Numerical simulation of nonclassical waves for immiscible three-phaseflow in heterogeneous porous media

Eduardo Abreu∗ 1

1 Instituto Nacional de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil.

In this paper we present accurate two-dimensional numerical results, based on a new fractional time-step method, for thenumerical solution of a highly nonlinear system of partial differential equations modeling three-phase immiscible water-gas-oil flow problems in heterogeneous petroleum reservoirs.

The new numerical procedure was used to extend theoretical results for one-dimensional three-phase flow available inthe literature by providing numerical evidence of the existence of nonclassical transitional shock in heterogeneous two-dimensional three-phase flows.

1 Introduction

In this paper we present numerical results for three-phase flow in heterogeneous petroleum reservoirs based on a two-leveloperator splitting technique that allows the use of distinct time steps for the three problems defined by the splitting procedure:convection, diffusion and pressure-velocity. A system of hyperbolic conservation laws modeling the convective transport of thefluid phases is approximated by a high resolution, nonoscillatory, second-order, conservative central difference scheme in theconvection step [2, 10]. This scheme is combined with locally conservative mixed finite elements for the numerical solutionof the parabolic and elliptic problems [2] associated with the diffusive transport of fluid phases and the pressure-velocityproblem, respectively. The time discretization of the parabolic problem is performed by means of the implicit backward Eulermethod. For more details and preliminary results see [1, 2].

The mathematical model for the three-phase flow considered in this work takes into account capillary forces and generalexpressions for the relative permeability functions, variable porosity and permeability fields, and the effect of gravity. Thechoice of general expressions for the relative permeability functions may lead to the loss of strict hyperbolicity and, therefore,to the existence of an elliptic region or umbilic points for the system of nonlinear hyperbolic conservation laws describingthe convective transport of the fluid phases [4]. As a consequence, the loss of hyperbolicity may lead to the existence ofnonclassical shocks (also called transitional shocks [8] or undercompressive shocks [12]) in three-phase flow solutions.

Different approaches for solving numerically the three-phase flow equations can be found in [5, 6, 7].

2 Concluding Remarks

Numerical experiments, including the study of WAG injection strategies [1], indicate that the proposed new numerical pro-cedure leads to computational efficiency and accurate numerical results. The delicate balance between the focusing effectsof nonlinear convection, which lead to the formation of shocks, and the smoothing effect of diffusion are captured by ourmethod. Moreover, the new simulator was used to extend theoretical results for one-dimensional three-phase flow availablein the literature [9] by providing numerical evidence of the existence of nonclassical transitional shocks in heterogeneoustwo-dimensional flows [1, 3, 2]. Based on the theoretical results for one-dimensional three-phase flow problems [9] and adetailed examination of the numerical results in heterogeneous formations shown in Figure 1 one can see three wave groups;the new kind nonclassical transitional shock is between the slow and fast wave groups. The slow wave group comprises astrong slow rarefaction and an adjoining slow shock wave. The fast wave group is a Buckley-Leverett shock wave. Figure 1shows the solution of the three-phase flow system of equations (see, e.g., [9, 2, 1]) when the left state is 72.1% water, 27.9 %

gas mixture and the right state is 80 % oil and 5% water.As a first step in the investigation of the scale-up problem for three-phase flows the author is currently investigating the

stability (with respect to viscous fingering) of transitional shocks in heterogeneous formations; stable transitional shocks couldimprove drastically oil production.

Acknowledgements EA thanks the following university, agencies and institute for their financial support: IPRJ/UERJ, CAPES and CNPq,and IMPA. The author wishes to especially thank the financial support handled by ICIAM07 to attend the 6th ICIAM in Zurich, Switzerland.

∗ Corresponding author: e-mail: eabreu@impa.br, Phone: +55 21 2529 5055, Fax: +55 21 2529 5075

PAMM · Proc. Appl. Math. Mech. 7, 2100011–2100012 (2007) / DOI 10.1002/pamm.200700077

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Fig. 1 From top to bottom are shown oil and gas saturation surface plots for two-dimensional three-phase flow problems in heterogeneousreservoirs with aspect ratio X/Y = 4 in the absence of gravity; variable porosity and permeability fields are considered. A nonclassicaltransitional shock is simulated.

References

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© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Contributed Papers 2100012