Homogenized Model of Two-Phase Flow with Local Nonequilibrium in Double Porosity Media

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.

Thermodynamically consistent modeling of two-phase incompressible flows in heterogeneous and fractured media

“…The boundary oX of the spatial domain X is decomposed into the Dirichlet part C D and Neumann part C N , where oX ¼ C D S C N . The boundary conditions associated to equations (40) and (41) are given by,…”

“…In the dual-porosity/dual-permeability model [34,37,39,40], empirical functions to describe the matrix-fracture mass transfer incorporate ad hoc shape factors, and moreover, no theory is available for the calculation of shape factors in two-phase flows with capillary [3]. A twophase flow model with local nonequilibrium in double porosity media has been derived in [41] using the homogenization approach, and this model includes the long-memory effects of the microscopic disequilibrium and the mass transfer between fractures and blocks. The discrete fracture model [35,36] is based on the conception that the fracture aperture is very small compared to the matrix blocks, and as a consequence, we can simplify the fracture as the lower dimensional domain to reduce the contrast of geometric scales occurring in the single-porosity model.…”

New non-equilibrium matrix imbibition equation for Kondaurov's double porosity model

“…where Remark 3 Notice that in contrast to the classical case (see, e.g., [20] and the references therein) or the case of the global Barenblatt model [5], equation (4.33) is integro-differential. This fact shows explicitly the impact of the capillary non-equilibrium on the mass exchange between the fracture system and blocks via the source terms Q w , Q n in (3.13).…”

The Global Kondaurov Double Porosity Model

“…It is important to notice that the mesoscopic models of single-and two-phase flow in double porosity media considered in these works are locally equilibrium ones. However, the homogenized models obtained exhibit the non-equilibrium behavior due to the additional "source terms" describing the memory effect of the corresponding model (see [2,5,12,15]). Moreover, as it follows from [1,6], it is possible to show that the capillary pressure of the global model is, in fact, a non-equilibrium one.…”

“…However, we observe that only in a few papers the authors start with a locally non-equilibrium model. This is done in [2,13], where the homogenization of Barenblatts and Hassanizadehs flow models (see [3]) and in [10,11], where the homogenization of Kondaurov's model is considered. Notice that the Barenblatt, Hassanizadeh, and Kondaurov flow models are discussed in details in the papers [9,10].…”