Numerical models for variable-density flow and solute transport (VDFST) are widely used to simulate seawater intrusion and related problems, including submarine groundwater discharge (SGD). The mathematical model for VDFST is a coupled nonlinear system written in state-space and time form, so the numerical discretization in time and space are usually required to be as fine as possible. As a result, such large space and time transient models are computationally very demanding, which is disadvantageous for state estimation, forward prediction or inverse calculation. The purpose of this research was to develop mathematical and numerical methods to simulate variable-density flow and salt transport via a model reduction technique called “proper orthogonal decomposition” (POD) for both linear and nonlinear models. It was showed that this method can generate representations of data that contain general information about the solution of the original partial differential equations. Data analysis using POD was conducted to extract dominant “model features” (basis functions) through singular value decomposition from experimental data or detailed simulations of high-dimensional systems (snapshots). These basis functions were then used in the Galerkin projection procedure that yielded low-dimensional reduced models. The original full numerical models were presented by the Galerkin finite-element method. The implementation of the POD reduced method was straightforward referring to the complex full model.

The developed POD method was applied to solve two classic VDFST problems, the Henry problem and the Elder problem, to investigate the accuracy and efficiency of the POD method. The reduced model can reproduce and predict the full model results very accurately with much less computational time in comparison with the full model. It was showed that the accuracy and efficiency of the POD reduced model is mainly determined by the optimal selection of snapshots and POD bases.