Estimating Agricultural Groundwater Withdrawals with Energy Data

Angiola provided 56 pump test reports for 30 well pumps. Pump test reports were produced in either 2015 or 2017 and included—as measured at the time of the test—groundwater depth, drawdown, discharge flow, OPPE, measured kilowatt power input, and more. The data presented a range of values from 75 to 250 horsepower (hp) pumps, TDH from 52 to 150 m (178 to 490 ft), tested discharge flows from 1,930 to 8,215 lpm (511 to 2,170 gpm), and reported OPPE from 27% to 75%.

Pump test report data were used to develop input data for two of five scenarios. Groundwater depth—the most sensitive data point—was applied on a well-by-well basis for the first scenario and averaged for the second. The remaining components of TDH (including pressure head, drawdown, and friction losses) and OPPE were determined through a linearly weighted blending of measurements reported in pump test reports per well pump. Some pumps in the study did not have any pump test reports available. In this case, averaged pump parameters were used. Pump operations were assumed to not change significantly during the study period. Thus, minor losses, drawdown, and OPPE were treated as predictable or constant in accordance with pump test reports.

When site-specific pump test reports were not available, other data sources were needed to establish the groundwater depth. CWEE collected publicly available CASGEM data within a $50\text{-}\mathrm{km}$ radius from Angiola (Fig. 2). The selected wells had at least two measurements of groundwater depth per year during the study period. CWEE used these measurements to generate well depth estimates for each well in Angiola for each month in the study period. CASGEM wells spatially closer to an Angiola well received a higher weight for the estimated groundwater depth for that well, and CASGEM well measurements that occurred closer in time to a given month received a higher weight for the estimated groundwater depth in that month.

CWEE also accessed groundwater depth contour maps managed by DWR and available through the web-based GICIMA tool. The benefit of these contour maps is that, within covered areas, groundwater depth estimates are already generated (i.e., do not require further processing) and can be extracted from the downloaded data source. However, these contour maps also have some limitations. There are many areas within the state with poor coverage by these maps, and most of Angiola Water District is not covered. Fortunately, the wells in the study area were mostly concentrated in the northeastern portion of the district, which is the area with the best coverage for DWR contour data (Fig. 3).

Estimation methods using both DWR depth contour and CASGEM well data produced inaccurate measurements on a per-well basis. On average, however, the estimates from CASGEM wells significantly outperformed the DWR contour estimates. We found that the DWR contours generally appeared to significantly underestimate well depth, whereas the CASGEM method did not consistently overestimate or underestimate (Fig. 4). Because the DWR contour estimates produced such consistently poor results—underestimating groundwater depth by more than an average of 30 m (100 ft)—they were not used to produce the results in this study.

Drawdown can be calculated using the Theis equation (Theis 1935), as follows:where $S$ = drawdown (m); $Q$ = discharge (${\mathrm{m}}^3/\mathrm{s}$); $T$ = transmissivity (${\mathrm{m}}^2/\mathrm{day}$); $t$ = time (days); $s$ = storage coefficient (dimensionless); and $r$ = radial observation distance from the pumped well (m).

To calculate the drawdown of a well at a particular time, values must be generated for the discharge rate of the pump ($Q$), the length of time the pump has been operating ($t$), the radius of the pipe ($r$), and the aquifer geological parameters for transmissivity ($T$) and the storage coefficient ($s$). The geological parameters for the Tulare Basin aquifer, the study area, were assumed to be $T=\mathrm{8,000}{\mathrm{m}}^2/\mathrm{day}$ and $s=0.5$ (Page 1983). The pipe radius, which is effectively equivalent to the radial observation difference, was assumed to be approximately $0.15\mathrm{m}$. $\mathrm{W}(\mathrm{x})$ is a symbolic representation of an exponential integral that is further detailed in Theis (1935). This approach provides a theoretical estimation of drawdown under ideal conditions, but many additional factors can impact the rate at which water and air pass between the well assembly and the surrounding soil formation. Drilling fluid residuals, corrosion, and fouling of the screens and well casing can reduce well productivity. Maintenance and rehabilitation can reverse these effects. For the purposes of this study, these variables are taken to be unobservable. Angiola did not provide maintenance records for their wells or any other data that could be used to determine the condition of any well screen or casing. However, pump test reports provided by Angiola provide precise drawdown measurements that can be used to determine the accuracy of the Theis equation in this context.

Energy billing data were used to estimate the pump discharge rate. The month with the highest total energy use was assumed to represent nearly continuous pump usage. Flow in that month was estimated using the ELM and regional average pump operating parameters. When comparing the estimated pump flow values to the measured pump flow from the provided pump test reports, a strong correlation exists with a slight bias toward overestimation (Fig. 5).

The value $t$, or the number of days of continuous pumping, was also estimated using energy data. Months with the highest energy usage were assumed to represent 30 days of continuous pumping, whereas months with the lowest energy use were assumed to represent 0 days of pumping. For each month, the $t$ value was estimated by summing the number of preceding months with high energy usage and multiplying by 30 days, then adding 15 days for the current month.

Given that all values are known, derived, or estimated, the drawdown for each pump and each month was then calculated. When compared with the measured data, the results indicated a general correlation and a mean absolute error of 7.7 m (25.4 ft), and a mean error of 49% (Fig. 6). Inaccuracies in our flow estimates explain some of this error, but our estimates of soil transmissivity, the storage coefficient, and the impacts of unobservable soil bore conditions are all likely to contributed to the error. Although these results produced a relatively high error on a per-pump basis, the mean estimated drawdown of 18.1 m (59.5 ft) was relatively close to the mean measured drawdown of 15.5 m (50.7 ft). This method for estimating drawdown was used in the Calculated scenario (Table 1: Summary of ELM scenarios).

OPPE is largely a function of the quality and condition of a pump; therefore, there is no way to determine OPPE using only energy data. However, correlations between OPPE and other pump operating parameters provide an opportunity for estimation. Pérez Urrestarazu and Burt (2012) presented several such correlations in their analysis of Fresno State’s pump test report database. We extended their work using a linear model to estimate OPPE based on TDH (in m), well energy usage (in $\mathrm{MW}·\mathrm{h}/\mathrm{year}$), and whether or not the pump was “high flow” ($>100\mathrm{L}/\mathrm{s}$), including an interaction term on usage and flow. The resulting model explained 75% of the variation in OPPE of the summary dataset with a high significance ($p<0.001$) for all model parameters. Because this model was not constructed using the original survey dataset, it does not represent true correlations between OPPE and pump operating parameters. Instead, it is a predictive tool to estimate OPPE using other observable pump data.

OPPE was estimated for each pump using CASGEM-estimated TDH, annual energy usage from PG&E data, and flow estimates, as described in the previous section. The resulting mean OPPE was 56.0%, slightly higher than the measured mean OPPE of 54.2%. There was a linear correlation of measured versus estimated OPPE for each well, but it was not a $1:1$ relationship (Fig. 7).

When none of these other data sources are available, regional averages may serve as a data source of last resort. Using regional averages is also the simplest approach to estimating pump operating parameters. Burt (2011) collected data from more than 15,000 pump test reports and published averages for three major valleys in California. In the case of Angiola in the San Joaquin Valley, the corresponding TDH was listed as 79 m (260 ft) and OPPE at 57%, the first underestimating the actual site-specific TDH average of 101 m (330 ft) and the second slightly overestimating the actual average site-specific OPPE of 54.2%.

ELM input data were aggregated to an annual basis in an attempt to improve the estimates of groundwater extraction from individual wells. Aggregation of the data reduced the error associated with each scenario. For example, on a well-by-well basis, the mean absolute percentage error (MAPE) for the Individual Tests approach improved to 5.0% on an annual basis (Fig. 9) from 13.5% (Fig. 8), a difference that was statistically significant ($\mathrm{p}<0.001$).

This reduction in error comes from two potential sources. The first is the reduction in the error from the variation in pump operating parameters during each year. Whereas some of this variation is random, increased overestimation occurred in July and August. When all months in a year are combined, the impact of random and seasonal variations in pump operating parameters is reduced. The second reduction in the error comes from the smoothing of errors introduced from standardizing the billing data into calendar month periods. The electricity billing data used in this study did not correspond to calendar months, whereas the measured flow data did. Standardizing the billing data to correspond to calendar months resulted in a significant error. However, standardizing the billing data to correspond to calendar years has a much lower potential for error. The scatterplot in Fig. 10 indicates a strong correlation between the estimated and actual water use when the data are annualized.

Annualizing monthly estimates of groundwater extraction for individual wells significantly improves the accuracy of the results. Annualized estimates based on data from individual pump test reports produced error rates similar to flow meters, indicating that the ELM may be a suitable alternative to flow meters in some cases.

Summing individual well estimates into an aggregate extraction estimate for a collection of wells reduced the average error associated with some scenarios. The Averaged Tests scenario showed the largest reduction in average annual error, falling from 8.5% on an individual well basis to 3.3% on an Angiola-wide basis (Fig. 11). The Individual Tests and Calculated scenarios showed little change in the error rate, and the CASGEM and All Regional scenarios had increased errors.

The Averaged Tests scenario produced inaccurate results when estimating individual well extractions because it failed to capture the significant variation in OPPE and TDH between pumps. However, this variation is less important when estimating extractions from a collection of wells. Estimates of OPPE and TDH that most closely reflect the true average of all pumps in the collection are all that are needed to produce an accurate estimate of aggregate groundwater extraction.

Because all wells were aggregated together for this analysis, the sample sizes were reduced, leading to reduced statistical significance of the perceived differences. The MAE produced using individual pump tests was 4.35 ha-m ($35.3\mathrm{acre}\text{-}\mathrm{ft}$) per month, which was slightly significantly higher than the MAE produced using averaged pump tests at 2.64 ha-m (21.4 acre-ft) ($\mathrm{p}<0.2$). Of the scenarios that do not use the pump test report data, the Calculated approach was again the best, followed closely by the CASGEM scenario. Again, using all regional data produced the highest error of all scenarios ($\mathrm{p}<0.001$). See Table 3 for a summary of these findings.

Collecting individual pump test reports and matching each report with a specific well’s energy data may not be necessary if the ELM is to be used to estimate groundwater extraction at a large spatial scale. Instead, generating an accurate estimate of the average pump operating parameters may be sufficient.

Because the Averaged scenario applies the same values for OPPE and TDH for each well, there is no additional value in using pump-level electricity data. The same estimates could be produced if all of the electricity data were aggregated into total monthly consumption and would be just as accurate. This information is particularly significant for entities such as GSAs, which may not be able to access individual-level electricity data but is permitted to request aggregated electricity data from energy utilities through the EDRP. Therefore, GSAs could perform this estimate by requesting aggregated data for all wells in their territory and collecting a representative sample of pump test reports to calculate the average values of TDH and OPPE.

When estimating groundwater withdrawals from wells on a monthly basis, the estimates indicate a strong correlation to the actual water use (Fig. 12). These errors in the Individual Test scenario are approximately normally distributed, indicating that random variations in physical conditions are the most likely drivers of errors rather than issues in the underlying data. This assumption was further tested by comparing the measured water-energy ratio of pumps to the modeled estimates. The measured water-to-energy ratio was obtained by dividing the measured well groundwater extractions using the PG&E hourly data for the same period. The modeled ratio was calculated by dividing the estimated OPPE by 27.2285 times the TDH for a well in a given month, where 27.2285 is the energy required to lift one ha-m of water a vertical distance of one meter at 100% efficiency in kW·h. Although these values typically followed a similar trend, the measured water-to-energy ratio displayed higher variance and less consistency (Fig. 13).

The general overestimation trend may have been an artifact of this particular subset of wells; alternatively, it could point to a problem with the model assumptions. The overestimation of water extractions could result from a bias toward an overestimation of OPPE and/or an underestimation of TDH. If the average OPPE calculated by the pump test report is higher than the average OPPE of the pump, then a bias toward overestimation is expected. Similarly, although efforts were taken to account for seasonal changes in groundwater depth, the groundwater depth or drawdown may have increased beyond what this model expected during the summer months.

Finally, all of the pump tests in this study display a very low pressure head, meaning that during the pump test, the irrigation system beyond the well itself was either unpressurized or was pressurized by a booster pump. If the pump test reports were conducted in a situation that does not reflect the system’s normal downstream pressure demands, then this could also be a potential cause for the overestimation. In addition to potential errors with TDH and OPPE, the energy data itself could cause overestimation if there were other energy demands on the same meter as the well pump.

Overestimation disproportionately occurred in late summer and early fall months, which can be visualized by comparing measured groundwater extractions over time to modeled extractions (Fig. 14). According to a regression analysis on errors in the dataset, the months of July, August, and September had statistically significant ($\mathrm{p}<0.001$) overestimation trends, with expected monthly overestimations of 1.47, 1.8, and 1.16 ha-m (11.9, 14.6, and $9.4\mathrm{acre}\text{-}\mathrm{ft}$) per well, respectively. A one-sample t-test was performed on the vector of all monthly errors, and the overestimation was statistically significant, with a 95% confidence interval of 0.3–0.75 ha-m ($2.4–6.1\mathrm{acre}\text{-}\mathrm{ft}$) per well per month. Repeating the t-test with July through September removed resulted in no significant overestimate. May and June also had statistically significantly higher errors than average with expected overestimates of 0.79 and 0.72 ha-m (6.4 and $5.8\mathrm{acre}\text{-}\mathrm{ft}$) per well per month, respectively.

Given that overestimation is concentrated in July through September and, to a lesser extent, May and June, the error likely has to do with conditions unique to these months. Because these months have the hottest temperatures and lowest rainfall, the overestimations might be correlated to the total pumping. Increased pumping likely increases drawdown, which this model does not capture. In addition, increased irrigation demand may lead to pumps being operated beyond their set points, resulting in inefficiencies not captured in the model. Alternatively, the groundwater depth could drop further than the model expects during these months. Without more data on pump operating conditions, determining the cause of these discrepancies with any certainty is impossible. However, given the significance of this seasonal overestimation trend, such questions should be a priority for future investigations.

All of the errors that impact the accuracy of the previously described individual well estimates also impact the estimates for a collection of wells, although the errors caused by the random variations captured in the normal distribution of errors can average out when aggregating all of the results, as indicated by the overall reduction in error in Figs. 8, 9, and 11. However, the overall bias toward overestimation is not removed by aggregation.

One major source of error that differs between each scenario was the error in the TDH estimate. The similarity of the average TDH used in each scenario to the average TDH of the pump test reports was a good indicator of the overall accuracy of regional estimates (Fig. 15). The Averaged Tests scenario, which by design exactly matches the averages from the pump test reports, had the lowest MAPE, whereas scenarios 4 and 5 had the largest difference in TDH and, correspondingly, had the largest difference from the regional water extraction estimate. These results indicate that accurate results for regional estimates of groundwater extraction do not necessarily require site-specific data or models to estimate the change in the groundwater level over time but do require accurate regional average estimates of TDH and OPPE.

The measured groundwater extraction data used in the production of these results may not represent true water demand. Changes in pump operating conditions and flow meter maintenance can significantly impact the accuracy of a flow meter; therefore, some of the differences between the measured and estimated extractions could be explained by measurement error.