4 research outputs found
A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity
Mathematical models for flow through porous media typically enjoy the
so-called maximum principles, which place bounds on the pressure field. It is
highly desirable to preserve these bounds on the pressure field in predictive
numerical simulations, that is, one needs to satisfy discrete maximum
principles (DMP). Unfortunately, many of the existing formulations for flow
through porous media models do not satisfy DMP. This paper presents a robust,
scalable numerical formulation based on variational inequalities (VI), to model
non-linear flows through heterogeneous, anisotropic porous media without
violating DMP. VI is an optimization technique that places bounds on the
numerical solutions of partial differential equations. To crystallize the
ideas, a modification to Darcy equations by taking into account
pressure-dependent viscosity will be discretized using the lowest-order
Raviart-Thomas (RT0) and Variational Multi-scale (VMS) finite element
formulations. It will be shown that these formulations violate DMP, and, in
fact, these violations increase with an increase in anisotropy. It will be
shown that the proposed VI-based formulation provides a viable route to enforce
DMP. Moreover, it will be shown that the proposed formulation is scalable, and
can work with any numerical discretization and weak form. Parallel scalability
on modern computational platforms will be illustrated through strong-scaling
studies, which will prove the efficiency of the proposed formulation in a
parallel setting. Algorithmic scalability as the problem size is scaled up will
be demonstrated through novel static-scaling studies. The performed
static-scaling studies can serve as a guide for users to be able to select an
appropriate discretization for a given problem size
On numerical stabilization in modeling double-diffusive viscous fingering
A firm understanding and control of viscous fingering (VF) and miscible
displacement will be vital to a wide range of industrial, environmental, and
pharmaceutical applications, such as geological carbon-dioxide sequestration,
enhanced oil recovery, and drug delivery. We restrict our study to VF, a
well-known hydrodynamic instability, in miscible fluid systems but consider
double-diffusive (DD) effects---the combined effect of compositional changes
because of solute transport and temperature. One often uses numerical
formulations to study VF with DD effects. The primary aim of the current study
is to show that popular formulations have limitations to study VF with DD
effect. These limitations include exhibiting node-to-node spurious
oscillations, violating physical constraints such as the non-negativity of the
concentration field or mathematical principles such as the maximum principle,
and suppressing physical instabilities. We will use several popular stabilized
finite element formulations---the SUPG formulations and three modifications
based on the SOLD approach---in our study. Using representative numerical
results, we will illustrate two critical limitations. First, we document that
these formulations do not respect the non-negative constraint and the maximum
principle for the concentration field. We will also show the impact of these
violations on how viscous fingers develop. Second, we show that these
stabilized formulations, often used to suppress numerical instabilities, may
also suppress physical instabilities, such as viscous fingering. Our study will
be valuable to practitioners who use existing numerical formulations and to
computational mathematicians who develop new formulations
Composable block solvers for the four-field double porosity/permeability model
The objective of this paper is twofold. First, we propose two composable
block solver methodologies to solve the discrete systems that arise from finite
element discretizations of the double porosity/permeability (DPP) model. The
DPP model, which is a four-field mathematical model, describes the flow of a
single-phase incompressible fluid in a porous medium with two distinct
pore-networks and with a possibility of mass transfer between them. Using the
composable solvers feature available in PETSc and the finite element libraries
available under the Firedrake Project, we illustrate two different ways by
which one can effectively precondition these large systems of equations.
Second, we employ the recently developed performance model called the
Time-Accuracy-Size (TAS) spectrum to demonstrate that the proposed composable
block solvers are scalable in both the parallel and algorithmic sense.
Moreover, we utilize this spectrum analysis to compare the performance of three
different finite element discretizations (classical mixed formulation with
H(div) elements, stabilized continuous Galerkin mixed formulation, and
stabilized discontinuous Galerkin mixed formulation) for the DPP model. Our
performance spectrum analysis demonstrates that the composable block solvers
are fine choices for any of these three finite element discretizations. Sample
computer codes are provided to illustrate how one can easily implement the
proposed block solver methodologies through PETSc command line options
A global sensitivity analysis and reduced order models for hydraulically-fractured horizontal wells
We present a systematic global sensitivity analysis using the Sobol method
which can be utilized to rank the variables that affect two quantity of
interests -- pore pressure depletion and stress change -- around a
hydraulically-fractured horizontal well based on their degree of importance.
These variables include rock properties and stimulation design variables. A
fully-coupled poroelastic hydraulic fracture model is used to account for pore
pressure and stress changes due to production. To ease the computational cost
of a simulator, we also provide reduced order models (ROMs), which can be used
to replace the complex numerical model with a rather simple analytical model,
for calculating the pore pressure and stresses at different locations around
hydraulic fractures. The main findings of this research are: (i) mobility,
production pressure, and fracture half-length are the main contributors to the
changes in the quantities of interest. The percentage of the contribution of
each parameter depends on the location with respect to pre-existing hydraulic
fractures and the quantity of interest. (ii) As the time progresses, the effect
of mobility decreases and the effect of production pressure increases. (iii)
These two variables are also dominant for horizontal stresses at large
distances from hydraulic fractures. (iv) At zones close to hydraulic fracture
tips or inside the spacing area, other parameters such as fracture spacing and
half-length are the dominant factors that affect the minimum horizontal stress.
The results of this study will provide useful guidelines for the stimulation
design of legacy wells and secondary operations such as refracturing and infill
drilling