A groundwater model using a geographic information system (GIS) has many distinct advantages over traditional models. The most important benefit of using GIS is that everything is defined within a spatial context. The distribution of precipitation, groundwater recharge and pumping, and reservoirs can all be seen on a single map. Moreover, relationships between these components are easily determined because GIS is essentially a database that can combine this information in an infinite number of ways. Traditional models are usually confined to a specific process, location, or set of conditions. In GIS, however, a single model can perform analyses for a variety of situations by simply having a map of a region and knowledge concerning its physical parameters. This report is intended as a brief overview of how this important tool can be used for groundwater modeling.
The majority of work performed under this study used a program that was developed by Ye Zichuan of the Center for Research in Water Resources at the University of Texas at Austin. This program is referred to as The Map-Based Groundwater Model throughout the report. Thanks to Ye for providing the program and guidance needed to complete this project.
A short description of the course of this study is provided below:
The major advantage of a map-based groundwater model is the capacity to display various types of information within a spatial framework. Being able to visualize the magnitude and direction of flow within an aquifer, for example, strengthens the analysis of the underlying physical processes that are occurring. Relationships between the representation and associated data are easily established through the use of geographic information systems (GIS) and object-oriented programming (OOP). Consequently, generic simulation programs can be applied to any region given the digital elevation model (DEM) and knowledge regarding the system parameters.
Using OOP, a map is divided into a series of line and polygon objects. A groundwater cell, for example, is represented by a polygon object. An object can be described by its state and behavior. The state of an object is described by various types of data that are stored and managed within a GIS database. Tables for groundwater cell objects specify hydraulic conductivity and whether the aquifer is confined or phreatic, amongst other things. An object’s behavior is simulated through scripts, which are programs that work within the GIS environment.
All of the information stored in a GIS database is referenced to specific features (objects). These pointers provide the linkage between the map and model. Scripts retrieve the data needed to solve equations through reference identification (ID). In this manner, GIS and OOP provide the framework for modeling groundwater processes within a well-defined spatial context. The equations governing groundwater movement and the management of associated data are described in more detail in the following sections.
There are two main physical principles that must be taken into account when modeling groundwater movement: continuity and momentum. The momentum equation is used to determine mass flux across a cell’s boundaries. Conservation of mass is satisfied by calculating all inflows and outflows for a cell and performing a mass balance to determine the storage. In order to incorporate these principles into the model, the continuous area representing the study region is divided into an array of points. Continuity and momentum equations are discretized over time and space. The Map-Based Groundwater Model in GIS addresses these equations separately, as described below.
The Darcy flux for a porous medium can be calculated using Darcy’s law,
where:
q is the Darcy flux (L/T)
h is the total head (L)
x is a distance in the x-direction (L)
K is the hydraulic conductivity (L/T)
This equation accounts for movement in one direction but can be expanded to a three-dimensional (3D) system. It also assumes that the medium is saturated. (Chow, Maidment, Mays 1988)
The Map-Based Groundwater Model calculates flow across the boundary between a left-polygon and a right-polygon using a discretized form of Darcy’s law,
where:
qlr is the mass flux across a boundary line (L/T)
Klr is the hydraulic conductivity (L/T)
hl is the total head (L)
SLlr is the distance between centroid points of the cells
The equation assumes that the flux across a boundary line is uniformly distributed and parallel. This assumption can be inaccurate when the groundwater cells are large. It becomes and issue when one-to-one data relationships are created between subwatershed objects and cells. This situation is described in the section describing interactions between surface and groundwater flows.
In most cases, boundary lines are not mutually parallel or perpendicular to each other and direction must be considered. Taking direction into account, the volumetric flux can be calculated from the Darcy flux in the following manner,
where:
Qs is the volumetric flowrate across line section S (L3/T)
l is the boundary line vector for section S (L)
hs is the average depth of the aquifer on line section S (L)
The volume of water crossing a boundary line during a given time interval is,
where:
Vs is the total volume of water exchanged (L3)
The continuity equation for a homogenous and isotropic medium in two dimensions (2D) is as follows:
where:
S is the storativity
R is the recharge to the aquifer (L/T)
P is the pumpage from the aquifer (L/T)
This relationship represents the combined form of the continuity equation, including Darcy’s Law. Many self-contained models solve the combined equation but it is computationally intensive and difficult to incorporate into GIS. The Map-Based Groundwater Model solves the momentum and continuity equations separately which simplifies calculations.
A finite difference form of the continuity equation is used to calculate the change in total head for a groundwater cell,
where:
V is the volume of water calculated from the flux equations (L3)
A is the area of the cell (L2)
A groundwater simulation occurs over time and space. A time loop, boundary-line loop, and cell-polygon loop are created to account for all dimensions. Initially (time step zero) the total head is known for each cell. Flow across all boundaries is determined using the discretized momentum equation. The sum of all the flows across a cell’s boundary, including flux, recharge, and pumping are then used to determine new values for total head. The time step is then incremented and calculations over the lines and polygons are repeated.
The equations above simulate groundwater dynamics. For a cell object or boundary-line object, the movement of groundwater is considered a behavior under OOP. The governing equations are defined in scripts which retrieve the information they need from the GIS database using ID’s. The various types of information used by the model are explained in the next section.
Knowledge of the physical characteristics of a system is needed to accurately simulate groundwater movement. These parameters can be static or variable. As an illustration, the initial total head for each cell is known, but changes over the course of the model period. The storativity of a given cell, on the other hand, is less variable and can often be considered static. Parameters can also control the calculations in some way. For example, a line object is subject to different processes depending on whether it is on the boundary of an aquifer or inside.
GIS has the ability to store parameters and manage information using spatial references. The database consists of several feature-attribute tables (FTABs) with ID fields that link characteristics to specific map features. The data for a given feature defines its state, to be used in modeling behaviors under OOP. An FTAB usually contains one value for a parameter associated with a given feature. This regime works well for single-valued parameters such as a groundwater cell’s bottom elevation or area. A FTAB from ArcView is shown in Figure 1.
Figure 1. A Feature-Attribute Table from ArcView
An alternate database structure must be used to address parameters that have multiple values for a single feature, such as time-series data. The model must be able to quickly store and retrieve this information as the simulation progresses. The Map-Based Groundwater Model solves this problem by creating a single table for each attribute (parameter). The table is organized such that the columns correspond to specific features and the rows correspond to different time increments. As a result, the information for a specific feature during a given time step is one-to-one, allowing the program to retrieve the data as if it were a single-valued parameter. In Figure 2, the time-series table shows the recharge rate for different groundwater cells over a ten-day period.
Figure 2. A Time-Series Attribute Table
Initial and boundary conditions must also be considered in a groundwater model. Additional parameters can be included to set constraints or modify calculations to provide a closer approximation to the actual process. Logical variables can affect the way and object is treated by the model. A boundary type variable, for example, can be used to indicate whether a line is a non-flow boundary or a constant-head boundary. Initial conditions, such as the total head for a cell, can also be specified using additional parameters. Altogether, the various FTABs contain information that is used for multiple purposes, including the physical properties used by the equations and the initial and boundary conditions that govern the simulation, as a whole.
In order to accurately represent groundwater processes, an aquifer’s interactions with the surface must be considered. This can be accomplished by modifying the GIS database to transfer and store information that the surface and groundwater models have in common. Data linkages must be established because the two models generally utilize different conceptual and spatial relationships. In this manner, surface and groundwater models can be integrated through a shared interface, the base map.
Object relationships between a surface water model (subwatersheds) and a groundwater model (cells) are usually not one-to-one. For example, if several groundwater cells below a subwatershed object are generating spring flow, each of their contributions must be added together and applied to the surface as a single value. The groundwater cells attached to each watershed object must be specified in the model so that recharge contributions can be distributed amongst them. This type of data relationship is computationally intensive.
Distributing data between a variable number of objects is not an issue if a one-to-one relationship is created. The Map-Based Groundwater Model equates subwatershed objects and groundwater cells by combining their physical properties into a single FTAB. This equivalence can affect the model’s accuracy if the assumption of uniformly distributed flux across a boundary line is no longer valid because of the size of a groundwater cell. However, a one-to-one relationship allows all information to be stored in one place, making retrieval and associated calculations more efficient. This is not an absolute requirement. Multiple data relationships can be created at the expense of simulation run-time and memory.
Once data relationships are established, the distribution of water between the two models is taken into account by additional tables and parameters. Two databases are used to keep track of recharge contributions to groundwater cells and spring flow contributions to the surface. A parameter specifying the percentage of surface runoff infiltrating into the groundwater system accounts for subwatershed losses. Because the two models only share volumes of water, they can be run separately. The surface model can be run first to determine the contributions of recharge and losses to the aquifer. Subsequently, the groundwater model can be run to calculate spring flow contributions to the surface.
In order to see how groundwater theory and concepts are incorporated into a GIS environment, an exercise was developed using a simple grid-cell model. The exercise is included in the Appendix. The exercise was based on a model previously developed by Ye Zichuan to verify that the Map-Based Groundwater Model gave accurate results with those predicted by theory. In order to apply the model as a learning tool, a review of the academic literature was performed. Some of the books included in the review were, Hydraulics of Groundwater (Bear 1979), Groundwater (Freeze 1979), and Groundwater Hydrology (Bouwer 1978).
In the exercise, the movement of groundwater was represented within a spatial reference. A major benefit of a map-based learning tool is the ability to visualize the physical processes that are occurring as a result of fundamental principles. Continuity and momentum were the major principles included in the exercise. The study region consisted of a five-by-five grid representing a plan-view of a 2500 km2 aquifer, as shown in Figure 3. The magnitude and direction of groundwater movement, along with the total head were, illustrated for each cell.
Figure 3. Grid-Cell Representation of an Aquifer
Based on the concepts of mass and momentum balances, the Map-Based Groundwater Model was able to calculate the steady-state conditions in the aquifer. Vectors representing moisture flux and piezometric head were shown for each time step as the program converged on a solution. Subsequently, students were asked to compare the values of these parameters that were determined by the model with those predicted by theoretical solutions. Discrepancies between the two values revealed the errors associated with approximating a complex hydrologic system. For example, the model assumed that flow everywhere was horizontal. However, "in the vicinity of the water table peak and under the streams, this assumption is incorrect," (Bear 1979). Flow in these regions is essentially vertical. Consequently, larger model errors were seen within the vicinity of the water table peak.
Two cases were modeled in the groundwater exercise. The physical characteristics of the system were different for each case. The first case involved an unconfined aquifer bounded by two rivers with a continuous, uniformly-distributed recharge over the surface. In the second case, a continuous drawdown was simulated at a point centered within the grid. Students were able to modify the conditions by changing specific parameters. Since GIS stores parameters using spatial references, specific features could be selected directly from the map and their parameters would be highlighted in the FTAB for editing. An FTAB with parameters highlighted for a specific feature is shown in Figure 4. In addition, the model determined values for parameters that varied over the course of the simulation. These were stored within time-series data tables, as previously described.
Figure 4. Editing Parameters by Selecting Features
The ultimate goal of a regional model for a complex hydrologic system is to be able to match the simulation results with actual observations under a given set of conditions. A groundwater model that uses GIS and OOP makes the objective more obtainable because every aspect of the simulation can be modified as part of an integrated whole. The representation and parameters can be changed easily through the base map and FTABs. Closer approximations of physical processes, such as groundwater movement, can be made by altering the scripts (programs) used by the model. Finally, interactions between different phases of a hydrologic system can be adjusted for accuracy. For example, interactions between surface water flows and groundwater flows can be adjusted by specifying the percentage of water that is distributed across the boundaries.
An early objective of this study was to develop a regional groundwater model of the Souss Basin, Morocco that was calibrated to accurately predict the movement of water within the aquifer. It became apparent, however, that a real-world model is extremely complex. A representation, or coverage, was created by the Center for Research in Water Resources at the University of Texas at Austin. The physical characteristics of the aquifer, including hydraulic conductivity, top and bottom elevations and storativity were also necessary. Because of time constraints, these parameters were arbitrarily selected to produce a simple hydrologic model that illustrated movement of water within the aquifer and changes in total head for each of the cells. The results from a run using The Map-Based Groundwater Model are shown in Figure 5, below.
Figure 5. Groundwater Model of the Souss Basin, Morroco
The simulation was modified in different ways to see the changes in groundwater movement that resulted. Initially, a uniformly-distributed recharge was applied over the surface of the aquifer. Logical variables were modified for subwatershed objects that were located above the aquifer to integrate the two models. Afterwards, time-series data tables for total head within the groundwater cells were used in a transient-state simulation. The tables were generated from a surface water model of the Souss Basin using precipitation records from 1987. The tables were provided by Kwabena Asante, who is a fellow student in the Civil Engineering department at the University of Texas. In the model, groundwater cells were equated with subwatershed objects to create a one-to-one spatial and data relationship. The results were rough approximations that served to illustrate the use of GIS in a groundwater model that was more complex than the grid-cell model used for the exercise.
This document has provided a brief overview of how groundwater modeling is incorporated into GIS. Modeling in GIS has a major advantage over traditional methods because of it's ability to describe relationships between the different components of a hydrologic system within a spatial context. In the report, the simulation of groundwater movement through scripts was described. The program retrieves the parameters necessary to solve equations in the scripts using references to specific map features. The manner in which interactions between surface and groundwater flows are accounted for was also explained. Finally, the groundwater exercise and the regional model that were developed over the course of this study were summarized. GIS is an important resource for any hydrologic study and will definitely have an increasingly prominent role in the future.
Bear, J. 1979. Hydraulics of Groundwater, McGraw-Hill Book Company.
Chow, V.T., Maidment, D.R., Mays, L.W. 1988. Applied Hydrology, McGraw-Hill Book Company.
Freeze, R.A., Cherry, J.A. 1979. Groundwater, Prentice-Hall, Inc.
Bouwer, H. 1978, Groundwater Hydrology, McGraw-Hill Book Company.
Click here to go to the exercise: Groundwater Flow Between Two Rivers
Click here to go to my personal home page: Kris Martinez