Evaluation of numerical schemes for capturing shock waves in modeling proppant transport in fractures

In petroleum engineering, the transport phe-nomenon of proppants in a fracture caused by hydraulicfracturing is captured by hyperbolic partial differentialequations （PDEs）. The solution of this kind of PDEs mayencounter smooth transitions, or there can be large gradi-ents of the field variables. The numerical challenge posedin a shock situation is that high-order finite differenceschemes lead to significant oscillations in the vicinity ofshocks despite that such schemes result in higher accuracyin smooth regions. On the other hand, first-order methodsprovide monotonic solution convergences near the shocks,while giving poorer accuracy in the smooth regions.Accurate numerical simulation of such systems is a chal-lenging task using conventional numerical methods. In thispaper, we investigate several shock-capturing schemes.The competency of each scheme was tested against one-dimensional benchmark problems as well as publishednumerical experiments. The numerical results have showngood performance of high-resolution finite volume meth-ods in capturing shocks by resolving discontinuities whilemaintaining accuracy in the smooth regions. Thesemethods along with Godunov splitting are applied to modelproppant transport in fractures. It is concluded that theproposed scheme produces non-oscillatory and accurateresults in obtaining a solution for proppant transportproblems.