The main aim of this study is to develop a suitable method for the calibration and validation of mathematical models of large and complex aquifer systems. Since the calibration procedure depends on the nature of the model to be calibrated and since many kinds of models are used for groundwater, the question of model choice is broached first. Various aquifer models are critically reviewed and a table to compare them as to their capabilities and limitations is set up. The need for a general calibration method for models in which the flow is represented by partial differential equations is identified from this table. The calibration problem is formulated in the general mathematical framework as the inverse problem. Five types of inverse problems that exist in modeling aquifers by partial differential equations are identified. These are, to determine (1) parameters, (2) initial conditions, (3) boundary conditions, (4) inputs, and (5) a mixture of the above. Various methods to solve these inverse problems are reviewed, including those from fields other than hydrology. A new direct method to solve the inverse problem (DIMSIP) is then developed. Basically, this method consists of transforming the partial differential equations of flow to algebraic equations by substituting in them the values of the various derivatives of the dependent variable (which may be hydraulic pressure, chemical concentration or temperature). The parameters are then obtained by formulating the problem in a nonlinear optimization framework. The method of sequential unconstrained minimization is used. Spline functions are used to evaluate the derivatives of the dependent variable. Splines are functions defined by piecewise polynomial arcs in such a way that derivatives up to and including the order one less than the degree of polynomials used are continuous everywhere. The natural cubic splines used in this study have the additional property of minimum curvature which is analogous to minimum energy surface. These and the derivative preserving properties of splines make them an excellent tool for approximating the dependent variable surfaces in groundwater flow problems. Applications of the method to both a test situation as well as to real-world data are given. It is shown that the method evaluates the parameters, boundary conditions and inputs; that is, solves inverse problem type V. General conditions of heterogeneity and anisotropy can be evaluated. However, the method is not applicable to steady flows and has the limitation that flow models in which the parameters are functions of the dependent variable cannot be calibrated. In addition, at least one of the parameters has to be preassigned a value. A discussion of uncertainties in calibration procedures is given. The related problems of model validation and sampling of aquifers are also discussed.

Photocopy of typescript. / Digitized by Kansas Correctional Industries

FLUMP, a recently developed mixed explicit -implicit finite -element program, was calibrated against a data base obtained from a portion of the Tucson Basin aquifer, Arizona, and represents its first application to a real -world problem. Two previous models for the same region were constructed (an electric analog and a finite -difference model) in which calibration was based on prescribed flux boundary conditions along stream courses and mountain fronts. These fluxes are not directly measured and estimates are subject to large uncertainties. In contrast, boundary conditions used in the calibration of FLUMP were prescribed hydraulic heads obtained from direct measurement. At prescribed head boundaries FLUMP computed time - varying fluxes representing subsurface lateral flow and recharge along streams. FLUMP correctly calculated fluctuations in recharge along the Santa Cruz River due to fluctuations in storm runoff and sewage effluent release rates. FLUMP also provided valuable insight into distributions of recharge, discharge, and subsurface flow in the study area.Properties of FLUMP were compared with those of two other programs in current use: ISOQUAD, a finite -element program developed by Pinder and Frind (1972), and a finite- difference program developed by the U.S. Geological Survey (Trescott, et al., 1976). It appears that FLUMP can handle a larger class of problems than the other two programs, including those in which the boundary conditions and aquifer parameters vary arbitrarily with time and /or head. FLUMP also has the ability to solve explicitly when accuracy requires small time steps, the ability to solve explicitely in certain parts of the flow region while solving implicitly in other parts, flexibility in mesh design and numbering of nodes, computation of internal as well as external fluxes, and global as well as local mass balance checks at each time step.