36 research outputs found
Geometric model of the fracture as a manifold immersed in porous media
In this work, we analyze the flow filtration process of slightly compressible
fluids in porous media containing man made fractures with complex geometries.
We model the coupled fracture-porous media system where the linear Darcy flow
is considered in porous media and the nonlinear Forchheimer equation is used
inside the fracture. We develop a model to examine the flow inside fractures
with complex geometries and variable thickness, on a Riemannian manifold. The
fracture is represented as the normal variation of a surface immersed in
. Using operators of Laplace Beltrami type and geometric
identities, we model an equation that describes the flow in the fracture. A
reduced model is obtained as a low dimensional BVP. We then couple the model
with the porous media. Theoretical and numerical analysis have been performed
to compare the solutions between the original geometric model and the reduced
model in reservoirs containing fractures with complex geometries. We prove that
the two solutions are close, and therefore, the reduced model can be
effectively used in large scale simulators for long and thin fractures with
complicated geometry
Exact Subdomain and Embedded Interface Polynomial Integration in Finite Elements with Planar Cuts
The implementation of discontinuous functions occurs in many of today's
state-of-the-art partial differential equation solvers. However, in finite
element methods, this poses an inherent difficulty: efficient quadrature rules
available when integrating functions whose discontinuity falls in the element's
interior are for low order degree polynomials, not easily extended to higher
order degree polynomials, and cover a restricted set of geometries. Many
approaches to this issue have been developed in recent years. Among them one of
the most elegant and versatile is the equivalent polynomial technique. This
method replaces the discontinuous function with a polynomial, allowing
integration to occur over the entire domain rather than integrating over
complex subdomains. Although eliminating the issues involved with discontinuous
function integration, the equivalent polynomial tactic introduces its problems.
The exact subdomain integration requires a machinery that quickly grows in
complexity when increasing the polynomial degree and the geometry dimension,
restricting its applicability to lower order degree finite element families.
The current work eliminates this issue. We provide algebraic expressions to
exactly evaluate the subdomain integral of any degree polynomial on parent
finite element shapes cut by a planar interface. These formulas also apply to
the exact evaluation of the embedded interface integral. We provide recursive
algorithms that avoid overflow in computer arithmetic for standard finite
element geometries: triangle, square, cube, tetrahedron, and prism, along with
a hypercube of arbitrary dimensions
Fracture Model Reduction and Optimization for Forchheimer Flows in Reservoir
In this study, we analyze the flow filtration process of slightly
compressible fluids in fractured porous media. We model the coupled fractured
porous media system, where the linear Darcy flow is considered in porous media
and the nonlinear Forchheimer equation is used inside the fracture.
Flow in the fracture is modeled as a reduced low dimensional BVP which is
coupled with an equation in the reservoir. We prove that the solution of the
reduced model can serve very accurately to approximate the solution of the
actual high-dimensional flow in reservoir fracture system, because the
thickness of the fracture is small. In the analysis we consider two types of
Forchhemer flows in the fracture: isotropic and anisotropic, which are
different in their nature.
Using method of reduction, we developed a formulation for an optimal design
of the fracture, which maximizes the capacity of the fracture in the reservoir
with fixed geometry. Our method, which is based on a set point control
algorithm, explores the coupled impact of the fracture geometry and
beta-Forchheimer coefficient