Subsurface Hydrology of the Lake Chad Basin from Convection Modelling and Observations

Abstract

In the Lake Chad basin, the quaternary phreatic aquifer (named hereafter QPA) presents large piezometric anomalies referred to as domes and depressions whose depths are ~15 and ~60 m, respectively. A previous study (Leblanc et al. in Geophys Res Lett, 2003, doi:10.1029/2003GL018094) noticed that brightness temperatures from METEOSAT infrared images of the Lake Chad basin are correlated with the QPA piezometry. Indeed, at the same latitude, domes are ~4–5 K warmer than the depressions. Leblanc et al. (Geophys Res Lett, 2003, doi:10.1029/2003GL018094) suggested that such a thermal behaviour results from an evapotranspiration excess above the piezometric depressions, an interpretation implicitly assuming that the QPA is separated from the other aquifers by the clay-rich Pliocene formation. Based on satellite visible images, here we find evidence of giant polygons, an observation that suggests instead a local vertical connectivity between the different aquifers. We developed a numerical water convective model giving an alternative explanation for the development of QPA depressions and domes. Beneath the depressions, a cold descending water convective current sucks down the overlying QPA, while, beneath the dome, a warm ascending current produces overpressure. Such a basin-wide circulation is consistent with the water geochemistry. We further propose that the thermal diurnal and evaporation/condensation cycles specific to the water ascending current explain why domes are warmer. We finally discuss the possible influence of the inferred convective circulation on the transient variations of the QPA reported from observations of piezometric levels and GRACE-based water mass change over the region.

Appendices

Appendix 1: Governing Equations, Parameters and Equations of State

The mass conservation equation for a fluid of variable density within a fluid-saturated porous medium (rock matrix) without an internal fluid source is:

$$\frac{{\partial (\varepsilon \cdot \rho_{\text{f}} )}}{\partial t} = - \nabla \cdot (\rho_{\text{f}} \cdot \vec{u})$$

(1)

where ε is the porosity, ρ _f the fluid density (kg m⁻³), t the time and $\vec{u}$ the Darcy velocity vector (m s⁻¹). We write $\vec{u} = (U,V)$, U and V being the fluid velocity component parallel to the x and z directions, respectively. The fluid is incompressible with a constant chemical composition, and its density is temperature dependent.

Darcy’s law is used to describe the fluid velocity field $\vec{u}$:

$$\vec{u} = - \frac{K}{\mu }\left( {\overrightarrow {\nabla p} - \rho \vec{g}} \right),$$

(2)

where K is the permeability, µ is the viscosity of the fluid, $\vec{g}$ is the gravity vector and p is the fluid pressure. The fluid density varies linearly with temperature T in °C:

$$\rho = \rho_{0} (1 - \alpha (T - T_{0} ))$$

(3)

where α is the thermal expansion coefficient and ρ ₀ is the fluid density at T ₀ = 20 °C, the surface temperature of the model.

Heat transport is achieved by both conduction and advection and is described for an incompressible single-phase fluid by:

$$C_{\text{eq}} \left( {\frac{\partial T}{\partial t}} \right) = \nabla \cdot \left( {\lambda_{\text{eq}} \overrightarrow {\nabla T} } \right) - C_{\text{L}} \vec{u} \cdot \overrightarrow {\nabla } T$$

(4)

where C _L is the volumetric heat capacity defined by C _L = ρ _f C _P, C _P is the specific heat capacity, C _eq and λ _eq are the weighted average volumetric heat capacity and equivalent thermal conductivity, respectively, as defined in saturated porous media of porosity ϕ:

$$C_{\text{eq}} = \phi \rho_{\text{f}} C_{{{\text{p}}_{\text{f}} }} + (1 - \phi )\rho_{\text{s}} C_{\text{ps}}$$

(5)

where f and s are subscripts for the fluid and the porous matrix, respectively. We assume that C _L remains approximately constant, a reasonable assumption since the decrease in density with temperature roughly balances the increase in the specific heat capacity with temperature. The equivalent thermal conductivity is written as:

$$\lambda_{\text{eq}} = \lambda_{\text{f}}^{\phi } \lambda_{\text{s}}^{1 - \phi }$$

(6)

where λ _f and λ _s are the thermal conductivities of the fluid and the porous matrix, respectively.

The vertical Rayleigh number Ra characterises the vigour of the convection within the stratified porous medium and is defined as follows:

$$Ra = \frac{{\alpha \rho gC_{\text{L}} K_{z} }}{{\mu \lambda_{\text{eq}} }}\gamma h^{2}$$

(7)

where K _z represents the vertical permeability, g is the acceleration due to gravity, h is the height of the layer and γ is the vertical temperature gradient.

Appendix 2: Parameter values

In our calculations, the parameter describing the temperature dependence, α, follows from the results obtained by Irvine and Duignan (1985). The viscosity of the fluid μ (given in units of Pa s) also varies with temperature, based on the following analytical expression (ASME 1968; Kestin et al. 1978; Rabinowicz et al. 1998a, b):

$$\mu = 0.00002414 \times 10^{{\left( {\frac{247.8}{T + 133}} \right)}}$$

(8)

The pressure dependence of viscosity is not included, because it is small in the range of depth considered here (Norton 1984). Equation (8) shows that the fluid viscosity decreases from 17.5 × 10⁻⁵ to 13.5 × 10⁻⁵ Pa s in the 2–200 °C temperature interval. Above 200 °C, the viscosity remains approximately constant. In our model, the thermal conductivity λ _eq varies with temperature for each type of formation (crystalline rock, sandstone, shale). We used the experimental data provided by Clauser and Huenges (1995) for the thermal conductivities of sandstone, shale and quartz-rich crystalline rock between 0 and 300 °C. Finally, the other parameters such as the porosity ϕ, the heat capacity C of each phase (rock and fluid) and the density of each phase at 20 °C assigned to each geological formation are listed in Table 1.

Appendix 3: Method and Boundary Conditions

The simulations are performed using the Comsol Multiphysics™ finite-element code, which has already been tested and successfully implemented for convective process simulations in various configurations (Holzbecher 2004; Eldursi et al. 2009; Guillou-Frottier et al. 2013). A two-dimensional grid constituted of square elements of size d = 200 m was considered. Thus, given the dimensions of the basin, the grid is composed of 40 elements in the z direction and 1500 elements in the x direction. The ratio 0.8d/V _max, where V _max represents the maximal amplitude of the Darcy velocity, determines the optimal time step at which the time evolution of the convective process can be realistically simulated.

The top of the model represents the surface topography of the Lake Chad basin, which is impermeable (the vertical velocity at the interface is zero) and maintained at a constant temperature of 20 °C. The bottom of the model consists of a quartz-rich substratum. It is maintained at a constant temperature of 247 °C and has a permeability low enough to be considered as impermeable (~10⁻¹⁸ m²). The flow and temperature along the lateral boundaries of the model are symmetric. All these chosen boundary conditions are the usual ones, except the flow condition along the top interface. Indeed, a real free condition would be mathematically more elegant. The latter condition is needed to adjust the depth of the piezometer (i.e. of the saturated zone) at each iteration, in order to obtain a zero pressure along that interface. That method is particularly costly and difficult to handle, because it does not necessarily ensure the incompressibility of the fluid. Besides, it is not necessarily justified as the depth of variation of the piezometer is small (~60 m) compared to the depth of the basin. Therefore, the pressure perturbation due to convection and surface topography is small in comparison with the lithostatic pressure variations in the basin. Accordingly, in our model, the coincidence of the surface topography with the piezometer results to be an acceptable first-order approximation (McKenzie et al. 1974). Besides, it is extremely cost-effective as it does not require internal iterations between two time steps of the temperature equation. As the program retrieves the pressure after each time step, we can a posteriori compute the height of the piezometer with first-order precision. Because the topography along the borders of the basin is not flat, local flows are generated and eventually combine with those developed by convection. Actually, they have a notable impact on the circulation recovered in the model over the Kadzell and Kanem area (Sect. 2).

Appendix 4: Evaluation of the Hydrological and Thermal Characteristics of Our Model

The maps of the fluid viscosity of the thermal expansion and of the effective thermal conductivity of the basin of our convective model are displayed in Fig. 16a–d. From these maps, we deduce that, within the Bima horizon, at the location of the P1 profile: γ = 57 °C km⁻¹, K = 2 × 10⁻¹³ m², h = 400 m, α = 2.9 × 10⁻⁴ °C⁻¹, λ _eq = 2.3 W °C⁻¹ m⁻¹, μ = 5 × 10⁻⁴ Pa s, and thus, the Rayleigh number is equal to 21 (Eq. 1). Alternatively, around P2: γ = 62 °C km⁻¹, h = 900 m, α = 3.8 × 10⁻⁴ °C⁻¹, and μ = 3 × 10⁻⁴ Pa s, and the Rayleigh number is equal to 230. Finally, to evaluate the vertical Rayleigh number of the whole basin above faults 5 and 6, we chose the following numbers: γ = 43 °C km⁻¹, K = 10⁻¹⁴ m², h = 4000 m, α = 2.6 × 10⁻⁴ °C⁻¹, k = 2 W °C⁻¹ m⁻¹, μ = 10⁻³ Pa s, and thus, the Rayleigh number is equal to 31. These numbers indicate that (a) convection likely develops in the Bima horizon outside the block separated by faults 1 and 2, and (b) the basin-wide cell is not split into smaller ones in the graben delimited by faults 5 and 6. Below Kanem, the evaluation of the effective Rayleigh number for the entire depth of the basin is more problematic, due to the permeability variations along the trajectory of the warm current. First, at the eastern border of the basin, the permeability of the contiguous Bima blocks is 2 × 10⁻¹³ m² in a channel extending from 4 km to ~700 m depth. In that channel, the vertical Rayleigh number is ~380 (i.e. much greater than 4π ²) and the mean vertical velocity is 10 cm year⁻¹.

Fig. 16

Maps of the different physical parameters used for the modelling of the convective circulation in the Lake Chad basin: a permeability fields assigned to the different formations of the basin. Dashed ellipses highlight the stratigraphic throws of faults 7 and 8 that impact the Fika formation and vertical black lines highlight the location of temperature profiles P1, P2, P3 displayed in Fig. 10, b fluid viscosity field (in Pa s), c variations of the thermal expansion coefficient (in K⁻¹) with the temperature and d distribution of the effective thermal conductivity (in W m⁻¹ K⁻¹) within the basin

Actually, in a real 3D geometry, hydrothermal convective currents parallel to the walls of the basin may develop (Rabinowicz et al. 1998a, b). Despite the ~150-m-thick Fika formation (located eastward of fault 8, Fig. 3a), the warm current reaches the QPA through the zone of the Fika formation shift induced by throws of faults 7 and 8 (ellipses A and B in Fig. 3a). In the last ~700-m-thick upper sediments, the intensity of the flow remains strong as the mean vertical velocity is ~4 cm year⁻¹. The strength of this warm current explains the intensity of the backflow of the basin-wide convection, which reaches a horizontal velocity of 39 cm year⁻¹ over the Kanem, and down to ~7 cm year⁻¹ at Kadzell. The resulting top pressure drop from Kanem to Kadzell explains the piezometric level topography.

Finally, let us suppose that the ascending and descending convective velocities along the lateral border of the basin-wide cell are constant and have an intensity equal to V. Then, an analytical 1D solution for the steady temperature (Eq. 4) shows that:

$$T\left( z \right) = T_{\text{bottom}} + \gamma \frac{{\left( {1 - \exp \left( { \pm Pe\frac{z}{h}} \right)} \right)}}{{\left( {1 - \exp \left( { \pm Pe} \right)} \right)}}$$

(9)

and

$${\text{d}}T\left( z \right)/\partial z = - \pm Pe\,\gamma \frac{{\exp \left( { \pm Pe\frac{z}{h}} \right)}}{{\left( {1 - \exp \left( { \pm Pe} \right)} \right)}}$$

(10)

where Pe represents the Péclet number, which is written as:

$$Pe = \frac{{C_{\text{L}} hV}}{{\lambda_{\text{eq}} }}$$

(11)

In Eqs. (9) and (10), z represents the height in the column of total thickness h, where the flow is ascending or descending, respectively. Over Kanem, +V ~ 4 cm year⁻¹, h ~ 700 m and λ _eq ~ 2 W °C⁻¹ m⁻¹ and thus Pe ~ 1.9. If we assume that Pe ≫ 1, we see that:

$${\text{d}}T\left( {z = h} \right)/\partial z \approx Pe\,\gamma \approx 1.9\,\gamma$$

(12)

Alternatively, below Kadzel, where V ~ −4 mm year⁻¹, h ~ 1000 m and λ _eq ~ 2 W °C⁻¹ m⁻¹, Pe ~ 0.3. Therefore, assuming that Pe ≪ 1, we see that:

$${\text{d}}T\left( {z = h} \right)/\partial z = \exp \left( { - Pe} \right)\gamma \approx 0.75\,\gamma$$

(13)

For Kadzell, close to the top of P1 profile, where the axis of the descending current associated with the basin-wide cell is approximatively located, the observed conductive heat flow is ~110 mW m⁻² and the convective heat flow is ~70 mW m⁻². These values show that the approximation deduced from Eq. (13) is valid. Over the Kanem, close to the top of the P3 profile, the conductive heat flow is ~100 mW m⁻² and the convective one is ~200 mW m⁻². The approximation given by Eq. (12), which is specifically valid for a fast ascending flow, leads to a good match with this last pair of heat flow values.

In Sect. 2, we emphasise that the mean heat flows at the top of our models are not significantly different when fluids circulate by convection and when the transport of heat at the surface is purely conductive: here 110 mW m⁻², instead of 90 mW m⁻². Actually, this result can be rationalised as follows. Along the interface of the sediment and the basement, inside the Bima formation, the mean horizontal velocity V of the L ~ 240 km basin-wide cell is ~10 cm year⁻¹. Besides, the temperature contrast ΔT between the descending current along P1 and the basement is ~30 °C (Fig. 10). For a boundary layer approximation of the mean heat flow ϕ along a cooling plate (Turcotte and Schubert 2002):

$$\phi = 2\lambda_{\text{eq}}\Delta T\sqrt {\frac{{C_{\text{L}} V}}{{\pi \lambda_{\text{eq}} L}}} \approx 13\,{\text{mW}}\,{\text{m}}^{ - 2}$$

(14)

This expression correctly accounts for the difference of heat flows between the convective and conductive models.