Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Abstract

International audienceFully implicit time-space discretizations applied to the two-phase Darcy flow problem lead to the systems of nonlinear equations, which are traditionally solved by some variant of Newton's method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton's method is not invariant with respect to a nonlinear change of variable. In this regard the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two phase water-gas Darcy flow in fractured porous media for which the fractures are modeled as interfaces of co-dimension one. The problem is discretized using an extension of Vertex Approximate Gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom

A Hybrid High-Order Method for Darcy Flows in Fractured Porous Media

“…The model could be extended to fracture networks. In this case, additional coupling conditions enforcing the mass conservation and pressure continuity at fracture intersections should be included; see e.g., [17,16].…”

Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media

“…The capillary pressures including different rocktypes at the matrix fracture interface can be taken into account in the framework of the VAG scheme following the usual phase based upwinding of the mobilities as in [25]. An alternative approach is proposed in [31] for two-phase flows in order to capture the jump of the saturations at the matrix fracture interface Γ due to discontinuous capillary pressure curves. These two choices are compared in [32] to a reference solution provided by an equi-dimensional model in the fractures.…”

“…To focus on compositional nonisothermal features, capillary pressures are not considered in this paper. They could be included following the usual phase based upwinding approach as in [25] or recent ideas developed in [31] for two-phase flows. Let us refer to [32] for a comparison of both approaches in the case of an immiscible two-phase flow using a reference solution provided by the equi-dimensional model in the fractures.…”

Non-isothermal Compositional Two-Phase Darcy Flow: Formulation and Outflow Boundary Condition

“…In order to account for a non zero entry pressure for the capillary function P c (S g ), let us choose P c as primary unknown rather than S g and denote by S g (P c ) the inverse of the monotone graph extension of the capillary pressure. As detailed in [2], a switch of variable between S g and P c could also be used in order to account for non invertible capillary functions.…”