Events
Department of Mathematics and Statistics
Texas Tech University
We develop and analyze a 2x2 dynamical system describing flow through a single pore to study the
dynamics of the appearance and dissolution of gas bubbles during two-component (CO2, H2O), two-phase
(gas, liquid) flow. Our analysis indicates that three regimes occur at conditions pertinent to petroleum
reservoirs. These regimes correspond to a critical point changing type from an unstable node to an unstable
spiral and then to a stable spiral as flow rates increase. Only in the stable spiral case do gas bubbles achieve
a steady-state finite size. Otherwise, all gas bubbles that form undergo, possibly oscillatory, growth and then
dissolve completely. Under steady flow conditions, this formation and dissolution repeats cyclically.
Background Introduction: Compositional flow involving a dissolved gas is of importance in many areas,
including oil reservoir production, pipeline transport, CO2 sequestration, and the disposal of radioactive
waste. Such flow involves the inherent possibility of creation of a gas phase and its subsequent transport.
Excluding specific tertiary recovery practices such as CO2 foam flooding, keeping potential gas components
dissolved in fluid phases is important for efficient extraction in reservoirs; the presence of gas bubbles and the
resultant fluid-gas menisci complicates flow and can compete with fluid movement. The ability to control
formation pressure or flow rates to prevent bubble formation is important for extraction efficiency. A challenge
to the numerical simulation of compositional flow in porous media is the change in the system of equations
that accompanies the appearance or disappearance of the gas phase. This difficulty has been addressed by
several approaches. In our computations of two-phase, two-component (H2O, CO2) flow in a 3D pore
network, we noted the periodic appearance and dissolution of the gas phase in certain pores. Intensive
evaluation of our algorithms led us to conclude that the phenomenon was not numerical in origin. In reviewing
the literature on gas transport in porous media at reservoir scales, in micromodel studies, specific studies on
gas bubble formation, and mathematical studies of gas phase disappearance in water-hydrogen systems, we
have been unable to find any mention of this periodic phenomenon. We have therefore pursued a
mathematical investigation. In this article, we extract a 2x2 dynamical system from the mathematical model
upon which our computations were based in order to study the mechanics of this phase-cycling phenomenon.
In this talk I will:
- present the mathematical model and derive the dynamical system,
- summarize the direction fields, critical points and solution trajectories,
- report the results of numerical solutions to the dynamical system
- provide summary critique of the work
Particle and mesh-free methods offer significant computational advantages in settings where quality mesh
generation required for many compatible PDE discretizations may be expensive or even intractable. At
the same time, the lack of underlying geometric grid structure makes it more difficult to construct meshfree methods mirroring the discrete vector calculus properties of mesh-based compatible and mimetic
discretization methods. In this talk we survey ongoing efforts at Sandia National Laboratories to develop
new classes of locally and globally compatible meshfree methods that attempt to recover some of the key
properties of mimetic discretization methods.
We will present two examples of recently developed ``mimetic’’-like meshfree methods. The first one is
motivated by classical staggered discretization methods. We use the local connectivity graph of a
discretization particle to define locally compatible discrete operators. In particular, the edge-to-vertex
connectivity matrix of the local graph provides a topological gradient, whereas a generalized moving leastsquares (GMLS) reconstruction from the edge midpoints defines a divergence operator.
The second method can be viewed as a meshfree analogue of a finite volume type scheme. In this
method, the metric information that would be normally provided by the mesh, such as cell volumes and
face areas, is reconstructed algebraically, without a mesh. This reconstruction process effectively creates
virtual cells having virtual faces and ensures a local conservation property matching that of mesh-based
finite volumes. In contrast to similar recent efforts our approach does not involve a solution of a global
optimization problem to find the virtual cell volumes and faces areas. Instead, we determine the
necessary metric information by solving a graph Laplacian problem that can be effectively preconditioned
by algebraic multigrid.
Several numerical examples will illustrate the mimetic properties of the new meshfree schemes. The talk
will also review some of the ongoing work to build a modern software toolkit for mesh-free and particle
discretizations that leverages Sandia’s Trillinos library and performance tools such as Kokkos.
This is joint work with N. Trask, M. Perego, P. Bosler, P. Kuberry, and K. Peterson.
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We develop and analyze a 2x2 dynamical system describing flow through a single pore to study the dynamics of the appearance and dissolution of gas bubbles during two-component (CO2, H2O), two-phase (gas, liquid) flow. Our analysis indicates that three regimes occur at conditions pertinent to petroleum reservoirs. These regimes correspond to a critical point changing type from an unstable node to an unstable spiral and then to a stable spiral as flow rates increase. Only in the stable spiral case do gas bubbles achieve a steady-state finite size. Otherwise, all gas bubbles that form undergo, possibly oscillatory, growth and then dissolve completely. Under steady flow conditions, this formation and dissolution repeats cyclically. In this talk I will: - present the mathematical model and derive the dynamical system,
- summarize the direction fields, critical points and solution trajectories,
- report the results of numerical solutions to the dynamical system
- provide summary critique of the work.
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