# 6 Mathematical Model Applications in Karst

Understanding karst aquifers, for purposes of their management and protection, poses unique challenges. Since the 1960s, advances have been made in the development of modeling software and their use for characterization and management of karst aquifers. Modeling is applied to synthesize data related to the aquifer. The model represents the aquifer, and that representation is adjusted until the simulated model values match the equivalent field observations. The process of creating even simple models of the system helps investigators improve their understanding of observed data and the flow system within the karst aquifer.

Different types of mathematical models can be applied to hydrologic and environmental problems in karst aquifers that have complex dual, or triple, porosity. The preferred mathematical techniques depend on the type of available data and the problems that need to be solved. Names of the types of mathematical models applied to karst aquifers can be confusing because terminology has not been used consistently in the literature. This section categorizes the mathematical approaches into three major groups based on the methods applied in the calculations: 1) fitting models; 2) lumped parameter models; and 3) distributed parameter models (Figure 62). There is some overlap in the mathematical methods employed in the categories of “lumped parameter models” and “fitting models” as both categories involve mathematical techniques that come from the fields of control theory (sometimes called operations research) or linear systems theory (Doo, 1973).

• The first category, fitting models, refer to mathematical methods that involve statistical regression, fitting shape functions, statistical transfer functions, or use of pattern recognition functions, such as artificial neural networks (ANN) to recreate field observations (Long and Putnam, 2002, 2007; Hu et al., 2007). These models have no physical basis and only require a known correlation between the system response (that is, the variable that will be predicted with the model) and system drivers or inputs (that is, independent variables used to define the shape function, neural network, or statistical transfer function). Most of the applications require pairs of observed data for dependent and independent variable(s) to calculate the coefficients for the function(s) and may not require detailed characterization of the karst aquifer.
• The second category, lumped parameter models, is based on mathematical methods that combine a physically based equation (an ordinary differential equation) with control theory or operations research techniques (Takahashi et al., 1972; Hillier and Lieberman, 1967; Dooge, 1973). This second category has some physical basis, but generally lumps the detailed properties of the karst aquifer into a few large basins for water management problems and is not meant to simulate details of flow in the system. A lumped parameter model does not distinguish between different system compartments and different flow processes. For example, many hydrogeochemical mixing models that are used to estimate average age of water are based on simple expressions of physical flow, such as binary, piston, exponential, dispersion, or ordinary differential equations (Maloszewski, 2000; Katz, 2004; Jurgens et al., 2012). For water supply problems, some concept of the aquifer system is required, such as groundwater sub-basins (tanks) and their connection to each other.
• The third category, distributed parameter models are based on physical processes defined by partial differential equations describing water flow, which are solved numerically usually with finite-difference or finite-element methods (Anderson et al., 2015; Kresic, 2007). In these models, hydraulic properties are defined for many blocks of varying size that are joined together to represent every volume within the groundwater system at a level of detail defined by the modeler. Distributed parameter models are used to simulate all types of flow within karst aquifers as inferred from hydrogeologic investigations. This is the most-applied type of model for synthesizing all data from investigations in order to test concepts of how water moves into, through and out of the aquifer. Such modeling helps determine the most useful types and locations of additional data collection and can lead to adjustment of the conceptual model of the aquifer system. There are many numerical model approaches that use different formulations of the groundwater flow equations and techniques for solving the equations.

Figure 62  Examples of three categories of groundwater models: a) fitting models in which a mathematical function is used to express groundwater head variation with time (depending on the complexity of the fitting function, the software and computations can be simple and set up in a spreadsheet or can be more complex, involving many lines of computer code or use of statistical or custom software applications); b) lumped parameter models in which a series of ordinary differential equations for tanks are solved simultaneously to represent flow between and storage in subbasins as well as the final discharge to a spring given historical recharge, pumpage, and spring discharge data (often lumped parameter problems are solved with simple arithmetic in a spreadsheet); and, c) distributed parameter models showing the color-coded distribution of hydraulic conductivity in cells of one cross section and one layer for which the partial differential groundwater flow equations are solved using finite differences (the software codes for distributed parameter models are complex and requiring thousands of lines of code and large computers).

In this section, the general categories of modeling approaches are described with some examples. The mathematical methods are described in words and figures. The examples are intended to help non-mathematicians understand their application. Readers wishing to pursue mathematical modeling of aquifers are encouraged to study fluid mechanics and mathematics (including probability and statistics, calculus, differential equations, linear algebra, and numerical methods), as well as develop skills for manipulation of large geospatial datasets and computer programming.

To apply any mathematical model to a karst system, many of the characterization steps described in previous sections of this book need to be undertaken. Characterization provides the data needed to develop a conceptual model of the karst aquifer, create the mathematical model and calibrate the model to represent field conditions. A conceptual model of an aquifer is a description of 1) where water enters and exits the aquifer and 2) the hydrogeologic structure (that is, shape and size of geologic units and faults; and for karst, the location of dissolution features and porosity types). Distributed parameter models require the most complete data describing hydrogeologic framework and detailed definition of the quantity and location of water moving into and out of the system. A lumped parameter model requires less data. For example, many lumped parameter geochemical models for age dating require only environmental tracer data from a public supply well or spring for estimation of travel time and water age (Jurgens et al., 2012). A fitting model, such as a linear regression may use little data (for example, a few water levels in a well and spring discharge collected at the same time) to predict spring discharge given a measured well water level (Knochenmus and Yobbi, 2001). In order to use the model to estimate future spring discharge (dependent variable) from the water level in a nearby well (independent variable), the observed data used to develop the model needs to represent the full range of discharge to be estimated.

The steps involved in modeling are somewhat circular in that the model is developed, evaluated, revised and reevaluated numerous times before it is put to use (Figure 63). The process includes:

1. defining the purpose of the model (for example, what is to be learned or understood about the system);
2. compiling data available in the literature;
3. visiting the field area;
4. conceptualizing the aquifer framework and movement of water through that framework;
5. applying initial models to test hypotheses about the system in order to guide plans for data collection and model selection;
6. collecting data at the field site;
7. calibrating the model;
8. if the model does not acceptably match the field observations after calibration, then using what was learned from the process to return to step 4 and revising the conceptual model or to step 6 and collecting more data to develop a more complex model that better represents the system; and when ready;
9. putting the model to use.

Figure 63  The circular modeling process. First develop a conceptual model and compile all existing data about the karst aquifer. Then collect or compile more data if required for the problem at hand. Decide on a modeling approach and calibration strategy, then test your hypothesis. If model is adequate for the problem or proves the conceptual model is correct, then no further study is needed. If not and funding is available, revise the conceptual model, collect more data and incorporate the data into a more complex model, recalibrate, and test the new hypothesis.

Many distributed parameter model projects are started after years of aquifer studies including development of some fitting, lumped-parameter, and geochemical age-dating models. The distributed parameter model is then used to synthesize all the data and better understand the entire aquifer system.