Interface spaces based on physics for multiscale mixed methods applied to flows in fractured-like porous media

Abstract

It is well known that domain-decomposition-based multiscale mixed methods rely on interface spaces, defined on the skeleton of the decomposition, to connect the solution among the non-overlapping subdomains. Usual spaces, such as polynomial-based ones, cannot properly represent high-contrast channelized features such as fractures (high permeability) and barriers (low permeability) for flows in heterogeneous porous media. We propose here new interface spaces, which are based on physics, to deal with permeability fields in the simultaneous presence of fractures and barriers, accommodated respectively, by the pressure and flux spaces. Existing multiscale methods based on mixed formulations can take advantage of the proposed interface spaces, however, in order to present and test our results, we use the newly developed Multiscale Robin Coupled Method (MRCM) [Guiraldello, et al., J. Comput. Phys., 355 (2018) pp. 1-21], which generalizes most well-known multiscale mixed methods, and allows for the independent choice of the pressure and flux interface spaces. An adaptive version of the MRCM [Rocha, et al., J. Comput. Phys., 409 (2020), 109316] is considered that automatically selects the physics-based pressure space for fractured structures and the physics-based flux space for regions with barriers, resulting in a procedure

A multiscale Robin-coupled implicit method for two-phase flows in high-contrast formations

“…Therefore the use of physics-based interface spaces is advantageous in comparison with polynomial spaces also for problems with gravity. This observation increases the importance of the aMRCM-PBS, which has shown accurate results, presenting error reductions up to one order of magnitude in cases with strong channelized structures [17,40]. We remark that typical values of saturation error attained by multiscale methods are in the order of 10%.…”

“…It is well known that the classical polynomials are not optimal for high-contrast channelized permeability fields [16]. Here we recall novel interface spaces based on physics to deal with permeability fields containing highly-permeable channels and barriers, that has been introduced recently [17,40]. These interface spaces are particularly relevant when the channels and barriers are relatively large as happens in karst reservoirs [41,42].…”

“…It is a generalization of the Multiscale Mixed Method (MuMM [12]) that allows for the independent choice of pressure and flux interface spaces through local Robin boundary conditions. For improved accuracy when the permeability field has high contrast and is channelized, we incorporate adapted interface spaces which behave better than classical polynomials [16,17].…”

“…In the literature the coupling of multiscale flow and transport problems has been treated by explicit operator splitting techniques [18,19] and implicit formulations [20,21]. We have considered operator splitting techniques with explicit approximations for the transport problem in previous works [15,17]. Here, we propose to combine the MRCM with a Sequential Implicit (SI) scheme [22] that allows for the use of large time steps for the coupled flow and transport problem, in contrast to explicit time integration approaches where a CFL-type condition restricts the size of the time step for the transport calculation.…”

The Multiscale Perturbation Method for Two-Phase Reservoir Flow Problems

“…However, for high-contrast channelized permeability fields, such as the ones considered here, polynomial based spaces are not adequate to capture these types of features. Alternatives are informed spaces, as in [17], or the use of recently developed spaces based on physics [28,29], which are capable of accurately capturing homogeneities such as channels and barriers, as happens in fractured karstified reservoirs [30,31].…”