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

PVWPS operationThe 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 borehole14,17. 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 resources14,29. When pumping starts, a cone of depression of radius rc is formed and there is a drawdown Hb,d in the borehole (see Fig. 1). The higher the pumped flow rate, the higher the drawdown Hb,d and therefore the deeper the water in the borehole Hb. If Hb reaches the position of the motor-pump Hmp, the motor-pump automatically switches off, therefore preventing the motor-pump from running dry44. The motor-pump remains shut down during a period Δtshut, after which it makes an attempt to restart44.Input data processingWe 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 Hb,s, transmissivity T and saturated thickness Hst 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 Hst 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 Hb,s is higher than 250 m, we consider 300 m (same for Hst); and, when Hb,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-modelFor each location, the irradiance on the plane of the PV modules Gpv at time t can be deduced from satellite data by47,48:$${G}_{{{{{{rm{pv}}}}}}}left(tright)={G}_{{{{{{rm{bn}}}}}}}left(tright){{cos }}left({{{{{rm{AOI}}}}}}left(t,theta ,alpha right)right)+{G}_{{{{{{rm{gh}}}}}}}left(tright)kappa frac{1-{{cos }}left(theta right)}{2}+{G}_{{{{{{rm{dh}}}}}}}left(tright)frac{1+{{cos }}left(theta right)}{2}$$

(1)

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 Gpv. The azimuth α is taken equal to51:$$alpha =left{begin{array}{c}180^circ quad {{{{{rm{if}}}}}},phi , > ,0\ 0^circ quad {{{{{rm{if}}}}}},phi , < ,0end{array}right.$$
(2)
where ϕ is the latitude of the location. The tilt is taken equal to51:$$theta =left{begin{array}{c}{{max }},(10,1.3793+(1.2011+(-0.014404+0.000080509phi )phi )phi )quad{{{{{rm{if}}}}}},phi > ,0\ {{min }},(-10,-0.41657+(1.4216+(0.024051+0.00021828phi )phi )phi )quad{{{{{rm{if}}}}}},phi , < ,0end{array}right.$$
(3)
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-modelConsidering 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:$$Pleft(tright)=frac{{G}_{{{{{{rm{pv}}}}}}}left(tright)}{{G}_{0}}{P}_{{{{{{rm{p}}}}}}}left(1-{c}_{{{{{{rm{pv}}}}}},{{{{{rm{loss}}}}}}}right)$$
(4)
where G0 is the reference irradiance (1000 W m−2), Pp is the peak power of the PV modules in standard test conditions (STC) and cpv,loss is a coefficient that represents the losses (e.g., soiling, temperature, mismatch, wiring52,53) at the level of the PV modules. For the sake of simplicity, and as we consider a generic PVWPS, we consider that cpv,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-modelThe total dynamic head TDH between the motor-pump and the pipe output is given by54:$${TD}Hleft(tright)={H}_{{{{{{rm{b}}}}}}}(t)+{H}_{{{{{{rm{p}}}}}}}(t)$$
(5)
where Hb is the water depth in the borehole and Hp is the additional head due to pressure losses in the pipe.The water depth in the borehole Hb is given by (see Fig. 1)42:$${H}_{{{{{{rm{b}}}}}}}(t)={H}_{{{{{{rm{b}}}}}},{{{{{rm{s}}}}}}}+{H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}(t)$$
(6)
where Hb,s is the static water depth and Hb,d is the drawdown. The drawdown is composed of two parts:$${H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}(t)={H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}(t)+{H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{b}}}}}}}(t)$$
(7)
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 rb, and a length parameter rc 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:$${H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{a}}}}}}}left(tright)=frac{{{{{{rm{ln}}}}}}left(frac{{r}_{{{{{{rm{c}}}}}}}}{{r}_{{{{{{rm{b}}}}}}}}right)}{2pi T}Qleft(tright)$$
(8)
where rc 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 rc 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), rc is equal to 1000 m; for the highest one (0.2947 m year−1), rc is equal to 100 m; in-between, rc is obtained linearly from the recharge (rc = 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:$${H}_{{{{{{rm{b}}}}}},{{{{{rm{d}}}}}}}^{{{{{{rm{b}}}}}}}left(tright)=beta {Q}{left(tright)}^{2}$$
(9)
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 Hp is given by57:$${H}_{{{{{{rm{p}}}}}}}left(tright)={H}_{{{{{{rm{p}}}}}},{{{{{rm{ma}}}}}}}left(tright)+{H}_{{{{{{rm{p}}}}}},{{{{{rm{mi}}}}}}}left(tright)$$
(10)
where Hp,ma(t) corresponds to losses that occur along the pipe length (also called “major losses”) and Hp,mi(t) corresponds to losses at junctions such as elbows and curvatures (also called “minor losses”). Hp,ma(t) is given by57:$${H}_{{{{{{rm{p}}}}}},{{{{{rm{ma}}}}}}}left(tright)=frac{8f}{{pi }^{2}g{D}_{{{{{{rm{p}}}}}}}^{5}}{L}_{{{{{{rm{p}}}}}}}Q{left(tright)}^{2}$$
(11)
where g is the gravitational acceleration (9.81 m s−2), Dp is the pipe diameter, Lp 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 Lp to be equal to the depth of the motor-pump Hmp (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 More