A Petrov-Galerkin spectral element technique for heterogeneous porous media flow

We apply minimum kinetic energy principles from classic mechanics to heterogeneous porous media flow equations to derive and evaluate rotational flow components to determine bounding homogenous representations. Kelvin characterized irrotational motions in terms of energy dissipation and showed that minimum dynamic energy dissipation occurs if the motion is irrotational; i.e., a homogeneous flow system. For porous media flow, reductions in rotational flow represent heterogeneity reductions. At the limit, a homogeneous system, flow is irrotational. Using these principles, we can find a homogenous system that bounds a more complex heterogeneous system. We present mathematics for using the minimum energy principle to describe flow in heterogeneous porous media along with reduced special cases with the necessary bounding and associated scale-up equations. The first, simple derivation involves no boundary differences and gives results based on direct Kelvin-type minimum energy principles. It provides bounding criteria, but yields only a single ultimate scale-up. We present an extended derivation that considers differing boundaries, which may occur between scale-up elements. This approach enables a piecewise less heterogeneous representation to bound the more heterogeneous system. It provides scale-up flexibility for individual model elements with differing sizes, and shapes and supports a more accurate representation of material properties. We include a case study to illustrate bounding with a single direct scale-up. The case study demonstrates rigorous bounding and provides insight on using bounding flow to help understand heterogeneous systems. This work provides a theoretical basis for developing bounding models of flow systems. This provides a means to justify bounding conditions and results.

Download Full-text

Computational poroelasticity — A review

Geophysics ◽

10.1190/1.3474602 ◽

2010 ◽

Vol 75 (5) ◽

pp. 75A229-75A243 ◽

Cited By ~ 97

Author(s):

José M. Carcione ◽

Christina Morency ◽

Juan E. Santos

Keyword(s):

Porous Media ◽

Wave Propagation ◽

Rock Physics ◽

Spectral Element ◽

Seismic Modeling ◽

Loss Mechanism ◽

Diffusion Wave ◽

Interface Waves ◽

Element Technique

Computational physics has become an essential research and interpretation tool in many fields. Particularly in reservoir geophysics, ultrasonic and seismic modeling in porous media is used to study the properties of rocks and to characterize the seismic response of geologic formations. We provide a review of the most common numerical methods used to solve the partial differential equations describing wave propagation in fluid-saturated rocks, i.e., finite-difference, pseudospectral, and finite-element methods, including the spectral-element technique. The modeling is based on Biot-type theories of dynamic poroelasticity, which constitute a general framework to describe the physics of wave propagation. We explain the various techniques and discuss numerical implementation aspects for application to seismic modeling and rock physics, as, for instance, the role of the Biot diffusion wave as a loss mechanism and interface waves in porous media.

Download Full-text

An OpenMP implementation of the MuMM solver for porous media flow problems

Revista Mundi Engenharia Tecnologia e Gestão (ISSN 2525-4782) ◽

10.21575/25254782rmetg2020vol5n61333 ◽

2020 ◽

Vol 5 (6) ◽

Author(s):

Sérgio Felipe Ferreira Silva ◽

Hanna Thaina Prates Arimatéia ◽

Alexandre Santos Francisco ◽

Weslley Luiz da Silva Assis

Keyword(s):

Porous Media ◽

Domain Decomposition ◽

Iterative Procedure ◽

Application Programming Interface ◽

Heterogeneous Porous Media ◽

Computational Effort ◽

Multiscale Methods ◽

Porous Media Flow ◽

Flow Problems ◽

Second Order Elliptic Problems

Multiscale methods are usually developed for solving second-order elliptic problems in which coefﬁcients are of multiscale heterogeneous nature. The Multiscale Mixed Method (MuMM) was introduced aiming at the efﬁcient and accurate approximation of large ﬂow problems in highly heterogeneous porous media. In the MuMM numerical solver, ﬁrst mixed multiscale basis functions are constructed, and next global domain decomposition iterations are performed to compute the discrete solution of the problems. However, this iterative procedure is a time-consuming step. In this paper, the authors improve the MuMM solver through the implementation of parallel computations in the step concerning the global iterative procedure. The parallel version of the solver employs the application programming interface Open Multi-Processing (OpenMP). The implementation with the OpenMP reduces signiﬁcantly the computational effort to perform the domain decomposition iterations, as indicated by the numerical results.

Download Full-text