Multiscale data integration using coarse-scale models☆
Introduction
With the increasing interest in accurate prediction of subsurface properties, subsurface characterization based on limited dynamic data, such as production and pressure transient data, as well as static data takes on greater importance. Uncertainties on the detailed description of reservoir lithofacies, porosity, and permeability are large contributors to uncertainty in reservoir performance forecasting. Reducing this uncertainty can be achieved by integrating additional data in subsurface modeling. In general, we have static data such as well logs, cores, seismic traces, and dynamic data such as multiphase production history, pressure transient tests and tracer tests etc. Integration of the data from different sources is a non-trivial task because different data sources scan different length scales of heterogeneity and can have different degree of precision. For example, well logs can resolve heterogeneity at the scale of a few feet whereas production data usually scan the length scales comparable to the inter-well distances.
The goal of this paper is to combine a multiscale data integration technique introduced in [19] with upscaling techniques for spatial modeling of permeability in petroleum reservoirs. Reservoir properties such as permeability data may be available at various scales. The fine-scale data represents point measurements such as well logs and cores. The coarse-scale data reflect the effects of the heterogeneities of the fine-scale permeability field and may be obtained based on production and/or seismic data. Such coarse-scale information will typically be associated with higher precision compared to the fine-scale data. In this paper we propose an approach to spatial modeling of permeability conditioned on coarse-scale permeability measurements and some fine-scale conditioning points. The coarse-scale permeability field is obtained from the inversion of the production data.
In previous findings several methods have been introduced which incorporate multiscale data with a primary focus on integrating seismic and well data. These methods include conventional techniques such as cokriging and its variations [6], [26], [27], [14], Sequential Gaussian Simulation with Block Kriging [1], and Bayesian updating of point kriging [1], [7]. Most kriging-based methods are restricted to multi-Gaussian and stationary random fields [6], [26], [27], [14], [1], [7], [5]. Thus, they require variogram construction which can be difficult because of limited data set. Improper variograms can lead to errors and inaccuracies in the estimation. Thus, one might also need to consider the uncertainty in the variogram models during the estimation [8]. Furthermore, most of the multiscale integration algorithms assume a linear relationship between the scales [19]. We would like to note that approximate and coarse-scale models have been frequently used to reduce computational cost [21], [22].
In this paper, we employ a Bayesian framework which allows us to formulate a posterior distribution of the fine-scale permeability field in terms of the coarse-scale permeability measurements. The likelihood function in this posterior distribution describes the stochastic link between the coarse-scale and the fine-scale permeability data. To calculate these kinds of likelihood functions we use upscaling procedures that involve the local solutions of the flow equation. Upscaling operators introduced in this way are non-local and, thus, the obtained posterior distributions are non-explicit. To draw the permeability from the posterior distribution rejection sampling or importance sampling can be used, but due to complicated structure of the posterior distribution they may be very inefficient. In this paper, Markov chain Monte Carlo (MCMC) ([15]) method with Metropolis-Hasting rule is used. The main idea of MCMC is create Markov chain with some proper choice of transition matrix such that the stationary distribution of the chain becomes the desired posterior distribution.
Some representative numerical examples are presented in the paper. For the first set of numerical examples the coarse-scale permeability field is computed using a reference true fine-scale permeability field. Using only the coarse-scale permeability and a few sampling points for the fine-scale permeability field we sample the fine-scale permeability field from the posterior distribution. The coarse-scale permeability field is not imposed exactly in the likelihood function for the following two reasons. First, in practice the coarse-scale permeability fields are often obtained from indirect information, for example seismic data that are affected by noise or artifacts or correlations that may have large scatter. The second reason for not imposing coarse-scale permeabilities exactly is due to the fact that the inversion of the production data on the coarse grid does not take into account the adequate form of the coarse-scale models. Indeed, the inversion on the coarse grid for flow problems often involves the same flow equations as the underlying fine ones, for example, the same relative permeabilities are used for the coarse-scale problems as those for the fine-scale problems or the effects of macrodispersion are neglected. It is known that [10], [12], [11], [13] the flow equations at the coarse level may have different form from the underlying fine-scale equations. In general this form depends on the detailed nature of the heterogeneities which are very difficult to obtain in solving inverse problems. Thus, calculating the coarse grid permeability fields by matching the production history introduces some errors. Since it is very difficult to assess these kinds of errors in general, we assume this error to be Gaussian in the paper. In the first set of numerical examples, we study the various degrees of coarsening and precisions for the coarse-scale fields. For the second set of our numerical examples we use the coarse-scale production data calculated from the inversion of the production data in [19] with 15 injectors and 27 producers. Two coarse-scale data, medium scale and large scale, are considered. We compare the production histories in the producing wells. Finally, in the paper we analyze the effects of upscaling errors on the posterior distribution.
Section snippets
Methodology
In this section we discuss the Bayesian framework used in generating realizations of fine-scale permeability field conditioned on a coarse-scale permeability data and some sampling points. We assume that k1, k2, …, kN are permeability fields at different scales (1—finest (or, simply, fine), N—coarsest). Using Bayes theorem we havewhere P(ki+1∣k1, …, ki) represents the conditional distributions of permeability field at i + 1 coarse level with
Numerical results
For our first numerical test a 40 × 40 reference field (see left plot of Fig. 2) is chosen as a true field. The spatial interaction coefficients are chosen β = 1 for all nearest neighbors. We pick only three points near left bottom corner and right upper corner (the well locations) as the fine-scale sampling points. The coarse-scale permeability field is calculated using the upscaling procedure described earlier on 5 × 5 coarse grid. As for the precision of the coarse-scale data (see (2.10)) we
Conclusions and future work
In this paper we propose a method for reconstructing the fine-scale permeability field from a coarse-scale permeability data and some sampling points. The upscaling methods have been used in the calculation of the posterior distribution. Since the upscaling operators are non-local and non-explicit, Markov chain Monte Carlo methods are used to draw samples from the posterior distribution. The proposed method allows us to integrate as many scales as required by the available data. Moreover, using
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This work is supported by NSF grant DMS-0327713.