29 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
A family of steady two-phase generalized Forchheimer flows and their linear stability analysis
We model multi-dimensional two-phase flows of incompressible fluids in porous
media using generalized Forchheimer equations and the capillary pressure.
Firstly, we find a family of steady state solutions whose saturation and
pressure are radially symmetric and velocities are rotation-invariant. Their
properties are investigated based on relations between the capillary pressure,
each phase's relative permeability and Forchheimer polynomial. Secondly, we
analyze the linear stability of those steady states.
The linearized system is derived and reduced to a parabolic equation for the
saturation. This equation has a special structure depending on the steady
states which we exploit to prove two new forms of the lemma of growth of
Landis-type in both bounded and unbounded domains. Using these lemmas,
qualitative properties of the solution of the linearized equation are studied
in details. In bounded domains, we show that the solution decays exponentially
in time. In unbounded domains, in addition to their stability, the solution
decays to zero as the spatial variables tend to infinity. The BernsteinComment: 33 page
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
Fluid Flows of Mixed Regimes in Porous Media
In porous media, there are three known regimes of fluid flows, namely,
pre-Darcy, Darcy and post-Darcy. Because of their different natures, these are
usually treated separately in literature. To study complex flows when all three
regimes may be present in different portions of a same domain, we use a single
equation of motion to unify them. Several scenarios and models are then
considered for slightly compressible fluids. A nonlinear parabolic equation for
the pressure is derived, which is degenerate when the pressure gradient is
either small or large. We estimate the pressure and its gradient for all time
in terms of initial and boundary data. We also obtain their particular bounds
for large time which depend on the asymptotic behavior of the boundary data but
not on the initial one. Moreover, the continuous dependence of the solutions on
initial and boundary data, and the structural stability for the equation are
established.Comment: 33 page