Simultaneous Estimation of Relative Permeability and Porosity/Permeability Fields by History Matching Production Data

Abstract

Abstract This paper presents a procedure to estimate relative permeability curves together with gridblock porosities and permeabilities by automatic history matching three-phase flow production data using the LBFGS algorithm for optimization and the adjoint method for generating sensitivity coefficients. Both power law and B-splines are considered for the representation of relative permeability curves. Power law functions provide a simple representation; however, B-splines have the advantage of being able to accurately represent any set of relative permeability curves. In the B-spline representation, we provide a simple procedure based on a transformation of parameters (control points) defining the B-splines to ensure that monotonic curves are obtained. Without this transformation, the automatic history-matching process fails. If endpoint saturations are included as parameters, we show how to modify the adjoint procedure to account for the fact that initial conditions are sensitive to endpoint saturations. While the history-matching process is inherently non-unique, we show that reasonable estimates of relative permeability curves and porosity/ permeability fields can be obtained for a synthetic reservoir example. We also provide a quantification of the uncertainty in model parameters and reservoir performance prediction using the randomized maximum likelihood (RML) method. Introduction Usually, the relative permeability curves are obtained from labs through coreflood tests. However, they can also be incorporated into the history matching procedure as model parameters. Archer and Wong(1) and Yang and Watson(2) were among the early researchers to consider the estimation of relative permeability curves by history matching laboratory coreflood data. A detailed literature review is given in Reynolds et al.(3). Reynolds et al.(3) also discussed the estimation of three-phase relative permeabilities by history matching production data. The current work is a natural extension of Reynolds et al.(3). In this paper, we provide an alternative method for relative permeability estimation based on B-spline approximation. Compared to power law representation, B-splines have the advantage of being able to accurately represent any set of relative permeability curves.