ABSTRACT
The mathematical simulation of the hydraulic behaviour of a groundwater system requires the analytical or numerical integration of two simultaneous partial differential equations, the governing equations, connecting one or more partial derivatives of the dependent variable head H with respect to the independent variables (x1, x2, x3, t). Occasionally, as for Dupuit’s assumption for unconfined flows, these partial differential equations involve the dependent variable H itself. The order of a partial differential equation corresponds to the order of its highest derivative. They are linear if the dependent variable H and the partial derivatives ∂H/∂xi and ∂H/∂t, including their products, are only raised to the first power, but they are considered quasi-linear, if their highest derivatives remain linear. The continuity equation, resulting from the combination of these two governing
equations, associates the spatial and temporal coordinates (r, t) of the points of the flow domain to their hydraulic heads H, via the hydraulic conductivity tensor [k]. The integration of the continuity equation gives the spatial-temporal description of the hydraulic head H on the flow domain as a scalar field. The derivatives of the hydraulic head H give the hydraulic gradients vectors J as a vector field. Finally, the empirical relationships between [k] and J yields the specific discharges vectors q as another vector field. Domains typified by simple structures and properties favour analytical solutions
but for most groundwater problems, particularly those involving fractured rocks, a closed analytical solution does not exist. In these cases, numerical techniques, may lead to approximate solutions. Table 2.1 lists the essential features of numerical and the analytical solutions, pointing out their merits and disadvantages (Chung-Yau, 1994). Fractured rock masses are inhomogeneous and anisotropic. Their hydraulic proper-
ties may vary enormously over the flow domain. Fortunately, they may be correlated to descriptors of the geological features of the flow domain, duly calibrated by field and laboratory tests. However, a consistent knowledge of these properties alone does not lead to a particular solution for a specific groundwater system. Information about its history and the external influences on the spatial and temporal boundaries must be furnished to the “modeller’’. In fact, every particular solution for the continuity equation depends on knowledge of some supplementary equations: the boundary conditions and initial values. The problem is said to be well posed if for every set of supplementary
equations one can find a unique solution continuously dependent on this set. When boundary conditions are partially unknown, it may be possible to estimate the missing data by plausible hypotheses about the past and about the hydraulic behaviour on the borders of the groundwater system.
