Systematic investigation of non-Boussinesq effects in variable-density groundwater flow simulations

• Dimensionless form of the governing equations describing variable-density flow are presented.
• Three mathematical models (low, medium and high physical accuracy) for variable-density flow and transport simulations are systematically investigated using the Boussinesq parameter (ερ) as main parameter. The ερ = βω Δω, where βω is the solutal expansivity coefficient and Δω is the characteristic difference is solute mass fraction. Differences between total solute mass in the system, plume penetration depth, center of mass and solute mass fluxes for the three models are quantified at different ερ values for free and mixed convection problems.
• For both free and mixed convection, the Oberbeck-Boussineq approximation (OB) is still a valid assumption at low-density flow.
• At medium- to high-density flow the Oberbeck-Boussineq approximation is valid for free and mixed convection problems at early times. At later times differences between the models is observed. Using a low physical accuracy will underestimate the total solute mass in the system and the vertical plume position and overestimate mass flux.
• Based on the results the following classification of variable-density flow and transport problems is proposed: ερ 0.10: high density contrast.

The validity of three mathematical models describing variable-density groundwater flow is systematically evaluated: (i) a model which invokes the Oberbeck–Boussinesq approximation (OB approximation), (ii) a model of intermediate complexity (NOB1) and (iii) a model which solves the full set of equations (NOB2). The NOB1 and NOB2 descriptions have been added to the HydroGeoSphere (HGS) model, which originally contained an implementation of the OB description. We define the Boussinesq parameter ερ = βω Δω where βω is the solutal expansivity and Δω is the characteristic difference in solute mass fraction. The Boussinesq parameter ερ is used to systematically investigate three flow scenarios covering a range of free and mixed convection problems: 1) the low Rayleigh number Elder problem (Van Reeuwijk et al., 2009), 2) a convective fingering problem (Xie et al., 2011) and 3) a mixed convective problem (Schincariol et al., 1994). Results indicate that small density differences (ερ ≤  0.05) produce no apparent changes in the total solute mass in the system, plume penetration depth, center of mass and mass flux independent of the mathematical model used. Deviations between OB, NOB1 and NOB2 occur for large density differences (ερ > 0.12), where lower description levels will underestimate the vertical plume position and overestimate mass flux. Based on the cases considered here, we suggest the following guidelines for saline convection: the OB approximation is valid for cases with ερ <  0.05, and the full NOB set of equations needs to be used for cases with ερ >  0.10. Whether NOB effects are important in the intermediate region differ from case to case.