Analytical Solutions to the Linearized Boussinesq Equation for Assessing the Effects of Recharge on Aquifer Discharge¹

Abstract

Boggs, Kevin G., Robert W. Van Kirk, Gary S. Johnson, Jerry P. Fairley, and P. Steve Porter, 2010. Analytical Solutions to the Linearized Boussinesq Equation for Assessing the Effects of Recharge on Aquifer Discharge. Journal of the American Water Resources Association (JAWRA) 46(6):1116–1132. DOI: 10.1111/j.1752‐1688.2010.00479.x Abstract: There is a need to develop a general understanding of how variations in aquifer recharge are reflected in discharge. Analytical solutions to the linearized Boussinesq equation governing flow in an unconfined aquifer provide a unified mathematical framework to quantify relationships among lag time, attenuation and distance between aquifer recharge and discharge and the effect of an up‐gradient no‐flow boundary. We applied this framework to three types of recharge: (1) instantaneous, (2) periodic, and (3) constant rate for a finite duration. When the temporal scale of recharge exceeds the diffusive aquifer time scale, recharge will be reflected in discharge quickly and with little attenuation. When aquifer time scale is large, most recharge events are shorter in scale than that of the aquifer, resulting in large attenuation. Attenuation is more sensitive to boundary effects than lag time, and boundary effects increase as recharge time scale increases. Boundary effects can often be ignored when the recharge source is farther than 1/3 of the domain length away from the no‐flow boundary. We illustrate analytical results with application to the economically critical Eastern Snake River Plain Aquifer in Idaho. In this aquifer, detectable annual and decadal cycles in discharge can result from recharge no farther than 20 and 60 km away from the discharge point, respectively. The effects of more distant, long‐term recharge can be detected only after a time lag of several decades.

Forecast of Natural Aquifer Discharge Using a Data‐Driven, Statistical Approach

“…The fact that these environmental predictors generally have stronger correlations with spring discharge at larger moving-average windows compared to predictors that are reflective of irrigation impacts is consistent with analytical model results, which indicate that (1) there is a direct relationship between the duration of spring discharge impact and the distance between the recharge and discharge location, (2) ESPA recharge from sources extending over the entire aquifer affect the spring discharge for more than one year, and (3) impacts of ESPA recharge associated with the irrigation season near the Thousand Springs area contribute to spring discharge variability for no more than one water year (Boggs et al 2010). The fact that these environmental predictors generally have stronger correlations with spring discharge at larger moving-average windows compared to predictors that are reflective of irrigation impacts is consistent with analytical model results, which indicate that (1) there is a direct relationship between the duration of spring discharge impact and the distance between the recharge and discharge location, (2) ESPA recharge from sources extending over the entire aquifer affect the spring discharge for more than one year, and (3) impacts of ESPA recharge associated with the irrigation season near the Thousand Springs area contribute to spring discharge variability for no more than one water year (Boggs et al 2010).…”

“…We used time-series regression models in which the predictors are potentially lagged in time and averaged over several water years to reflect that aquifer recharge is lagged in time and attenuated in magnitude before it is discharged from the aquifer (Townley 1995;Knight et al 2005;Criss and Winston 2008;Boggs et al 2010). We divided the data into calibration (1950 through 1999) and validation (2000 through 2010) sets and used Akaike's information criterion (AIC; Akaike 1973; Burnham and Anderson 2002;Anderson 2008) to select the optimal model to predict aquifer discharge.…”

“…Previous analytical work suggests that we need to focus on recharge variability close to the Thousand Springs Complex to predict its discharge (Boggs et al 2010). Previous analytical work suggests that we need to focus on recharge variability close to the Thousand Springs Complex to predict its discharge (Boggs et al 2010).…”

“…Because the time scale of attenuation varies with spatial and temporal properties of the recharge (Boggs et al 2010), we expected that some moving-average windows would result in higher correlations with discharge than others. Annual increment in Box Canyon discharge was most strongly correlated with increment in annual precipitation at lag 1 and movingaverage window width 5.…”

“…We proposed a priori a set of candidate models (Table 2), based on the water budget and results of analytical models (Boggs et al 2010). We used the same set of candidate models in both the Model 1 and Model 2 procedures.…”