Aquifer conditions, not irradiance determine the potential of photovoltaic energy for groundwater pumping across Africa

PVWPS operation

The motor and the pump are built in together¹⁴ and the motor-pump set is submersed in the borehole under the water⁴³. Control equipment is also installed between the PV modules and the motor-pump and/or integrated to the motor-pump set in the borehole^14,17. This equipment allows the motor-pump to stop and also to operate the motor-pump and the PV modules at their best operating points¹⁴. Once the water is pumped, it might then be stored in a water tank to mitigate the variability of solar resources^14,29. When pumping starts, a cone of depression of radius r_c is formed and there is a drawdown H_b,d in the borehole (see Fig. 1). The higher the pumped flow rate, the higher the drawdown H_b,d and therefore the deeper the water in the borehole H_b. If H_b reaches the position of the motor-pump H_mp, the motor-pump automatically switches off, therefore preventing the motor-pump from running dry⁴⁴. The motor-pump remains shut down during a period Δt_shut, after which it makes an attempt to restart⁴⁴.

Input data processing

We observe in Table 1 that the datasets have varying spatial resolutions. In the article, we use the spatial resolution of the irradiance map, 0.2° (~22 km). Indeed, this resolution is sufficient for the purposes of this article and it allows to divide computing time and memory requirements by ~16 in comparison to the 0.05° resolution. At this 0.2° resolution, the total area of Africa of 30 million km² is divided into 62,000 pixels. We apply this resolution of 0.2° to all datasets by nearest interpolation.

No exact value of the static water depth H_b,s, transmissivity T and saturated thickness H_st are provided for each location by the original source but only a range of variation. For instance, for −15.8° (lat) & 21.9° (lon), the saturated thickness H_st is comprised between 25 and 100 m. In most cases, we consider the middle of the range (e.g., 62.5 m in the example). The only two exceptions are: when H_b,s is higher than 250 m, we consider 300 m (same for H_st); and, when H_b,s is between 0 and 7 m, we consider 7 m⁴⁵. Due to the lack of available information, the input groundwater data provided in Table 1 are considered to remain constant over time.

Reference⁴⁶ provides complete irradiance data with a time step of 15 min from 2013 to 2020 across Africa. In this article, except mentioned otherwise, we use irradiance data from 2020 with a 30-min time step (by taking one point every two 15-min points), instead of all the available complete irradiance data. It divides computing time and memory requirements by ~16. Additionally, it produces reduced and acceptable deviations on the results. Indeed, for 100 randomly chosen locations, we simulated the pumped volume V, for the three considered PVWPS sizes, using (1) irradiance data from 2013 to 2020 with a 15-min time step and (2) irradiance data from 2020 with a 30-min time step. For these locations, the absolute error on volume V is systematically lower than 7.9% and the average absolute error is 2%. These results are coherent with the observed low influence of irradiance on the pumped volume in comparison to groundwater resources. Thanks to the consideration of this reduced irradiance vector, the random access memory (RAM) and the computing time required to obtain a map of final results (such as Fig. 5b) are respectively 38 Gb and 10 h (time for Intel Xeon E5-2643 3.3 GHz processors and 96 GB RAM, running on Debian 4.19.194-2), which is more reasonable.

Atmospheric sub-model

For each location, the irradiance on the plane of the PV modules G_pv at time t can be deduced from satellite data by^47,48:

where κ is the albedo of the surrounding environment, θ and α are the tilt and azimuth of the PV modules and AOI is the angle of incidence between the sun’s rays and the PV modules. The albedo κ is taken equal to 0.2 because it corresponds to the albedo of cropland, which is a common environment in the rural areas considered⁴⁹. In any case, additional simulations show that the value of the albedo has a negligible effect on the pumped volume V. AOI is computed using the MATLAB toolbox PVLIB developed by the Sandia National Laboratories⁵⁰.

For each location, the azimuth α and the tilt θ of the PV modules are chosen to maximize the irradiance on the plane of the PV modules G_pv. The azimuth α is taken equal to⁵¹:

where ϕ is the latitude of the location. The tilt is taken equal to⁵¹:

As evidenced by Eq. (3), the tilt should be higher than 10° or lower than −10°, so that the PV modules are tilted enough to be cleaned when it rains.

Photovoltaic modules sub-model

Considering that the maximum power point tracking of the PV modules is correctly performed, a simplified model to compute the power P produced by the modules is used:

where G₀ is the reference irradiance (1000 W m⁻²), P_p is the peak power of the PV modules in standard test conditions (STC) and c_pv,loss is a coefficient that represents the losses (e.g., soiling, temperature, mismatch, wiring^52,53) at the level of the PV modules. For the sake of simplicity, and as we consider a generic PVWPS, we consider that c_pv,loss is independent of the operating point of the PV modules, of the time, and of the location. We take it constant, equal to a single value (see Table 2).

Hydraulic sub-model

The total dynamic head TDH between the motor-pump and the pipe output is given by⁵⁴:

where H_b is the water depth in the borehole and H_p is the additional head due to pressure losses in the pipe.

The water depth in the borehole H_b is given by (see Fig. 1)⁴²:

where H_b,s is the static water depth and H_b,d is the drawdown. The drawdown is composed of two parts:

where ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}(t)) is the head loss due to aquifer losses and ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{b}}}}}}}(t)) is the head loss due to borehole losses.

The head loss due to aquifer losses ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}(t)) depends on the pumping flow rate Q, the aquifer transmissivity T, the borehole radius r_b, and a length parameter r_c representing the distance of water travel to replace the water pumped out. From dimensional analysis, we expect that (tfrac{{H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}left(tright)cdot T}{Qleft(tright)}) should be a function of (tfrac{{r}_{{{{{{rm{c}}}}}}}}{{r}_{{{{{{rm{b}}}}}}}}). We thus propose the following model for ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}), which is derived from Thiem equation⁵⁵:

where r_c can be considered the effective radius of the cone of depression. This model satisfies horizontal, radial and steady Darcy flow in a uniform, homogeneous and isotropic aquifer. It captures the essential features for aquifer losses: ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}(t)) proportional to pumped flow rate and inversely proportional to transmissivity⁴⁵. Though the flow is transient, only simplified steady-state models, as the one of Eq. (8), can be applied with the available information as dynamic models would require pumping tests. Furthermore, we consider that the radius of the cone of depression r_c is comprised between 100 and 1000 m and, to correlate it to a measured quantity, that it depends linearly on the groundwater recharge R: for the lowest recharge (0 m/year), r_c is equal to 1000 m; for the highest one (0.2947 m year⁻¹), r_c is equal to 100 m; in-between, r_c is obtained linearly from the recharge (r_c = 1000–3054 · R). Thus, groundwater recharge R is used to constrain the size of the cone of depression.

The head loss due to borehole losses ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{b}}}}}}}(t)) is given by⁵⁶:

where β is a coefficient related to the borehole design. For the yields considered in this article, ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{b}}}}}}}(t)) usually remains lower than a few meters but, as ({H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{b}}}}}}}(t)) depends on the square of the pumped flow rate, it may be more important for larger abstraction capacities.

The additional head due to pipe losses H_p is given by⁵⁷:

where H_p,ma(t) corresponds to losses that occur along the pipe length (also called “major losses”) and H_p,mi(t) corresponds to losses at junctions such as elbows and curvatures (also called “minor losses”). H_p,ma(t) is given by⁵⁷:

where g is the gravitational acceleration (9.81 m s⁻²), D_p is the pipe diameter, L_p is the pipe length, Q is the pumped flow rate, and f is the friction coefficient between the water and the pipe. We approximate the pipe length L_p to be equal to the depth of the motor-pump H_mp (see Fig. 1). The expression of f depends on the value of the Reynolds number ({{{{{rm{Re}}}}}}=tfrac{4Q}{pi {D}_{{{{{{rm{p}}}}}}}w}), where w is the water kinematic viscosity (taken equal to 1 × 10⁻⁶ m² s⁻¹)⁵⁷:

• for Re <3 × 10³, (f=tfrac{64}{{{{{{rm{Re}}}}}}});

• for Re ≥3 × 10³, f is the solution of (tfrac{1}{surd f}=-2{{{{{rm{ln}}}}}}left(tfrac{epsilon }{3.7{D}_{{{{{{rm{p}}}}}}}}+tfrac{2.51}{{{{{{rm{Re}}}}}}sqrt{f}}right)), where ϵ is the pipe roughness.

This complex formulation for f very importantly complicates the resolution of Eq. (15). To avoid this problem, we started by computing the head loss due to major losses H_p,ma using Eq. (11), for various:

• pipe diameters D_p between 0.04 m and 0.1 m, which is an usual range for PVWPS^58,59;

• pipe roughnesses ϵ between 0 and 1.5 × 10⁻⁴ m, which is an usual range for PVWPS^57,58;

• flow rates Q between 0 and 5 × 10⁻³ m³ s⁻¹, which is an usual range for PVWPS^29,60;

• pipe lengths L_p between 10 and 500 m, which corresponds to possible motor-pump depths.

We then fitted H_p,ma, as a function of L_p and Q, and according to²⁹:

where ν is a coefficient that depends on ϵ and D_p. For all the considered combinations of D_p and ϵ, we always obtained a fitting R² higher than 0.99. For instance, for D_p = 0.052 m and ϵ = 1.5 × 10⁻⁶ m, we obtained ν = 8.9 × 10² s²m⁻⁶ with R² = 0.996. Thus, we approximate major losses with Eq. (12) and determine ν through fitting.

The head due to minor losses H_p,mi(t) is given by:

where k_i is the coefficient associated to each junction i (values for the different junction types are provided in article⁵⁷). We neglect the dependency of k_i on the Reynolds number, as usually done⁵⁷.

Motor-pump sub-model

To determine the pumped flow rate Q, we first suppose that the motor-pump is operating, which is the case when the input power to the motor-pump P is higher than the starting power of the motor-pump P_mp,0 and when the water depth in the borehole H_b does not reach the motor-pump position H_mp. When the motor-pump is operating, the pumped flow rate Q is given by^61,62,63:

where g is the gravitational acceleration (9.81 m s⁻²), ρ is the water density (1000 kg m⁻³) and η_mp is the motor-pump efficiency. We consider that η_mp is a constant for the same reasons as for the PV modules loss coefficient c_pv,loss. By integrating relations (5), (6), (7), (8), (9), (10), (12) and (13) into Eq. (14), we obtain Q by solving:

We take the only physically feasible solution of the equation for obtaining Q.

Once Q is determined, we compute H_b thanks to Eqs. (6)–(9). If H_b is found to be higher than H_mp, then we in fact set Q to 0 (i.e., the motor-pump stops) for a period Δt_shut. After this period Δt_shut, the motor-pump attempts to restart.

Once Q for each 30-min time step of the year is determined, we deduce the average daily pumped volume V as following:

Thus, when we use input irradiance data for 2020 with a 30-min time step, for each pixel, the average daily pumped volume V is obtained from 17568 (=2 × 24 × 366) values of pumped flow rate Q.

Source: Resources - nature.com