Homogenized model with memory for two-phase compressible flow in double-porosity media

Abstract

A completely averaged model of two-phase flow of compressible fluids in a medium with double porosity is developed. The variational asymptotic two-scale averaging method with splitting the nonlocality and nonlinearity is presented. Several mechanisms of delay are detected, as the nonequilibrium capillary redistribution of phases, pressure field relaxation caused by the compressibility, and the cross effects of fluid extrusion from pores due to the rock com-paction and fluid expansion. A generalized non-equilibrium capillary equation is obtained. All characteristic times of delay are explicitly defined as functions of saturation. water injection in heterogeneous reservoirs, 20 . With re-51 spect to the flow in fractures, the spontaneous imbibition 52 in blocks is very slow, which causes the capillary delay 53 effects in terms of the averaged saturation of water (the 54 term "saturation" means the volume fraction). 55 The capillary imbibition can be represented in terms of 56 the propagation of the wave of water saturation from the 57 block boundary towards the block centre. The dynam-58 ics of this wave is nonlinear, proper to two-phase flow in 59 general. In classical two-phase model the counter-current 60 imbibition is described by a nonlinear diffusion equation 61 with nonlinear boundary conditions. We are, thus, faced 62 with a situation where delay/memory (or time nonlocal-63 ity) at the macroscale is caused by the propagation of a 64 nonlinear wave at the local scale. 65 In the flow is two-phase and compressible simultane-66 ously, the two types of the memory mentioned above, 67 caused by the capillarity and the compressibility, should 68 interact in some way, which could generate new physical 69 cross phenomena. The detection of such phenomena is 70 the main objective of the present paper. For this, it was 71 necessary to develop the macroscale model of two-phase 72 compressible flow in double-porosity media. 73 Attempts to develop the model of two-phase flow in 74 double porosity media have been undertaken essentially 75 for incompressible fluids 4 , 7 , 8 , 10 , 12 , 13 , 19 , 22 , 28 . The ap-76 pearance of nonlocality in a nonlinear system, mentioned 77 above, is a strong obstacle for obtaining averaged model. 78 This is why the problem remains open for two-phase case, 79 even non-compressible. Two papers devoted to compress-80 ible two-phase flow in double-porosity media, 2 , 8 , in which 81 one phase was an ideal gas while the second one was in-82 compressible, have illustrated even deeper problems of 83 interference between the nonlocality and twice nonlinear-84 ity. The attempts to overcome this difficulty by lineariz-85 ing the equations of capillary imbibition, as, for instance, 86 in 6 , 7 , 10 , mean physically that the flow in blocks becomes 87 single-phase. 88 As a result of such an imposition of nonlocality and 89 nonlinearity, the models obtained are not completely 90 averaged. Namely, macroscopic equations and the cell 91 problems make up a coupled system of equations, which

Continuous upscaling and averaging

“…A particularly developed example is a theory of homogenization in spatially periodic media (e.g. Bloch and Auriault, 1998;Royer, 2019;Panfilov, 2019). Fundamental differences between averaging in ordered and disordered porous media were discussed by Quintard and Whitaker, 1994a,b,c,d;Wood, 2013. The approaches related to the assumptions about microstructure of the medium or the flow will not be considered in detail for the reason explained below.…”