Introduction

In Africa, >300 million people use unimproved water sources for domestic use, mostly in rural off-grid areas1. Irrigation is also limited and there are calls to increase irrigation to improve food security given increased climate variability2,3. Groundwater and surface water are the main water sources. Even though surface water is often shallower and cheaper to extract, groundwater constitutes the largest and most widely distributed store of freshwater in Africa4 and, unlike surface water, often does not require treatment5. Groundwater is also suitable for irrigation because it responds more slowly to meteorological conditions and thus provides a natural buffer against climate variability6,7. Currently, most rural groundwater pumping in Africa is undertaken by community handpumps8,9, which have proved easily repaired and resilient to drought10,11. However ongoing functionality rates can be low due to installation and maintenance issues12. The Sustainable Development Goals call for a higher level of service, with safe water available reliably at individual households13. There is therefore a considerable challenge to move service levels beyond community handpumps.

Pumping systems powered by photovoltaic energy are a promising solution to improve water access in many off-grid areas without importantly increasing greenhouse gas emissions. They are already economically competitive in many contexts14, technological advances have improved their longevity15 and local case studies (e.g.,16,17,18) have shown promising results. However, these results may not be encountered in other locations due to management issues4,19 and the spatial variability of groundwater and solar resources.

Some studies have investigated the potential of photovoltaic water pumping systems (PVWPS) over continuous geographical areas. These studies have been conducted in Ethiopia20, Ghana21, Egypt22,23, Algeria24, Spain and Morocco25, China26,27 and for locations with shallow groundwater (static water depth <50 m) in sub-Saharan Africa28. However, articles20,21,22,23,24,26 do not use a technical model of PVWPS, which prevents consideration of the relative importance of groundwater and solar resources. The other studies25,27,28 consider a technical PVWPS model. Nevertheless, they use monthly average irradiance values instead of hourly/sub-hourly time series, which influences PVWPS operation and performances18,29. Additionally, they do not account for the saturated thickness of the aquifer, which restricts the maximum possible drawdown and thus pumped flow rate. Finally, existing studies do not provide results for the whole African continent, which limits comparison between different countries and regions.

Here we propose a model that uses openly available groundwater and irradiance data to simulate the abstraction capacities of PVWPS across Africa. The first contribution of this work is the detailed design of the large-scale PVWPS model to include realistic geological constraints on the depth of pumping, through the aquifer saturated thickness notably, and sub-hourly irradiance time series. Considering, for each pixel, sub-hourly irradiance time series instead of monthly average irradiance values has several advantages. First, the pumped flow rate varies non-linearly with irradiance, therefore, results from sub-hourly irradiance data are different from using averages. Secondly, in numerous cases, for high irradiance values (e.g., in the middle of the day) pumping will cease due to high drawdowns reaching the motor-pump, thus reducing the total pumped volume. This would not be observed if monthly average irradiance values were used. This is where the other specificity of the model, which consists in including a realistic geological constraint on the depth of pumping, also plays an important role. Finally, considering sub-hourly irradiance time series enables simulations for critical days of the year (notably very low irradiance days) for which the pumped volume may be very low, which can impact smooth water consumption. The second contribution of this work is the provision of results for the whole continent, including for North Africa and for locations where groundwater is deeper than 50 m, where PVWPS have particular relevance as groundwater is very difficult to access through handpumps4. The results are provided for three PVWPS sizes, for the whole year as well as for extreme periods of the year (e.g., low irradiance consecutive days) and compared with groundwater recharge. Providing results for the whole continent allows comparing regions between themselves and identifying regions where PVWPS have the highest potential and thus help target investments. Our results notably reveal that for 27% of the locations the largest considered system does not yield the highest volume because of too important drawdowns which reach the motor-pump thus forcing the system to stop. They also show that the main determinant of the spatial variations of the pumped volume is the aquifer conditions rather than the irradiance.

System, input data and model overview

The considered PVWPS architecture is shown in Fig. 1 and its operation is presented in the methods section. This architecture is common for groundwater abstraction with photovoltaic energy16,30. In this article, we consider a generic PVWPS and with the size of the motor-pump proportional to the peak power of the photovoltaic (PV) modules. The peak power of the PV modules is therefore used as a proxy for the size of the PVWPS.

Fig. 1: Considered PVWPS architecture.
figure 1

All the lengths are defined as positive. rb: borehole radius, Hbb: borehole depth, Hmp: motor-pump depth, Hst: aquifer saturated thickness, Hb,s: static water depth (depth when there is no pumping), Hb,d: drawdown, Hb: water depth in the borehole, rc: radius of the cone of depression.

Full size image

The characteristics of the irradiance and groundwater resources input data considered in this study are presented in Table 1. Information about the processing carried out on input data are given in the methods section. In Fig. 2, we plot the average over the year of the global horizontal irradiance Ggh for 2020, static water depth Hb,s, aquifer transmissivity T, saturated thickness Hst and groundwater recharge R.

Table 1 Original input location-dependent data.
Full size table
Fig. 2: Representation of selected input data.
figure 2

a global horizontal irradiance Ggh (average over the year for 2020, yellow scale), b static water depth Hb,s (blue scale), c aquifer transmissivity T (blue scale), d aquifer saturated thickness Hst (blue scale), e groundwater recharge R (blue scale). Data sources are shown in Table 1.

Full size image

The proposed model allows to simulate, for each location of Africa, the average daily pumped volume V by a PVWPS of peak power Pp. The block diagram of the model is presented in Supplementary Fig. 1. To compute the model results, two types of elements are required. The first one is the design parameters, which are listed in Table 2. They are set by the designer depending on the components chosen for the PVWPS. For the rest of the article, we choose the values presented in Table 2 for the design parameters. The second one is the location-dependent data (in particular irradiance and groundwater resources input data). Except for the albedo of the surrounding environment κ, the data are provided by the references of Table 1. We discuss about the choice of κ in the methods section. The proposed model is composed of several sub-models (see Supplementary Fig. 1). The atmospheric sub-model computes the irradiance on the plane of the PV modules Gpv from input irradiance data. The PV modules sub-model computes the power produced by the PV modules P from Gpv and the peak power of the PV modules Pp. The motor-pump sub-model computes the pumped flow rate Q from P and the total dynamic head that the motor-pump has to overcome TDH. The hydraulic sub-model computes TDH from Q, accounting for the response of the water level in the borehole to pumping and for pipe losses. The pumped flow rate Q is integrated over time to obtain the average daily pumped volume V. These sub-models are detailed in the methods section. In Fig. 3, we present the average over the year of the irradiance on optimally tilted PV modules across Africa, which is an intermediary result of the model and which is relevant for the analysis of the final results.

Table 2 Values considered for the design parameters.
Full size table
Fig. 3: Irradiance Gpv on optimally tilted PV modules.
figure 3

Average over the year for 2020 (yellow scale).

Full size image

Results

Detailed pumped volume results for a single location

We start by detailing the model results for one given location (GPS: latitude = 6.2°, longitude = −5.2°), which is in the Ivory Coast. In Fig. 4, we plot the evolution of the pumped flow rate Q and of the water depth in the borehole Hb for three sizes of systems (100 Wp, 1000 Wp and 3000 Wp) and for this location. 100 Wp corresponds to the smallest considered size of PVWPS. Indeed, the majority of PVWPS for groundwater abstraction are larger than 100 Wp14,31,32. As a general order of magnitude, 1000 Wp is a typical PVWPS size for domestic water access14,31 and 3000 Wp is a typical size for irrigation14,32 (the size may of course vary depending on the considered specific system). Logically, we observe that the pumped flow rate Q and thus the borehole water depth Hb follow the evolution of the irradiance. In Fig. 4c, we observe the shutdown of the motor-pump (Q = 0) for a long part of the day, when the borehole water depth Hb would have reached the position of the motor-pump Hmp (see ‘PVWPS operation’ in the methods section). This leads to a decrease of the pumped volume V. For instance, in this case, the average daily pumped volume V for a 1000 Wp PVWPS is 15.9 m3 and only 5.2 m3 for a 3000 Wp system.

Fig. 4: Evolution of the pumped flow rate and of the borehole water depth, at the location 6.2° (lat) & −5.2° (lon), for three PVWPS sizes.
figure 4

a Pp = 100 Wp, b Pp = 1000 Wp, c Pp = 3000 Wp. The pumped flow rate is plotted in solid blue, the borehole water level in solid orange and the motor-pump depth with a dashed gray line.

Full size image

Pumped volume maps for the whole Africa

The results for all of Africa are shown in Fig. 5 presenting the average daily pumped volume V for PVWPS of 100, 1000 and 3000 Wp. Results indicate that the pumped volume values vary importantly from one area to another. These values have to be compared to the water requirements for domestic use and irrigation. For domestic use, daily water requirements are ~15 L/person/day for basic access33. Consequently, a PVWPS which extracts 5 m3 per day meets the basic water needs of a village of ~330 inhabitants, although if water was available on premises individual use would rise considerably34. Regarding agriculture, as a general order of magnitude, farmer led or community irrigation would typically require low yields (10–100 m3 per day) and commercial irrigation would require larger yields, often >200 m3 per day4,35 (these numbers of course vary depending on the irrigated area and the crop type).

Fig. 5: Average daily pumped volume V for three PVWPS sizes.
figure 5

a Pp = 100 Wp, b Pp = 1000 Wp, c Pp = 3000 Wp for 2020. The results are plotted in a blue scale.

Full size image

As highlighted by the detailed results on the single location, the pumped volume maps (Fig. 5) also indicate that, in certain areas (e.g., for a large portion of Zimbabwe), smaller PVWPS have greater pumping capacities than larger systems. This is notably the case where the transmissivity T and the saturated thickness Hst are low, and thus where an increased drawdown may reach the motor-pump depth for high irradiance levels during the day. In order to better illustrate this, we present in Fig. 6a, for each pixel, the size of the system (amongst the three considered sizes) which yields the highest pumped volume V. In Fig. 6b, we show the pumped volume associated to the most effective size. We observe that, even though for the majority of cases (73%) the system of 3000 Wp extracts the highest volume, there are still a large number of cases where the systems of 100 Wp and 1000 Wp extract the most water (13 and 14% respectively).

Fig. 6: Best performing PVWPS size and associated pumped volume for 2020.
figure 6

Size values are given in a and volume values in b. The results are plotted in a blue scale.

Full size image

Influence of temporal irradiance variations

In addition to the spatial variation of irradiance (highlighted for instance by Fig. 3), temporal irradiance variations also occur, which may have an impact on the pumped volume. In order to investigate the influence of inter-annual variation of irradiance, we first computed and plotted the average yearly irradiance Gpv and the associated pumped volume V for 2014 and 2017 for a 1000 Wp system (see Supplementary Fig. 2). We observe that the results for 2014, 2017 and 2020 are similar both in terms of average yearly irradiance and pumped volume. When we compute, for each pixel, the difference in absolute value between the 2014 results and the 2020 results and we average the difference over Africa, we obtain differences of 2.5 % for the irradiance and 2.7 % for the pumped volume. When we compare 2017 results to 2020 results, differences of 2.5% for the irradiance and 2.8% for the pumped volume are obtained.

We then studied the influence of the variation of irradiance during the year (intra-annual variations) for 2020. First, for each pixel, we computed the monthly average irradiance for each month; and then applied the PVWPS model for a 1000 Wp system, to compute the pumped volume, using the time series of the month with the highest (Supplementary Fig. 3a, b) and lowest (Supplementary Fig. 3c, d) average irradiance. Then, we repeated for consecutive three days periods with highest (Supplementary Fig. 3e, f) and lowest (Supplementary Fig. 3g, h) average irradiance. We considered periods of three days because typically water storage systems for domestic water access are sized to hold for several low pumping days36,37. Supplementary Fig. 3 reveals that, while some influence is noticed for the most and least irradiated month, the most important impact occurs for the most and least irradiated three days period. The differences in absolute value (averaged over Africa) on the pumped volume in comparison to the results for the whole 2020 (see Fig. 5b) are 16.2% and 19.7% for the best and worst month respectively; and 29.7% and 53.6% for the best and worst 3 days period respectively. This highlights the importance of considering extreme periods days when investigating the variation of the system performance over the year and its ability to enable smooth water consumption and when sizing the storage component (when it is present). For systems not limited by the drawdown reaching the motor-pump, the period with the lowest average irradiance is going to be relevant and oppositely for the other systems.

Parametric analysis and drivers of the pumped volume spatial variations

In addition to spatial and temporal irradiance variations, the values of input groundwater parameters and design parameters may also vary, which may then influence the pumped volume V. The variation of input groundwater parameters may be due to the uncertainty in these parameters, or using local data rather than spatially averaged data. The variation of design parameters would be due to design choices. In order to investigate how a variation of input groundwater parameters and design parameters will affect the pumped volume, we consider a variation of +/− 15% and +/− 30% of input groundwater parameters and of the relevant design parameters. For each new value of the parameter, we apply the PVWPS model for 2020 for the three considered PVWPS sizes (100, 1000 and 3000 Wp) and for 100 random locations (3 × 100 = 300 cases considered in total). We then compare the average pumped volume for these 300 cases to the average pumped volume for these same cases with all the parameters being at their nominal value. The results are provided in Supplementary Fig. 4a. We observe that the most influential parameters are the static water depth Hb,s and the motor-pump efficiency ηmp. The transmissivity T, the saturated thickness Hst and the PV modules loss coefficient cpv,loss also have a notable influence. Considering only locations where the largest system does not yield the highest pumped volume, we observe that the transmissivity T and the saturated thickness Hst have the highest influence on the pumped volume (see Supplementary Fig. 4b). Indeed, for these locations, the pumped volume is strongly affected by the drawdown (which is driven by T) reaching the motor-pump position (which is set depending on Hst). This is also coherent with the visual inspection of the African wide maps where we see that area where the largest system does not yield the highest volume correspond to low transmissivity T and/or low saturated thickness Hst areas. For locations where the largest system yields the highest volume, we observe that the static water depth Hb,s and the motor-pump efficiency ηmp (which has the same influence as the irradiance Gpv(t) on the pumped flow rate Q(t), see Eq. 14 and Eq. 4) have the highest impact on the pumped volume (see Supplementary Fig. 4c). Variations in water depth Hb,s dominate the African wide pumped volume maps for this type of locations since the spatial variability of the static water depth (between 7 and 300 m) is much higher than the one of the irradiance (average over the year between 169 and 329 W m−2).

Comparison of pumping performances to recharge

To provide an initial estimate of the sustainability of pumping in comparison to the groundwater recharge, in Fig. 7, we plot, for each location, the ratio between the volume pumped by 50 PVWPS of most efficient capacity for 2020 (see Fig. 6) and 25% of the yearly volume recharged in a square of 22 km × 22 km. The yearly volume recharged is computed by multiplying the recharge R in the pixel (in m/year) by the area of the square (22 km × 22 km). We choose 50 systems to be conservative, as its unlikely to have more than this number in a 22 km × 22 km rural area. Abstracting <25% of renewable freshwater represents little risk of environmental water stress according to the indicator developed for monitoring the Sustainable Development Goals38. We observe in Fig. 7 that, for the majority of locations (78%), the volume pumped by the 50 PVWPS is lower than 25% of the recharged volume. However, it is important to keep in mind that Fig. 7 does not account for potentially already existing pumping systems (e.g., handpumps, diesel pumps). Additionally, the number of PVWPS considered to plot Fig. 7 (50 PVWPS) is arbitrary and the real number of systems implemented would vary from one location to another depending on the needs. Using population density39 (see Fig. 8) as a proxy for demand indicates that in more populated rural areas recharge tends to be higher, therefore, risks of over-exploitation in these areas (where PVWPS are particularly relevant) are generally low. This is consistent with recent work which showed that groundwater recharge was sufficient to sustainably supply rural water supply across much of populated Africa40.

Fig. 7: Percentage of the exploitable recharge that would be extracted by 50 PVWPS of most efficient capacity.
figure 7

Red pixels indicate that the extraction exceeds the exploitable recharge and blue pixels that it does not.

Full size image
Fig. 8: Population density.
figure 8

The results are plotted in a blue scale. Data from ref. 39.

Full size image

Conclusions

We proposed a model that uses publicly and freely available groundwater and irradiance data to simulate the operation of PVWPS for any location in Africa. The model, which accounts for the different components of the PVWPS, notably simulates the evolution of the pumped flow rate and of the drawdown with a 30-min time step during a year and it considers the possible stops of the motor-pump due to excessive drawdowns. Using this model, we estimated, for all locations in Africa, the volume pumped by PVWPS of 100, 1000 and 3000 Wp. Results indicate a strong spatial variability of the pumped volume. We found that, for 27% of the positions, the largest system does not yield the highest pumped volume due to excessive drawdowns reaching the motor-pump. This is notably the case where the transmissivity and/or the saturated thickness are low. For the rest of the positions, the main driver of the pumped volume is found to be static water depth.

The first limitation of our study is the one inherent to approaches consisting of evaluating a model to a very large spatial scale (a whole continent). Both groundwater and irradiance input data have themselves been estimated and hide the specificity of the local resources. This is especially true for data regarding the static water depth, the transmissivity and the saturated thickness where important variations may occur over a few meters. Additionally, as we have used data available for the whole Africa, the proposed model is less accurate than the model of a PVWPS for which we can access and measure all the characteristics of the site (through pumping tests for instance). Finally, the results for the pumped volume at every location do not take into consideration possible interference with neighboring pumping systems. For all these reasons, the results of this large-scale analysis should be seen as approximate that provide an overview over the continent, and as complementary to results of local and therefore more accurate analyses. In particular, as groundwater resources have a strong impact on the pumped volume and may strongly vary locally, a coordinated and detailed local investigation and monitoring of groundwater resources41,42 is very beneficial.

Despite these limitations, the results of this study can help identify regions or countries where PVWPS have the highest potential. In particular, they provide information on the suitability of locations in terms of pumped volume, variations of this pumped volume for extreme periods (low and high irradiance) of the year, and sustainability of the pumped volume in comparison to the recharge. They also provide insight on the most effective PVWPS size for the considered location. It may then help target investments in large-scale PV water pumping programs and identify areas where pumping potential is low and will need additional investigation to de-risk investment. This may notably be of interest to funding institutions (e.g., the World Bank, the African Development Bank) and governments. Since the model has been applied to give an initial estimate of suitability for any location in Africa (without the need to collect additional data), it can be used as a screening tool to provide a first estimate of the pumping performances of PVWPS for water sector practitioners (e.g., local companies, non-governmental organizations). Additionally, our parametric analysis results provide information on the design parameters that may have the most influence on the pumped volume and which could thus be optimized first by local installers. Finally, it is possible to adapt the model to pumping systems powered by other energy sources (e.g., hand pumps, diesel pumps), by changing the “photovoltaic energy” sub-model.

Methods

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 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 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 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-model

For each location, the irradiance on the plane of the PV modules Gpv at time t can be deduced from satellite data by47,48:

$${G}_{{{{{{\rm{pv}}}}}}}\left(t\right)={G}_{{{{{{\rm{bn}}}}}}}\left(t\right){{\cos }}\left({{{{{\rm{AOI}}}}}}\left(t,\theta ,\alpha \right)\right)+{G}_{{{{{{\rm{gh}}}}}}}\left(t\right)\kappa \frac{1-{{\cos }}\left(\theta \right)}{2}+{G}_{{{{{{\rm{dh}}}}}}}\left(t\right)\frac{1+{{\cos }}\left(\theta \right)}{2}$$
(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:

$$\alpha =\left\{\begin{array}{c}180^\circ \quad {{{{{\rm{if}}}}}}\,\phi \, > \,0\\ 0^\circ \quad {{{{{\rm{if}}}}}}\,\phi \, < \,0\end{array}\right.$$
(2)

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

$$\theta =\left\{\begin{array}{c}{{\max }}\,(10,1.3793+(1.2011+(-0.014404+0.000080509\phi )\phi )\phi )\quad{{{{{\rm{if}}}}}}\,\phi > \,0\\ {{\min }}\,(-10,-0.41657+(1.4216+(0.024051+0.00021828\phi )\phi )\phi )\quad{{{{{\rm{if}}}}}}\,\phi \, < \,0\end{array}\right.$$
(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-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:

$$P\left(t\right)=\frac{{G}_{{{{{{\rm{pv}}}}}}}\left(t\right)}{{G}_{0}}{P}_{{{{{{\rm{p}}}}}}}\left(1-{c}_{{{{{{\rm{pv}}}}}},{{{{{\rm{loss}}}}}}}\right)$$
(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-model

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

$${TD}H\left(t\right)={H}_{{{{{{\rm{b}}}}}}}(t)+{H}_{{{{{{\rm{p}}}}}}}(t)$$
(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 \(\tfrac{{H}_{{{{{{\rm{b}}}}}},{{{{{\rm{d}}}}}}}^{{{{{{\rm{a}}}}}}}\left(t\right)\cdot T}{Q\left(t\right)}\) 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:

$${H}_{{{{{{\rm{b}}}}}},{{{{{\rm{d}}}}}}}^{{{{{{\rm{a}}}}}}}\left(t\right)=\frac{{{{{{\rm{ln}}}}}}\left(\frac{{r}_{{{{{{\rm{c}}}}}}}}{{r}_{{{{{{\rm{b}}}}}}}}\right)}{2\pi T}Q\left(t\right)$$
(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:

$${H}_{{{{{{\rm{b}}}}}},{{{{{\rm{d}}}}}}}^{{{{{{\rm{b}}}}}}}\left(t\right)=\beta {Q}{\left(t\right)}^{2}$$
(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:

$${H}_{{{{{{\rm{p}}}}}}}\left(t\right)={H}_{{{{{{\rm{p}}}}}},{{{{{\rm{ma}}}}}}}\left(t\right)+{H}_{{{{{{\rm{p}}}}}},{{{{{\rm{mi}}}}}}}\left(t\right)$$
(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:

$${H}_{{{{{{\rm{p}}}}}},{{{{{\rm{ma}}}}}}}\left(t\right)=\frac{8f}{{\pi }^{2}g{D}_{{{{{{\rm{p}}}}}}}^{5}}{L}_{{{{{{\rm{p}}}}}}}Q{\left(t\right)}^{2}$$
(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 <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 Hp,ma using Eq. (11), for various:

  • pipe diameters Dp between 0.04 m and 0.1 m, which is an usual range for PVWPS58,59;

  • pipe roughnesses ϵ between 0 and 1.5 × 10−4 m, which is an usual range for PVWPS57,58;

  • flow rates Q between 0 and 5 × 10−3 m3 s−1, which is an usual range for PVWPS29,60;

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

We then fitted Hp,ma, as a function of Lp and Q, and according to29:

$${H}_{{{{{{\rm{p}}}}}},{{{{{\rm{ma}}}}}}}(t)=\nu \cdot {L}_{{{{{{\rm{p}}}}}}}\cdot {Q}^{2}$$
(12)

where ν is a coefficient that depends on ϵ and Dp. For all the considered combinations of Dp and ϵ, we always obtained a fitting R2 higher than 0.99. For instance, for Dp = 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 Hp,mi(t) is given by:

$${H}_{{{{{{\rm{p}}}}}},{{{{{\rm{mi}}}}}}}\left(t\right)=K\cdot Q{\left(t\right)}^{2\quad}{{{{{\rm{with}}}}}} \quad K=\frac{8{\sum }_{i=1}^{N}{k}_{i}}{{\pi }^{2}g{D}_{{{{{{\rm{p}}}}}}}^{4}}$$
(13)

where ki is the coefficient associated to each junction i (values for the different junction types are provided in article57). We neglect the dependency of ki 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 Pmp,0 and when the water depth in the borehole Hb does not reach the motor-pump position Hmp. When the motor-pump is operating, the pumped flow rate Q is given by61,62,63:

$$Q\left(t\right)=\frac{P(t){\eta }_{{{{{{\rm{mp}}}}}}}}{\rho {gTDH}(t)}$$
(14)

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

$$\left(\beta +\nu {\cdot L}_{{{{{{\rm{p}}}}}}}+K\right)Q{\left(t\right)}^{3}+\frac{{{{{{\rm{ln}}}}}}\left(\frac{{r}_{{{{{{\rm{c}}}}}}}}{{r}_{{{{{{\rm{b}}}}}}}}\right)}{2\pi T}Q{\left(t\right)}^{2}+{H}_{{{{{{\rm{b}}}}}},{{{{{\rm{s}}}}}}}Q(t)-\frac{P(t){\eta }_{{{{{{\rm{mp}}}}}}}}{\rho g}=0$$
(15)

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

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

$$V=\frac{\int\limits_{2020}Q\left(t\right){dt}}{366}$$
(16)

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.