Uses of the Glover Equation
The Glover-Balmer Solution was developed in 1954 by Robert Ellsworth Glover and Glenn G. Balmer. This analytical solution for estimating impacts to streamflow from the pumping or recharge of alluvial aquifer systems is often simply referred to as the Glover Equation. (Sorry, Glenn!)
In Colorado Water Court cases, the Glover equation is used for applications like estimating lagged depletions to streams from well pumping, estimating lagged accretions to streams from recharge operations, and lagged accretions from irrigation return flows [see LWS October 25, 2020 blog]. While the Glover equation is in wide use, it is seen as a simplified estimate of lagged accretions and depletions. This is largely due to the number of assumptions inherent in the equation.
Inherent in the Glover equation are the assumptions that:
the aquifer is of uniform thickness;
the aquifer is homogenous and isotropic;
the aquifer is infinite in lateral extent;
streams fully penetrate the aquifer;
recharge and pumping is through a fully-penetrating well of infinitely small diameter;
streams are straight lines; and
the water table is flat.
While none of these assumptions are typically met in any real-world situation, several means are used to obtain representative results even when the basic equation assumptions are not met.
Since the Glover equation assumes uniform thickness of materials that are homogeneous and isotropic, to address real-world conditions such as changing aquifer transmissivity, i.e. the ease with which water can flow through the aquifer, different methods of averaging the aquifer characteristics are used. When the changes in the aquifer transmissivity are minor, a simple arithmetic mean can be used to estimate the overall transmissivity of the aquifer. When transmissivities vary widely, harmonic means are generally used to emphasize the impact of low transmissivity materials on the flow capabilities of the aquifer.
Since the Glover equation also assumes the aquifer is of infinite extent, to address the issue of an aquifer being finite in nature, “image wells” are used to simulate boundary conditions in the aquifer. Image wells are imaginary wells which are placed in the equation to pump or recharge the aquifer to create an effect similar to opposite waveforms which cancel each other out. The placement of the image wells can replicate the edge of an alluvial aquifer or a stream which does not allow for water level changes to occur on the opposite side of the stream from the modeled pumping or recharge.
While the Glover equation is able to simulate either the pumping or recharge of water at a well, i.e., a point source discharge or recharge, it is also widely used to simulate non-point source actions, which is not consistent with the underlying assumptions of the Glover equation. This use of the Glover equation generally relates to irrigation of fields and residential lawns. In these situations, a centroid of irrigation is estimated that serves as the “point” source for the Glover equation. While dispersed recharge from the irrigation of a large area is not the same as the water level response in the aquifer from recharge from a well, the Glover equation provides a simplified estimate of accretions to a stream system and has become institutionalized as the acceptable administrative tool for assessing the timing, amount, and location of stream accretions from a dispersed recharge related to irrigation.
Not specifically meeting the remaining assumptions is generally not regarded as particularly concerning for most basic applications of the Glover equation; however, the validity of the use of the Glover equation may be called into question if the geologic conditions are particularly complex or uncommon compared to a simple alluvial aquifer system.
The results of the Glover equation can estimate maximum depletion or accretion rates to the stream, the time it takes for the depletions or accretions to reach the stream, and the total volume of depletion or accretion to the stream at a particular time since pumping or recharge began. All of these results are important in Water Court as a means to demonstrate that injury to other water rights is being prevented, so it is easy to see why this simple analytical solution is used so widely for cases like augmentation plan cases, lawn irrigation return flow (LIRF) studies, change of use cases, aquifer storage and recovery cases, and more.
LWS has published several blogs addressing many aspects of the use of the Glover equation: lagged depletions and accretions, augmentation plans, change of use cases, LIRF studies.
If you are in need of water resources engineering, in or out of a courtroom, please reach out to Lytle Water Solutions and we will be happy to provide you a proposal tailored to your needs, free of charge.
Bruce Lytle: bruce@lytlewater.com
Chris Fehn: chris@lytlewater.com
Anna Elgqvist: anna@lytlewater.com
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