Estimating Heterogeneous Anisotropic Permeability Fields From Multiwell Interference Tests: A Field Example
Authors: Abacioglu Y. , Reynolds A.C. , Oliver D.S. 
Published: December 2007

Abstract
Previously, the last two authors and their coworkers have developed techniques based on inverse problem theory to generate maximum a posteriori estimates and realizations of rock property fields conditioned to geostatistical information and multiwell pressure data. The objectives of this work are to provide insight on the practical field implementation of these techniques and to apply them to a field example.

Introduction
Following pioneering work of Gavalas et al. and Shah et al. and the inverse problem theory philosophy of Tarantola, the last two authors and their coworkers have developed procedures for generating rock property fields conditioned to geostatistical models and multiwell pressure data. The emphasis of Refs. 4-10 is on generating a suite of realizations of the rock property fields conditioned to all available information and data by sampling the a posteriori probability density function (pdf) for the rock property fields. By predicting reservoir performance under proposed operating conditions with each realization of the rock property fields, one can evaluate the uncertainty in predicted reservoir performance by constructing statistics (histogram, mean, variance) for the set of outcomes for each predicted parameter of interest, e.g., cumulative oil production, water-oil ratio, or break through time.
Our previous comments on generating realizations notwithstanding, our primary objective here is to generate estimates (not a set of realizations) of areally anisotropic permeability fields that match pressure data. For the field example considered, no prior geostatistical model is available so we present an ad hoc procedure to estimate a prior model to use as a regularization term. Regularization is required because we wish to obtain permeability estimates at several hundred gridblocks but typically less than 100 independent pressure measurements are available. Thus a conventional least squares approach based on a Newton iteration procedure would fail because the Hessian matrix would be singular. Although standard techniques such as singular value decomposition are often employed to obtain a nonsingular Hessian matrix when analyzing well-test data with nonlinear regression based on simple analytical solutions, such a procedure is computationally expensive for the large problems of interest to us where we wish to estimate thousands of parameters so that the order of the Hessian matrix is several thousand.

Reservoir Model
The reservoir is assumed to be a rectangular parallelepiped of uniform thickness h. Reservoir boundaries are assumed to be no-flow boundaries. Two-dimensional (x-y) single-phase flow in a Cartesian coordinate system is considered since that is the model used for the field example considered. However, the basic techniques apply to three-dimensional flow problems. The reservoir is produced by any number of complete-penetration vertical wells. Each well is produced at a specified rate (negative for injection) where the rate may vary with time; pressure buildup (or falloff) at a well is simulated by setting the rate to zero subsequent to a producing (or injection) period. Interference or observation wells are simulated by setting the rate to zero at all times. Fluid properties are assumed to be known. We assume a slightly compressible fluid of constant compressibility and viscosity. For the field example considered, porosity is assumed to be known, however, the computational techniques applied can be used to estimate both permeability and porosity fields.
Permeability is considered to be heterogeneous and an isotropic, but we assume that the principal axes of permeability coincide with the directions of the coordinate system, i.e., the only permeabilities involved are kx and ky.
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