Benjamin Ganis

A lubrication fracture model in a poro-elastic medium

We present a non-planar fracture model in a poro-elastic medium. The medium in which the fracture is embedded is governed by the standard Biot equations of linear poro-elasticity and the flow of the fluid within the fracture is governed by the lubrication equation. We establish existence and uniqueness of the linearized coupled system under weak assumptions on the data. Two discretizations of the problem are formulated: one with a continuous Galerkin method and one with a mixed method for the flow in the reservoir and in the fracture. We perform the numerical analysis of the former and we provide an algorithm and numerical experiments for the latter.

Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method

We describe an algorithm for modeling saturated fractures in a poroelastic domain in which the reservoir simulator is coupled with a boundary element method. A fixed stress splitting is used on the underlying fractured Biot system to iteratively couple fluid and solid mechanics systems. The fluid system consists of Darcy’s law in the reservoir and is computed with a multipoint flux mixed finite element method, and a Reynolds’ lubrication equation in the fracture solved with a mimetic finite difference method. The mechanics system consists of linear elasticity in the reservoir and is computed with a continuous Galerkin method, and linear elasticity in the fracture is solved with a weakly singular symmetric Galerkin boundary element method. This algorithm is able to compute both unknown fracture width and unknown fluid leakage rate. An interesting numerical example is presented with an injection well inside of a circular fracture.

A Frozen Jacobian Multiscale Mortar Preconditioner for Nonlinear Interface Operators

We present an efficient approach for preconditioning systems arising in multiphase flow in a parallel domain decomposition framework known as the mortar mixed finite element method. Subdomains are coupled together with appropriate interface conditions using mortar finite elements. These conditions are enforced using an inexact Newton--Krylov method, which traditionally required the solution of nonlinear subdomain problems on each interface iteration. A new preconditioner is formed by constructing a multiscale basis on each subdomain for a fixed Jacobian and time step. This basis contains the solutions of nonlinear subdomain problems for each degree of freedom in the mortar space and is applied using an efficient linear combination. Numerical experiments demonstrate the relative computational savings of recomputing the multiscale preconditioner sparingly throughout the simulation versus the traditional approach.

A Stochastic Mortar Mixed Finite Element Method for Flow in Porous Media with Multiple Rock Types

This paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types. The governing equations are based on Darcys law with nonstationary stochastic permeability represented as a sum of local Karhunen-Loeve expansions. The approximation uses stochastic collocation on either a tensor product or a sparse grid, coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method. The latter method requires solving a coarse scale mortar interface problem via an iterative procedure. The traditional implementation requires the solution of local fine scale linear systems on each iteration. We employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration. In the stochastic setting, the basis is further reused over multiple realizations, leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude. Error analysis and numerical experiments are presented.

A Global Jacobian Method for Mortar Discretizations of a Fully Implicit Two-Phase Flow Model

We consider a fully implicit formulation for two-phase flow in a porous medium with capillarity, gravity, and compressibility in three dimensions. The method is implicit in time and uses the multiscale mortar mixed finite element method for a spatial discretization in a nonoverlapping domain decomposition context. The interface conditions between subdomains are enforced in terms of Lagrange multiplier variables defined on a mortar space. The novel approach in this work is to linearize the coupled system of subdomain and mortar variables simultaneously to form a global Jacobian. This algorithm is shown to be more efficient and robust compared to previous algorithms that relied on two separate nested linearizations of subdomain and interface variables. We also examine various upwinding methods for accurate integration of phase mobility terms near subdomain interfaces. Numerical tests illustrate the computational benefits of this scheme.

An Enhanced Velocity multIpoint Flux Mixed Finite Element Method for Darcy Flow on Non-matching Hexahedral Grids☆

Abstract This paper proposes a new enhanced velocity method to directly construct a flux-continuous velocity approximation with multipoint flux mixed finite element method on subdomains. This gives an efficient way to perform simulations on multiblock domains with non-matching hexa- hedral grids. We develop a reasonable assumption on geometry, discuss implementation issues, and give several numerical results with slightly compressible single phase flow.

A global jacobian method for mortar discretizations of nonlinear porous media flows

We describe a nonoverlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newton iteration; and (2) algebraically eliminate subdomain velocities (and optionally, subdomain pressures) to solve linear systems for the 1st (or the 2nd) Schur complements. Solving the 1st Schur complement system gives the multiscale solution without the need to solve an interface iteration. Solving the 2nd Schur complement system gives a linear interface problem for a nonlinear model. The methods are less complex than a previously developed nonlinear mortar algorithm, which requires two nested Newton iterations and a forward difference approximation. Furthermore, efficient linear preconditioners can be applied to speed up the iteration. The methods are implemented in parallel, and a numerical study is performed ...

A multiscale fixed stress split iterative scheme for coupled flow and poromechanics in deep subsurface reservoirs

Abstract In coupled flow and poromechanics phenomena representing hydrocarbon production or CO 2 sequestration in deep subsurface reservoirs, the spatial domain in which fluid flow occurs is usually much smaller than the spatial domain over which significant deformation occurs. The typical approach is to either impose an overburden pressure directly on the reservoir thus treating it as a coupled problem domain or to model flow on a huge domain with zero permeability cells to mimic the no flow boundary condition on the interface of the reservoir and the surrounding rock. The former approach precludes a study of land subsidence or uplift and further does not mimic the true effect of the overburden on stress sensitive reservoirs whereas the latter approach has huge computational costs. In order to address these challenges, we augment the fixed-stress split iterative scheme with upscaling and downscaling operators to enable modeling flow and mechanics on overlapping nonmatching hexahedral grids. Flow is solved on a finer mesh using a multipoint flux mixed finite element method and mechanics is solved on a coarse mesh using a conforming Galerkin method. The multiscale operators are constructed using a procedure that involves singular value decompositions, a surface intersections algorithm and Delaunay triangulations. We numerically demonstrate the convergence of the augmented scheme using the classical Mandels problem solution.

A Parallel Framework for a Multipoint Flux Mixed Finite Element Equation of State Compositional Flow Simulator

Mathematical models of physical problems are becoming increasingly complex and computationally intensive. At the same time, computing hardware is becoming more parallelized with an increasing number of cores promoting simultaneous tasks. In this work we present a parallel, equation of state (EOS), compositional flow simulator for evaluating CO

A multiscale mortar method and two-stage preconditioner for multiphase flow using a global jacobian approach

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