An analysis of the grid orientation effect in numerical simulation of miscible displacement

Shubin, Gregory R.; Bell, John B.

doi:10.1016/0045-7825(84)90047-1

Cited by 77 publications

(41 citation statements)

References 9 publications

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“…In two space dimensions harmonic averaging is also commonly used to deal with discontinuous coefficients [5], [25], now integrating over squares to obtain the harmonic average of (x, y). In this case, however, the method does not appear to give second-order accurate results and we find that our approach is greatly superior.…”

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confidence: 99%

“…Such problems arise frequently, for example, at the interface between two materials with different diffusion parameters in steady state heat diffusion or electrostatic problems. A Poisson problem with discontinuous coefficients also arises in multicomponent flow problems, e.g., the porous media equations used to model the interface between oil and an injected fluid in simulations of secondary recovery in oil reservoirs [2], [5], [25].…”

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confidence: 99%

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The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

LeVeque¹,

Li²

1994

SIAM J. Numer. Anal.

1,247

1,032

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Abstract. The authors develop finite difference methods for elliptic equations of the formin a region in one or two space dimensions. It is assumed that gt is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in which there is an irregular surface F of codimension contained in fl across which , a, and f may be discontinuous, and along which the source f may have a delta function singularity. As a result, derivatives of the solution u may be discontinuous across F. The specification of a jump discontinuity in u itself across F is allowed. It is shown that it is possible to modify the standard centered difference approximation to maintain second order accuracy on the uniform grid even when F is not aligned with the grid. This approach is also compared with a discrete delta function approach to handling singular sources, as used in Peskin's immersed boundary method.

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confidence: 99%

mentioning

confidence: 99%

The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

LeVeque¹,

Li²

1994

SIAM J. Numer. Anal.

1,247

1,032

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“…U nsplit schemes are customarily used in a variety of applications, including petroleum reservoir simulation [18], ionospheric physics [24], and Lagrangian hydrodynamics [1]. Thus, one of our goals is to provide algorithms which have the same robustness and resolution as the existing operator split algorithms, but which have the same unsplit structure as the existing algorithms used in the applications codes in those areas.…”

Section: Introductionmentioning

confidence: 99%

Multidimensional upwind methods for hyperbolic conservation laws

Colella¹

1990

Journal of Computational Physics

769

630

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We present a class of second-order conservative finite difference algorithms for solving numerically time-dependent problems for hyperbolic conservation laws in several space variables. These methods are upwind and multidimensional, in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge, and that all fluxes are evaluated and differenced at the same time; in particular, operator splitting is not used. Correct behavior at discontinuities is obtained by the use of solutions to the Riemann problem, and by limiting some of the second-order terms. Numerical results are presented, which show that the methods described here yield the same high resolution as the corresponding operator split methods.

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“…We model the incompressible, miscible displacement of a resident fluid with viscosity res µ by an injected fluid with viscosity inj µ using the following nonlinear coupled system of equations (Shubin and Bell, 1984) ).…”

Section: Experimentation and Discussionmentioning

confidence: 99%

Industrial Compositional Streamline Simulation for Efficient and Accurate Prediction of Gas Injection and WAG Processes

Gerritsen¹

2008

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Executive SummaryGas-injection processes are widely and increasingly used for enhanced oil recovery (EOR). In the United States, for example, EOR production by gas injection accounts for approximately 45% of total EOR production and has tripled since 1986. The understanding of the multiphase, multicomponent flow taking place in any displacement process is essential for successful design of gas-injection projects. Due to complex reservoir geometry, reservoir fluid properties and phase behavior, the design of accurate and efficient numerical simulations for the multiphase, multicomponent flow governing these processes is nontrivial. In this work, we developed, implemented and tested a streamline based solver for gas injection processes that is computationally very attractive: as compared to traditional Eulerian solvers in use by industry it computes solutions with a computational speed orders of magnitude higher and a comparable accuracy provided that cross-flow effects do not dominate. We contributed to the development of compositional streamline solvers in three significant ways: improvement of the overall framework allowing improved streamline coverage and partial streamline tracing, amongst others; parallelization of the streamline code, which significantly improves wall clock time; and development of new compositional solvers that can be implemented along streamlines as well as in existing Eulerian codes used by industry.We designed several novel ideas in the streamline framework. First, we developed an adaptive streamline coverage algorithm. Adding streamlines locally can reduce computational costs by concentrating computational efforts where needed, and reduce mapping errors. Adapting streamline coverage effectively controls mass balance errors that mostly result from the mapping from streamlines to pressure grid. We also introduced the concept of partial streamlines: streamlines that do not necessarily start and/or end at wells. This allows more efficient coverage and avoids the redundant work generally done in the near-well regions. We improved the accuracy of the streamline simulator with a higher order mapping from pressure grid to streamlines that significantly reduces smoothing errors, and a Kriging algorithm is used to map from the streamlines to the background grid. The higher accuracy of the Kriging mapping means that it is not essential for grid blocks to be crossed by one or more streamlines. The higher accuracy comes at the price of increased computational costs, but allows coarser coverage and so does not generally increase the overall costs of the computations. To reduce errors associated with fixing the pressure field between pressure updates, we developed a higher order global time-stepping method that allows the use of larger global time steps. Third-order ENO schemes are suggested to propagate components along streamlines. Both in the two-phase and three-phase experiments these ENO schemes outperform other (higher order) upwind schemes. Application of the third order ENO scheme leads to ii...

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An analysis of the grid orientation effect in numerical simulation of miscible displacement

Cited by 77 publications

References 9 publications

The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

Multidimensional upwind methods for hyperbolic conservation laws

Industrial Compositional Streamline Simulation for Efficient and Accurate Prediction of Gas Injection and WAG Processes

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