This paper presents a functional formulation of the groundwater flow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a “forward operator” which describes the measurement process. In the particular case considered here, the parameter of interest is the large‐scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwater flow equation. The inverse algorithm seeks the “most probable” or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness‐of‐fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well‐posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss‐Newton or gradient‐based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer‐Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.

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A numerical model is used to examine groundwater flow in vertical section near surface water bodies, such as lakes, wetlands, ponds, rivers, canals, and drainage and irrigation channels. Solutions are generated partly by superposition to achieve computational efficiency. A large number of flow regimes are identified, with their characteristics controlled by regional water table gradients, recharge to the aquifer, water body length, aquifer anisotropy, and the hydraulic resistance of the bottom sediments. Different flow regimes are distinguished by the presence and nature of groundwater mounds or depressions near the edges of a surface water body and by corresponding stagnation points. Ranges of values for dimensionless flow parameters over which particular regimes occur are determined for six representative geometries and presented in the form of transition diagrams. Increasing anisotropy or sediment resistance and decreasing the length of a water body relative to aquifer thickness are shown to have similar effects on flow geometry, the main effect being an increasing tendency for stagnation points to form in the interior of the aquifer. Flow-through behavior becomes more prevalent with decreasing anisotropy and sediment resistance and increasing water body length. land surface also plays a role, as does negative recharge, i.e., net evapotranspirative discharge from the land surface.This paper provides a general framework for surface 1Now at Nield Consulting, Nedlands, Western Australia.Paper number 94WR00796. 0043-1397/94/94 WR-00796505.00 water-groundwater interaction, but is sti!l limited by a number of simplifying assumptions. In particular, we assume steady saturated groundwater flow, a shallow water body (relative to aquifer thickness), a long water body (such that a two-dimensional approach in vertical section is valid), and homogeneity in hydraulic conductivity. These simplification• reduce the number of site-specific components of our model, allowing some general features of surface watergroundwater interaction to emerge. Since a primary objective of this paper is to provide guidance for preliminary assessment of water bodies which have not been extensively monitored, the simplifying assumptions are probably justified. Application of the paper to a wide variety of real-word settings is facilitated by describing the geometry, aquifer properties, and flow conditions in terms of a small number of nondimensional parameters. Previous WorkOver the past two decades, increasing recognition of the need to address problems affecting lakes has led to numerous field and numerical investigations of lake-aquifer systems. Born et al. [1979] compiled hydrogeological data for numerous lakes in NorthAmerica and identified three broad classes of groundwater flow patterns near lakes: recharge (if the lake recharges the aquifer over the entire lake bed), discharge (if the aquifer discharges water to the lake over the entire lake bed), and flow-through (if water in different areas of the lake bed moves through the lake bed in...

Abstract. Steady flow regimes for three-dimensional lake-aquifer systems are studied via idealized mathematical models that are extensions of earlier simplified vertical section models of interaction between shallow lakes and underlying aquifers. The present models apply to a shallow circular lake at the surface of a rectangular aquifer of finite depth, yielding a truly three-dimensional representation of the resulting flow system. Flux boundary conditions are applied at the ends of the aquifer, with net vertical recharge or evapotranspiration at the water table. The lake is defined by a region with constant head. By determining and visualizing solutions to the discretized saturated flow equations, a range of possible flow regimes is identified, and their topological properties are studied. Tools for analyzing flow regimes are described, including a method for locating and mapping three-dimensional dividing surfaces within steady flow fields. Results show strong similarities between two-and three-dimensional systems, including a large number of flowthrough, recharge, and discharge regimes and reverse flow cells. Flow lines calculated on a vertical plane through the middle of a lake resemble but are not identical to twodimensional streamlines for a range of aquifer flow and recharge conditions. Estimates of the widths and depths of capture and release zones for various lake-aquifer geometries are asymptotic to earlier results for two-dimensional systems. Numerical predictions are compared with analytical results for certain limiting flow regimes.

Finite difference and finite element methods are frequently used to study aquifer flow' however, additional analysis is required when model parameters, and hence predicted heads are uncertain. Computational algorithms are presented for steady and transient models in which aquifer storage coefficients, transmissivities, distributed inputs, and boundary values may all be simultaneously uncertain. Innovative aspects of these algorithms include a new form of generalized boundary condition' a concise discrete derivation of the adjoint problem for transient models with variable time steps' an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation; and a new efficient first-order second-moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values. The techniques are presented in matrix form, and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations. Details of matrix structures are provided .for a two-dimensional linear triangular finite element model. INTRODUCTION Modeling of regional groundwater flow involves more than the straightforward calculation of piezometric heads. An investigation may also require (1) the estimation of model parameters using both historical head measurements and prior information (i.e., the inverse problem) and (2) an assessment of prediction uncertainty. Many papers have addressed the largescale inverse problem [e.g., Neuman, 1980a; Cooley, 1982], and others have addressed the estimation of the covariance of predicted heads due to uncertain parameters [e.g., Delhomme, 1979; Smith and Freeze, 1979; Dettinger and Wilson, 1981; Clifton and Neuman, 1982]. The purpose of this paper is to describe a systematic approach to the design of computational procedures for carrying out both of these steps. The proposed procedures have evolved from the observation that sensitivities of heads to parameters are fundamental to all the necessary calculations. Sensitivities are simply the first derivatives of heads with respect to parameters. They are required for traditional sensitivity analysis, for calculation of the gradient of the objective function in the inverse problem, for second derivative calculations during line searches in the inverse problem, for calculation of the posterior covariance of parameters estimated in the inverse problem, and finally for calculating the covariance of predicted heads due to uncertainty in the valu es of parameters. The sensitivities appear time and time again, but in many cases it is possible to use alternative computational procedures which avoid thei r explicit evaluation. In these cases, the sensitivities are implicitly contained in other matrices, and the use of alternative procedures can greatly improve the speed of calculations. The matrices which implicitly contain sensitivities are defined by (18) and (20) below and are denoted by [K•h(0)] and [A•,•,h(k)] for steady state and tra...