A Regression-Based Method for Estimating Relative Permeabilities From Displacement Experiments

Abstract

Summary We address the process of estimating relative permeability curves from data collected during unsteady-state displacement experiments. We introduce B-splines for use as functional representations of relative permeability curves. With B-splines, polynomial splines of any order can be conveniently implemented in the estimation procedure. This feature allows greater flexibility for incorporating inequality constraints into the parameter-estimation process. We present a method for choosing an appropriate number of spline partitions for representing permeability curves. The method is based on considerations for obtaining the most accurate estimates of the relative permeability curves. We illustrate the method with both hypothetical and actual experimental data. We demonstrate that serious estimation errors may be encountered when relative permeability curves are represented with functions containing too few parameters - e.g., power-law functions. Introduction Relative permeability curves are critical for many reservoir engineering calculations. Frequently, these properties are estimated on the basis of laboratory displacement experiments. They are not measured directly, but instead are inferred from flow data measured during the displacement experiments through the use of some interpretive method. The methods of estimating relative permeability and capillary pressure functions from measured data may be classified as explicit or implicit methods. The Johnson-Bossler-Naumann (JBN) and related methods1–3 are explicit methods, in that point values of relative permeabilities are calculated from the measured data. One disadvantage of the explicit method is that such physical effects as capillary pressure cannot easily be taken into account in the interpretive procedure. Another disadvantage is that the process of differentiating measured data tends to magnify the effects of error present in the measured data.4 Such disadvantages can be avoided with an implicit interpretive method. In such a method, one chooses estimates of the properties so that data simulated with a mathematical model of the displacement experiment match, in some sense, the measured data. This method was implemented in a qualitative manner by Archer and Wong5 in estimating relative permeability curves from unsteadystate coreflood data. Sigmund and McCaffery 6 used nonlinear regression to estimate exponents in power-law model representations for the unknown relative permeability curves. It has been determined that significant errors in relative permeability estimates may be encountered when power-law representations are used.7,8 Kerig and Watson8 determined that relative permeabilities can be accurately estimated when cubic splines are used to represent the relative permeability curves. They also developed an algorithm that can be used to estimate the parameters associated with the cubic splines. If desired, inequality constraints can be specified to ensure that the relative permeability curves are convex downward at each step of the iterative procedure. The constraints substantially improve the robustness of the estimation procedure. This paper presents two significant improvements over Kerig and Watson's paper.7 The first is the development of a method for choosing an appropriate number of partitions in spline representations of the relative permeability curves. Splines can be broken into any number of segments. The number of segments used does impact the accuracy with which the relative permeability curves are estimated. A method for choosing an appropriate number of partitions has not previously been reported. The second significant improvement is the use of B-spline representations for the relative permeability curves. B-splines provide for a convenient representation for polynomial splines of any order. The rather. complicated procedure for determining the independent parameters in the piecewise polynomial representation of cubic splines7 is not necessary because they are given explicitly with the B-spline representation. Furthermore, we show how monotonicity constraints may be used with parabolic splines. In this way, the robustness previously achieved with inequality constraints7 can be obtained without requiring that the estimated curves be convex downward. Theory In this section, we review the use of parameter estimation for determining, from measured data, coefficients in assumed functional representations of relative permeability curves. Next, we introduce the use of B-splines as functional representations of relative permeability curves. Finally, we discuss our method for choosing the appropriate number of knots to be used for representing the relative permeability curves.