Relative-Permeability Estimation From Displacement Experiments: An Error Analysis
Authors: Kerig P.D. , Watson A.T. 
Published: December 2007

Summary
Parameter estimation can be a superior method of estimating relative permeabilities from displacement (unsteady-state) coreflood experiments. To use the parameter-estimation method, one must choose functional representations of the relative-permeability curves. In previous work, only a simple exponential function was used, which can result in very large errors in estimation of relative permeabilities. In this work, we extend the parameter-estimation method to consider the use of several alternative functional forms, including the highly flexible cubic splines, which should be better able to represent a wide variety of relative-permeability curves. In this way, the error associated with the assumption of a simple functional representation can be greatly decreased.
We also develop a procedure to analyze the accuracy with which relative permeabilities may be estimated by parameter estimation. The analysis is used to evaluate the performance of different functional representations in the parameter-estimation method. Furthermore, it can be used to assess quantitatively the effect of experimental design on the accuracy of estimates of relative permeability.

Introduction
Relative permeabilities frequently are estimated from flow data gathered from laboratory displacement-type coreflood experiments in which one fluid (e.g., water) is pumped through a sample of the reservoir rock that is saturated with another fluid or fluids (e.g., oil and connate water). The flow data typically consist of measurements of two time-dependent quantities - the pressure drop across the core and the volume of displaced fluid recovered.
There are two basic approaches for estimating relative permeability curves from displacement data. The methods of Johnson et al.1 and Jones and Roszelle2 are explicit methods in which saturation values and corresponding relative-permeability values at the end of the core sample are estimated directly. Alternatively, an implicit approach may be used. In this approach, relative-permeability curves are chosen so that the quantities simulated with the mathematical model. of the experiment match, in some sense, the experimental data.
The explicit methods are suitable only for systems that are described adequately by the Buckley-Leverett model3; thus they are not appropriate for low-flow-rate experiments in which the effects of capillary pressure are important. Furthermore, they require the numerical or graphical differentiation of experimental data. It is well known that inaccuracies in data measurement become amplified by the process of differentiation. Tao and Watson4,5 have quantified errors in relative-permeability estimates caused by differentiation of data in explicit methods and found that they can be quite large.
The use of an implicit method overcomes the two limitations of explicit methods - that is, differentiation of data is not required, and general mathematical models of the experimental process can be used (such as those that include the effects of capillary pressure). While trial-and-error adjustment6 could be used to choose estimates for the relative-permeability curves, a more appropriate way is to make use of parameter estimation (or nonlinear regression) techniques. Because relative permeabilities are curves (Le., functions of saturation), parameter estimation is applied by representing the relative-permeability curves by a set of parameters. This can be done by assuming a functional representation, in terms of a set of parameters (or adjustable coefficients), for each relative permeability curve. Then, parameter estimation may be used to estimate the parameters in those functional representations of the relative-permeability curves.