D. Marchesin

Front tracking for hyperbolic systems

A method of front tracking yields a zero diffusion calculation of fluid interface discontinuities. The method is applied to the problem of petroleum reservoir simulation. Unstable interfaces with multiple fingers can be resolved by this method even on a coarse grid.

Transitional waves for conservation laws

A new class of fundamental waves arises in conservation laws that are not strictly hyperbolic. These waves serve as transitions between wave groups associated with particular characteristic families. Transitional shock waves are discontinuous solutions that possess viscous profiles but do not conform to the Lax characteristic criterion; they are sensitive to the precise form of the physical viscosity. Transitional rarefaction waves are rarefaction fans across which the characteristic family changes from faster to slower.This paper identifies an extensive family of transitional shock waves for conservation laws with quadratic fluxes and arbitrary viscosity matrices; this family comprises all transitional shock waves for a certain class of such quadratic models. The paper also establishes, for general systems of two conservation laws, the generic nature of rarefaction curves near an elliptic region, thereby identifying transitional rarefaction waves. The use of transitional waves in solving Riemann problems...

Characterisation of deep bed filtration system from laboratory pressure drop measurements

Abstract During the injection of sea/produced water, permeability decline occurs, resulting in well impairment. Solid and liquid particles dispersed in the injected water are trapped by the porous medium and may increase significantly the hydraulic resistance to the flow. We formulate a mathematical model for deep bed filtration containing two empirical parameters—the filtration coefficient and the formation damage coefficient. These parameters should be determined from laboratory coreflood tests by forcing water with particles to flow through core samples. A routine laboratory method determines the filtration coefficient from expensive and difficult particle concentration measurements of the core effluent; then, the formation damage coefficient is determined from inexpensive and simple pressure drop measurements. An alternative method would be to use solely pressure difference between the core ends. However, we prove that given pressure drop data in seawater coreflood laboratory experiments, solving for the filtration and formation damage coefficients is an inverse problem that determines only a combination of these two parameters, rather than each of them. Despite this limitation, we show how to recover useful information on the range of the parameters using this method. We propose a new method for the simultaneous determination of both coefficients. The new feature of the method is that it uses pressure data at an intermediate point of the core, supplementing pressure measurements at the core inlet and outlet. The proposed method furnishes unique values for the two coefficients, and the solution is stable with respect to small perturbations of the pressure data.

A numerical method for two phase flow with an unstable interface

Abstract The random choice method is used to compute the oil-water interface for two dimensional porous media equations. The equations used are a pair of coupled equations: the (elliptic) pressure equation and the (hyperbolic) saturation equation. The equations do not include the dispersive capillary pressure term and the computation does not introduce numerical diffusion. The method resolves saturation discontinuities sharply. The main conclusion of this paper is that the random choice is a correct numerical procedure for this problem even in the highly fingered case. Two methods of inducing fingers are considered: deterministically, through choice of Cauchy data and heterogeneity, through maximizing the randomness of, the random choice method.

Subgrid resolution of fluid discontinuities, II

Abstract In computation of discontinuities in solutions of hyperbolic equations, the random choice method gives a zero viscosity numerical solution with perfect resolution but first-order position errors ∼±2.5Δx. The Lax-Wendroff scheme gives very small first-order position errors, but resolution errors ∼±2.5Δx. We propose two very simple tracking methods in the context of the random choice method, which combine the best features of both methods: perfect resolution and good accuracy. We compare the above with tracking in the context of the Lax-Wendroff scheme. The latter method is morre complicated, but much more accurate than any of the other methods considered here.

Nonuniqueness of solutions of Riemann problems

We investigate a general mechanism, utilizing nonclassical shock waves, for nonuniqueness of solutions of Riemann initial-value problems for systems of two conservation laws. This nonuniqueness occurs whenever there exists a pair of viscous shock waves forming a 2-cycle, i.e., two statesU1 andU2 such that a traveling wave leads fromU1 toU2 and another leads fromU2 toU1. We prove that a 2-cycle gives rise to an open region of Riemann data for which there exist multiple solutions of the Riemann problem, and we determine all solutions within a certain class. We also present results from numerical experiments that illustrate how these solutions arise in the time-asymptotic limit of solutions of the conservation laws, as augmented by viscosity terms.

A global formalism for nonlinear waves in conservation laws

We introduce a unifying framework for treating all of the fundamental waves occurring in general systems ofn conservation laws. Fundamental waves are represented as pairs of states statisfying the Rankine-Hugoniot conditions; after trivial solutions have been eliminated by means of a blow-up procedure, these pairs form an (n+1)-dimensional manifold, the fundamental wave manifold. There is a distinguishedn-dimensional submanifold of containing a single one-dimensonal foliation that represents the rarefaction curves for all families. Similarly, there is a foliation of itself that represent shock curves. We identify othern-dimensional submanifolds of that are naturally interpreted as boundaries of regions of admissible shock waves. These submanifolds also have one-dimensional foliations, which represent curves of composite waves. This geometric framework promises to simplify greatly the study of the stability and bifurcation properties

A RIEMANN PROBLEM IN GAS DYNAMICS WITH BIFURCATION

Abstract We construct the solution of the Riemann problem for 2 × 2 isothermal gas dynamics in a duct with discontinuous diameter. Besides shocks and rarefaction, there are standing waves. The solution exists globally. It is obtained as an asymptotic solution for an appropriate Cauchy problem with continuous data. In certain cases bifurcation occurs and there are three solutions, one of which is unstable. This is an example of a Riemann problem whose solution depends discontinuously on the initial data.

A fast inverse solver for the filtration function for flow of water with particles in porous media

Models for deep bed filtration in the injection of seawater with solid inclusions depend on an empirical filtration function that represents the rate of particle retention. This function must be calculated indirectly from experimental measurements of other quantities. The practical petroleum engineering purpose is to predict injectivity loss in the porous rock around wells. In this work, we determine the filtration function from the effluent particle concentration history measured in laboratory tests knowing the inlet particle concentration. The recovery procedure is based on solving a functional equation derived from the model equations. Well-posedness of the numerical procedure is discussed. Numerical results are shown.

Technical Communique: Upper bounds for the solution of the discrete algebraic Lyapunov equation

New upper bounds for the solution of the discrete algebraic Lyapunov equation (DALE) P=APA^T+Q are presented. The only restriction on their applicability is that A be stable; there are no restrictions on the singular values of A nor on the diagonalizability of A. The new bounds relate the size of P to the radius of stability of A. The upper bounds are computable when the large dimension of A make direct solution of the DALE impossible. The new bounds are shown to reflect the dependence of P on A better than previously known upper bounds.