Simultaneous Estimation of Absolute and Relative Permeability by Automatic History Matching of Three-Phase Flow Production Data
Authors: Reynolds A.C. , Li R. , Oliver D.S. 
Published: March 2010

Abstract
This paper focuses on the simultaneous estimation of the absolute permeability field and relative permeability curves from three-phase flow production data. Irreducible water saturation, critical gas saturation, and residual oil saturations are assumed to be known. The two-phase relative permeability curves for an oil-gas system and the two-phase relative permeability curves for an oil-water system are represented by power law models. The three-phase oil relative permeability curve is calculated from the two sets of two-phase curves using Stone's Model II. The adjoint method is applied to three-phase flow problems to calculate the sensitivity of production data to the absolute permeability field and the parameters defining the relative permeability functions. Using the calculated sensitivity coefficients, absolute permeability, and relative permeability fields are estimated by automatic history matching of production data.

Introduction
The main objective of this paper is to consider the feasibility of estimating absolute permeability fields and parameters that define relative permeability functions by automatic history matching of production data obtained under multiphase flow conditions. While the topic is not new, to the best of our knowledge, no paper in the petroleum engineering literature has considered this problemunder three-phase flow conditions.
It appears that Archer and Wong(1) were the first authors to consider the estimation of relative permeability curves by applying a reservoir simulator to history match laboratory core flood data. They estimated only parameters that define the shape of relative permeability curves for simple empirical relative permeability models and adjusted relative permeabilities by a trial and error method during the history matching.
Sigmund and McCaffery(2) were the first to apply nonlinear regression to the problem of history matching laboratory core flood data. They used power law expressions to model relative permeability curves and estimated only the two exponential parameters in these formulas. Kerig and Watson(3) considered a similar problem. They calculated predicted data from the Buckley- Leverett model, used cubic splines to parameterize relative permeability curves and compared relative permeability estimates obtained with such a representation to those obtained using a power law functional form. They showed that, in general, power law models do not contain enough degrees of freedom to represent the truth well, whereas cubic splines with a small number of knots appear to be sufficiently flexible to yield more accurate estimates of true relative permeability curves. In their results, they assume absolute permeability is known.
Lee and Seinfeld(4) considered the simultaneous estimation of the absolute permeability field and relative permeabilities for a two-dimensional, two-phase flow oil-water system. They assumed power law relative permeability curves and assumed that the end point values of relative permeabilities were known. Thus, only the two exponents in the power law relative permeability functions were estimated. They modelled the two-dimensional isotropic heterogeneousermeability field using bi-cubic B-splines. In the specific xamples considered, they matched pressure and water cut data at wells producing from an oil reservoir under waterflood. Tikhonov(5) regularization was used to stabilize the nonlinear least squares problem.