Calibrating the exchange coefficient in the modified coupled continuum pipe-flow model for flows in karst aquifers
Nan Chen, Max Gunzburger, Bill Hu, Xiaoming Wang and Celestine Woodruff
We investigate the validity of the popular coupled-continuum pipe-flow (CCPF) model for flow in a karst aquifer. The (Navier) Stokes-Darcy model is used as the ``true model'' for calibrating the exchange coefficient in the CCPF model by minimizing the relative differences between results from the two models or at least by having those differences being below a prescribed threshold value. We find that although the CCPF model is never in perfect agreement with the Stokes-Darcy model, there is an almost universal choice for a nearly optimal exchange coefficient such that the relative error is below one percent. Our numerics suggest that the nearly optimal choice of the exchange coefficient should be sufficiently large instead of being a small quantity that is proportional to the hydraulic conductivity, as suggested in existing literatures. We also show that this nearly optimal choice of exchange coefficient is robust under a wide range of model parameters. This result demonstrates that the CCPF model is a valid approximation for flows in karst aquifers as long as we set the fluid exchange coefficient sufficiently large and at least in the simple two-dimensional setting that we consider.