# 6.2 Lumped Parameter Models

The control theory approach has been successfully applied to simulate karst aquifers using mathematical methods for solving sets of ordinary differential equations. For example, the control theory approach is used to evaluate the filling and mixing of water in a tank, or a network of tanks, and a karst basin can be conceptualized as a group of interconnected tanks. The tanks are not assigned spatial coordinates as is done for cells of distributed parameters models, rather flow between tanks is based only on connections assigned by the modeler. Coefficients (that is, lumped parameters) for each tank are developed by calibrating a system of equations describing the connections between tanks (Wanakule and Anaya, 1993; Hartmann et al., 2014). Data preparation for lumped parameter models is simpler than that required for distributed parameter models, and computational time is shorter. However, detailed representation of the aquifer is not possible using this approach.

Groundwater withdrawals and recharge are summed together for each tank which represents a geographic area (for example, a spring basin). Withdrawals from the model simulate spring discharge and inputs to the model may represent distributed infiltration, sink holes or sinking streams. Lumped parameter models may be adequate for providing gross estimates of the effects of changes in pumpage and/or recharge rates on spring discharge, as well as, for estimating recharge given natural discharge and pumpage. Mathematical filters can be applied to some of the input data, for example, to divide the annual basin recharge into varying recharge for each month, in order to achieve a better fit between observed and simulated spring discharge (Wanakule and Anaya, 1993; Dreiss, 1982). Many of the geochemical mixing models, such as piston flow, binary mixing, or exponential mixing models applied to atmospheric environmental tracers are also used for estimation of groundwater age. These are forms of lumped parameter models (Böhlke and Denver, 1995; Cook and Herzog, 2000).

# Example Application of a Lumped Parameter Model

The lumped parameter modeling method is demonstrated in a simulation of spring flow at Comal and San Marcos Springs, Texas. The model input and calibration data were based on annual estimates of recharge and pumpage in nine surface watershed basins and an index water level in each sub-basin of the San Antonio part of the Edwards Aquifer (Wanakule and Anaya, 1993). The lumped parameter model of Wanakule and Anaya (1993) was one of the earliest applications of such models to the Edwards Aquifer. Barret and Charbeneau (1997) applied the lumped parameter method to the Barton Springs part of the Edwards Aquifer, but it was simpler because there were fewer basins, so the earlier model is used as an example in this book. Conceptually, each of the nine watershed basins is treated as an interconnected tank. The lumped parameter mathematical relation was developed much like a statistical regression model where recharge and pumpage in each lumped parameter block (tank) were treated as input to a set of linked tanks that transport water to the major springs. The mathematical description of each tank was formulated with an ordinary differential equation. Figure 67 shows the Edwards Aquifer and catchment area as well as the conceptualized tanks.

Figure 67  a) Map of the Edwards Aquifer and its recharge area with basin numbers in maroon. b) Schematic diagram of the lumped parameter model tanks and connections between tanks. Modified from Wanakule and Anaya (1993). View a larger version of this figure in a separate tab.

Surface runoff from the catchment area infiltrates the Edwards Aquifer across its outcrop. Wanakule and Anaya (1993) used the recharge and pumpage data as inputs to the system. They developed filters to break up the annual estimates of recharge into monthly estimates using stream gage data and annual recharge estimates. Additionally, the monthly pumpage by county was reapportioned to each sub-watershed (Wanakule and Anaya, 1993). These values were then matched to average monthly groundwater levels in each basin and to the historical spring discharge at Comal and San Marcos Springs by calibrating values of parameters related to storage and transmissivity for each tank. The filtering method for the disaggregation of recharge falls into category 1 (fitting models), which includes time series techniques. Wanakule and Anaya (1993) were successful in refining estimates of monthly recharge and pumpage for each basin and simulating the spring discharge at Comal and San Marcos Springs (Figure 68) as well as water levels in each basin. which are not shown here, but were presented by Wanakule and Anaya (1993).

Figure 68  Simulated and observed spring flow at Comal and San Marcos Springs, Texas, USA. Modified from Wanakule and Anaya (1993).

Schulman and others (1995) developed equations for stochastically generating recharge volumes for the watershed basins. The equations have the same statistical properties as the historical (since 1934) recharge data for each basin. The wellbeing of endangered species at the springs are a concern, and the courts have required minimum discharges to be maintained at both springs. Given these minimums, groundwater use is restricted at times during summer months even as the population continues to grow. Thus, water-resource planners recognize that water may need to be imported to maintain discharge at the springs. Because of its computational speed and simplicity, the calibrated lumped parameter model of Wanakule and Anaya (1993), combined with the generated recharge scenarios, were used to screen water-supply options for the Edwards Aquifer (Watkins and McKinney, 1999).