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Abstract

Summary

We investigate the numerical simulation of polymer flooding model without IPV law [1] and with the IPV percolation law [2]. The two mathematical models (with and without the percolation law) are weakly hyperbolic. They include a resonance region where strict hyperbolicity is lost.

Providing exact solution to Riemann problems and devising accurate numerical schemes is a challenging task.

Without IPV law, the mathematical model coincides with the Keyfitz-Kranzer model [3]. For all initial data, a unique solution to the Riemann problem can be defined thanks to Isaacson and Temple’s entropy condition [1], which is to be imposed in addition to Lax’s one. Our theoretical contribution is to prove that the two models (with and without the IPV percolation law) are equivalent for both smooth and discontinuous solutions, up to a change of variables. Finally, we are able to provide a unique solution of the Riemann problems for both models.

Regarding our numerical simulation contributions, we propose second order finite volume schemes based on the Godunov scheme and a new Suliciu-type [4] relaxation scheme which can be applied to any IPV law. For the two mathematical models, we perform a mesh convergence and compute the errors of approximate solutions relatively to exact solutions and then determine the effective order of those schemes. Because of the system resonance and non-linearity, the so-called first and second order schemes have respectively an effective order of about 0.24 and 0.33 in contact discontinuities, both 0.5 in shocks and, 0.66 and 0.86 in rarefaction waves.

Because of this lack of accuracy, we implement the Glimm scheme for the two mathematical models. The obtained results are in good agreement with the exact solutions. Shocks and contact discontinuities are resolved with three points at most.

[1] E. L. Isaacson, J. B. Temple (1986), “Analysis of a singular hyperbolic system of conservation laws,” J. Diff. Eqs., vol 2, no. 65, pp 250–268.

[2] G. A. Bartelds, J. Bruining, J. Molenaar (1997), “The modeling of velocity enhancement in polymer flooding,” Transp. Porous Media, vol 1, no 26, pp 75–88.

[3] B. L. Keyfitz and H. C. Kranzer (1980), “A system of non-strictly hyperbolic conservation laws arising in elasticity theory,” Archive for Rational Mechanics and Analysis, vol 72, no 3, pp 219–241.

[4] I. Suliciu (1988), “On the thermodynamics of fluids with relaxation and phase transitions fluids with relaxation,” Int. J. Engin. Sci., vol 36, pp 921–947.

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2020-09-14
2024-04-28

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References

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Glimm and Finite Volume Schemes for Polymer Flooding Model with and Without Inaccessible Pore Volume Law
Conference Proceedings 2020, 1 (2020); https://doi.org/10.3997/2214-4609.202035090
/content/papers/10.3997/2214-4609.202035090
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