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**The solution of the Riemann problem for a hyperbolic system of conservation laws modeling polymer flooding.**
*(English)*
Zbl 0658.35061

This paper solves the global Riemann problem for a nonstrictly hyperbolic \(2\times 2\)-system of conservation laws arising in polymer flooding of oil reservoirs. In particular the model contains a nonlinear term which models absorbtion of the polymer on the porous rock in which the fluids reside.

By a travelling wave analysis an entropy condition for the model is obtained and this yields uniqueness of the solution in state space as well as in the physical space. The existence proof is constructive. Based on this construction a computer programme has been developed and the paper presents calculated examples demonstrating that the solution is continuous in \(L_ 1\)-norm with respect to the right state of the Riemann problem.

Furthermore, comparisons are made between analytical solutions and approximate solutions based on finite difference methods. These results demonstrate that the behaviour of the exact solution is not easily detected by these approximations, though it clearly indicates the convergence of the upwind scheme.

In forthcoming papers the construction referred to above is extended to solve \(n\times n\)-systems of conservation laws modelling multicomponent processes in two phase flow.

By a travelling wave analysis an entropy condition for the model is obtained and this yields uniqueness of the solution in state space as well as in the physical space. The existence proof is constructive. Based on this construction a computer programme has been developed and the paper presents calculated examples demonstrating that the solution is continuous in \(L_ 1\)-norm with respect to the right state of the Riemann problem.

Furthermore, comparisons are made between analytical solutions and approximate solutions based on finite difference methods. These results demonstrate that the behaviour of the exact solution is not easily detected by these approximations, though it clearly indicates the convergence of the upwind scheme.

In forthcoming papers the construction referred to above is extended to solve \(n\times n\)-systems of conservation laws modelling multicomponent processes in two phase flow.

Reviewer: T.Johansen

### MSC:

35L65 | Hyperbolic conservation laws |

76S05 | Flows in porous media; filtration; seepage |

35B40 | Asymptotic behavior of solutions to PDEs |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

35A35 | Theoretical approximation in context of PDEs |