Deepankar Biswas's dissertation
by
Deepankar Biswas, Ph.D.
University of Texas at Austin, 1997
Supervisors: Graham F. Carey
Kamy Sepehrnoori
Mathematical models for the transport of constituent components
of a multicomponent system in a porous medium play an important part in
petroleum reservoir analysis and production. The success of such models
is dependent on (1) how well the relevant physical and chemical processes
controlling subsurface transport is represented by mathematical equations
and their parameters and (2) how accurately and efficiently the equations
are approximated with numerical methods. The existing models of multicomponent
transport employ two basic sets of equations. The transport of solutes
is described by a set of partial differential equations (mass conservation)
with the corresponding constitutive relationships and the phase behavior
is described by algebraic expressions.
This study proposes a new least-squares mixed finite element method
(LSFEM) for approximating multiphase flow equations and coupled multicomponent
transport equations. The problem is first recast as a system of first-order
partial-differential equations. Then a least-squares residual functional
is constructed. The least-squares problem is then posed as a mixed finite-element
model. This implies that a C0 basis can be used. Also, the least-squares
formulation leads to a symmetric algebraic system. Since derivatives of
the field variables enter explicitly in the system, greater accuracy in
the computed fluxes will be realized by this mixed method compared to the
standard (non-mixed) Galerkin method. The accuracy and performance of the
least-squares FEM for transport problems is studied. The effect of varying
Peclet number on the accuracy and stability of the solutions is also investigated.
Numerical dissipation and other properties of the scheme are examined.
The method is verified for model problems by comparisons with analytic
solutions and results in the literature.
A thermodiffusion model is developed for areal compositional variations
in hydrocarbon reservoirs from the fundamental equations of change in a
multicomponent system. Under conditions of stationary state and utilizing
concepts of non-equilibrium thermodynamics to compute the fluxes, this
model is solved using the least-squares finite-element method. Its usefulness
and applicability are demonstrated by means of comparison to observed variations
in a real gas field.
Finally, given a system of equations describing transport, the computational
cost of numerical simulation is dependent on the domain size and the resolution
required to minimize numerical errors. However, the local resolution required
for accurate solutions can vary over space and time as regions with rapid
changes of concentration gradients propagate through the solution domain.
Thus, such problems are suited for adaptive numerical strategies that seek
to determine where increased numerical resolution is required and then
provide more accurate approximations in those regions. Adaptive strategies
based on the element residual as an error indicator in conjunction with
unstructured remeshing are developed and tested for representative problems
of subsurface transport.
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