Stephanie Schmidt, LRCWE
March 31, 2009
The Glover equation was developed by Robert Glover to estimate the timing and magnitude of the impact to a river from a nearby pumping or injection well. The Glover equation is very accurate if an aquifer meets many simplifying assumptions, but no aquifer is so simple in real life. The farther away the aquifer is from these ideal conditions, the more error there will be in the solution to the Glover equation. For the results to be accurate the aquifer should be isotropic, homogenous, of uniform thickness, and infinite. The transmissivity should be the same everywhere and not change with time. It is also assumed that drawdown is negligible compared to aquifer thickness, the water table is initially flat, the stream is straight, infinite in length, and fully penetrates the aquifer, the pumping rate is constant for any pumping period, and water is released instantly from storage.
The equation can be modified to overcome some of these simplifications. For example, the impacts can be calculated for several different pumping rates and the result can be added if the pumping rate is not constant. Results from image wells can be added to estimate the effects of no-flow boundaries or other rivers on the impacts. When an aquifer is not homogeneous, average properties are typically used. Typically the harmonic mean of transmissivity along the shortest distance path from the well to the river is the value of transmissivity used in the equation. Sometimes, the transmissivity along the distance from the stream to some negative boundary is used in calculating the harmonic mean. This talk examined whether the harmonic mean is an appropriate value to use and over what area it should be calculated.
The harmonic mean is calculated as the total distance divided by the sum of the length of each unit divided by the transmissivity of that unit. This method was originally developed for pipes because as the pipe diameter goes to zero, the harmonic mean goes to zero. The harmonic mean places a higher weight on lower values.
This presented a problem in a project Leonard Rice Consulting Water Engineers (LRCWE) was working on. LRCWE was originally asked to build a ground water model to estimate the amount of water captured by drains under a new development. After the model was completed, LRCWE was asked to estimate the timing of these impacts to the South Platte River. In this situation, because of a narrow zone of low transmissivity, the harmonic mean intuitively seemed too low. A simplified MODFLOW model was created to estimate the depletions, and it was determined that using a value of over four times the harmonic mean for transmissivity in the Glover equation produced results comparable to the model.
This problem was examined further, and six simple models were created to compare the results to the results of the Glover equation. Initially, LRCWE examined whether variability in transmissivity values and the order of the values of highest to lowest or lowest to highest affected the efficacy of the use of the harmonic mean. Each of the six models had four zones of transmissivity ranging from 10 to 10,000, 125 to 1,000, and 200 to 500.
LRWCE ran the models and calculated a depletion curve for each scenario. Depletion curves were then created using the Glover equation. The shape of the curves was slightly different, making it difficult to match them exactly. The value of transmissivity used in the Glover equation was varied until the curves matched at 20 years. This value of transmissivity used in the Glover equation to make the depletion curve match the modeled depletion curve at 20 years, was referred to as “the Glover Transmissivity”. We compared the Glover transmissivities from each scenario to various types of average transmissivities for each scenario such an arithmetic average, weighted arithmetic average, and the weighted harmonic mean. These averages were calculated both from the well to the stream and from the negative boundary to the stream.
We found that in all of the scenarios, the Glover Transmissivity was closer to the weighted harmonic mean from the well to the river than any other average we compared it to. We also calculated the percent difference between the weighted harmonic mean of transmissivity from the well to the river and the Glover transmissivity. There was no correlation between when harmonic mean did and didn’t work. Neither variability within the system nor order of hydraulic conductivity zones affected the use of the harmonic mean.
From this study, we cannot generalize when the harmonic mean of transmissivity is and isn’t appropriate to use in the Glover equation. However, our results show that the harmonic mean should always be used with care. With the advancement of computer programs, it may be more effective to create simple numerical models where analytical models would have been used in the past. The following factors should be considered: the level of precision necessary for the situation, the availability of input data, and limitations associated with time and budget. With these considerations in mind, we have more options and tools at our disposal than ever before.
The Powerpoint presentation from the talk can be accessed by clicking below: