A new solution to transient single-well push-pull test with low-permeability non-Darcian leakage effects

An accurate solute transport model is critical to the interpretation of single-well push-pull (SWPP) test. Previous studies of SWPP test generally consider solitary aquifer that is confined by impermeable layers. Also, existing solutions for solute transport in aquifer-aquitard systems only consider the injection phase and over-simplify the flow field by assuming uniformly distributed velocity in the aquitard. In this study, we developed a numerical model with Dirichlet boundary condition for SWPP test affected by leakage described by a low-permeability non-Darcian expression involving a threshold pressure gradient (I₀). Our SWPP test model considered transient flow in multi-phases, which include injection, chase, rest and extraction phases. Finite-difference scheme was adopted to solve the models of flow and solute transport. The results indicate that an increasing hydraulic diffusivity leads to a greater peak value of breakthrough curve (BTC) while a medium with larger grain size results in less estimation error when using steady-state flow model to interpret the transient SWPP test. Additionally, A greater I₀ makes the solute stored in aquitard more difficult to be extracted out due to dispersion dominance, which results in higher BTC values. For the purpose of application, a lumped dimensionless index called the non-Darcian index (NDI) was proposed to quantify the overestimation degree by neglecting leakage, and the underestimation degree by accounting for Darcian leakage, when interpreting the SWPP test with low-permeability non-Darcian leakage. The long-term slope of breakthrough curve coupled with the NDI can be employed to determine the cases in which the low-permeability non-Darcian leakage should be considered.