Abel Bustos’s research while affiliated with Pontificia Universidad Javeriana - Cali and other places

Hysteresis phenomenon plays an important role in fluid flow through porous media and exhibits convoluted behavior that are often poorly understood and that is lacking of rigorous mathematical analysis. We propose a twofold approach, by analysis and computing to deal with hysteretic, two-phase flows in porous media. First, we introduce a new analytical projection method for construction of the wave sequence in the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows. Using the projection method we show the existence by analytical construction of the solution. The proposed computational method is based on combining locally conservative hybrid finite element method and finite volume discretizations within an operator splitting formulation to address effectively the stiff relaxation hysteretic system modeling fundamental two-phase flows in porous media.

We introduce a novel modeling of phase transitions in thermal flow in porous media by using hyperbolic system of balance laws, instead of system of conservation laws. We are interested in two different behaviors of the balance system: the long time behavior, in which we study the solution with fixed relaxation term and very large time; and the behavior of the solution when the relaxation term is taken to zero and the time is fixed. We also are interested in solving the question: “Does this balance system tend to the conservation system under equilibrium hypothesis?”. To answer this question, we introduce a projection technique for the wave groups appearing in the system of equations and we study the behavior of each group. For a particular Riemann datum, using the projection method, we show the existence of a decaying traveling profile supported by source terms and we analyze the behavior of this solution. We corroborate our analysis with numerical experiments.

We developed an unsplitting finite volume scheme to account the delicate nonlinear balance between numerical approximations of the hyperbolic flux function and the source linked to balance laws. The method is Riemann-solver-free and no upwinding technique is used. By means of this new approach, we conducted an analysis for two new models of balance laws linked to compositional and thermal flow in porous media problems, under and without a thermodynamic equilibrium hypothesis. For concreteness, we adopt the nitrogen and steam injection models in a porous media. To this model we found an interesting behavior linked to the relaxation term, which is the existence of a non-monotonic traveling wave. We applied this numerical technique to others well-known differential models with relaxation terms available in the literature. Qualitatively we were able to reproduce the expected results.

We study the existence of non-monotone traveling wave solutions and its properties for an isothermal Euler system with relaxation describing the perfect gas flow. In order to confront our results, we first apply a mollification approach as an effective regularization method for solving an ill-posed problem for an associated reduced system for the Euler model under consideration, which in turn is solved by using the method of characteristics. Next, we developed a cheap unsplitting finite volume scheme that reproduces the same traveling wave asymptotic structure as that of the Euler solutions of the continuous system at the discrete level. The method is conservative by construction and relatively easy to understand and implement. Although we do not have a mathematical proof that our designed scheme enjoys the asymptotic preserving and well-balanced properties, we were able to reproduce consistent solutions for the more general Euler equations with gravity and friction recently published in the specialized literature, which in turn are procedures based on a Godunov-type scheme and based on an asymptotic preserving scheme, yielding good verification and performance to our method.