## A characteristic domain decomposition technique for two-phase flows with interfaces

**A 1997 Preprint by
B. Ersland and
H. Wang
**

- 1997:16
The mathematical model that describes the process of an immiscible displacement of oil by water in reservoir production or other two-phase fluid flows in porous media leads to a strongly coupled system of a degenerated nonlinear advection-diffusion equation for saturation and an elliptic equation for pressure and velocity. The hyperbolic nature, strong coupling, and nonlinearity of the system and the degeneracy of the diffusion makes numerical simulation a challenging task. Many numerical methods suffer from serious non-physical oscillations, excessive numerical dispersion, and/or a combination of both [CJ86, Ewi84]. Previously, Espedal, Ewing, and coworkers developed a characteristic, operator-splitting technique in effectively solving two-phase fluid flow problems [DEES90, EE87]. In practice, a reservoir often consists of different subdomains with different porosities and permeabilities. In the case of single-phase fluid flows the concentration and total flux are continuous across the interfaces between different subdomains since the diffusion never vanishes. Our earlier work addressed numerical simulation to linear transport equations arising in single-phase flows with interfaces [?]. However, in the case of two-phase fluid flows the saturation equation itself is nonlinear and different subdomains have different capillary pressure curves. The continuity of capillary pressure across interfaces implies a jump discontinuity of the water saturation at the same locations. The jump discontinuity of the saturation at the interfaces might incur some oscillations around the interfaces, which can be propagated into the domain and destroy the overall accuracy. Hence, great care has to be taken in the development of an effective solution procedure for the simulation of two-phase fluid flows in porous media with interfaces.

This paper describes a characteristic-based, non-overlapping domain decomposition algorithm for solving the saturation equation in two-phase fluid flows with interfaces. First, with the known saturation at the previous time step one obtains an approximate Dirichlet boundary condition at the outflow domain interface by integrating thesaturation equation (ignoring the capillary pressure term) along characteristics. With the approximate outflow Dirichlet boundary condition at the domain interface and the given boundary condition at the physical inflow boundary one can close the system on the current subdomain and applies the characteristic operator-splitting procedure [DEES90, EE87] to solve the full saturation equation (including the capillary pressure effect). Second, one uses the continuity of capillary pressure across the domain interface to pass the value of saturation as an approximate inflow Dirichlet boundary condition to the next subdomain, one then applies the same characteristic operator-splitting procedure to solve the saturation equation on the current subdomain. Third, according to the overall loss or gain of mass one adjusts the approximate outflow Dirichlet boundary condition at the domain interface to iterate between different subdomains until the algorithm converges. Finally, a mixed method is adopted to solve the pressure equation due to its accurate approximation to the velocity field and its local mass conservation property.

The rest of the paper is organized as follows: In Sections 2 and 3 we formulate the problem and discuss related solution techniques. In Section 4 we present a domain decomposition algorithm for the two-phase fluid flow problems with interfaces. In Section 5, we present some numerical results to show the promise of the method.