numerical method. For two-dimensional problems, the area of the groundwater flow either in the x-y plane or the x-z plane is subdivided by a square grid (Fig. 7.9). The nodes of the grid are the points at which the values of head are required. At each node, the h value is assumed to be representative of a block centred on the node. If the value of h at A is unknown, it can be calculated from considerations of the flow pattern around the block. Using the lettering in Fig. 7.9, and applying Darcy’s Law, flow per unit thickness from the neighbouring node B is given by:
Similarly, values of Q from C, D and E towards A can be stated, but from the conservation of mass for steady state flow, the sum of these flows must be zero. Thus for an isotropic, homogeneous aquifer with all the Ks equal and since the grid is square (∆x=∆y) then:
Such an equation can be used to determine h values at all inner nodes within a grid. The boundary nodes often have known h values for certain boundary conditions. They are the starting points of sequential h determination at all nodes across an area by the method of relaxation. In this method, after initial estimates of h have been made at all nodes, a residual error (R) at A viz:
is removed by adjusting hA (resulting in changes to R values at B, C, D and E). Such
adjustments are made systematically until all R values have fallen to an acceptable tolerance. Treatments are available for boundaries where h is not known or where their alignments are not grid lines.
With the wide capabilities of the digital computer, complex problems incorporating anisotropic conditions and transient flows can be solved using this and similar finite difference methods. Package programs for such computations are readily available.
The finite-element method is a more recently developed numerical technique. It is being used in many branches of engineering. The computations resemble finite- differencing, but a mesh of more flexible geometry is used to model the problem area. Variable anisotropy can easily be accommodated by adjusting the local axes to the principal directions of hydraulic conductivity, which reduces computational time. The method normally gives greater accuracy than finite differences, but usually at greater expense. Considerable research is being undertaken with the finite-element method and further details are to be found in Pinder and Gray (1977).
7.5 Groundwater Measurement
It has already been emphasized that within the hydrological cycle in temperate regions of the world, groundwater constitutes the largest storage of fresh water. In the previous sections of this chapter, the theory of groundwater movement has been described, but
little attention has been given to the practical aspects of finding or measuring groundwater. The relatively slow but varying water movement through the heterogeneous mixture of unconsolidated sands and gravels and consolidated rocks ensures that a continuous slowly varying base flow is maintained in most rivers. Part of this contribution to surface flow can be measured at permanent spring sites, the methods used depending on the quantities involved. Small springs can be sampled volumetrically over a measured time interval to give discharge in l s−1 or m3 s−1, but more usually small thin
plate weirs are installed (see Chapter 6). The lateral contribution of groundwater through seepage to surface streams cannot be measured directly, and calculated estimates must be made using knowledge of the water table and of the flow characteristics of the geological strata.
The basic measurement for assessing groundwater is the depth of the water table in unconfined aquifers or the position of the piezometric surface in confined aquifers. According to the nature of the terrain, penetration of the ground can be made by methods ranging from hand-dug wells in unconsolidated surface deposits to the high-speed drilling of deep boreholes. The latter are usually employed for the exploitation of confined aquifers at great depths. In localities where every village or separate farmstead had its own well-water supply, there are often disused wells that can be made available for water-table measurements. When the groundwater storage is to be exploited for large public water supplies, special narrow observation wells are sometimes bored in addition to the large-diameter supply wells destined to be pumped. Hydrogeologists assemble well data from all possible sources to obtain knowledge of the groundwater behaviour. In the UK, some records of well levels are available for hundreds of years, especially in the chalk lands of Southern England.
The slow movement of water in the ground results in slow natural changes in groundwater levels, and thus well measurements are in general made regularly on a monthly basis. Where more rapid movement is expected or detailed information is required, the sounding of wells may be made weekly or even over shorter periods. In the expansion of hydrometric studies in the UK for the evaluation of water resources, well- level recorders have been recommended. These are labour saving and therefore, in developed countries, provide an economical method of obtaining data and of ensuring the continuation of valuable long records at old established measuring wells. Some of the various types of level recorders used in river flow measurements (Chapter 6) are readily adapted for recording well levels.
The standing-water level in a well is dependent on the character of the well, whether it is lined or unlined, and its depth. The structure and composition of the ground are most important too. A shallow well may penetrate through only one or two layers of rock and, if it is unlined, the water level may be considered to represent the level of the local water table. A deep bore-hole could pass through many different geological strata and the well- water level may be affected by pressures in confined aquifers, thus giving a piezometric level rather than the level of the water table. A cased or lined well certainly gives the piezometric level representing the water pressures at the bottom of the well.
To assess natural groundwater contribution to streams and other surface channels, it is the measurements of the water table from the shallower wells that are required.
7.5.1 Data Processing
From the nature of the material covered so far in this chapter, it is clear that the study of groundwater requires the recording, processing and storing of a wide variety of information. Qualitative descriptive material pertaining to geological structure and composition of the strata particularly of the aquifers are surveyed in all three dimensions, the areal distributions on the surface and the occurrences at depths in the ground. In this context, there are certain numerical parameters to be noted, such as rock properties and grain sizes. These are usually identified or measured during the digging of wells or the drilling of boreholes. However most of the quantitative data are related to the movement of the groundwater and hence changes with time in the storage of the aquifers. Allied to the physical location of the water is its chemical composition dependent on the aquifer type and the water’s original source.
A comprehensive GRoundwater Information Processing System (GRIPS) has been developed by the Institute of Hydrology. GRIPS has been designed to manage the systematic storage and presentation of the data using microcomputers and is particularly geared to the collection of data during groundwater investigations. The system operates using a choice of menus and up to 13 types of data can be handled for up to a maximum of 100000 stations. The data can be input from a keyboard, from existing computer files or from solid state loggers. Resultant outputs may be in the form of maps, standard graphs and time series diagrams for the changing variables and there are facilities for editing and updating the data archives. Of major importance is the ability to record the lithology and borehole construction data and produce logarithmic plots of water level drawdowns from pumping tests. Listings, graphs and maps of the results of chemical analyses for eight common major ions and up to eleven trace elements can also be produced. The system can also store isotope information, results of grain analyses and rock properties such as porosity, permeability and specific yield. GRIPS is well able to handle the wide variety of data and to replace dependence on mainframe computers.
7.6 Unconfined Flow
The lateral seepage from a river bank into a river produced by unconfined flow from a porous aquifer with a well defined water table is portrayed in Fig. 7.10. The aquifer is assumed to be isotropic and homogeneous, and it is underlain by an impermeable stratum. The flow pattern in the combined unsaturated and saturated layers of such an aquifer is not so easily identified as in a completely saturated confined aquifer. The free surface of the water table has an increasingly downward slope to meet the river bank at point A, the top of a seepage surface. The water-table slope reflects a hydraulic gradient and the line of the water table describes a stream line if no percolation occurs from the ground surface. In effect, the water table is the upper boundary of a flow net in the saturated part of the aquifer. The river bed/bank interface is an equipotential line and hence all streamlines meeting it must turn to meet BC at right angles. The seepage surface AB is not an equipotential and the water table flowline is tangential to the river bank at A (Raudkivi and Callendar, 1976).
To be able to calculate the flow to the river, several approximations are generally made. These approximations, developed by Dupuit and Forchheimer, make the major assumptions:
(a) the hydraulic gradient dh/dx is equal to the slope of the water table; (b) the related specific discharge is constant throughout the depth of flow; and