3D coupled fluid and heat transport simulations of the Northeast German Basin and their sensitivity to the spatial discretization: different sensitivities for different mechanisms of heat transport

Abstract

Based on a numerical model of the Northeast German Basin (NEGB), we investigate the sensitivity of the calculated thermal field as resulting from heat conduction, forced and free convection in response to consecutive horizontal and vertical mesh refinements. Our results suggest that computational findings are more sensitive to consecutive horizontal mesh refinements than to changes in the vertical resolution. In addition, the degree of mesh sensitivity depends strongly on the type of the process being investigated, whether heat conduction, forced convection or free thermal convection represents the active heat driver. In this regard, heat conduction exhibits to be relative robust to imposed changes in the spatial discretization. A systematic mesh sensitivity is observed in areas where forced convection promotes an effective role in shorten the background conductive thermal field. In contrast, free thermal convection is to be regarded as the most sensitive heat transport process as demonstrated by non-systematic changes in the temperature field with respect to imposed changes in the model resolution.

Appendix

The finite element method-based simulator FEFLOW^® (Diersch 2009a) was used to solve the following set of governing equations:

$$\frac{\partial h}{\partial t} + \nabla \cdot \left( {{\mathbf{q}}^{f} } \right) = 0 $$

(1)

$${\mathbf{q}}^{f} = {\mathbf{q}}_{\text{linear}}^{f} + {\mathbf{q}}_{\text{non - linear}}^{f} = - {\mathbf{K}}\nabla h - {\mathbf{K}}\frac{{\rho^{f} \left( {p,T} \right) - \rho_{0}^{f} }}{{\rho_{0}^{f} }}\frac{{\mathbf{g}}}{{\left| {\mathbf{g}} \right|}} $$

(2)

$$\left\{ {\phi \left( {\rho_{0}^{f} c^{f} } \right) + \left( {1 - \phi } \right)\rho^{s} c^{s} } \right\}\frac{\partial T}{\partial t} + \rho_{0}^{f} c^{f} \left\{ {\nabla \cdot \left( {{\mathbf{q}}_{\text{linear}}^{f} T} \right) + \nabla \cdot \left( {{\mathbf{q}}_{\text{non - linear}}^{f} T} \right)} \right\} - \nabla \cdot \left( {{\mathbf{q}}^{T} } \right) = Q_{T} $$

(3)

Equation (1) represents the continuity equation of fluid mass conservation with $h$ being the hydraulic head (m), ${\mathbf{q}}^{f}$ (ms⁻¹) the specific discharge (Darcy’s velocity). Darcy’s law is given by Eq. (2). It may be split into two terms, a linear (advective) and a non-linear (thermal convective) flow component. The linear component expresses the contribution to the fluid velocity as coming from changes in groundwater head. The non-linear part takes into account the additional contribution of the fluid-density $\rho^{f} \left( {T,p} \right)$ (kg m⁻³) with respect to its reference state $\rho_{0}^{f}$ (kg m⁻³) along the unit vector of gravity ${\mathbf{g}}/\left| {\mathbf{g}} \right|$. The fluid-density in the non-linear buoyancy terms in the Darcy equation (2) is calculated by an Equation of State (Magri et al. 2005, 2009); Magri 2009:

$$\rho^{f} (T,p) = \rho_{0}^{f} (1 + \gamma (T,p)\Updelta p - \beta (T,p)\Updelta T), $$

(4)

where $\gamma$ (Pa⁻¹) and $\beta$ (K⁻¹) are the coefficients for fluid compressibility and thermal expansion accounting for temperature and pressure effects on the fluid-density $\rho_{0}^{f}$ at reference condition T ₀ = 20 °C.

${\mathbf{K}} = \frac{{{\mathbf{k}}\rho_{0}^{f} g}}{{\mu^{f} }}$ is the hydraulic conductivity tensor of the porous media consisting of the permeability k (m²), gravity acceleration $g$ (ms⁻²) and temperature-dependent viscosity $\mu^{f} (T^{f} )$ (Pas). Under thermal equilibrium between the fluid and the solid, i.e. $T^{f} = T^{s} \equiv T$, and within the frame provided by the Oberbeck–Boussinesq approximation (Holzbecher 1998; Kolditz et al. 1998; Nield and Bejan 2006), Eq. (3) denotes the heat transport equation where $\phi \left( {\rho_{0}^{f} c^{f} } \right)$ [kJ/(kg K)] and $\left( {1 - \phi } \right)\rho^{s} c^{s}$ [kJ/(kg K)] are the specific heat capacity of the fluid and the solid phase, respectively, $\phi$ (–) stands for the effective porosity, T (K) the temperature, $Q_{T}$ (μW/m) the radiogenic heat production. The last term of the left hand side of Eq. 3 expresses the contribution to the internal thermal budget from the conductive (Fourier’s law) and the thermodispersive processes as:

$${\mathbf{q}}^{T} = - {\varvec{\lambda}} \cdot \nabla T, $$

(5)

where ${\varvec{\lambda}} = {\varvec{\lambda}}_{\text{DISP}} + {\varvec{\lambda}}_{\text{COND}}$ [W/(mK)] is the thermal conductivity of the saturated porous medium involving dispersive ${\varvec{\lambda}}_{\text{DISP}}$ [W/(mK)] and conductive ${\varvec{\lambda}}_{\text{COND}} = \phi {\varvec{\lambda}}^{f} + (1 - \phi ){\varvec{\lambda}}^{s}$ effects.