Steam injection into water-saturated porous rock

Bruining, J.; Marchesin, D.; Duijn, C.J. van

doi:10.1590/s0101-82052003000300004

Cited by 5 publications

(19 citation statements)

References 15 publications

(9 reference statements)

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“…Then, considering a generalized Riemann problem (the situation is not symmetric with respect to u and v), we prove that system (6) can be solved thanks to the solution of two nonlinear scalar hyperbolic equations in u, the nonlinear functions in each of these equations being linked in order to provide the same shocks and characteristic velocities (theorem 2.5). Note that such a generalized Riemann problem has been studied in a case of multiphase flow in a porous medium, leading to the determination of the shocks and the rarefaction waves in some physical situations [5]. Then, the Liu condition of admissibility of the shocks happens to result from a simple linear combination of the entropy inequalities for both nonlinear hyperbolic equations, using Krushkov entropy pairs [12].…”

Section: Introductionmentioning

confidence: 99%

Mathematical and numerical study of a system of conservation laws

Eymard¹,

Tillier²

2007

J. evol. equ.

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The system of equations (f (u))t − (a(u)v + b(u)) x = 0 and ut − (c(u)v + d(u)) x = 0, where the unknowns u and v are functions depending on (x, t) ∈ R × R+, arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of ut − ((cg + d)(u))x = 0 and simultaneously w = f (u) is the entropy solution of wt − ((ag + b)(f (−1) (w)))x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.Keywords: nonlinear hyperbolic equations, existence of a solution, Riemann problem, admissible shocks, entropy solutions, convergence of a finite volume scheme.

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Section: Introductionmentioning

confidence: 99%

Mathematical and numerical study of a system of conservation laws

Eymard¹,

Tillier²

2007

J. evol. equ.

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“…In [1], cases I and II are correct, but there are some mistakes that influence the solution in case III. The first relevant mistake is the statement that the saturation S † maximizes v SC F (Remark 11 , which is corrected here in Figure 2.…”

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confidence: 99%

“…The Riemann problem for steam injection at boiling temperature into a porous medium saturated with water was solved in [1]. Here, we correct the Riemann solution for case III and redraw the speed diagrams 3.1, 3.4 and 4.5.…”

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confidence: 99%

Erratum: (Computational and Applied Mathematics, Vol. 22, N. 3, pp. 359-395) Steam injection into water-saturated porous rock

Lambert¹,

Bruining²,

Marchesin³

2005

Mat. apl. comput.

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“…al. [2], where the condensation shock appears. It is a step towards obtaining a general method for solving Riemann problems for a wide class of balance equations with phase changes (see [8]).…”

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“…This class of balance equations has appeared in mathematical models for clean-up, see [2]. Soil and groundwater contamination due to spills of non-aqueous phase liquids (NAPL's) have received a great deal of attention from society, because, in general, these components can cause damage to the ecosystem and environmental impact to a large area around the spills.…”

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confidence: 99%

On the Riemann Solutions of the Balance Equations for Steam and Water Flow in a Porous Medium

Lambert¹,

Marchesin²,

Bruining³

2005

Methods and Applications of Analysis

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Abstract. Conservation laws have been used to model a variety of physical phenomena and therefore the theory for this class of equations is well developed. However, in many problems, such as transport of hot fluids and gases undergoing mass transfer, balance laws are required to describe the flow.As an example, in this work we obtain the solutions for the basic one-dimensional profiles that appear in the clean up problem or in recovery of geothermal energy. We consider the injection of a mixture of steam and water in several proportions in a porous rock filled with a different mixture of water and steam. We neglect compressibility, heat conductivity and capillarity and present a physical model for steam injection based on the mass balance and energy conservation equations.We describe completely all possible solutions of the Riemann problem. We find several types of shock between regions and develop a scheme to find the solution from these shocks. A new type of shock, the evaporation shock, is identified in the Riemann solution. This work generalizes the work of Bruining et. al. [2], where the condensation shock appears. It is a step towards obtaining a general method for solving Riemann problems for a wide class of balance equations with phase changes (see [8]).

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Steam injection into water-saturated porous rock

Cited by 5 publications

References 15 publications

Mathematical and numerical study of a system of conservation laws

Mathematical and numerical study of a system of conservation laws

Erratum: (Computational and Applied Mathematics, Vol. 22, N. 3, pp. 359-395) Steam injection into water-saturated porous rock

On the Riemann Solutions of the Balance Equations for Steam and Water Flow in a Porous Medium

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