High-Order Bound-Preserving Finite Difference Methods For Miscible Displacements In Porous Media And Incompressible Wormhole Propagation

In this paper,we develop high-order bound-preserving finite difference methods for the coupled system of compressible miscible displacements and incompressible wormhole propagation.We apply WENO reconstruction to construct high-order finite difference schemes for these two models.Meanwhile,to construct the bound-preserving technique,we also construct the first-order finite difference schemes of these two models,and prove that under the appropriate time step constraint,the first order schemes can preserve the positivity of the numerical approximations.Therefore,we can construct bound-preserving technique by using the linear combination of the first-order numerical flux and the high-order numerical flux with applying the parameterized positivity-preserving flux limiter.For compressible miscible displacements,we consider the problem with multi-component fluid mixture and the(volumetric)concentration of the th component,,should be between and.It is well known that does not satisfy a maximum-principle.Hence most of the existing BP techniques cannot be applied directly.The main idea in this paper is to construct the positivity-preserving techniques to all and enforce simultaneously to obtain physically relevant approximations.By doing so,we must treat the time derivative of the pressure as a source in the concentration equation and choose suitable “consistent” numerical fluxes in the pressure and concentration equations.For incompressible wormhole propagation,the important physical properties of acid concentration and porosity involve their boundness between and as well as the monotonically increasing porosity.High-order bound-preserving finite difference methods can maintain the high-order accuracy and keep these important physical properties,simultaneously.The main idea is to choose a suitable time step size in the bound-preserving technique and construct a consistent flux pair between the pressure and concentration equations to deduce a ghost equation.Therefore,we can apply the positivity-preserving technique to the original and the deduced equations.There have been prior studies on the high-order bound-preserving discontinuous Galerkin methods for miscible displacements and incompressible wormhole propagation.However,the bound-preserving technique for discontinuous Galerkin methods is not straightforward extendable to high-order finite difference schemes.There are two main difficulties.Firstly,it is not easy to determine the time step size in the BP technique.In finite difference schemes,we need to choose suitable time step size first and then apply the flux limiter to the numerical fluxes.Secondly,the general treatment for the diffusion term,e.g.centered difference,in miscible displacements may require a stencil whose size is larger than that for the convection term.It would be better to construct a new spatial discretization for the diffusion term such that a smaller stencil can be used.In this paper,we will solve both problems and numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.