5. Aquifer Characteristics and Eigenmodel Parameters
In order to use the parameter values of the eigenmodel to assist with assessment of a groundwater resource, it is important to appreciate the meaning of these parameters in terms of the physical characteristics of an aquifer. We will be concerned primarily with the eigenvalues ki and the gain coefficients gi(x,y), shown in Figure 3. The values of these parameters are provided by the eigenmodel calibration procedure described in Section 4.4.
5.1 Eigenvalues of a Simple Aquifer
Although the eigenmodel theory can be applied to any aquifer, a useful insight is obtained by considering the eigenvalues of a rectangular-shaped aquifer with homogeneous properties of transmissivity T and storativity S. The horizontal dimensions are Lx, Ly, and there is a fixed-head boundary on all edges (such as a surface water boundary all around the aquifer). This simple case can be solved analytically, and the general formula for the eigenvalues is:
(16)
We now examine some of the implications of equation (16) for groundwater assessment.
5.1.1 Significance of Eigenvalues
The relative magnitudes of the eigenvalues given by equation (16) depends on the squares of the integers 1,2,... This means that the second eigenvalue is at least four to five times the magnitude of the first, depending on the relative magnitudes of Lx and Ly. The storage residence time (Section 3.1.3) of an aquifer, which is the inverse of the eigenvalue, will therefore be dominated by the first eigenvalue. The larger eigenvalues can be important for more accurate simulation of the dynamics of piezometric response, but they do not contribute very significantly to water storage.
5.1.2 Effect of Horizontal Scale
Equation (16) states that the eigenvalues are inversely proportional to the square of the horizontal dimensions. This means that, for example, an aquifer 16 km by 10 km has four times the storage residence time (inverse of eigenvalue) of an aquifer 8 km by 5 km that has similar geological properties and boundary conditions.
5.1.3 Effect of Aquifer Boundaries
Equation (16) has been derived for an ideal simple aquifer that drains to all four boundaries (fixed head). If this aquifer is square, Lx = Ly = L, and the first eigenvalue is:
(17)
If the aquifer drains only to two opposite boundaries, this is equivalent to the 1-D case where Lx = L and Ly → ∞, in which case equation (16) gives:
(18)
If the aquifer drains to only one boundary, it can be shown that:
(19)
Comparison of equations (17) to (19) shows that changing only the boundary conditions of this simple aquifer results in an eight-fold variation of storage residence time.
5.1.4 Effect of Geological Properties
The ratio T/S appears in each of the above equations for the eigenvalues of the homogeneous aquifer. Analysis of heterogeneous aquifers by eigenvalue methods has the potential to provide estimates of the large-scale value of this ratio, if aquifer geometry and boundaries are well defined. However, it is unlikely that this value would be the same as properties derived from pumping tests, because the up-scaling process is quite complex. It is worth noting that the T/S value appears only to the first power in the above equations, and the likely effect of geological properties needs to be kept in perspective with the effects of horizontal scale and aquifer boundaries.
5.2 Relevance of the Gain Coefficients
The gain coefficients gi(x,y) at each well location not only indicate the relative significance of the aquifer eigenvalues at that location, but can also provide information about aquifer boundaries and storativity. The total steady-state gain ssg(x,y) at a location is given by:
(20)
and can also be derived from the difference equation (10) as:
(21)
The average piezometric effect of land surface recharge (LSR effect in the eigenmodel results), ha(x,y), is given by:
(22)
where Ra is the long-term average of the recharge magnitude series R(t).
5.2.1 Effect of Location and Boundaries
The value of ssg(x,y) is zero at fixed-head boundaries, and increases with distance from those boundaries. This property can be used, for example, to indicate whether a river is interacting directly with an aquifer or is perched above the groundwater surface. In the latter case, the value of ha(x,y) will be relatively large.
5.2.2 Estimating Aquifer Storativity
If the availability of data allows, then a plot of ha(x,y) would permit estimation of the average thickness, Ha, of the aquifer that is occupied by the average amount of water originating from land surface recharge. This average amount of water can be reasonably estimated by means of the average recharge Ra and the average storage residence time 1/k1. Then the large-scale aquifer storativity S is given by:
(23)
In cases where there are few observation well records, there is an approximation that may be useful. This relies on the assumption that the shape of the groundwater surface under steady-state conditions is a paraboloid, which degenerates to a parabola for 1-D groundwater flow. The average height of a paraboloid is 4/9 the maximum height. The corresponding ratio for a parabola is 2/3. If the value of ha(x,y) is known for a well that is furthest from a fixed-head boundary then, as a first approximation, the average thickness Ha is half this "maximum" value.
5.3 The "Base" Parameter and River Recharge
The Base parameter value in the eigenmodel results (Figure 4) is the piezometric effect of river recharge r(x,y) at the particular well location, which is assumed to be steady. If there are sufficient locations with piezometric records, these base values can provide a map of the river recharge effect as an additional set of information about the aquifer. This piezometric surface could be used to estimate the quantity of river recharge if values of transmissivity are available. However, it would be simpler to express river recharge as a proportion of land surface recharge, which is more accurately calculated, by means of the relative piezometric gradients, because the transmissivity in both cases is the same.
Contact for Enquiries
MAF Information Services
Pastoral House
25 The Terrace
PO Box 2526
Wellington, NEW ZEALAND
Fax: +64 4 894 0721
Contact this person
