unfortunately, most oil and gas reservoirs are inconvenientlyburied beneath thousands of feet of overburden. Direct obser-vations of physical properties of the reservoir are available onlyat a few well locations. Additionally, we have some indirectobservations known as the production data, which are typicallymade at the surface, either at the wellhead or at the distributed

© 2014 American Statistical Association andthe American Society for Quality

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Color versions of one or more of the figures in the article can be foundonline at www.tandfonline.com/r/tech.

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382 ANIRBAN MONDAL ET AL.

locations. The main intention is to determine the plausible phys-ical properties of the reservoir given these direct and indirectobservations. This is an inverse problem and the solution of theinverse problem provides an estimate of the characteristics of thesubsurface media, which is usually a spatial or spatiotemporalfield. To solve this inverse problem, the mismatch between sim-ulated (from the numerical reservoir simulator) and observedmeasurements of production data is minimized. This method isknown as “history matching” in petroleum engineering.

Classical statistical approaches to inverse problems have usedregularization methods to impose wellposedness, and solvethe resulting deterministic problems by optimization and othermeans (Vogel 2002). Here we focus on the Bayesian approach,which contains a natural mechanism for regularization in theform of prior information and can incorporate information fromheterogeneous sources and provide a quantitative assessment ofuncertainty in the inverse solution (Kaipio and Somersalo 2004;Stuart 2010). Indeed, the Bayesian setting casts the inverse so-lution as a posterior probability distribution over the modelparameters. Learning unknown inputs by using some observeddata is known as calibration. The principles of Bayesian cali-bration for computer codes are set out in Kennedy and O’Hagen(2001).

In this article, we consider the inverse problems whose solu-tions are unknown functions (say spatial or temporal fields)(Ramsey and Silverman 2005; Tarantola 2005). Estimatingfields rather than parameters typically increase the ill-posednessof the inverse problem, since one is recovering an infinite-dimensional object from a finite amount of data. Most existingstudies explored the value of the field on a finite set of gridpoints and then employed the Gaussian process or Markov ran-dom field priors on them (Lee et al. 2000; Ferreira et al. 2002;Lee et al. 2002). That way the dimension of the posterior is tiedto the discretization of the field, and computational methodsfor similar problems are developed by several authors (Higdon,Swall, and Kern 1998; Banerjee et al. 2008).

On the contrary, we use dimensional reduction in the Bayesianformulation of inverse problems, and allow the dependence ofthe dimensionality on both prior and the data. We employ theKarhunen-Loeve (K-L) expansion of this unknown field (Loeve1977). The number of terms in the K-L expansion determineshow much information is truly required to capture variationamong realizations of the unknown field. Usually, this numberof terms as well as the parameters in the covariance function areassumed to be known (see Efendiev et al. 2005; Marzouk andNajim 2009). On the other hand, we treat them as additionalmodel unknowns and use reversible jump Metropolis (RJM)algorithm to handle this random dimension situation. Since theparameters of the covariance function are unknown, at eachstep of the reversible jump MCMC procedure, we have to usethe K-L expansion of the covariance function, which is verycomputationally demanding. Hence, we propose an alternativeapproach in which we have precomputed the K-L expansion fora given set of the parameters and then use linear interpolation tofind the respective eigenvalues and eigenvectors for a proposednew value of the parameters. This linear interpolation makes thecomputation much faster. Using the matrix perturbation theory,we show that if the interpolating grid is small, the approximatedeigenvalues and eigenvectors are very close to the true ones.

Furthermore, to obtain physically meaningful results, we in-corporate additional information on the unknown field throughspatially smoothing priors as well as additional multiscale data.Several methods have been previously introduced to incorporatemultiscale data with a primary focus on integrating seismic andwell data. These methods include conventional techniques suchas cokriging or block-kriging and its variations as described byBehrens (1998), Deutsch, Srinivasan, and Mo (1996), and Xuet al. (1992). Most kriging-based methods require variogramconstruction that can be difficult because of limited availabilityof data from the unknown field. We employ a Gaussian processprior for the unknown field and use a hierarchical Bayes modelto incorporate multiscale data.

In this multiscale Bayesian framework, we show that the in-verse problem is well posed by proving that the posterior mea-sure is Lipschitz continuous with respect to the data in total vari-ation norm. In our model, the likelihood function contains theforward solver equations (several differential equations), whichis not explicitly available and very expensive to compute. Hence,instead of the reversible jump MCMC algorithm, we proposetwo-stage reversible jump MCMC. In this algorithm, the propos-als are screened in the first stage using the forward solver in anupscaled coarse-grid, which is inexpensive due to the small di-mensions of the coarse-grid. Then, it is passed to the final stage;only it is accepted at the first stage. Thus the two-stage algorithmreduces the computational effort by rejecting the bad propos-als at the initial stage. We show that this proposed two-stagereversible jump MCMC satisfies the detailed balance condition.

Numerical results are presented for the estimation of two-dimensional permeability fields obtained from petroleum reser-voirs. The permeability field is characterized by two-point cor-relation functions with an unknown mean. We assume that thevalues of the fine-scale permeabilities are known at the wells andthe permeability data in a coarse-scale are available. Our numer-ical results illustrate that the proposed method can adequatelypredict this permeability field.

The article is organized as follows. In Section 2 we formulatethe inverse problem and discuss various examples. In Section 3,we discuss the hierarchical Bayes’ model and formulate the pos-terior distribution. In Section 4, Section 5, and Section 6 we dis-cuss the Metropolis Hastings, reversible jump MCMC, and two-stage reversible jump MCMC technique, respectively, to samplefrom the posterior. Finally, in Section 7, we present numericalresults. Section 8 concludes the article with a brief discussion.

2. FORMULATION

This section introduces the forward model and the corre-sponding inverse problem. As stated in the Introduction, we wantto estimate a random field Y (x, ω), x ∈ D, and ω ∈ �, where� is a sample space in a probability space (�,U,P ) withsigma algebra U over �, P is a probability measure on U andD ∈ Rn be a bounded spatial domain. Y is treated as the modelvariable or the input variable. If Y is known then outcomes (out-put variables, response) can be predicted, usually by runninga numerical simulator that solves a discretized approximationto a system of nonlinear partial differential equations. Differentnames are used in different fields for this model, for exam-ple, in reservoir simulations the nonlinear function that maps

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Figure 1. The figure describes the forward simulator. On the left, a typical permeability spatial field is shown, which is the input for theforward simulator. On the right, the output from the forward simulator is shown, which is the fractional flow (water-cut) versus the pore volumeinjected (PVI).

the input variables Y to the output G(Y ) is called the forwardsimulator and the concerned modeling problem is called theforward problem. Due to model discrepancy, the mathematicalmodel G may not represent the physical system in the real worldperfectly. Moreover, due to presence of measurement error andother sources of uncertainty, the observed output responses (sayd) will be different from those that can be produced from thisforward model. In an additive model framework, we can relatethe observations d to the unknown field Y as

d = G(Y ) + ε, (1)

where ε can be (roughly) viewed as the combined modeldiscrepancy error and measurement error. We assume ε ∼MVN(0, σ 2

d I). The problem considered here is the inverse ofthis forward problem, where we want to estimate the model pa-rameter, that is, the random field Y , based on the observations d.A limited number of direct data are available on the spatial fieldY (x,w) on a fine-grid which will be denoted as yo. The observeddata on the fine-scale spatial field yo is extremely sparse. Fur-thermore, additional data on Y may be available on a relativelycoarser grid, say yc. We desire to solve the inverse problem ofestimating Y given the data from the output d, the coarse-scaledata yc on the spatial field and the data yo on the fine-scalespatial field. First we discuss our application to reservoir char-acterization, then we move to the general Bayesian formulationof this problem.

2.1 Reservoir Characterization

Petroleum reservoirs are complex geological formations thatexhibit a wide range of physical and chemical heterogeneities.These heterogeneities span over multiple-length scales and areimpossible to describe in a deterministic fashion. Geostatisticsand, more specifically, stochastic modeling of reservoir het-erogeneities are being increasingly considered by reservoir andpetroleum engineers for their potential in generating more accu-rate reservoir models together with realistic measures of spatialuncertainty. The goal of reservoir characterization is to providea numerical model of reservoir attributes, such as hydraulicconductivities (permeability), storativities (porosity), fluid satu-ration, etc. These attributes are then used as inputs in the forward

model represented by various flow simulators to forecast futurereservoir performance and oil recovery potential.

In most flow situations, the single most influential input is thepermeability spatial field, k in our notation. Permeability is animportant concept in porous media flow (such as flow of the un-derground oil). Physically, permeability arises both from the ex-istence of pores and from the average structure of the connectiv-ity of pores. As permeability takes positive values, we transformY = log(k) for our modeling convenience. The main availableresponse is the fractional flow or the water-cut data, which isthe fraction of water produced in relation to the total productionrate in a two-phase oil–water flow reservoir and denoted by d.The forward simulator operator G (see Figure 1), which mapsthe water-cut data with the permeability field through a logittransformation, is given by d = logit[G(Y )] + ε.

G is obtained from Darcy’s law which contains several par-tial differential equations. The specification of G is described inSection 7.1. We obtain permeability data in different scales. Thefine-scale data represent point measurements such as well logsand cores, where as the coarse-scale data can be obtained fromseismic traces. Our intention is to infer about the fine-scale per-meability field using the data from the output (fractional flow)and the coarse-scale data. In this article, our simulated exam-ples and the practical oil-field example deal with the reservoircharacterization, but our method can be easily adopted for otherexamples.

3. BAYESIAN FRAMEWORK

We shall explain the general Bayesian framework to solvethe inverse problem to infer about the random field Y from theequation

d = logit(G(Y )) + ε. (2)

We have the response data d, some observations on Y at the fine-scale denoted by yo, and some coarse-scale observations of Ysay yc. The Bayesian solution of the inverse problem will be theposterior distribution of Y conditioned on all the observationswhich is P (Y |d, yc, yo). We express this posterior distributionusing Bayes’ theorem as

P (Y |d, yc, yo) ∝ P (d|Y, yc, yo)P (yc|Y, yo)P (yo|Y )P (Y ). (3)

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We need to specify each of the probabilities on the right-hand side of the expression to develop the hierarchical Bayesianmodel. Therefore, the steps to develop the hierarchical Bayes’model will be to specify (i) P (Y ): the prior model for the un-known random field Y where we use the Karhunen-Loeve expan-sion to parameterize Y . (ii) P (yo|Y ): the conditional probabilityof the fine-scale observation given the field Y . (iii)P (yc|Y, yo):modeling the coarse-scale observation yc conditioning on thefine-scale observations yo and the Y using the upscaling tech-nique. (iv) P (d|Y, yc, yo): the likelihood function, which will beobtained from (2). In following sections, we provide the detailsof each of these modeling parts.

3.1 Modeling the Prior Process P(Y)

One of the commonly used stochastic descriptions of spatialfields is based on a two-point correlation function of the spatialfield. For spatial fields described by a two-point correlationfunction, it is assumed that R(x, x∗) = E [Y (x, ω)Y (x∗, ω)] isknown, where E[·] refers to the expectation (i.e., average overall realizations) and x, x∗ are points in the spatial domain. Inapplications, the spatial fields are considered to be defined ona discrete grid. In this case, R(x, x∗) is a square matrix with Nrows and N columns, where N is the number of grid blocks in thedomain. For spatial fields described by a two-point correlationfunction, one can use the Karhunen-Loeve expansion (KLE),following Wong (1971), to obtain a spatial field description withpossibly fewer degrees of freedom. This is done by representingthe spatial field in terms of an optimal L2 basis. By truncatingthe expansion, we can represent the spatial matrix by a smallnumber of random parameters.

We briefly recall some properties of the KLE. For sim-plicity, we assume that E[Y (x, ω)] = 0. Suppose Y (x, ω) is asecond-order stochastic process with E[Y 2(x, ω)] < ∞, ∀x ∈D. Given an orthonormal basis {�i} in L2, we can expandY (x, ω) as a general Fourier series Y (x, ω) = ∑

i Yi(ω)�i(x),where Yi(ω) = ∫

DY (x, ω)�i(x)dx. We are interested in the

special L2 basis {�i} that makes the random variables Yi uncor-related. That is, E(YiYj ) = 0 for all i �= j . The basis functions{�i} satisfy E[YiYj ] = ∫

D�i(x)dx

∫D

R(x, x∗)�j (x∗)dx∗ =0, i �= j. Since {�i} is a complete basis in L2, it follows that�i(x) are eigenfunctions of R(x, x∗):∫

D

R(x, x∗)�i(x∗)dx∗ = λi�i(x), i = 1, 2, . . . , (4)

where λi = E[Y 2i ] > 0. Furthermore, we have R(x, x∗) =∑

i λi�i(x)�i(x∗). Denote θi = Yi/√

λi , then θi satisfy E(θi) =0 and E(θiθj ) = δij . It follows that

Y (x, ω) =∑

i

√λiθi(ω)�i(x), (5)

where �i and λi satisfy (4). We assume that the eigenvaluesλi are ordered as λ1 ≥ λ2 ≥ . . .. The expansion (5) is called theKarhunen-Loeve expansion. In the KLE (5), the L2 basis func-tions �i(x) are deterministic and resolve the spatial dependenceof the spatial field. The randomness is represented by the scalarrandom variables θi . After the domain D is discretized by a rect-angular mesh, the continuous KLE (5) is reduced to finite termsand �i(x) are discrete fields. Generally, we only need to keep

the leading order terms (quantified by the magnitude of λi) andstill capture most of the energy of the stochastic process Y (x, ω).For an NKL-term KLE approximation YNKL = ∑NKL

i=1

√λiθi�i ,

define the energy ratio of the approximation as

e(NKL) = E‖YNKL‖2

E‖Y‖2=

∑NKLi=1 λi∑∞i=1 λi

. (6)

If λi, i = 1, 2, . . . , decay very fast, then the truncated KLEwould be a good approximation of the stochastic process in theL2 sense.

There are different types of spatial covariance functionsR(x, x∗) considered in spatial statistics, for example, spheri-cal, exponential, squared exponential or Gaussian and Maternclass covariance functions. In our examples we assume that theunknown spatial field is smooth so that we can use the squaredexponential covariance structure, although the method is notrestricted to this particular covariance structure. R(x, x∗) in

this case is defined as R(x, x∗) = σ 2 exp (−|x1−x∗1 |2

2l21

− |x2−x∗2 |2

2l22

),where l1 and l2 are the correlation lengths in each direction andσ 2 is the variance. We reparameterize the spatial field Y by K-Lexpansion and keep the leading m terms in the KLE. For anm-term, KLE approximation

Ym = θ0 +m∑

i=1

√λiθi�i,= B(l1, l2, σ

2)θ, (7)

where B = [√

λ1�1,√

λ2�2 . . .√

λm�m] and θ = (θ0, . . . ,

θm). Here, B depends only on l1, l2, and σ 2. Consequentlywe have a parametric representation of the field Y through(l1, l2, σ 2,m, θ0, θ1, . . . θm)′, and we can evaluate Y if we knowthese parameter values. First, we develop the model and thecomputation schemes for a fixed m; afterward extend them forunknown m in Section 7. Therefore using Bayes’ theorem wecan write the posterior P (Y |d, yc, yo) as given in (3) in terms ofthis set of parameters as

P (θ, l1, l2, σ2|d, yc, yo)

∝ P (d|θ, l1, l2, σ2, yc, yo)P (yc|θ, l1, l2, σ

2, yo)

×P (yo|θ, l1, l2, σ2)P (θ )P (l1, l2)P (σ 2)

∝ P (d|θ, l1, l2, σ2)P (yc|θ, l1, l2, σ

2)P (yo|θ, l1, l2, σ2)

×P (θ )P (l1, l2)P (σ 2). (8)

3.2 Modeling the Fine-Scale Data P(yo|Y)

The fine-scale observations are obtained at some loca-tions of the field Y and we specify a model P (yo|Y ) orP (yo|σ, θ, l1, l2) as yo = yp + εk , where yp is the the fine-scalespatial field at the given well locations xobs obtained from theK-L expansion described in Section 3.1. εk is the model errorfor the K-L approximation. We assume, εk follows a multivari-ate normal distribution with mean 0 and covariance σ 2

k I , thatis, yo|θ, l1, l2, σ

2, σ 2k ∼ MVN(yp, σ 2

k ). The prior for σ 2k is as-

sumed to be σ 2k ∼ InverseGamma(ak, bk). After integrating out

σ 2k we obtain

P (yo|θ, l1, l2, σ2) ∝ (ak + Nobs/2)[

bk + 12 (yo − yp)′(yo − yp)

](ak+Nobs/2) ,

(9)

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where Nobs is the number of observations of the fine-scale per-meability field.

3.3 Modeling P(yc|Y, yo) Through Upscaling

In many cases the coarse-scale data are readily available thatcontain important information to reduce the uncertainty in theestimation of the fine-scale spatial field. Moreover, solving theforward problem in a coarse-grid is always much faster and weexploit it in our multistage MCMC algorithm. The upscalingprocedure is a way to link the coarse and the fine-scale data.The simplest way to think about the upscaling procedure in thespatial domain is the use of spatial block averages of the fine-scale spatial data to obtain the coarse-scale data. The idea ofMarkov random field and linear link equation have been usedto model multiscale data (Ferreira et al. 2002; Ferreira and Lee2007). We need to modify this linear link idea in a way so that theforward equations (and the corresponding boundary conditions)remain valid in this upscaling scheme. We upscale the spatialfield Y on the coarse-grid, then solve the original system on thecoarse-grid with upscaled spatial field (Christie 1996; Durlofsky1998).

The main theme of the procedure is that given a fine-scalespatial field Y , we can use an operator L (it can be averagingor more complicated integrations with boundary conditions) sothat the coarse data yc can be expressed as yc = L(Y ) + εc,where εc is a random error term, which explains the variationsfrom deterministic upscaling procedures. As we have param-eterized the spatial field Y using the K-L expansion the finalequation is given as yc = L(Y ) + εc = Lc(θ, l1, l2, σ

2) + εc,where Lc can be looked upon as an operator whose inputis the fine-scale spatial field or the parameters of the modelθ, l1, l2, and σ 2, and output is the coarse-scale value at a givenlocation. We assume that the error εc follows a multivariatenormal distribution with mean 0 and covariance σ 2

c I , that is,yc|θ, l1, l2, σ

2, σ 2c ∼ MVN(Lc(θ, l1, l2, σ

2), σ 2c I ). We assume

the prior distribution of σ 2c as σ 2

c ∼ InverseGamma(ac, bc). Fur-thermore, after integrating out σ 2

c , we obtain the marginal dis-tribution as

P (yc|θ, l1, l2, σ2)

∝ (ac + N∗/2)[bc + 1

2 ||(yc − Lc(θ, l1, l2, σ 2)||2](ac+N∗/2) , (10)

where N∗ is the number of observations of the coarse-scale per-meability field. The choice of the upscaling operator Lc dependson the forward solver related to the scientific problem. The de-tails about the choice of Lc for the reservoir simulation problemare provided in Section 7.

3.4 The Likelihood and Prior Distributions

The likelihood is derived from (2) and (7) asd = G((B(l1, l2, σ 2)θ )) + εf = Ff (θ, l1, l2, σ

2) + εf , whereFf can be looked upon as a realization from the forward simula-tor whose input variables are the parameters θ, l1, l2 and σ 2. Thisrealization Ff is obtained from the forward simulator throughsolution of several differential equations. We assume the errordistribution as εf ∼ MVN(0, σ 2

f I ), that is, d|θ, l1, l2, σ2, σ 2

f ∼

MVN(Ff (θ, l1, l2, σ2), σ 2

f I ). The prior distribution for σ 2f is

assumed to be σ 2f ∼ InverseGamma(af , bf ). Then, after inte-

grating out σ 2f , we have the marginal likelihood as

P (d|θ, l1, l2, σ2)

∝ (af + n/2)[bf + 1

2 ||(d − Ff (θ, l1, l2, σ 2)||2](af +n/2) . (11)

We need to assign prior distribution for the parameters of thecovariance kernel. The prior distribution for θ is given byθ |σ 2

θ ∼ MVN(0, σ 2θ I ) and σ 2

θ ∼ InverseGamma(a0, b0). Again,after integrating out σ 2

θ we obtain the marginal prior distributionas

P (θ ) ∝ (a0 + m/2)[b0 + 1

2θ ′θ](a0+m/2) . (12)

Additionally, the prior distribution for σ 2 is taken to beGamma(as, bs). We assume uniform priors for l1 and l2.

3.5 The Posterior Distribution and its Continuity

From (8) we obtain the posterior distribution of the spatialfield Y given the output data d, coarse-scale data yc and theobserved fine-scale data yo using the Bayes’ theorem as:

P (θ, l1, l2, σ2|d, yc, yo) ∝ P (d|θ, l1, l2, σ

2)P (yc|θ, l1, l2, σ2)

×P (yo|θ, l1, l2, σ2)P (θ )P (l1, l2)

×P (σ 2). (13)

Each part of the expression on the right-hand side are spec-ified in Section 3. For simplicity, all the model unknowns(l1, l2, σ 2, θo, θ1 . . . θm)′ are denoted as τ , Ff (θ, l1, l2, σ

2) isdenoted by Fτ and Lc(θ, l1, l2, σ

2) is denoted by Lτ in furtherdiscussions.

Using (9), (10), (11), (12), (13), the posterior distribution isgiven by

π (τ ) = P (τ |d, yc, yo)

∝ 1[bf + 1

2 ||d − Fτ ||2](af +n/2)

× 1[bc + 1

2 ||yc − Lτ ||2](ac+N∗/2)

× 1[bk + 1

2 ||yo − yp||2](ak+Nobs/2) × 1[b0 + 1

2θ ′θ](a0+m/2)

× (σ 2)as−1 exp(σ 2/bs), (14)

where ‖d − Fτ‖2 = ∑ni=1(di − Fτ (i))2 for n output observa-

tions.In our Bayesian hierarchical model, the likelihood term con-

tains the forward model, G, which is highly nonlinear and,hence, creates an ill-posed inverse problem. Hence, the sen-sitivity of the solutions to slight perturbations in the data is un-acceptably high (O’Sullivan 1986). In the Bayesian framework,this solution of the inverse problem is the posterior distributionof the unknowns. As a result to show that the Bayesian inverseproblem is well-posed, we have to prove that under some reg-ularity conditions small perturbations of the given data do notlead to a large perturbation of the posterior distribution of theunknowns. In other words, we have to show that the posterior

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distribution is continuous in a suitable probability metric withrespect to changes in data, that is, there exists a unique posteriordistribution that depends continuously on the observations. Thusin the Bayesian framework, if we can show that the posteriormeasure is Lipschitz continuous with respect to the data in thetotal variation distance, then it guarantees that this Bayesian in-verse problem is well-posed (see, e.g., Cotter et al. 2009; Stuart2010). We prove this result for our multiscale Bayesian hier-archical model. To show the continuity of the posterior withrespect to the data, we define

πz(τ ) = 1

Zg(τ, z)π0(τ ), (15)

where z is the concatenating dataset, that is, z = (d, yc, yo)T ,

g(τ, z) = 1[bf + 1

2 ||d − Fτ ||2](af +n/2)

× 1[bc + 1

2 ||yc − Lτ ||2](ac+N∗/2)

× 1[bk + 1

2 ||yo − yp||2](ak+Nobs/2)

π0(τ ) = 1[b0 + 1

2θ ′θ](a0+m/2) × (σ 2)as−1 exp(σ 2/bs) and

Z =∫

g(τ, z)π0(τ )dτ

Theorem 1. ∀r > 0, ∃C = C(r) such that the posteriormeasures π1 and π2 for two different datasets z1 andz2 with max (‖z1‖2, ‖z2‖2) ≤ r , satisfy ‖π1 − π2‖T V = 1

2

∫|Z−1

1 g(τ, z1) − Z−12 g(τ, z2)|π0(τ )dτ ≤ C‖z1 − z2‖2, where

Z1 and Z2 are defined by (15) for z1 and z2, respectively.

The proof is given in the supplementary materials. Note that itcan also be shown that the above Lipschitz continuity conditionis also valid for Hellinger distance, that is,

dHell(π1 − π2)

= 1

2

(∫ (√Z−1

1 g(τ, z1) −√

Z−12 g(τ, z2)

)2π0(τ )dτ

)−1/2

≤ C‖z1 − z2‖2

Furthermore, this proof can be extended for a general G withsome additional conditions.

4. BAYESIAN COMPUTATION USING MCMC

As the posterior is not analytically tractable, hence, we use anMCMC-based computation method to simulate the parametersfrom the posterior distribution. First, we consider the case, wherewe fix the number of terms retained in the K-L expansion.We solve the eigenvalue problem for the fine-scale spatial fieldbeforehand and select an m, number of terms retained in the K-Lexpansion, such that the energy ratio defined in (6) is at least90%. For a constant m we use the standard Metropolis HastingsMCMC to sample from the posterior.

Algorithm (Metropolis-Hastings MCMC), Robert andCasella (2004). Suppose at the rth step we are at the state τr ,then

• Step 1. Generate τ ∗ from q(τ ∗|τr ).• Step 2. Accept τ ∗ with probability

α(τr , τ∗) = min

⎧⎪⎪⎨⎪⎪⎩1,

P (d|τ ∗)P (yc|τ ∗)P (yo|τ ∗)

P (d|τr )P (yc|τr )P (yo|τr )︸ ︷︷ ︸likelihood ratio

× P (τ ∗)

P (τr )︸ ︷︷ ︸prior ratio

× qτ ∗ (τr |τ ∗)

qτ ∗ (τ ∗|τr )︸ ︷︷ ︸proposal ratio

⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

Starting with an initial parameters of the spatial sample τ0, theMCMC algorithm generates a Markov chain {τr}. The targetdistribution π (τ ) is the stationary distribution of the Markovchain τr , so τr represent the samples generated from π (τ ) afterthe chain converges and reaches a steady state. As an example wecan use standard random walk Metropolis-Hastings algorithmto generate samples from the posterior distribution. Then at therth step, we propose τ ∗ = τr + hτuτ , where uτ is generatedfrom a N (0, I ) distribution.

Here at each iteration step after we propose a new θ, l1, l2, σ2,

we have to solve the eigenvalue problem for the K-L expansionto get the fine-scale spatial realizations, which is very expensive.To speed up the computation, we compute the eigenvalue prob-lem (K-L expansion) for a certain number of pairs of l1, l2 be-forehand and interpolate them to find the eigenvalues and eigen-vectors at each step in the Metropolis-Hastings MCMC.

Note that change of σ doesn’t change the eigenvectors, it onlychanges the magnitude of the eigenvalues which can be adjustedby a scale factor. We can show that this approximation is validif the interpolation grid of the correlation length is sufficientlysmall. Since the magnitude of σ doesn’t effect the interpolationwithout loss of generality we can assume σ 2 = 1. Also here weprove only the isotropic case, that is, l1 = l2 = l.

Theorem 2. Suppose Al be the covariance matrix for agiven correlation length, l. Let λ1, λ2 . . . , λm be the m orderedeigenvalues considered in the K-L expansion of Al and letφ1, φ2 . . . , φm be the corresponding orthonormal eigenvectors.Suppose Al+δl be the covariance matrix if we perturb the corre-lation length l by a small quantity δl. Let λ′

1, λ′2 . . . , λ′

m be them ordered eigenvalues considered in the K-L expansion and letφ′

1, φ′2 . . . , φ′

m be the corresponding orthonormal eigenvectors.then, λ′

i = λi + O(δl), and φ′i = φi + O(δl), ∀i.

The proof is given in supplementary materials.

5. EXTENSION TO MODEL WITH UNKNOWN M

In Section 4, m, the dimension of θ remained fixed, so thenumber of terms retained in K-L expansion is taken to be aconstant. Usually it is estimated by using (6). This method onlyuses the fine-scale direct data yo but ignores the output data dand the coarse-scale data yc. That way this approach may notcapture the actual heterogeneity of the spatial field very well.We extend our previous model by treating m as an additionalmodel unknown and obtain its posterior distribution by condi-tioning on all the available data. In this situation, using Bayesian

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BAYESIAN UNCERTAINTY QUANTIFICATION FOR SUBSURFACE INVERSION USING A MULTISCALE HIERARCHICAL MODEL 387

hierarchical model the posterior can be written as

π (τ,m) = P (θ, l1, l2,mσ 2|d, yc, yo)

∝ P (d|θ, l1, l2, σ2,m)P (yc|θ, l1, l2, σ

2,m)

×P (yo|θ, l1, l2, σ2,m)P (l1, l2)P (σ 2)P (θ |m)P (m).

(16)

We keep all the model specifications same as in Section 3 butuse a truncated Poisson prior for P (m). We need to modify theMCMC computation procedure due to this unknown dimen-sion. If we vary the number of terms in the K-L expansionthen the dimension of θ will also change in each step. Thisjumping between different dimensions in the parameter spacecan be achieved through reversible jump Markov chain MonteCarlo methods as proposed by Green (1995). We describe the re-versible jump MCMC procedure in our case following the gen-eral approach of the reversible jump MCMC (Waagepetersenand Sorensen 2001).

We assume the prior for m|λ as truncated Poisson(λ), trun-cated at mmax, where λ ∼ Gamma(ν, β). Integrating out λ weget P (m) ∝ 1

(1/β+1)(m+ν+1) . All the other terms in (16) remainsame as in (14).

Algorithm (Reversible Jump MCMC as Birth and DeathProcess). Suppose at the rth step we are at the state (mr, τr ),then we have three possible steps:

• Birth Step: Propose to add the (mr + 1)th term in the K-Lexpansion with probability pb

mr. Propose θ ′ from q(.) and

hence θ∗ = (θr , θ′). The acceptance probability is given by

αmr,mr+1(θr , θ∗) = min{1,

π(θ∗,mr+1)pdmr +1

π(θr ,mr )pbmr

q(θ ′) }.• Death Step: Propose to delete the (mr )th term with prob-

ability pdmr

. So here (θ∗, θmrr ) = θr . The acceptance

probability is given by αmr,mr−1(θr , θ∗) = min{1,

π(θ∗,mr−1)pbmr −1q(θmr

r )π(θr ,mr )pd

mr

}.• Jump Step: Propose a new θ with the same dimension along

with l1, l2, σ2 with probability ps

mr. In other words generate

τ ∗ from q(τ ∗|τr ). The acceptance probability is given byα(τr , τ

∗) = min{1, π(τ ∗)q(τr |θ∗)π(τr )q(τ ∗|τr ) )}.

Here, pbmr

+ pdmr

+ psmr

= 1,∀mr .

6. TWO-STAGE REVERSIBLE JUMP MCMC

The main disadvantage of the above reversible jump MCMCalgorithm is the high computational cost of solving the forwardmodel on the fine-grid to compute G in the target distributionπ (τ,m). Typically, in our simulations, reversible jump MCMCconverges to the steady state after several iterations. That way,a large amount of CPU time is spent on simulating the rejectedsamples, making the direct (full) reversible jump MCMC simu-lations very expensive.

The direct reversible jump MCMC can be improved by adapt-ing the proposal distribution q(τ,m|τn,mn) to the target distri-bution using a coarse-scale model. This can be achieved bya two-stage reversible jump MCMC method, where at firstwe compare the output from the forward model on a coarse-grid. If the proposal is accepted by the coarse-scale test, thena full fine-scale computation will be conducted and the pro-

posal will be further tested as in the direct reversible jumpMCMC method. Otherwise, the proposal will be rejected bythe coarse-scale test and a new proposal will be generated fromq(τ,m|τn,mn). The coarse-scale test filters the unacceptableproposals and avoids the expensive fine-scale tests for thoseproposals. The filtering process essentially modifies the pro-posal distribution q(τ,m|τn,mn) by incorporating the coarse-scale information about the problem. The algorithm for a gen-eral two-stage MCMC method was introduced in Christen andFox (2005). Our hierarchical model can also take an advan-tage of inexpensive upscaled simulations to screen the propos-als. Here we extend the algorithm to two-stage reversible jumpMCMC method. Let F ∗

τ be the output computed by solvingthe forward model on a coarse-scale for the given fine-scalespatial field with parameters (τ,m). In the case of Reservoircharacterization (Section 2.1) this is done either with upscalingmethods or mixed MsFEM. The fine-scale target distributionπ (τ,m) is approximated on the coarse-scale by π∗(τ,m). Hereall the terms in the expression of π∗(τ,m) are same as that ofπ (τ,m) except only the likelihood term 1

[bf + 12 ||d−Fτ ||2](af +n/2) is

replaced by 1[bf + 1

2 H ||d−F ∗τ ||2](af +n/2) , where, the function H is esti-

mated based on offline computations using independent samplesfrom the prior. More precisely using independent samples fromthe prior distribution, the spatial fields are generated. Then boththe coarse-scale and fine-scale simulations are performed and‖d − Fτ‖ versus ‖d − F ∗

τ ‖ are plotted. This scatterplot datacan be modeled by ‖d − Fτ‖ = H (‖d − F ∗

τ ‖) + w, where w isa random component representing the deviations of the true fine-scale error from the predicted error. Using the coarse-scale dis-tribution π∗(τ ) as a filter, the two-stage reversible jump MCMCcan be described as follows.

Algorithm (Two-stage Reversible Jump MCMC as Birth andDeath Process). Suppose at the nth step we are at the state νn.Let kn be the corresponding fine-scale permeability field. Hereνn = (τn,mn)

• Step 1. This step is the same as the reversible jump MCMCmethod described earlier. The only difference is the frac-tional flow F ∗

ν is computed by solving the coarse-scalemodel. At νn, generate a trial proposal ν from distributionq(ν|νn) the same way as in the reversible jump MCMC de-scribed earlier, that is, this step is same as doing reversiblejump MCMC on π∗(ν).

• Step 2. Take the proposal as

ν ={

ν with probability αp(νn, ν),

νn with probability 1 − αp(νn, ν),

If we are at birth step, then the acceptance probability

is given by αp(νn, ν) = min{1,π∗(τ ,mn+1)pd

mn+1

π∗(τn,mn)pbmn

q(θ ′) }. If we are

at death step, then the acceptance probability is given by

αp(νn, ν) = min{1,π∗(τ ,mn−1)pb

mn−1q(θmnn )

π∗(τn,mn)pdmn

}. If we are going to

have only jumps then the acceptance probability is givenby αp(νn, ν) = min{1, π∗(τ ,mn)q(τn|τ )

π∗(τn,mn)q(τ |τn) )}.• Step 3. Accept ν as a sample with acceptance probability,αf (νn, ν) = min(1, π(ν)π∗(νn)

π(νn)π∗(ν) ).

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To show that the two stage reversible jump MCMC samplinggenerates a Markov chain, whose stationary distribution is thecandidate distribution it is sufficient to show that the transitionkernel satisfies the detailed balance condition.

Theorem 3. If K(νn, ν) is the transition kernel of the Markovchain νn generated by the two-stage reversible jump MCMC,then π (νn)K(νn, ν) = π (ν)K(ν, νn).

The proof is given in supplementary materials.

7. SIMULATION AND REAL EXAMPLES FROMRESERVOIR MODEL

The main goal in reservoir modeling is to infer about theimportant physical properties of the reservoir such as perme-ability, porosity, fluid saturation, oil-water and gas-oil contact,etc., which are the major contributors to the uncertainties inreservoir performance forecasting, using the direct and indirectobservations. In the following examples, we are particularly in-terested in quantifying and reducing the uncertainties for one ofthe major characteristics of subsurface property, permeability.Specifically, our goal is to infer about the fine-scale permeabilityspatial filed using the few fine-scale permeability data obtainedas well logs and cores, the coarse-scale data obtained fromseismic traces and the indirect observation from the productionhistory such as water-cut or fractional flow.

7.1 The Mathematical Model and Specification of G

The model is described in Section 2.1 as d = logit[G(Y )] + ε,where d is the watercut data, Y is the fine-scale permeabilityfield expressed in a logarithmic scale, that is, Y = log(kf ) andG is simulator output by using the log-permeability field Y .We consider a two-phase flow in a subsurface formation overa bounded set D ⊂ R2 under the assumption that the fluiddisplacement is dominated by viscous effects. For clarity ofexposition, we neglect the effects of gravity, compressibility,and capillary pressure, although our proposed approach is in-dependent of the choice of physical mechanisms. Furthermore,porosity φ will be considered to be known constant.

G is determined by combining the Darcy’s law with a state-ment of conservation of mass and thus solving the pressure andsaturation equations which are a couple of partial differentialequations. More details about the pressure and saturation equa-tions can be found on Efendiev et al. (2005); Efendiev, Hou,and Luo (2006). The fractional flow or water-cut F depends onthe total velocity v and the water saturation S, which are thesolutions of the pressure and saturation equations for a givenspatial permeability field kf (x) = exp(Y (x)) with some bound-ary conditions on S and p. In other words Y (x) is the input andF is the output for the forward simulator. So F can be writtenas F = G(Y (x)). Since F is always between 0 and 1 we takea logit transformation on G and write the forward model as:d = logit(G(Y (x))) + ε.

7.2 The Upscaling Procedure

Consider the fine-scale spatial field which is defined in thedomain with underlying fine grid as shown in Figure 2. On the

same graph, we illustrate a coarse-scale partition of the domain.Here we consider a single-phase flow upscaling procedure fortwo-phase flow in heterogeneous porous media. The main ideaof the calculation of coarse-scale permeability is that it deliversthe same average response of the forward model as that of theunderlying fine-scale problem locally in each coarse-block (seeChristie 1996; Durlofsky 1998; Efendiev et al. 2005; Efendiev,Hou, and Luo 2006). For each coarse domain K, we solve thelocal pressure equations in the fine-grid with some coarse-scaleboundary conditions. The approach considered here is to replacekf with upscaled coarse permeabilities, kc, which is constant ineach fine-grid within the same coarse block. By definition kc isa discrete quantity relying on the discretization of the medium.In particular, kc depends on the location and geometry of thegrid-block in which it is computed. The essential requirementof kc is that it leads to pressure and velocity solutions withdesired accuracy so that the average response of the forwardmodel in each coarse domain is almost the same as the responsefrom the fine-scale model. In our numerical examples, we takethe logarithm of the observed coarse-scale permeability as ourcoarse data, that is, yc = log(kc).

7.3 Numerical Results for Simulated Reservoirs

In our first example, we consider a simulated reservoir modelin which the unknown fine-scale permeability field is taken tobe a smooth spatial field on a 50 × 50 grid on the unit square.We consider only the isotropic case, that is, we take l1 = l2 = l.We generate 15 fine-scale permeability fields from a Gaussianfield with the squared exponential covariance structure withl = 0.25 and σ 2 = 1. The reference permeability field is takento be the average of these 15 permeability fields. In this model,water is injected at 6 injector wells along the two edges andoil and water is produced in one producing well at the center.The fractional flow or water cut data are generated by usingthe reference permeability field as inputs in the eclipse soft-ware (Eclipse 2010) and were validated by the petroleum en-gineering department at Texas A&M University. The observed

Figure 2. Schematic description of fine- and coarse-grids. Solidlines illustrate a coarse-scale partitioning, while dotted lines show afine-scale partitioning within coarse-grid cells.

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BAYESIAN UNCERTAINTY QUANTIFICATION FOR SUBSURFACE INVERSION USING A MULTISCALE HIERARCHICAL MODEL 389

Figure 3. Numerical results using two-stage reversible jumpMCMC for the simulated example. (a) The true (reference) fine-scalelog permeability field. (b) Initial fine-scale log permeability field. (c)The observed coarse-scale log permeability field. (d) The median ofthe sampled fine-scale log permeability field.

coarse-scale permeability field is calculated using the upscalingprocedure in a 5 × 5 coarse-grid. Our goal is to infer aboutthe fine-scale permeability field using the data at the welllocations, coarse-scale data, and the water cut data, and seehow closely the predicted field resembles the reference per-meability field. The prior for l is taken to be truncated Uni-form Distribution over (0.1, 0.5). The prior for σ 2 is assumedto be a Gamma distribution with hyperparameters as = 3 andbs = 2. The prior distribution for m is taken to be truncatedPoisson distribution truncated at 30, with hyperprior λ fol-lowing a Gamma distribution with hyperparameters ν = 4 andβ = 4.

First, we implement the reversible jump MCMC algorithmand draw 250,000 samples from the posterior. After 30,000 burnin period we retain every 10th sample from the posterior. Themode of the posterior distribution of m is 19. The posteriormedian of fine-scale permeability field is very close to the ref-erence permeability field. The mode of the posterior density ofl is near 0.25 and the posterior density of σ 2 is centered at 1,which are the corresponding original parameter values of the

Figure 5. Numerical results for the simulated example. (a) The firstquartile of the sampled posterior fine-scale log permeability field. (b)The third quartile of the sampled posterior fine-scale log permeabilityfield.

generated reference permeability field. Then we implement thetwo-stage reversible jump MCMC algorithm with the same ref-erence field and water cut data as we have used in the directreversible jump MCMC method. The two-stage reversible jumpMCMC produced the same results as the direct reversible jumpMCMC (see Figures 3 and 4). The two-stage algorithm is muchfaster as it rejects the bad samples in the first-stage, where wesolve the partial differential equations on a coarse-grid. The ef-fective acceptance rate of the two-stage algorithm increases toalmost 80%, whereas the regular reversible jump MCMC hasan acceptance rate of nearly 10%. We can see that although wehave taken almost flat priors for the parameters of our models,the posterior densities have peaks at the corresponding origi-nal values of the generated reference permeability field. Thus,we can conclude that integrating data from different sources inthe hierarchical model helped to reduce the uncertainties in theunknown fine-scale permeability field. To visualize the uncer-tainties in the prediction, we plot the first and third quartiles ofthe sampled fine-scale permeability field from the posterior inFigure 5.

In the next example, we consider a case where we assumethat coarse-scale data are not available but 10% of the fine-scaledata are available at equidistant points. We proceed with thesame reference permeability field and water-cut data. We cansee from Figure 6 (b) that the posterior median is not close to thetrue permeability field. Moreover, the posterior distribution of lis centered around 0.42 with standard deviation approximately0.01. The same procedure is replicated assuming 25% data avail-able (see Figure 6(c)). In this case, the posterior distributionof l is centered around 0.25, the true value of l, with standard

15 20 25 300

100

200

300

400

500

freq

uenc

y

m

(a)

0 0.1 0.2 0.3 0.40

5

10

15

20

l

pdf

(b)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

σ2

pdf

(c)

Figure 4. Posterior distributions using two-stage reversible jump MCMC for the simulated example. (a) Histogram of the posterior distributionof m for two-stage reversible jump MCMC. (b) Posterior density of l. (c) Posterior density of σ 2.

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Figure 6. (a) The true fine-scale log permeability field. (b) The median of the sampled fine-scale log permeability field with only 10%fine-scale data observed and no coarse-scale data available. (c) The median of the sampled fine-scale log permeability field with only 25%fine-scale data observed and no coarse-scale data available.

Figure 7. Results from two-stage reversible jump MCMC samplingfor the PUNQ-S3 model. (a) The true fine-scale log permeability field.(b) Initial fine-scale log permeability field. (c) The observed coarse-scale log permeability field. (d) The median of the sampled fine-scalelog permeability field.

deviation approximately 0.01. The posterior distribution of σ 2 iscentered around 1 with standard deviation approximately 0.17.Thus we can conclude that, if coarse-scale data are not available,10% fine-scale data are not enough to capture the parameters ofthe model; we need at least 25% of the fine-scale data to inferabout the model parameters.

7.4 Numerical Results for a Real Field Example

In this section, we apply our model on a real field exam-ple, that is, the PUNQ-S3 model dataset. The PUNQ-S3 caseis from a reservoir engineering study on a real field performedby Elf Exploration Production. It is qualified as a small-sizeindustrial reservoir engineering model. The model contains19 × 28 × 5 grid blocks. The PUNQ-S3 dataset was an exper-imental study where the true permeability was actually knownon the 19 × 28 × 5 grid, but the researchers were asked notto use the permeability data for their modeling purpose. Theywere asked to use the production history to infer about the truepermeability field and then compare how their model resem-bles the actual permeability field. For our example, we consideronly the top of the five layers in the dataset and follow thesame guidelines. We have used the production history, that is,the water-cut data, the permeability data on a 5 × 5 coarse-grid and the true fine-scale permeability data only on thewell locations to infer the fine-scale permeability field. The

20 22 24 26 28 300

200

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600

800

1000

1200

m

freq

uecy

(a)

0 0.1 0.2 0.3 0.40

2

4

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lx

pdf

(b)

4 6 8 10 12 14 160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

σ2

(c)

pdf

Figure 8. Posterior distributions for the PUNQ-S3 model. (a) Histogram of the posterior distribution of m. (b) Posterior density of l.(c) Posterior density of σ 2.

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BAYESIAN UNCERTAINTY QUANTIFICATION FOR SUBSURFACE INVERSION USING A MULTISCALE HIERARCHICAL MODEL 391

Figure 9. (a) The first quartile of the sampled posterior fine-scalelog permeability field. (b) The third quartile of the sampled posteriorfine-scale log permeability field for the PUNQ-S3 model.

permeability measurements are expressed in the unit ofmD, where 1mD = 10−3 Darcy = 10−12m2. We use a log-arithmic transformation of the permeability data and a logittransformation of the fractional flow data in our model. Thespatial locations of the field were given to the researchers ina transformed Cartesian coordinate system with each grid of180 × 180 square unit starting from the origin; that is, coordi-nate of the top-left grid block is (0,0) and that of the bottom-rightgrid block is (3420, 5040). For simplification, we make anothertransformation on the coordinate system to a (0,1) scale. Soin the transformed spatial domain the coordinate of the bot-tom right grid block is (0.6786,1) and each grid block is ofsize 0.0357 × 0.0357 square unit. The fine-scale permeabilityfield is taken to be known at six injector well locations alongx = 0 and x = 1 boundaries, that is, on the coordinates (0, 0),(0, 0.5), (0, 1), (0.678, 0), (0.678, 0.5), and (0.678, 1), and alsoat the producer well location at the center, that is, on thecoordinate (0.339, 0.5). The other inputs, such as pore vol-ume injected, porosity, water saturations, etc., are taken to beknown. We use a squared exponential structure for the priordistribution of the fine-scale log-permeability field while doingthe Karhunen-Loeve transform. We assume a proper prior forthe correlation length, which is uniform on a truncated space.We draw 200,000 samples from the posterior distribution usingtwo-stage reversible jump MCMC method. After 20,000 burn inperiod, we retain every 10th sample from the posterior. We cansee from Figure 7 that the posterior median of the fine-scale per-meability is very close to the true permeability field. The modeof the number of coefficients to be retained in KLE expansion isfound to be 24. The posterior mode of l is nearly 0.25 and thatof σ 2 is nearly 9 (see Figure 8). To visualize the uncertainties inthe prediction, we plot the first and third quartiles of the sampled

Figure 10. Results for the PUNQ-S3 model with no coarse-scaledata. (a) The true fine-scale log permeability field. (b) The median ofthe sampled fine-scale permeability field.

fine-scale permeability field from the posterior in Figure 9. Nextwe consider the model assuming no coarse-scale data available.From Figure 10 we can see that the posterior median is not veryclose to the true PUNQ-S3 model. Hence, we can conclude thatintegrating coarse-scale data in the model helps us to predictthe uncertainties in the reservoir more efficiently. The sum ofsquared errors between the true fine-scale permeability and theposterior median when we use the available coarse-scale datais 542.3. In contrast, when we only use the fine-scale data at afew well locations but no coarse-scale data, the sum of squarederrors becomes 1192.4.

8. CONCLUSIONS

We have developed a Bayesian multiscale hierarchical modelfor large-scale spatial inverse problems. Data from differentsources are integrated in the hierarchical model to reduce theuncertainties in the unknown spatial field. We have proven thatthe posterior is Lipschitz continuous with the data in total vari-ation norm, which ensures that the Bayesian inverse problemis well-posed. A two-stage MCMC technique is exploited forcomputational efficiency. We have applied our methodology tosimulated datasets as well as a real dataset.

Alternatively, statistical interpolation techniques like an em-ulator (Kennedy and O’Hagen 2001; Higdon et al. 2004; Oakleyand O’Hagan 2004) can be used in this problem. Developmentof a multiscale emulator (see, e.g., Craig et al. 1996, 1997;Cumming and Goldstein 2009; Vernon, Goldstein, and Bower2010) for our inverse problem, where the forward model canbe approximated by a Gaussian process regression or spline re-gression using some training sample of simulation runs, will beour challenging future project.

In the simulated and real field applications, we have assumedthat the unknown spatial field is stationary and smooth, whichmay not be true in many practical examples. For example, per-meability fields in reservoir models may have major faults orhigh permeability channel may be embedded in a nearly imper-meable background resulting in discontinuities and channelizedstructure of the spatial field. In such cases, our method can beextended to include a partition of the spatial field and then use apiecewise Gaussian process for the input spatial field (see Kim,Mallick, and Holmes 2005). For a channelized spatial field, theuncertainties in the channel boundaries can be quantified us-ing level-set parameterization (see Mondal et al. 2010) or othermeans and then within each channel K-L expansion of the spatialfield can be used.

Another important topic, which is not considered in this ar-ticle, is that the forward simulator G may not represent thephysical system perfectly. In that situation, the assumption ofindependent errors will not hold, and we need to add the modeldiscrepancy term as in Kennedy and O’Hagen (2001), Higdon,Lee, and Holloman (2003), Goldstein and Rougier (2006), andGoldstein and Rougier (2009). We have also assumed that thecombined model discrepancy error and measurement errors areindependent, which may not be true in general, for example, theoutputs may be correlated over time, in that situation we haveto account for the autocorrelation in the model.

Finally, in the real physical world, of course, we have to dealwith the three-dimensional reservoir model. One of the very

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popular and easiest way to tackle that problem is to model mul-tiple slices of the two-dimensional model. In that case, we canrun our algorithm for multiple slices simultaneously and thencombine the slices to get a three-dimensional model. However,the spatial correlation in such situations should have three com-ponents: one in each direction. Combining the slices in a rectan-gular grid may create difficulties in defining the edges especiallyfor faults, cross-stratified beds, etc. A more efficient way to builda three-dimensional model is to consider corner point geometry,which can represent complex reservoir geometries by specify-ing the corners of each grid block in grid building and we intendto build such three-dimensional models in our future projects.

SUPPLEMENTARY MATERIALS

Data Analysis and Proof of Theorems: A supplementary pdffile containing the data analysis and the proof of theorems(pdf file)

Data Set and Codes: A zip file containing the data and matlabcodes (zipped tar file).

ACKNOWLEDGMENTS

The authors acknowledge NSF-CMG. This work is partlysupported by Award No. KUS-C1-016-04, made by King Ab-dullah University of Science and Technology.

[Received December 2011. Revised August 2013.]

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