Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter

Authors


Abstract

[1] Model-based predictions of flow in porous media are critically dependent on assumptions and hypotheses that are not always based on first principles and that cannot necessarily be justified on the basis of known prevalent physics. Constitutive models, for instance, fall under this category. While these predictive tools are usually calibrated using observational data, the scatter in the resulting parameters has typically been ignored. In this paper, this scatter is used to construct stochastic process models of the parameters which are then used as the cornerstone in a novel model validation methodology useful for ascertaining the confidence in model-based predictions. The uncertainties are first quantified by representing the unknown model parameters via their polynomial chaos decompositions. These are descriptions of stochastic processes in terms of their coordinates with respect to an orthogonal basis. This is followed by a filtering step to update these representations with measurements as they become available. In order to account for the non-Gaussian nature of model parameters and model errors, an adaptation of the ensemble Kalman filter is developed. Instead of propagating an ensemble of model states forward in time as is suggested within the framework of the ensemble Kalman filter, the proposed approach allows the propagation of a stochastic representation of unknown variables using their respective polynomial chaos decompositions. The model is propagated forward in time by solving the system of partial differential equations using a stochastic projection method. Whenever measurements are available, the proposed data assimilation technique is used to update the stochastic parameters of the numerical model. The proposed method is applied to a black oil reservoir simulation model where measurements are used to stochastically characterize the flow medium and to verify the model validity with specified confidence bounds. The updated model can then be employed to forecast future flow behavior.

1. Introduction

[2] Numerical models that describe subsurface multiphase flows greatly enhance the process of reservoir characterization and management, as they improve the accuracy of production forecast. The validity of these reservoir models hinges on access to information regarding the geological and petrophysical features of the actual field. Automatic history matching is a commonly used methods for reservoir characterization whereby reservoir parameters such as porosities and permeabilities are estimated so as to minimize a measure of discrepancy between observations and predictions. The heterogeneities of the geological formations and the uncertainties associated with the medium properties contribute to errors incurred in any history matching effort.

[3] Recently, the ensemble Kalman filter (EnKF) [Evensen, 2003] has been shown to be an efficient tool for history matching of reservoir models [Gao et al., 2006; Gu and Oliver, 2005; Liu and Oliver, 2005; Naevdal et al., 2003] enabling useful estimation of various reservoir parameters. The EnKF uses a Monte Carlo simulation scheme for characterizing the noise in the system, and therefore allows the representation of non-Gaussian perturbations. The performance of the EnKF has been assessed in a number of efforts. For example, in the assimilation of data for land surface models [Zhou et al., 2006] conditional marginal distributions and moments estimated from the EnKF have been compared with estimates obtained from a sequential importance resampling (SIR) particle filter. It has been found, expectedly, that the accurate estimation of higher-order statistics requires a large ensemble size that is often computationally prohibitive. Moreover, current data assimilation techniques provide a map between the probability measure of input parameters and that of predicted observables. This distributional mapping does not lend itself to sensitivity analysis and uncertainty apportionment, as the functional dependencies are smeared out when probability measures are estimated. Furthermore, current procedures do not permit the straightforward estimation of modeling errors associated with the structure of the forward model.

[4] This paper presents a history matching methodology whereby the functional relationship between parameters (described as stochastic processes) and predictions (also described as stochastic processes) is itself approximated as a stochastic mapping. A description of this functional relationship permits the rapid assessment of stochastic sensitivities. Furthermore, the structured construction of an approximation to this functional relationship provides a methodology for model validation that will also be explored in the paper. Moreover, relying on concepts of Hilbertian projections and L2 convergence from approximation theory, it can be shown that a polynomial chaos description [Ghanem and Spanos, 2003] of this functional relationship ensures that the complete probabilistic content of the predicted solution is well approximated, including higher-order statistics and joint distributions. A polynomial chaos expansion of a random variable or process is a series expansion of the random quantity in terms of orthogonal (i.e., uncorrelated) random variables, a stochastic basis, each multiplied by a deterministic coefficient (the polynomial chaos coefficients). Thus, by integrating polynomial chaos representations of stochastic processes with the EnKF methodology, it becomes possible to mitigate the computational cost of the standard EnKF while yielding a robust methodology for exploring the effect of errors in model structure on model-based predictions. Efforts along similar lines have already been reported whereby polynomial chaos representations have been used to accelerate the convergence of Markov chain Monte Carlo methods used in Bayesian estimation [Marzouk et al., 2007].

[5] The computational model for the stochastic response is first constructed by representing the unknown model parameters, as well as unknowns describing errors in model structure, via their polynomial chaos decompositions. A solution is then sought in the form of a polynomial chaos expansion. This yields a system of coupled nonlinear stochastic partial differential equations to solve for the polynomial chaos coefficients of the solution. The spectral stochastic finite element method (SSFEM) [Ghanem and Spanos, 2003], which relies on a stochastic Galerkin projection scheme, is employed to solve this system using a discrete time-marching scheme. The SSFEM is integrated within Sundance 2.0 [Long, 2004], a computational toolkit for the finite element formulation and solution of partial differential equations, for enhanced computational efficiency. Uncertainty propagation is then followed by a filtering step to update the polynomial chaos representations with measurements as they become available. During the filtering step, and since the predicted solution has been described by its polynomial chaos decomposition, its full joint probability distribution is readily computable, thus allowing data assimilation in the presence of non-Gaussian measurement and model errors, as well as non-Gaussian uncertainties in model and parameters. The computational load for a specified accuracy is comparable to that of the EnKF and the storage size is limited to the number of terms in the polynomial chaos expansion of the model states.

[6] The next section presents the details of the particular model of multiphase flow in porous media that is used to demonstrate the foregoing methodology. The equations governing multiphase flow in a random porous medium are derived using the polynomial chaos formulation. Section 3 gives a background of Kalman filtering techniques and their evolution from the standard Kalman filter [Kalman, 1960] to the EnKF. Then a detailed derivation of the polynomial chaos-based Kalman filter is presented. In section 4, the uncertainty representation is discussed leading to the derivation of the stochastic reservoir numerical model. Finally, the application of the polynomial chaos Kalman filter for reservoir characterization problems is presented, results are described, and a discussion of the validity of the proposed method is provided.

2. Multiphase Flow in Porous Media

[7] The general form of the equations describing mass conservation in a multiphase flow system can be expressed as

equation image

where i refers to a specific fluid phase, ϕ is the porosity of the medium, ρi is the density of phase i, Si is the phase saturation, equation imagei is the phase velocity, qi is a source-sink term, and t is time. Darcy's law is commonly used for calculating phase flow velocities. Although initially developed for single phase flows, it was experimentally shown [Muskat, 1949] that Darcy's law provides an acceptable model for each fluid separately when two or more immiscible fluids share the pore space. Darcy's law for phase velocities is given by

equation image

where equation image is the intrinsic permeability tensor, Kri is the relative permeability of phase i, μi is the viscosity of phase i, Pi is the pressure of phase i, g is the gravitational acceleration, and z is the depth of the fluid. Substituting equation (2) into equation (1) yields the general form of the flow equations for all phases,

equation image

[8] The latter set of equations signifies that flow in the porous medium is driven by gravity, pressure gradients, and viscous forces. It also incorporates the effects of porous matrix compressibility, fluid compressibility, capillary pressure, and spatial variability of permeability and porosity. The nonlinearity arises from the constitutive relations relating the phase relative permeabilities and capillary pressure to the phase saturations. In the present paper, the Brooks-Corey empirical model [Brooks and Corey, 1964] is used to define these relationships in the form

equation image
equation image

where Krw and Krn are the relative permeability of the wetting nonwetting phases, respectively, λ is a model fitting parameter related to the pore size distribution of the soil material, and Se is the reduced saturation given by

equation image

where Sm is the maximum wetting phase saturation and Swc is the critical wetting phase saturation. The capillary pressure is the pressure difference across the interface of the coexisting fluid phases in the subsurface. It is defined as

equation image

where pc is the capillary pressure, pnw is the pressure of the nonwetting phase, and pw is the pressure of the wetting phase. Brooks and Corey's equation for the capillary pressure is

equation image

where pd is the displacement or threshold pressure which first gives rise to the nonwetting phase permeability.

[9] The present study deals with the two-phases immiscible flow problem, typically used for modeling oil reservoirs. Initially the porous medium is assumed to be fully saturated with oil, and water is pumped through one well to push the oil out through other wells in the field. The governing flow equations consist of the water continuity equation,

equation image

and the oil continuity equation,

equation image

Assuming oil is the nonwetting phase, these equations are subject to the following constraints:

equation image
equation image

[10] The objective of this study is to present an efficient reservoir characterization technique to estimate the above model parameters, the medium's intrinsic permeability and porosity. Since these parameters are heterogeneous in nature, and possess a high range of variability, it is of utmost importance to properly characterize their associated uncertainties. The polynomial chaos methodology has proved to be an efficient uncertainty representation technique, and hence an automatic history matching technique based on coupling the ensemble Kalman filter with the polynomial chaos is proposed.

3. Stochastic Filtering

3.1. Background on Kalman Filters

[11] The Kalman filter (KF) [Kalman, 1960] is an optimal sequential data assimilation method for systems driven by linear dynamics and involving measurement processes with Gaussian errors. Subject to these assumptions of linearity and Gaussianity, the KF provides an unbiased, minimum variance estimate of the state of the system from noisy measurements. The methodology consists of a forecast step to propagate the model state forward in time and an assimilation step in which variables describing the state of the system are corrected to honor the observations. The linearity of the model and Gaussianity of the errors permits a closed form expression for the filtered (updated) state. Consider a linear system with a state space representation,

equation image

where yk represents the model state vector at time instant k, Ak is a constant known matrix, and ξk is a sequence of zero-mean Gaussian white noise representing the model errors. Under conditions of linear dynamics, a recursive equation for forecasting the error covariance, Pk = E[(yktyk)(yktyk)T] where ykt is the true value of the state vector at time step k, is obtained in the form

equation image

where the matrix Q = Var(ξk).

[12] For simplicity, the time index will be ignored in the sequel, and the model forecast and analysis will be referred to as yf and ya, respectively. Furthermore, the respective covariances for model forecast and analysis are denoted Pf and Pa. Consequently, the analysis step is given by

equation image
equation image

where H is the measurement operator relating the true model state yt to the observations d,

equation image

ε is the measurement error vector assumed to be a Gaussian white noise independent of the model error {ξk}, and R = Var(ε) is the measurement error covariance matrix.

3.2. Extended Kalman Filter

[13] For nonlinear systems, a linearization process can be used to approximate the nonlinear dynamics, thus paving the way for an application of Kalman filtering on the approximate model. A first-order Taylor approximation through which nonlinear functions are recursively linearized around the most recent estimate is typically used. The resulting filter is known as the extended Kalman filter (EKF) [Chui and Chen, 1991]. Note that a model nonlinearity also arises when solving parameter estimation problems even when the underlying dynamics is linear. In this case, the parameters to be estimated are typically appended to the state vector, and the filtering process carried out for the augmented state.

3.3. Ensemble Kalman Filter

[14] The ensemble Kalman filter [Evensen, 1994; Burgers et al., 1998] aims at resolving some of the drawbacks of the EKF. The EnKF is based on forecasting the error statistics using Monte Carlo sampling which turns out to be a computationally more efficient procedure than the EKF, which typically involves the inversion of a large matrix associated with the error covariance matrix. The EnKF was originally designed to resolve two major problems related to the use of the EKF with nonlinear dynamics in large state spaces. The first is the approximate closure scheme adopted by the EKF, and the second is the significant computational requirement associated with the storage and forward integration of the error covariance matrix [Evensen, 2003]. Although the EnKF resolves some of the problems of the EKF, it has some disadvantages depicted when small size ensembles are used to represent the model states. It has been shown [Houtekamer and Mitchell, 1998] that for ensemble sizes less than or equal to 100, the EnKF fails at approximating the model state statistics properly. The procedure developed in this paper addresses this very issue by providing a more efficient method for propagating uncertainties from parameters to model-based predictions.

3.4. Polynomial Chaos Expansion

[15] Uncertainties associated with model-based predictions can be attributed to two sources: (1) irregular phenomena which cannot be described deterministically by the mathematical instrument used to interpret physical evidence and (2) phenomena that are not necessarily inherently uncertain but for which paucity of experimental data can be described, to benefit, as a random effect [Ghanem and Spanos, 2003]. Let ξ(θ) be a vector of independent random variables, denoting the independent sources of uncertainty, where θ references the probabilistic character of ξi. It is noted that the identification of a suitable set of such ξ is a modeling decision that could be based on either first principles or expert opinion, and should address both sources of uncertainty mentioned above. The polynomial chaos (PC) representation then prescribes a mathematical representation for general, mean square bounded, nonlinear functionals of ξ(θ). Typically, nonlinear functionals of ξ are expanded with respect to a basis in the space [Soize and Ghanem, 2004] of square-integrable random variables. A convenient set of such bases is provided by multidimensional polynomials that are orthogonal with respect to the joint probability measure of the vector ξ. According to this construction, if the random variables ξ are Gaussian, the basis functions are obtained as the set of multidimensional Hermite polynomials. The deterministic coefficients in this expansion provide a parameterization of the probabilistic description of the function being expanded, and they uniquely characterize it in an L2 sense as well as in distribution. Thus, if we are provided with a construction procedure, for the coefficients in the PC expansion of some random quantity, that assures L2 convergence to some other random quantity, then we are also assured that the corresponding (multidimensional) probability distributions will also converge. In the case where some aspect of parametric uncertainty exhibit spatial fluctuations and is best modeled as a random field, then a truncated Karhunen-Loeve expansion [Loeve, 1977] is first used to discretize the random field using a finite set of orthogonal random variables which are then appended to the set ξ efficiency. To clarify the PC expansion further, consider random process u(x, θ) which depends on random variables ξ and assume that ∫∫Ruu(x, y)dxdy < ∞, where Ruu(x, y) denotes the covariance of u(x, θ), the PC expansion of u(x, θ) can then be written as

equation image

where Γn (ξi1, …, ξin) denotes the nth-order polynomial chaos in the Gaussian variables (ξi1, …, ξin). These are multidimensional Hermite polynomials of their arguments, and ai1,…,iN are deterministic coefficients in the expansion [Wiener, 1938]. Introducing a one to one mapping to a set of ordered indices and truncating the polynomial chaos expansion after the Pth term, the above equation can be rewritten as

equation image

These polynomials are orthogonal with respect to the Gaussian measure. As an example, the polynomials up to third order are explicitly given as

equation image
equation image
equation image
equation image

where δij denotes the Kronecker delta.

[16] A truncated polynomial chaos can be refined by either adding more random variables to the set ξ(θ) (increasing the random dimension) or by increasing the order of the polynomials in the polynomial chaos expansion. Note that the total number of terms P + 1 in a polynomial chaos expansion with order less than or equal to p in M random dimensions is given by

equation image

It should also be noted that the mean of the stochastic process is given by the zeroth-order term in the expansion, while the covariance of the process is readily obtained given the orthogonality of the PC basis. Our objective in the sequel will be to propagate the uncertainty in model-based predictions by relying on PC representations of random processes rather than their probability density functions. The task of filtering the state (which contains the dynamic state as well as the parameters to be identified) of the system will thus consist of filtering the coefficients in the PC expansions of the state. We will also develop an algorithm for updating the deterministic coefficients in the various PC expansions following observations.

3.5. Polynomial Chaos-Based Kalman Filter (PCKF)

[17] The filter developed in the present paper allows the propagation of a polynomial chaos representation of the unknown variables. The computational burden is thus shifted from the propagation of an ensemble of realizations, when using the EnKF, to that of estimating the evolution of the PC coefficients of the state. This task is carried out using the spectral stochastic finite element formulation [Ghanem and Spanos, 2003]. Since the dynamic state of the system is constructed as a function of system parameters, joint probabilistic description of the state of the filter (which contains both the dynamic state and the parameters to be estimated) is available at any time instant. Furthermore, this approach allows the representation of non-Gaussian measurement and parameter uncertainties in a simple manner without the necessity of managing a large ensemble.

3.5.1. Representation of Error Statistics

[18] It is usual in the Kalman filter and its many variants, to define the error covariance matrices of the forecast and analyzed estimate relative to the true state, in the following form:

equation image
equation image

where 〈〉 denotes the operation of mathematical expectation, y is the model state vector at a particular time, and the superscripts f, a, and t refer to the forecast, analysis, and true state, respectively. However, in the polynomial chaos-based Kalman filter (PCKF), the true state is not known, and therefore the error covariance matrices are defined using the polynomial chaos representations of the model state. In the PCKF, the model state is given by

equation image

where P + 1 is the number of terms in the polynomial chaos expansion of the state vector, and {ψi} is the set of Hermite polynomials. Consequently, the covariance matrices are defined around the mean of the stochastic representation and are given by

equation image
equation image

As indicated previously, the polynomial chaos representation captures all the information available through the joint probability density function, and therefore permits the evaluation of the probability of any event of interest. Under some general conditions on integrability, the moments of the random quantities of interest are also accurately estimated.

[19] As observations accumulate, corrupted by measurement errors, they contribute to lengthen the vector ξ and the forecast dynamics depends on a vector of increasingly greater length. This explosion in dimension of the functional dependence of the forecast is mitigated by introducing a finite memory window for the measurements, thus effectively forcing the forecast to only depend on a finite number of the most recent observations.

3.5.2. Analysis Scheme

[20] For computational and algorithmic simplicity, and without loss of generality, the dimension and order of the polynomial chaos expansion are fixed throughout forecast and filtering. The length of the vector ξ is thus kept fixed, as is the number of terms retained in the PC expansion of the state. The significance of an individual ξi, however, is allowed to vary to reflect new, independent, measurements. It will also be assumed that the model state and measurement vectors are statistically independent, leading to a decoupling of the multidimensional polynomial expansion into two sets of shorter lower-dimensional expansions.

[21] Let A be the matrix holding the chaos coefficients of the state vector yRn,

equation image

where P + 1 is the total number of terms in the polynomial chaos representation of y and n is the size of the model state vector. The mean of y is stored in the first column of A and is denoted by y0 and the state error covariance matrix PRn×n is obtained, as explained previously, as

equation image

Given a vector of measurements dRm, with m being the number of measurements taken at any given time, a polynomial chaos representation of the measurements is defined as

equation image

where the mean d0 is given by the actual measurement vector, and the higher-order terms represent the measurement uncertainties. As D depends only on a subset of the random variables ξ, the above representation is sparse. Moreover, a Gaussian model for measurement errors, if justified by evidence, can be easily rendered using only linear polynomials in Gaussian variables. The representation of D can be stored in the matrix

equation image

From equation (32), the measurement error covariance matrix can also be constructed as

equation image

The forecast step is carried out by employing a stochastic Galerkin scheme [Ghanem and Spanos, 2003]. This schemes ensures that the computed PC representation does indeed converge in mean square, as the length of the PC expansion increases, to the solution of the stochastic equation. The filtering step involves the a minimum mean square error estimator of the polynomial chaos coefficients of the model state vector,

equation image

Projecting on an approximating space spanned by the polynomial chaos {ψi}i=0P yields

equation image

In matrix form, the assimilation step is thus expressed as

equation image

where † denotes the pseudoinverse.

4. Stochastic Reservoir Model

[22] The heterogeneity in the porous medium is treated probabilistically by modeling the intrinsic permeability and porosity of the medium as stochastic processes [Ghanem and Dham, 1998; Rupert and Miller, 2007]. In this paper, two distinct models are used for representing the uncertain medium properties. In the first, the intrinsic permeability and porosity are represented as stochastic processes using their respective polynomial chaos expansions,

equation image
equation image

where {Ki (x)} and {ϕi (x)} are sets of deterministic functions to be estimated using the proposed sequential data assimilation technique.

[23] The second model represents the intrinsic permeability of the porous medium as a stochastic process while modeling the porosity as a random variable. Therefore the ratio, α(x, θ), of the intrinsic permeability and porosity of the medium is represented as

equation image

where {αi(x)} is a set of deterministic functions to be estimated using the proposed sequential data assimilation technique. Although both models provide suitable alternatives, the first requires additional assurances that the quantities K and ϕ remain positive. It is noted that an analytical form has been derived for the polynomial chaos expansion of lognormal random variables and processes [Ghanem, 1999], and could be used for representing K or ϕ if justified by experimental evidence.

[24] The solution of the resulting system of stochastic partial differential equations is also represented via polynomial chaos,

equation image
equation image

where {Pwk} and {Swk} are the deterministic nodal vectors to be solved for, P denotes the number of terms in the polynomial chaos expansion, and {ψk(ξ)} is a basis set consisting of orthogonal polynomial chaoses of consecutive orders.

[25] The relative permeabilities and the capillary pressure are functions of water saturation, and therefore, they are represented using their polynomial chaos expansion as well. Fourth-order polynomial approximations of the functional forms appearing in the Brooks-Corey model are adopted and the relative permeabilities and capillary pressure are thus approximated as

equation image
equation image
equation image

where η represents the modeling error, which is itself described by its polynomial chaos expansion, and the coefficients {ci} are numerically calculated on the basis of the value of the Brooks-Corey model fitting parameter, λ. These parameters are fitted once only, prior to the EnKF estimation. By introducing the model error term η(ξ), it becomes possible to express the forecast state in terms of η, and hence ascertain the sensitivity of the prediction with respect to linear perturbations to model structure. In addition, coefficients in the PC expansion of η are themselves updated as part of the filtering process.

[26] The error resulting from truncating the polynomial chaos expansions at a finite number of terms is minimized by forcing it to be orthogonal to the solution space spanned by the basis set of orthogonal stochastic processes appearing in the polynomial chaos expansion [Ghanem and Spanos, 2003]. This yields a system of nonlinear coupled partial differential equations to be solved for the deterministic coefficients of the water saturation and pressure [Hatoum, 1998]. If the first model is adopted, the system is represented as

equation image
equation image

where the expectations 〈ψiψjψm〉 and 〈ψiψjψkψm〉 are easily calculated [Ghanem and Spanos, 2003]. On the other hand, when the second model is employed, the resulting system is expressed as

equation image
equation image

[27] Evaluating the above expectations is greatly facilitated, for a lognormal model for α, by using the following relationship:

equation image

Applying a change of variable, ξα1 = u, and employing the following Hermite polynomial identity:

equation image

These expectations are transformed into linear combinations of terms of the form 〈ψiψjψm〉.

5. PCKF for Stochastic Reservoir Characterization

[28] Two synthetic sets of problems are designed to assess the efficiency of the proposed history matching technique. The first one explores the one-dimensional two-phase water flooding, while the second one is a two-dimensional two-phase problem. In both sets, the model state vector consists of all the reservoir variables that are uncertain and need to be specified. These include the phase saturations and pressures as well as the probabilistic representations of the medium properties described previously. The objective is to estimate the polynomial chaos coefficients of the medium properties through measurements of the water saturation at selected locations. The two problems are selected with an eye toward demonstrating the flexibility of the proposed procedure in terms of attributing modeling errors at intermediate components of the model and not just at the predictive level. The ability to handle non-Gaussian randomness is also highlighted.

[29] The modeling and measurement errors at any given time, are assumed to be independent stochastic processes, and are therefore represented using their respective polynomial chaos expansions. Unlike the EnkF where the model error is represented using an additive noise, in PCKF the model error is incorporated in the Brooks-Corey model. In order to accommodate the fact that the medium properties may deviate from Gaussianity, the unknown porosity and permeability of the medium are either modeled as dependent one-dimensional, second-order polynomial chaos expansions according to the first model proposed earlier, or their ratio is expressed as an exponential function of a multidimensional first-order polynomial chaos expansion as explained in the second model. Table 1 details the uncertainty representation in the numerical model.

Table 1. Uncertainty Representation Using Polynomial Chaos
Source of UncertaintyRepresentation
Parametric (1)ξi
Parametric (2)exp(ξi)
Modelingξj
Measurementξk

[30] The model state vector in all the following examples is an augmented vector consisting of the dynamic state vectors and the model parameters. The dynamic state consists of the water saturation and pressure for each grid in model, and the model parameters are the porosity and permeability also defined for each grid. The initial state PC representation is constructed on the basis of assumed prior geostatistical information.

[31] First a set of one-dimensional problems is analyzed. The goal behind these examples is twofold. First, it is intended to show the validity of the proposed approach by demonstrating convergence to the exact values of the parameters in the absence of modeling errors. Second, we capitalize on the ease of visualizing one-dimensional results, to demonstrate the flexibility of the method at capturing the water front for complex spatial profiles of model parameters. For all the one-dimensional cases, model 1 (equations (38) and (39)) is used to represent the model parametric uncertainties.

[32] The approach is also applied on a set of two-dimensional problems. Two particular problems are investigated. The first example uses equations (38) and (39) to represent the parametric uncertainties, while the second adopts equation (40). The aforementioned examples reveal the efficiency of the proposed methodology under different modeling assumptions.

[33] Last, to assess the importance of the different terms in the polynomial chaos expansion used to represent the unknown model parameters and to investigate the importance of interdependent noise sources, a variation of the second two-dimensional example employing a higher-dimensional representation of the parametric uncertainty is examined.

5.1. One-Dimensional Buckley-Leverett Problem

[34] The first test problem is that of the incompressible water flood Buckley-Leverett example. The relative permeabilities within the model are given by the Brooks-Corey model. The reservoir is horizontal with a length of 304.8 m, cross-sectional area of 929 m2, and constant initial water saturation Swi = 0.16. Oil is produced at x = 304.8 m at a rate of 12.1 m3/d, and water is injected at x = 0 at the same rate. Three case studies are developed around this problem. In all three cases, model one is adopted for representing the parametric uncertainties. It is known that for this problem, the flux function is independent of intrinsic permeability, and that observations of saturations will therefore contain no information regarding permeability. We note, however, that both saturation and pressure data are being assimilated in the present cases.

5.1.1. Case 1: Homogeneous Medium

[35] In the absence of modeling errors, it is expected that as more observations are assimilated, the exact parameters of the underlying model are recovered. To test the validity of the proposed approach under this limiting assumption, the same Brooks-Corey model parameter, λ, is assumed for both the forward and inverse analysis. The field is assumed to be homogeneous with an intrinsic permeability of 270md and a porosity of 0.275, and measurements of the water saturation and pressure are available each 15.24 m every 10 time steps (2 h).

[36] Three sources of uncertainty are built into the assimilation model. The modeling uncertainties are represented via a one-dimensional second-order PC representation in terms of ξ1. The measurement and parametric errors are modeled as independent one-dimensional, first-order PC representations, in terms of ξ2 and ξ3, respectively. Given that the same model used to generate the data is used for assimilation, it is anticipated that all uncertainty sources will decay as more measurement data is assimilated.

[37] Figure 1 gives the statistical properties of the estimated parameters; it represents the variation of these parameters with time. It is noticed that as more measurements are available, the mean estimate converges toward the true model parameters, and the polynomial chaos coefficients decay exponentially indicating a deterministic estimate. This is expected since the model used to estimate the reservoir state is identical to the model used for generating the measurements.

Figure 1.

Case 1 estimate of medium properties: (a) mean intrinsic permeability, (b) polynomial chaos coefficients of intrinsic permeability, (c) mean porosity, and (d) polynomial chaos coefficients of porosity.

5.1.2. Case 2: Continuous Intrinsic Permeability and Porosity Profiles

[38] The presence of modeling errors introduces uncertainties on the estimated parameters. In this case, as more data is assimilated, scatter will remain in the estimated parameters. Thus, without introducing any extrinsic uncertainties, in the present case, we estimate the PC expansion of the porosity and intrinsic permeability from which probability density functions are synthesized. In this case continuous profiles for the porosity and intrinsic permeabilities are assumed. Measurements of the water saturation and pressure are also assumed available at 15.24 m intervals every 10 time steps. Furthermore, the Brooks-Corey model used to generate the measurements has a fitting parameter λ equal to 2.2 while the model used in filtering assumes a value of λ equal to 2.0. This results in uncertainties in the estimate due to modeling errors.

[39] Modeling uncertainties are thus represented via a one-dimensional second-order PCE, represented by ξ1. Measurement errors are assumed Gaussian and independent of the modeling uncertainties and thus represented using a one-dimensional polynomial first-order PCE, in terms of ξ2. To investigate the interdependence between different noise sources, the parametric uncertainties are assumed fully dependent on both measurement and modeling errors.

[40] Figure 2 represents the polynomial chaos estimate of the medium properties obtained after 2000 updates. Figure 3 shows the probability density functions (pdf's) of the estimated porosity and intrinsic permeability at quarter span. It is clear from the obtained pdf's that the porosity and intrinsic permeability have non-Gaussian properties. In order to validate the stochastic model associated with estimated parameters, the forecast mean water saturation is plotted against the true model state in Figure 4. Also shown in Figure 4 are plots of the higher-order terms of the PC expansion of the water saturation. It is emphasized again that, in this example, the parameters of the “truth” model from which the observables were synthesized are deterministic. The model from which forecasts are generated, however, is different from the “truth” model (different values of λ). This deterministic modeling error engenders a scatter in the parameters of the assimilation model. This scatter is interpreted within a probabilistic context using a PC representation. The value of this probabilistic interpretation is in the resulting ability to calculate the corresponding scatter implied on the predictions.

Figure 2.

Case 2 estimate of medium properties: (a) mean intrinsic permeability, (b) polynomial chaos coefficients of intrinsic permeability, (c) mean porosity, and (d) polynomial chaos coefficients of porosity.

Figure 3.

Case 2 probability density function of the estimated medium properties: (a) intrinsic permeability and (b) porosity.

Figure 4.

Case 2 (a) mean water saturation and (b) polynomial chaos coefficients of the estimated water saturation at ΔT = 450.

5.1.3. Case 3: Discontinuous Porosity and Intrinsic Permeability Profiles

[41] Discontinuities in the parameters being estimated presents a challenge to many data assimilation and inversion techniques. This case study investigates the behavior of the proposed procedure under conditions of discontinuities in porosity and intrinsic permeability.

[42] In this example, the measurements of the water saturation and pressure are also assumed available at 15.24 m intervals every 10 time steps. Furthermore, the Brooks-Corey model used to generate the measurements has a fitting parameter λ equal to 2.2 while the model used in filtering assumes a value of λ equal to 2.0. To better represent the scatter in the estimated parameters, parametric uncertainties are represented using a one-dimensional second-order PCE, in terms of ξ1. On the other hand, modeling and measurement errors are assumed Gaussian and independent and therefore represented as functions of ξ2 and ξ3, respectively.

[43] Figure 5 presents the estimated medium properties. It is observed here that the location of the jumps is well delineated both in the estimated mean and higher-order chaos terms. It is also noted that the dependence on ξ1, the parameter uncertainty, is more significant than any other component. In Figure 6, the estimated water saturation profile is plotted along with the true model state at different times during the simulation. A visual inspection of these results indicates that the estimated stochastic model does have a predictive value. It is noted from Figure 6b that the modeling uncertainty ξ2 significantly contributes to the uncertainty in the estimated state. This is consistent with its interpretation as the uncertainty associated with the Brooks-Corey model.

Figure 5.

Case 3 estimate of medium properties: (a) mean intrinsic permeability, (b) polynomial chaos coefficients of intrinsic permeability, (c) mean porosity, and (d) polynomial chaos coefficients of porosity.

Figure 6.

Case 3 (a) mean water saturation and (b) polynomial chaos coefficients of the last estimated water saturation.

5.2. Two-Dimensional Water Flood Problem: Model 1

[44] Two example problems are studied to explore the two-dimensional water flood example. In these problems, the relative permeabilities are also described by a polynomial fit to the Brooks-Corey model. While the first example adopts the first model (equations (38) and (39)) to represent the parametric uncertainties, model 2 (equation (40)) is used in the second.

[45] The first example consists of a rectangular reservoir with a length of 18.3 m, width of 18.3 m, cross-sectional area of 0.093 m2, and constant initial water saturation Swi = 0.20. Oil is produced at x = 18.3 m at a rate of 0.0098 m3/d, and water is injected at x = 0 at the same rate. Measurements of the water saturation are available at 20 equidistant points within the domain at a frequency of 10 time steps.

[46] Measurement data is generated using a different fitting parameter λ for the Brooks-Corey model than the one used in the assimilation process. Modeling uncertainties are represented as one-dimensional second-order PCE in terms of ξ1. Measurement errors are assumed independent of modeling errors and thus modeled as a function of a new noise source, ξ2. An additional noise source, ξ3 is introduced to represent the initial parametric uncertainties, and therefore the estimated parameters are spatially varying functions of ξ1, ξ2, and ξ3.

[47] Figure 7 gives the means of the estimated medium properties. Although the estimated and true parameters are different, it can be noticed from Figure 8 that the predicted water front is captured accurately. This difference, also implied in the uncertainty in the estimated parameters which is visible in Figures 9 and 10, is mainly due to modeling assumptions made prior to solving the reservoir characterization problem. These assumptions are a reflection of the lack of knowledge relating to the constitutive laws governing the behavior of the true relative permeabilities and capillary pressure of the medium.

Figure 7.

(a) Mean of estimated intrinsic permeability, (b) true intrinsic permeability, (c) mean of estimated porosity, and (d) true porosity.

Figure 8.

(a) Mean of the estimated water saturation and (b) true water saturation.

Figure 9.

Polynomial chaos coefficients of the estimated intrinsic permeability: (a) K1 is the coefficient of ξ1, (b) K2 is the coefficient of ξ2, (c) K3 is the coefficient of ξ3, and (d) K4 is the coefficient of ξ12 − 1.

Figure 10.

Polynomial chaos coefficients of the estimated intrinsic porosity: (a) P1 is the coefficient of ξ1, (b) P2 is the coefficient of ξ2, (c) P3 is the coefficient of ξ3, and (d) P4 is the coefficient of ξ12 − 1.

5.3. Two-Dimensional Water Flood Problem: Model 2

[48] The second model (equation (40)) is used to represent the parametric uncertainty in the second group of problems. Furthermore, a more complex physical setup is modeled to reflect the efficiency of this approach. This example consists of a rectangular domain of a length of 30.48 m, width of 18.3 m, cross-sectional area of 0.093 m2, and constant initial water saturation Swi = 0.20. Oil is produced from one well at a rate of 0.2253/d, and water is injected from two different point sources at rates of 0.108 m and 0.116 m3/d, respectively. Figure 11 shows the location of these wells along with the measurement locations. In this example, measurements are also available at a frequency of 10 time steps. Figure 12 presents the medium properties used for generating the synthetic measurement data. Measurement data consists of water saturations and is generated using a different fitting parameter λ for the Brooks-Corey model than the one used in the assimilation process.

Figure 11.

Example 2 problem description.

Figure 12.

Example 2 medium properties (α = equation image).

[49] In order, to assess the importance of the different terms in the polynomial chaos expansion used to represent the unknown model parameters, two different approximations for the parametric and modeling uncertainties are assumed.

5.3.1. Case 1: Three Dimensions First-Order Approximation

[50] For this part, it is assumed that the unknown model parameters are represented as

equation image

where α0(x) and α1(x) are the unknown coefficients to be estimated using the filtering scheme. In this example, the modeling and measurement errors are considered Gaussian represented by ξ2 and ξ3, respectively. It is important to note that via this representation, the parametric uncertainty is independent of other noise sources.

[51] Figure 13 presents a comparison between the mean estimated and the actual water saturation profiles. Although the filter succeeds in capturing the flow front, discrepancies are noticed between the true and estimated water saturation profiles. Figure 14 represents the coefficients of the exponential chaos expansion used to represent the ratio of the permeability and porosity, and Figure 15 shows the mean and variance of the estimated ratio. Figures 14 and 15 indicate that the estimated model parameters have a large spacial variability, and are characterized with a high degree of uncertainty. Hence, the confidence associated with the characterized model is minimal and it is advisable to reconsider the uncertainty representations and modeling assumptions.

Figure 13.

Case 1 (left) true and (right) estimated water saturation profiles.

Figure 14.

Case 1 estimated chaos coefficients of the exponential representation of the medium properties.

Figure 15.

Case 1 mean and variance of estimated medium property α = equation image.

5.3.2. Case 2: Coupled Three Dimensions Second-Order Approximation

[52] For this part, it is assumed that the unknown model parameters are represented as

equation image

where α0(x), α1(x), α2(x) and α3(x) are the unknown coefficients to be estimated using the filtering scheme. In this example, the modeling error is considered Gaussian represented by ξ3, while the modeling uncertainties are modeled as a second-order one-dimensional expansion involving ξ1 and ξ12 − 1.

[53] Figure 16 shows that using the latter approximation, a better agreement between the mean estimated and the actual water saturation profiles is achieved. Figure 17 represents the coefficients of the exponential chaos expansion used to represent the ratio of the permeability and porosity, and Figure 18 shows the mean and variance of the estimated ratio. It is readily noticeable that although the estimated model parameters are still different from those associated with the forward problem, the estimated variability is acceptable and this ascertains the confidence in the henceforth characterized model.

Figure 16.

Case 2 (left) true and (right) estimated water saturation profiles.

Figure 17.

Case 2 estimated chaos coefficients of the exponential representation of the medium properties.

Figure 18.

Case 2 mean and variance of estimated medium property α = equation image.

6. Conclusion

[54] The combination of polynomial chaos with the ensemble Kalman filter provides an efficient data assimilation methodology that addresses methodically some of the challenges faced by most filtering techniques. The proposed method relies on the polynomial chaos decomposition of the stochastic fluctuations and on a stochastic Galerkin projection to propagate the uncertainty in the system and ascertain the confidence in the model predictions. This reduces the computational cost associated with the large ensemble sizes associated with the EnKF, and incorporates modeling errors directly within the system and not as additive noise excitations.

[55] Applying the proposed method for characterizing reservoir simulation models is an innovative approach that allows estimating the statistical properties of the model parameters and thus conveys the medium in a more realistic way. Furthermore, representing the medium properties as an exponential function of a polynomial chaos expansion proves to be an efficient proposal that guarantees the positiveness of the estimated parameters and thus maintains physically valid estimates. This enhances the production forecasting process and improves the reservoir management. The efficiency of the approach is assessed by applying it to a black oil reservoir model.

[56] The statistical descriptions of the estimated properties and pressures are parameterized by all the uncertainties in the predictive model. This provides a path for managing the uncertainty quickly and accordingly estimating the sensitivities of the stochastic predictions with respect to the modeling uncertainties.

Acknowledgments

[57] The authors are grateful for the research support of the Center for Interactive Smart Oilfield Technologies (CiSoft) and the National Science Foundation (NSF).

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