Analytical Solution to the Riemann Problem of Three-Phase Flow in Porous Media

Abstract

In this paper we study one-dimensional three-phase flow of immiscible, incompressible fluids through porous media. The model uses the common multiphase flow extension of Darcy's equation, and does not include gravity and capillarity effects. Under these conditions, the mathematical problem reduces to a 2 × 2 system of conservation laws, whose essential features are: (1) the system is strictly hyperbolic; (2) both characteristic fields are nongenuinely nonlinear, with single, connected inflection loci. We argue that these are necessary properties for the solution to be physically sensible, and show they are natural extensions of the two-phase flow model. We present the complete analytical solution to the Riemann problem (constant initial and 1 R. Juanes and T. W. Patzek: Analytical solution to the Riemann problem . . . 2 injected states) in detail, and describe the characteristic waves that may arise, concluding that only 9 combinations of rarefactions, shocks and rarefaction-shocks are possible. We demonstrate that assuming the saturation paths of the solution are straight lines may result in very inaccurate predictions for some realistic systems. Efficient algorithms for computing the exact solution are also given, making the analytical developments presented here readily applicable to the interpretation of lab displacement experiments, and to the implementation in streamline simulators.

On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions

“…We utilize here a semi-analytical approach, by which we understand an analytical solution that involves the numerical solution of ordinary differential equations and non-linear systems of equations. Semi-analytical methods have been applied to systems of conservation laws in various applications in numerous works including [1,27,28,40,42,43,49]. In [27], a general algorithm for the solution of Riemann problems is proposed and applied to three-phase flow in porous media (see also [28]).…”

“…Semi-analytical methods have been applied to systems of conservation laws in various applications in numerous works including [1,27,28,40,42,43,49]. In [27], a general algorithm for the solution of Riemann problems is proposed and applied to three-phase flow in porous media (see also [28]). However, the algorithm is restricted to 2 × 2 systems and general flux functions with one single manifold of linear degeneracy per characteristic family.…”

Stabilized numerical solutions of three-phase porous media flow using a multiscale finite element formulation

“…The first application is an oil filtration problem in a relatively dry medium, and the second reproduces water-gas injection in a hydrocarbon reservoir. Numerical solutions are compared with a general, newly developed, analytical solution [40]. These simulations illustrate the outstanding performance of the proposed methodology.…”

“…Finally, it is very valuable in practical applications, because many laboratory and field experiments reproduce in fact the conditions of the Riemann problem. The general analytical solution to the Riemann problem of capillarityfree three-phase flow is given in [40,45]. The system of conservation laws describing three-phase flow is a 2 × 2 system, which is strictly hyperbolic for all saturation paths of interest [43].…”

“…Based on the analysis of the wave structure in [40], a wave of the ifamily connecting two constant states may only be one of the following: an i-rarefaction (R i ), an i-shock (S i ), or an i-rarefaction-shock (R i S i ). Since the full solution to the Riemann problem is a sequence of two waves, W 1 and W 2 , there are only 9 possible combinations of solutions.…”