Conservation Laws of Mixed Type Describing Three-Phase Flow in Porous Media

Abstract

In this paper we examine the mathematical structure of a model for three-phase, incompressible flow in a porous medium. We show that, in the absence of diffusive forces, the system of conservation laws describing the flow is not necessarily hyperbolic. We present an example in which there is an elliptic region in saturation space for reasonable relative permeability data. A linearized analysis shows that in nonhyperbolic regions solutions grow exponentially. However, the nonhyperbolic region, if present, will be of limited extent which inherently limits the exponential growth. To examine these nonlinear effects we resort to fine grid numerical experiments with a suitably dissipative numerical method. These experiments indicate that the solutions of Riemann problems remain well behaved in spite of the presence of a linearly unstable elliptic region in saturation space. In particular, when initial states are outside the elliptic region the Riemann problem solution appears to stay outside the region. Furthermore, stable shocks are formed connecting states inside the nonhyperbolic regions with states outside.

Nonlinear interfacial dynamics in stratified multilayer channel flows

“…The nonlinearities are quadratic flux functions (the equations are quasilinear) and for certain initial conditions and physical parameters of the problem, they can themselves be a source of inertialess instability when the Jacobian matrix of the flux function possesses complex conjugate eigenvalues (the nonlinearities in the partial differential equations (PDEs) are of mixed hyperbolic-elliptic type). This feature has been studied in the context of systems of conservation laws arising in fluid dynamics problems such as stratified flows (Milewski et al 2004), (Chumakova et al 2009), jet flows (Papageorgiou & Orellana 1998), steady transonic flows (Cole & Cook 1996), magnetofluid dynamics (Kogan 1961), in fluids of van der Waals type (Slemrod 1983), and in three-phase convection-driven flow in porous media modelling fluid flows in petroleum reservoirs (Bell et al 1986). Additionally, these quadratic nonlinearities can be derived as an approximation of more general flux functions in the neighborhood of isolated singular points in the state space (Schaeffer & Shearer 1987).…”

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

“…It was long believed that, for negligible capillary forces, the system of equations would be strictly hyperbolic for any relative permeability functions. This is far from being the case and, in fact, most relative permeability models used today give rise to systems which are not strictly hyperbolic for the entire range of admissible saturations [8,15,18,19,39,40]. Loss of strict hyperbolicity typically occurs at bounded regions of the saturation triangle (the so-called elliptic regions), where the system is elliptic in character.…”

Multiple Viscous Solutions for Systems of Conservation Laws

“…Systems of conservation laws of mixed elliptic-hyperbolic type arise in the theory of three-phase flow in porous media (for example in Stone's permeability model) [4], widely employed in petroleum reservoir engineering [10]. They arise in many other applications as well, including transonic flow in gas dynamics [9].…”

“…The first is that the viscous profile entropy criterion which we employ is the most selective in common usage and takes into account effects usually neglected when physical systems are modeled by conservation laws. The second reason is that this nonuniqueness occurs in a relatively simple mixed-type model with quadratic polynomial flux functions, which captures essential features of Stone's permeability model ( [10], [4]). This model is a description of immiscible three-phase flow commonly used in petroleum reservoir engineering.…”