This paper summarizes the recent development of the generalized matrix formulation and its incorporation into a reservoir simulator as collaborative efforts conducted at Alberta Research Council (ARC) and Computer Modelling Group (CMG). The focus of this paper is on how the matrix coefficients, which are the generalization of relative permeability curves, can be determined from experimental data, in two-phase and three-phase systems. We studied different types of experimental conditions available in the literature to obtain these matrix coefficients for two phase flow. These include experiments by Bentsen and Manai and Boubiaux and Kalaydjian. A more general description was derived, leading to the disclosure of intrinsic connections between the above two types, and all other possible types. This allows the choice of various independent pairs of relative permeability curves as input for a reservoir simulator (CMG's STARS model). Selected experiments were then re-simulated with this matrix formulation. The concept was, for the first time, further generalized systematically to three-phase flow. A scheme and detailed formulations have been developed, allowing a reservoir simulator to deal with a matrix formulation for three-phase flow. These relations were also implemented in CMG's simulator STARS.
In the past few years, persistent efforts were made collaboratively at ARC and CMG in developing a generalized description of coupled multiphase flow in porous media, in terms of relative permeability matrix coefficients [1–6]. The fundamental concept of this new description can find its root in Prof. S. T. Yuster's profound work on multiphase flow in a capillary [7] and has been promoted by Prof. W. Rose [8] since 1969. Although non-traditional, this concept has been acceptedand well discussed in the area of fundamental research [9–20]. However, it has not been accepted for practical use because it is more complicated and has not yet proven more effective. Our focus, therefore, was to demonstrate that models developed based on the phase-coupling concept could be practical and used for reservoir simulation. Work done at ARC and CMG on this subject consists of several components:
(a) Development of the concept. For example, capillary coupling in addition to the commonly discussed viscous coupling has been considered as another possible mechanism. We have developed several schemes of relating experimental data to the relative permeability matrix coefficients, generalization of relative permeability curves.
(b) Identification of areas for field applications. For example, SAGD process has been identified as one of these applicable areas because it consists of a combination of co- and counter-current scenarios. (c) Incorporation of the concept into CMG's STARS. We believe that all these components are essential especially the last one, which is the key to the success of finally bringing this not-so-new-but-important concept into a practically useful tool for reservoir engineering problems.
Traditionally, in a porous medium with a two-phase flow, one expresses flow rates for each phase in the following way:
Equation (1) (Available in full paper)
Equation (2) (Available in full paper)
