A multiscale Darcy–Brinkman model for fluid flow in fractured porous media

Abstract

The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures. The Darcy’s law is assumed in the porous domain while the Stokes–Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one

Modeling flow in porous media with fractures; Discrete fracture models with matrix-fracture exchange

“…Another interesting problem is a model with Stokes flow in the fracture. Results for this problem have been obtain using a different type of model (see [11]; cf. also [13]).…”

First-order indicators for the estimation of discrete fractures in porous media

“…Remark 3.2 Unsurprisingly, the tentative adjoint model (29) and (30), or (29) and (33), makes sense for ε α , ε β ≥ 0, as the tentative direct model (16) does. When ε α = ε β = 0, the adjoint model reduces to the current adjoint model, that will be defined in (37), complemented with a limit fault adjoint model similar to (17) and whose solution is given by (31) on the candidate fault.…”

“…In the same way, when a Darcy velocity measurement U T,E is provided for element T ∈ T h and for edge E ∈ E T , the additional source term U T,E − U T,E appears on the right-hand side in either the second or the third equation in (29), following that E is a barrier edge or not.…”

“…To conclude this section, we recall that the tentative direct and adjoint systems (16), (29), and (33) define uniquely the direct and adjoint variables X and Λ for all ε α , ε β ≥ 0, including P N , λ N , V E,N , and ν E,N on the candidate fault. Hence, the gradient (27) or (35) of the objective function J ∆F h (ε α , ε β ) is also defined when the intensity parameters ε α and ε β vanish on the candidate fracture.…”

“…see [4,25,24,32,26,22], as well as in the mathematical literature, e.g. see [30,3,29,16,20,34,27,12,11], to name just a few. We use here the model developed in [30], where flow in the fracture as well as in the rock is governed by Darcy's law.…”