Abstract
We consider two-phase flow in a heterogeneous porous medium with highly permeable fractures and low permeable periodic blocks. The flow in the blocks is assumed to be in local capillary disequilibrium and described by Barenblatt’s relaxation relationships for the relative permeability and capillary pressure. It is shown that the homogenization of such equations leads to a new macroscopic model that includes two kinds of long-memory effects: the mass transfer between the blocks and fractures and the memory caused by the microscopic Barenblatt disequilibrium. We have obtained a general relationship for the double nonequilibrium capillary pressure which represents great interest for applications. Due to the nonlinear coupling and the nonlocality in time, the macroscopic model remains incompletely homogenized in general case. The completely homogenized model was obtained for two different regimes. The first case corresponds to a linearized flow in the blocks. In the second case, we assume a low contrast in the block-fracture permeability. Numerical results for the two-dimensional problem are presented for two test cases to demonstrate the effectiveness of the methodology.
Highlights
The classical theory of multiphase flow of liquids and gases through porous media is based on the hypothesis of the local thermodynamic equilibrium
If we displace oil by water, the displaced oil ahead of the front occupies all the pores including narrow, while the injected water behind the front moves in all the pores including the large ones. Such a fluid distribution does not correspond to the equilibrium state; a relaxation process spontaneously starts in the medium and tends to redistribute the two fluids over the pore network
We present numerical simulations to analyze the impact of the nonequilibrium effects for twophase flow in a fractured reservoir
Summary
The classical theory of multiphase flow of liquids and gases through porous media is based on the hypothesis of the local thermodynamic equilibrium. In such systems, despite the local equilibrium, another nonequilibrium phenomenon appears on the macroscale, which is caused by the high delay in capillary redistribution of the fluids between blocks and fractures This disequilibrium is described in terms of long-memory operators, which appear in the macroscopic model of flow. The differential microscopic flow equations transform into integrodifferential macroscopic equations Another completely homogenized model (always without Barenblatt’s nonequilibrium) was obtained in [10, 11] for the case of a lower contrast between the block and fracture permeability. The homogenization method is used to derive the upscaled model for two-phase flow with local Barenblatt’s nonequilibrium in double porosity media with periodic microstructure.
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