Section B – The Eigenmodel Method

1. Eigenmodel – a New Name

As a clear and convenient label for the approach presented in this report, we have coined the name "eigenmodel". The term "eigen-" is a German word that is long established in mathematical theory, for which it has the meaning of "characteristic". The eigenmodel approach is derived from the partial differential equations of groundwater flow by means of mathematical concepts called eigenvalues and eigenvectors (e.g. Sahuquillo, 1983), but has not previously had a specific name. Sloan (2000) applied the theory to groundwater discharge at catchment scale.

2. Assumptions

2.1 The Management Objective

The amount of groundwater stored in an aquifer at any instant of time depends on the dynamic relationship between recharge inputs, through the overlying land surface and from rivers, and outflow to surface waters and pumped abstraction. Aquifer storage provides a buffer between highly variable, climatically driven recharge processes and the less variable outflow that supports surface water ecology. Abstraction of groundwater for human use, and some kinds of land use changes, alters the dynamic balance between "natural" recharge and the state of surface waters. The resource management objective is to determine the regime of abstraction that results in acceptable environmental effects.

2.2 Causes of Environmental Effects

The eigenmodel method for aquifer management depends on three assumptions:

Recharge through the land surface overlying an aquifer can be estimated from water balance models based on climatic data and land-use parameters
Most of the temporal variation in piezometric levels throughout an aquifer is caused by temporal variations in land surface recharge, together with the effects of pumped abstraction
Environmental effects, such as the low-flow regime of streams are usually related to piezometric levels in the aquifer.

In summary, if the dynamic response of an aquifer to land surface recharge is quantified then environmental effects can be related to abstractive demand on the aquifer. Each of the above factors and the concepts of dynamic response will be explained in more detail.

3. Dynamic Behaviour of Aquifers

3.1 Linear Storage – the Building Block of Dynamic Response

The concept of a linear water storage element is illustrated in Figure 1. We will examine the dynamic behaviour of this simple element in some detail, because complex linear systems are analysed as assemblies of this basic unit. Subsequently, we will show that the dynamic response of an aquifer to recharge can be analysed as a complex linear system.

Figure 1: The linear water storage element

The water balance of the linear storage element (Figure 1) can be written as a differential equation:

(1)

Since this is a linear water storage, the outflow O(t) is proportional to the amount of stored water S(t). This relationship can be expressed with a proportionality constant k as:

(2)

By substituting equation (2) into equation (1):

(3)

The transition from equation (1) to equation (3) is quite significant. Equation (1) is a mathematical statement of the water budget without any other knowledge about processes, whereas equation (3) incorporates knowledge about the process expressed as the dynamic parameter k. Equation (3) no longer involves the outflow O(t). A further step is to incorporate the relationship between water depth h(t) and storage S(t) in the form:

(4)

The constant g, called the gain, could incorporate the cross-sectional area of the storage and other properties such as porosity of the storage medium, for example. By combining equations (3) and (4):

(5)

Equation (5) provides the relationship between temporal variations in water depth h(t) and temporal variations in water inflow I(t). Equations of this kind are the reason that groundwater management can be conducted without knowing all the recharge and outflow quantities, because these are substituted for by knowledge of dynamic behaviour. Now we examine some of the practical implications of equation (5).

3.1.1 Computational Equations

In order to use equation (5) in a spreadsheet calculation with time-series data, a solution of this differential equation is required for time intervals of ∆t. For the case where the input I(t) is averaged over the time interval from t - ∆t to t and the output h(t) is the instantaneous value at the end of the interval, the solution is:

(6)

for which:

(7)

Equation (6) is a difference equation suitable for use in a spreadsheet with data observed at discrete time intervals. The coefficients a and b are related to the process properties k and g by the relationships (7).

3.1.2 Steady-state Conditions

If the linear water storage receives a steady input of Is, then hn = hn-1 = hs. If these values are substituted onto equation (6):

(8)

The ratio b/(1-a) is called the steady-state gain (ssg). This concept will be used in the eigenmodel assessment of groundwater resources.

3.1.3 Storage Residence Time and Eigenvalue

The residence time of a water storage is defined as the ratio of storage volume to mean flow. In the case of a single linear storage, the outflow is always related to storage by the coefficient k, as shown by equation (2). Therefore the residence time TR of a single linear storage is 1/k.

If there is no input to the storage, then the initial contents will be reduced to half in a time equal to 0.69TR.

For this single linear storage, considered as a dynamic system, the eigenvalue is equal to k and has the dimensions of 1/time.

3.1.4 Combinations of Linear Storages

Single linear storages, each with different values of k and g, can be interconnected into networks to form complex linear systems. The mathematical techniques of eigenvalue analysis can be used to convert any network into a parallel set of linear elements. The resulting eigenvalues provide the k values of the elements. This is the approach that we use to analyse the dynamic behaviour of aquifers by means of eigenmodels.

3.2 The Aquifer as a Complex Linear System

Our approach to assessment and management of groundwater is based on a two-dimensional concept of aquifers, in which groundwater flow is essentially horizontal. Most applications of numerical models, such as MODFLOW, are for problems of this kind. In these numerical models, the horizontal extent of the aquifer is divided up into (usually) rectangular cells. Sets of equations are generated which specify the relationship between the piezometric heads on the cell corners, groundwater flow through the cell, abstraction and recharge in each cell, and aquifer properties of transmissivity and storativity.

These sets of equations that are the core of a numerical model are actually the mathematical description of a complex linear system, given the assumptions inherent in the 2-D concept of an aquifer. It is possible to convert these sets of equations into the mathematically equivalent eigenvalue-eigenvector form but there may be little computational advantage. The reason for this is mainly due to the ability of the numerical model to simulate localised variations in recharge and abstraction, which is the strength of these models. However, this modelling capacity demands appropriate levels of data. For many groundwater resource assessments these data are not available, and application of the numerical modelling packages may be inappropriate. If we are prepared to sacrifice some of this modelling flexibility then the eigenvalue-eigenvector analysis can yield models that are simple, have lower data requirements, and are suitable for implementation in spreadsheets. These are our eigenmodels.