Estimating multiphase flow functions in porous media from dynamic displacement experiments

Abstract

In this work, the estimation of relative permeability and capillary pressure from the data of dynamic displacement experiments was investigated. These properties are used extensively to predict the behavior of petroleum reservoirs. A new implicit method (i.e., based on matching experimental data with simulated values) was developed, which, for the first time, addresses the functional aspects of the implicit estimation problem. Using this regression-based method, those functional bias errors resulting from the specification of a finite dimensional representation for a infinite dimensional function can be recognized and essentially eliminated. The relative permeability and capillary pressure functions are represented with B-splines. Provided a sufficient number of knots is allowed, spline functions of a fixed order can be used to approximate any smooth function to an arbitrary degree of accuracy. The adjustable coefficients in the B-spline representation are chosen by minimizing a nonlinear weighted-least-squares objective function. A set of linear constraints on the parameters was designed to ensure monotone behavior of these estimates. A descent method in which the step vector is calculated using a trust-region strategy was used for this minimization. The constraints are observed using a sequential quadratic programming logic, and orthogonalization techniques are used to compute the step calculations instead of forming the normal equations. A modular design facilitated the investigation of estimates obtained from both the analytical noncapillary model and the finite difference solution to the model equations including capillary pressure effects. A linearized covariance analysis was used to investigate the accuracy of relative permeability estimates when capillary pressure was specified, and to investigate the accuracy of simultaneously estimated relative permeability and capillary pressure functions.