Self-adaptive finite element and finite difference methods for one-dimensional two-phase immiscible flow

Douglas, Jim; Darlow, Bruce L.; Wheeler, Mary F.; Kendall, Richard P.

doi:10.1016/0045-7825(84)90050-1

Cited by 15 publications

(8 citation statements)

References 5 publications

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“…Another reason is undoubtedly that h-methods fit more naturally into the framework of finite-difference approximations, which do not explicitly use trial functions. Quite a few ALGR schemes for finite differences have appeared; among them are the methods of von Rosenberg (157), Heinemann and van Handelmann (81), and Douglas et al (57), who present both finite-difference and finite-element schemes. A considerable amount of theoretical work and numerical experimentation has focused on finite-element schemes with ALGR (19,52,24,54).…”

Section: Adapative Local Grid Refinementmentioning

confidence: 99%

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Numerical modelling of multiphase flow in porous media

Allen

1985

Advances in Water Resources

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The simultaneous flow of immiscible fluids in porous media occurs in a wide variety of applications. The equations governing these flows are inherently nonlinear, and the geometries and material properties characterizing many problems in petroleum and groundwater engineering can be quite irregular. As a result, numerical simulation often offers the only viable approach to the mathematical modeling of multiphase flows. This paper provides an overview of the types of models that are used in this field and highlights -some of the numerical techniques that have appeared recently. The exposition includes discussions of multiphase, multispecies flows in which chemical transport and interphase mass transfers play important roles. The paper also examines some of the outstanding physical and mathematical problems in multiphase flow simulation. The scope of the paper is limited to isothermal flows in natural porous media; however, many of the special techniques and difficulties discussed also arise in artificial porous media and multiphase flows with thermal effects.

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Section: Adapative Local Grid Refinementmentioning

confidence: 99%

“…Douglas et al (57), for example, propose adding an artificial capillarity to the Buckley-Leverett equation to force convergence to the correct physical solution. An equivalent effect can be achieved by using certain numerical approximations whose lowest-order error terms mimic the desired dissipative phenomena (8).…”

Section: The Buckley-leverett Problemmentioning

confidence: 99%

Numerical modelling of multiphase flow in porous media

Allen

1985

Advances in Water Resources

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“…In particular, the method proposed by the authors and their colleagues [2] does give demonstrably convergent approximation for the examples treated in that paper.…”

Section: Conservation Of Mass For the Incompressible Fluids Requires mentioning

confidence: 99%

“…Select the grid 8"+ ' tentatively as follows. Let The algorithm above is essentially the one that was used in [2]. Experimentally, it was found that maintaining the constraint (4.3) placed a sufficiently fine grid around and immediately in front of the location of the approximate shock that it was better not to adjust the grid after each time step, but instead every few steps.…”

Section: J0 Jr Jrmentioning

confidence: 99%

Implicit, time-dependent variable grid finite difference methods for the approximation of a linear waterflood

Douglas¹,

Wheeler²

1983

Math. Comp.

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Abstract. An implicit, time-dependent variable grid finite difference method based on the addition of an artificial diffusivity is introduced and analyzed for approximating the solution of a scalar conservation law in a single space variable. No relation between the grids at successive time steps is required for convergence. Two adaptive grid selection procedures are shown to be covered by the analysis. Analogous results are also established for an implicit upwinding procedure.

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“…The selection of the grid 8" can be made in the following fashion [2]. First, select S1 = [xx0 = 0, x\,...}, with x) = iKxAt for small /' and set wx = 0 for the remaining ;'.…”

Section: J0 Jr Jrmentioning

confidence: 99%