768 research outputs found
Stable cell-centered finite volume discretization for Biot equations
In this paper we discuss a new discretization for the Biot equations. The
discretization treats the coupled system of deformation and flow directly, as
opposed to combining discretizations for the two separate sub-problems. The
coupled discretization has the following key properties, the combination of
which is novel: 1) The variables for the pressure and displacement are
co-located, and are as sparse as possible (e.g. one displacement vector and one
scalar pressure per cell center). 2) With locally computable restrictions on
grid types, the discretization is stable with respect to the limits of
incompressible fluid and small time-steps. 3) No artificial stabilization term
has been introduced. Furthermore, due to the finite volume structure embedded
in the discretization, explicit local expressions for both momentum-balancing
forces as well as mass-conservative fluid fluxes are available.
We prove stability of the proposed method with respect to all relevant
limits. Together with consistency, this proves convergence of the method.
Finally, we give numerical examples verifying both the analysis and convergence
of the method
Finite volume discretization for poroelastic media with fractures modeled by contact mechanics
A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law, whereas slip is described by a Coulomb‐type friction law. This physical model results in a nonlinear variational inequality problem. The variational inequality is rewritten as a complementary function, and a semismooth Newton method is used to solve the system of equations. For the discretization, we use a hybrid scheme where the displacements are given in terms of degrees of freedom per element, and an additional Lagrange multiplier representing the traction is added on the fracture faces. The novelty of our method comes from combining the Lagrange multiplier from the hybrid scheme with a finite volume discretization of the poroelastic Biot equation, which allows us to directly impose the inequality constraints on each subface. The convergence of the method is studied for several challenging geometries in 2D and 3D, showing that the convergence rates of the finite volume scheme do not deteriorate when it is coupled to the Lagrange multipliers. Our method is especially attractive for the poroelastic problem because it allows for a straightforward coupling between the matrix deformation, contact conditions, and fluid pressure.publishedVersio
Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
In this paper we develop adaptive iterative coupling schemes for the Biot
system modeling coupled poromechanics problems. We particularly consider the
space-time formulation of the fixed-stress iterative scheme, in which we first
solve the problem of flow over the whole space-time interval, then exploiting
the space-time information for solving the mechanics. Two common
discretizations of this algorithm are then introduced based on two coupled
mixed finite element methods in-space and the backward Euler scheme in-time.
Therefrom, adaptive fixed-stress algorithms are build on conforming
reconstructions of the pressure and displacement together with equilibrated
flux and stresses reconstructions. These ingredients are used to derive a
posteriori error estimates for the fixed-stress algorithms, distinguishing the
different error components, namely the spatial discretization, the temporal
discretization, and the fixed-stress iteration components. Precisely, at the
iteration of the adaptive algorithm, we prove that our estimate gives
a guaranteed and fully computable upper bound on the energy-type error
measuring the difference between the exact and approximate pressure and
displacement. These error components are efficiently used to design adaptive
asynchronous time-stepping and adaptive stopping criteria for the fixed-stress
algorithms. Numerical experiments illustrate the efficiency of our estimates
and the performance of the adaptive iterative coupling algorithms
Numerical methods for coupled processes in fractured porous media
Numerical simulations have become essential in the planning and execution of operations in the subsurface, whether this is geothermal energy production or storage, carbon sequestration, petroleum production, or wastewater disposal. As the computational power increases, more complex models become feasible, not only in the form of more complicated physics, but also in the details of geometric constraints such as fractures, faults and wells. These features are often of interest as they can have a profound effect on different physical processes in the porous medium. This thesis focuses on modeling and simulations of fluid flow, transport and deformation of fractured porous media. The physical processes are formulated in a mixed-dimensional discrete fracture matrix model, where the rock matrix, fractures, and fracture intersections form a hierarchy of subdomains of different dimensions that are coupled through interface laws. A new discretization scheme for solving the deformation of a poroelastic rock coupled to a Coulomb friction law governing fracture deformation is presented. The novelty of this scheme comes from combining an existing finite-volume discretization for poroelasticity with a hybrid formulation that adds Lagrange multipliers on the fracture surface. This allows us to formulate the inequalities as complementary functions and solve the corresponding non-linear system using a semi-smooth Newton method. The mixed-dimensional framework is used to investigate non-linear coupled flow and transport. Here, we study how highly permeable fractures affect the viscous fingering in a porous medium and show that there is a complex interplay between the unstable viscous fingers and the fractures. The computer code of the above contributions of the thesis work has been implemented in the open-source framework PorePy. The introduction of fractures is a challenge to the discretization and the implementation of the governing equations, and the aim of this framework is to enable researchers to overcome many of the technical difficulties inherent to fractures, allowing them to easily develop models for fractured porous media. One of the large challenges for the mixed-dimensional discrete fracture matrix models is to create meshes that conform to the fractures, and we present a novel algorithm for constructing conforming Voronoi meshes. The proposed algorithm creates a mesh hierarchy, where the faces of the rock matrix mesh conform to the cells of the fractures, and the faces of the fracture mesh conform to the cells of the fracture intersections. The flexibility of the mixed-dimensional framework is exemplified by the wide range of applications and models studied within this thesis. While these physical processes might be fairly well known in a porous medium without fractures, the results of this thesis improves our understanding as well as the models and solution strategies for fractured porous media
- …