Efficient Permeability Parameterization With the Discrete Cosine Transform

The inverse estimation of permeability fields (history matching) is commonly performed by replacing the original set of unknown spatially discretized permeabilities with a smaller (lower dimensionality) group of unknowns that captures the most important features of the field. This makes the inverse problem better posed by reducing redundancy. The Karhunen-Loeve Transform (KLT) is a classical option for deriving low dimensional parameterizations for history matching applications. The KLT can provide an accurate characterization of complex permeability fields but it can be computationally demanding. In many respects this approach provides a benchmark that can be used to evaluate the performance of more computationally efficient alternatives. The KLT requires knowledge of the permeability covariance function and can give poor results when this matrix does not adequately describe the actual permeability field. By contrast, the Discrete Cosine Transform (DCT) provides a robust parameterization alternative that does not require specification of covariances or other statistics. It is computationally efficient and in many cases is almost as accurate as the KLT. The DCT is able to accommodate prior information, if desired. Here we describe the DCT approach and compare its performance to the KLT for a set of geologically relevant examples. Introduction Reservoir characterization is generally based on localized borehole and outcrop observations that are interpolated to give regional descriptions of uncertain geological properties such as permeability. The interpolation process introduces uncertainty in the permeability field that translates directly into uncertainty about reservoir behavior. Incorporation of dynamic measurements during the production phase, i.e. history matching, provides a way to reduce permeability uncertainty. History matching identifies the permeability values that provide the best match, in terms of a specified performance measure, to observations of dynamic production variables such as bottom-hole pressure and fluid rates. This process can increase the accuracy and usefulness of model predictions if the estimated permeabilities provide a reasonable description of the true field. It is generally accepted that history matching methods work best when they incorporate geologically realistic facies information. Realistic facies representations should account for depositional continuity and connectivity since these properties have a significant effect on fluid flow within the reservoir [1]. When the permeability field is characterized by finely discretized block values the history matching problem can be ill-posed and result in non-unique solutions [2,3]. Ill-posed problems can produce reservoir models that honor observed measurements but provide incorrect predictions. Moreover, if estimated block permeabilities are not constrained to preserve facies connectivity, they may yield geologically inconsistent and unrealistic permeability fields. In order to deal with illposedness and to respect geological facies it is desirable to adopt a parametric description of permeability that is lowdimensional while also able to preserve important geological features and their connectivity. Several parameterization approaches with varying complexity have been proposed and implemented for reservoir history matching problems. A simple zonation approach is used by [4] in which an aggregate of block properties are assembled and assigned a single value. Adaptive versions of this approach have been adopted to perform the history matching in multiple steps with increasing resolution [5,6]. Other multi-resolution techniques have also been proposed for parameterization and history matching at different scales [7,8]. A particularly powerful parametrization approach suitable for history matching is the Karhunen-Loeve Transform (KLT), named after Karhunen [9] and Loeve [10]. This approach represents the permeability in any given block with a linear expansion (or transform) composed of the weighted eigenvectors of a specified block permeability covariance matrix. This matrix can, in turn, be derived from a specified continuous permeability covariance function. In practice, the covariances used to derive the KLT basis functions are often derived from permeability measurements. When this is done the KLT is data-dependent (i.e. its characterization of permeability depends on correlation properties of a particular SPE 106453 Efficient Permeability Parameterization with the Discrete Cosine Transform B. Jafarpour, SPE, D. B. McLaughlin, Massachusetts Institute of Technology