Finding the history of a groundwater contaminant plume from measurements of its current spatial distribution is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, Tikhonov regularization (Tikhonov and Arsenin, 1977) is used in numerical experiments to recover the release history of a plume that has originated from a known, single site. The recovered release history is then used to reconstruct the plume evolution history. The method is found to be insensitive to round off errors, but its accuracy is affected by plume measurement errors, the extent to which the plume has dissipated, and, to a lesser degree, the accuracy of the transport parameter estimates. A regularization approach may be effective at finding a plume history when there are adequate data. IntroductionConcern about the quality of the environment has led to increased efforts in cleaning and protecting groundwater resources. Remediation of groundwater contamination is expensive; by the time a cleanup strategy is devised and implemented and the seemingly inevitable litigation costs figured in, the total bill is often in the millions of dollars. Because of the high costs, an important step in any remediation project is deciding who is going to pay for cleanup operations.When groundwater contamination has resulted from the actions of more than one party, we would like to distribute the costs among the parties in a way that is consistent with their degree of culpability (the polluter pays principle). Records of handling hazardous substances are frequently insufficient for determining the spatial and temporal origin ofcontamination so its history must be reconstructed from observations of the existing contaminant plume [National Research Council, 1990]. Groundwater contaminant transport is a dispersive and therefore irreversible process; modeling it with reversed time is an ill-posed problem. The implications of this are twofold. First, ill-posed problems are extremely sensitive to errors in data, so small errors in the measurement of the existing plume may drastically change the calculated plume history. Second, ill-posed problems result in unstable numerical schemes making it impossible to run existing contaminant transport models with reversed time. The ill-posed nature of the problem makes obtaining an accurate estimate of a contamination history extremely difficult. Ill-Posed Problems The literature on ill-posed problems has grown substantially over the last 40 years. Typical ill-posed problems include numerical differentiation of noisy data [Anderssen and Bloomfield, 1974], the inverse heat conduction problem [Beck, 1985], interpretation of geophysical data [Backus and Gilbert, 1970], and the inverse problem of groundwater hydrology [Yeh, 1986; Neuman, 1973]. The general theory of ill-posed problems has been covered by Tikhonov and Arsenin [1977], Lavrent'ev et aI. [1986], and Payne [1975]. Formally, a problem is ill-posed if its solution does not satisfy general conditions of exist...

A new single‐borehole measurement technique for confined aquifers, the dipole flow test, yields the vertical distributions of the horizontal hydraulic conductivity, the vertical hydraulic conductivity, and the specific storativity when applied to different borehole intervals. The test utilizes straddle packers to isolate two chambers in the borehole, pressure transducers to monitor drawdown in them, and a small pump to create a dipole flow pattern in the aquifer by pumping water at a constant rate from the aquifer into one chamber, transferring it within the well to the next chamber, and finally injecting it back to the aquifer. A mathematical model describing the drawdown in each chamber is derived for the transient as well as the steady state cases. The aquifer parameters may be estimated from data produced by the dipole flow test alone or in conjunction with conventional pumping tests. The dipole flow regime reaches a steady state relatively quickly, especially in well permeable aquifers. A robust computational methodology for estimating the aquifer parameters, suitable for automatization, is based on the Newton‐Raphson algorithm applied to a system of up to three nonlinear equations, each describing the well drawdown at a different judiciously chosen time. Due to the relatively small drawdown it invokes, the dipole flow test may be applicable to unconfined aquifers as well.

The method of quasi-reversibility (QR) (Lattes and Lions, 1969) has been used previously to solve the diffusion equation with reversed time. We develop a quasireversible solution to a convection-dispersion equation by solving the QR diffusion operator in a moving coordinate system. The solution procedure is applied to the problem of recovering the history of a groundwater contaminant plume from observations of its present conditions. This approach to the plume history problem is potentially superior to the Tikhonov regularization approach used by Skaggs and Kabala (1994) because it is easier to implement and readily allows for space-and time-dependent transport parameters. However, our results for a few example problems suggest that the QR procedure is less accurate than the regularization technique. Thus the easy implementation and improved generality of the QR procedure come at the expense of accuracy; this trade-off will have to be weighed if the QR technique is to be used.

The cumulant expansion method, used previously by Sposito and Barry (1987) to derive an ensemble average transport equation for a tracer moving in a heterogeneous aquifer, is generalized to the case of a reactive solute that can adsorb linearly and undergo first‐order decay. In the process we also generalize the Van Kampen (1987) result for the cumulant expansion of a multiplicative stochastic differential equation containing a time‐dependent sure matrix. The resulting partial differential equation exhibits terms with field‐scale coefficients that are analogous to those in the corresponding nonstochastic local‐scale transport equation. There are also new terms in the third‐ and fourth‐order spatial derivatives of the ensemble average concentration. It is demonstrated that the effective solute velocity for the aqueous concentration, not that for the total concentration (aqueous plus sorbed), is relevant for a field‐scale description of solute transport. The field‐scale effective solute velocity, dispersion coefficient, retardation factor, and first‐order decay parameters, unlike their local‐scale counterparts, are time‐dependent because of autocorrelations and cross correlations among the random local solute velocity, retardation factor, and first‐order decay constant. It is shown also that negative cross correlations between the random tracer solute velocity and the inverse of the local retardation factor may produce both enhanced dispersion and a temporal growth in the field‐scale retardation factor. These effects are possible in any heterogeneous aquifer for which a stochastic description of aquifer spatial variability is appropriate.

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