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.

Thesis (M.Sc.) -- Adelaide University, Dept. of Applied Mathematics, 2001. / Bibliography: 80-82. Also available in a print form.

A stochastic analysis of groundwater flow leads to probability distributions on the predicted hydraulic head values. This variability reflects our uncertainty in the system being modeled due to the spatial heterogeneity of hydraulic conductivity. Monte Carlo techniques can be used to estimate the head distributions. This approach relies on the repetitive generation of discrete-block conductivity realizations. In this study, steady state flow through one and two-dimensional flow domains is investigated. A space law based on a first order, nearest neighbour stochastic process model is used to generate the multilateral spatial dependence in the hydraulic conductivity values within the block structure. This allows consideration of both statistically isotropic and anisotropic autocorrelation functions. It is shown that the probability distribution of hydraulic head and the head gradient or the flux across the boundaries of the flow domain, must be interpreted in terms of: 1) The spatial variation of expected head gradients. 2) The standard deviation in the conductivity distribution. 3) The ratio of the integral scale of the autocorrelation function for conductivity to the distance between boundaries on the flow domain. 4) The arrangement of stationary units within the flow domain. The standard deviations in hydraulic head increase with an increase in either the conductivity standard deviation or the strength of the correlation between neighbouring conductivity values. Provided the integral scales of the medium are preserved, the standard deviations in head show only a minor dependence on the discretization interval. The head standard deviations are approximately halved in a two-dimensional model from those in a one-dimensional model with an equivalent space law. Spatial trends in the mean conductivity can considerably alter the magnitude and spatial variation in the hydraulic head standard deviations. The geometric mean has been suggested by others as a suitable effective conductivity in a heterogeneous two-dimensional flow domain. This study shows that only in the case of uniform flow through a single stationary unit is this concept valid. If the mean gradient field is nonuniform, or if the mean conductivity has a spatial trend, predictions based on the geometric mean do not satisfy the necessary equivalence criteria. Direct comparisons cannot be made, but the Monte Carlo and spectral approaches to the solution of the stochastic flow equations predict a similar behavior. A first order, nearest neighbour model is matched to a data set collected from a relatively uniform but stratified, unconsolidated sand deposit. The data show statistically anisotropic autocorrelation functions, both in the integral scale and in the functional form of the correlation. A broader class of spatial models may need to be considered to describe the cyclic behavior of sedimentary sequences. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

The problem of evaluating the parameters of the mathematical model of an unconfined aquifer is examined with a view toward development of automated or computer -aided methods. A formulation is presented in which subjective confidence ranges for each of the model parameters are quantified and entered into an objective function as linear penalty functions. Parameters are then adjusted by a procedure which seeks to reduce the model error to acceptable limits. A digital computer model of the Tucson basin aquifer is adapted and used to illustrate the concepts and demonstrate the method.

The semi and climate at the Fordwah Eastern Sadiqia (South) Project, Pakistan, comprised of 105,000 ha of culturable command area, is characterized by large seasonal temperature fluctuations and a monsoon season. The canal system behaves as a recharge source to the regional groundwater and has caused waterlogging and salinity problems. The aquifer of the project area is unconfined and underlain by sediments deposited by the Sutleg-Hakra river system. / To quantify the rate of groundwater recharge in the project area, a numerical groundwater model was developed. A network of 125 observation wells was installed and watertable depth data were collected for the period of June 1994 to June 1997. Within this network, a distinction was made between internal and external nodes representing nodal areas and boundary conditions, respectively. Other data used in the model were aquifer characteristics obtained from seven historical and five newly performed pumped well tests. The aquifer analysis showed a regional decrease in aquifer transmissivity from the eastern region to the western. The hydraulic conductivity values obtained from these analyses were assigned to each side of each nodal area. / The aim of the present study was to develop a more reliable and time consuming methodology to determine the yearly, seasonal and monthly net-recharge occurring in the study area. (Abstract shortened by UMI.)

Thesis (Ph. D.)--University of Hawaii at Manoa, 1995. / Includes bibliographical references (leaves 174-180). / Microfiche. / xvii, 180 leaves, bound ill. 29 cm

Two digital- computer models of the ground -water reservoir of the Tucson basin, in south - central Arizona, were constructed to study errors in digital models and to evaluate the worth of additional basic data to models. The two models differ primarily in degree of detail -- the large -scale model consists of 1,890 nodes, at a 1/2 -mile spacing; and the small -scale model consists of 509 nodes, at a l -mile spacing. Potential errors in the Tucson basin models were classified as errors associated with computation, errors associated with mathematical assumptions, and errors in basic data: the model parameters of coefficient of storage and transmissivity, initial water levels, and discharge and recharge. The study focused on evaluating the worth of additional basic data to the small -scale model. A basic form of statistical decision theory was used to compute expected error in predicted water levels and expected worth of sample data (expected reduction in error) over the whole model associated with uncertainty in a model variable at one given node. Discrete frequency distributions with largely subjectively- determined parameters were used to characterize tested variables. Ninety -one variables at sixty - one different locations in the model were tested, using six separate error criteria. Of the tested variables, 67 were chosen because their expected errors were likely to be large and, for the purpose of comparison, 24 were chosen because their expected errors were not likely to be particularly large. Of the uncertain variables, discharge /recharge and transmissivity have the largest expected errors (averaging 155 and 115 feet, respectively, per 509 nodes for the criterion of absolute value of error) and expected sample worths (averaging 29 and 14 feet, respectively, per 509 nodes). In contrast, initial water level and storage coefficient have lesser values. Of the more certain variables, transmissivity and initial water level generally have the largest expected errors (a maximum of 73 per feet per 509 nodes) and expected sample worths (a maximum of 12 feet per 509 nodes); whereas storage coefficient and discharge/ recharge have smaller values. These results likely are not typical of those from many ground -water basins, and may apply only to the Tucson basin. The largest expected errors are associated with nodes at which values of discharge /recharge are large or at which prior estimates of transmissivity are very uncertain. Large expected sample worths are associated with variables which have large expected errors or which could be sampled with relatively little uncertainty. Results are similar for all six of the error criteria used. Tests were made of the sensitivity of the method to such simplifications and assumptions as the type of distribution function assumed for a variable, the values of the estimated standard deviations of the distributions, and the number and spacing of the elements of each distribution. The results are sensitive to all of the assumptions and therefore likely are correct only in order of magnitude. However, the ranking of the types of variables in terms of magnitude of expected error and expected sample worth is not sensitive to the assumptions, and thus the general conclusions on relative effects of errors in different variables likely are valid. Limited studies of error propagation indicated that errors in predicted water levels associated with extreme erroneous values of a variable commonly are less than 4 feet per node at a distance of 1 mile from the tested node. This suggests that in many cases, prediction errors associated with errors in basic data are not a major problem in digital modeling.

No description available.

wo digital-computer models of the ground-water reservoir of the Tucson basin, in south-central Arizona, were constructed to study errors in digital models and to evaluate the worth of additional basic data to models. The two models differ primarily in degree of detail -- the large-scale model consists of 1,890 nodes, at a 1/2-mile spacing; and the small-scale model consists of 509 nodes, at a 1-mile spacing. Potential errors in the Tucson basin models were classified as errors associated with computation, errors associated with mathematical assumptions, and errors in basic data: the model parameters of coefficient of storage and transmissivity, initial water levels, and discharge and recharge. The study focused on evaluating the worth of additional basic data to the small-scale model. A, basic form of statistical decision theory was used to compute expected error in predicted water levels and expected worth of sample data (expected reduction in error) over the whole model associated with uncertainty in a model variable at one given node. Discrete frequency distributions with largely subjectively-determined parameters were used to characterize tested variables. Ninety-one variables at sixtyone different locations in the model were tested, using six separate error criteria. Of the tested variables, 67 were chosen because their expected errors were likely to be large and, for the purpose of comparison, 24 were Chosen because their expected errors were not likely to be particularly large. Of the uncertain variables, discharge/recharge and transmissivity have the largest expected errors (averaging 155 and 115 feet, respectively, per 509 nodes for the criterion of absolute value of error) and expected sample worths (averaging 29 and 14 feet, respectively, per 509 nodes). In contrast, initial water level and storage coefficient have lesser values. Of the more certain variables, transmissivity and initial water level generally have the largest expected errors (a maximum of 73 per feet per 509 nodes) and expected sample worths (a maximum of 12 feet per 509 nodes); whereas storage coefficient and discharge/ recharge have smaller values. These results likely are not typical of those from many ground-water basins, and may apply only to the Tucson basin. The largest expected errors are associated with nodes at which values of discharge/recharge are large or at which prior estimates of transudssivity are very uncertain. Large expected sample worths are associated with variables which have large expected errors or which could be sampled with relatively little uncertainty. Results are similar for all six of the error criteria used. Tests were made of the sensitivity of the method to such simplifications and assumptions as the type of distribution function assumed for a variable, the values of the estimated standard deviations of the distributions, and the number and spacing of the elements of each distribution. The results are sensitive to all of the assumptions and therefore likely are correct only in order of magnitude. However, the ranking of the types of variables in terms of magnitude of expected error and expected sample worth is not sensitive to the assumptions, and thus the general conclusions on relative effects of errors in different variables likely are valid. Limited studies of error propagation indicated that errors in predicted water levels associated with extreme erroneous values of a variable commonly are less than 4 feet per node at a distance of 1 mile from the tested node. This suggests that in many cases, prediction errors associated with errors in basic data are not a major problem in digital modeling.

A finite difference model was developed specifically for analyzing the Grand Island, Nebraska aquifer test. Time-drawdown data for the aquifer test were fitted by least squares to an exponential type equation. To facilitate calibration of the model, interpolated distance-drawdown profiles also were fitted to an exponential type equation. The treatment of aquifer boundaries and the assumption of isotropic aquifer conditions affected the model computed water table profile. The effect was significant enough as to defy making accurate estimates of saturated hydraulic conductivity and specific yield. When the analysis was extended to long time periods of discharge, problems with the boundaries, particularly the distance to the lateral constant head boundary, led to unrealistic estimates of pumping level. The finite difference technique has its greatest application as a research method for analyzing short-duration aquifer tests provided that the aquifer conditions are well defined, measurements of pumping level are available and drawdown measurements have been secured for at least two observation wells within close proximity of the discharge well. Because of difficulties in maintaining convergence and model stability, the finite difference model reviewed in this study is too cumbersome to be considered a practical, field method for the analysis of unconfined aquifer parameters.