Approximate modeling of transient wells and line-sinks with analytic response functions

We propose a new approach for the modeling of groundwater head fluctuations due to wells and line-sinks with discharges that are highly variable in time. The approach is based on the use of impulse response functions, the head response due an impulse of discharge. Once the impulse response function is known, the response to any stress that varies with time may be obtained through convolution. Superposition and convolution of the classic impulse response function for a well allows for the simulation of well fields with complicated discharge distributions. This classic approach becomes cumbersome or may break down when other aquifer features are present. When the well field is located near a stream, for example, it is necessary to include the effect of the stream on the impulse response function, and thus on the heads in the aquifer. We have developed an approach where the impulse response function of a well or line-sink may be approximated by a parametric, analytic function. Impulse response functions may be viewed as scaled probability density functions and may be characterized by certain integral characteristics such as the area, mean, and standard deviation. Alternative integral characteristics are the temporal moments and the exponentially-scaled temporal moments (the mean and standard deviation may be expressed in terms of moments). Temporal moments are specifically useful, as they fulfill steady, Poisson-type differential equations, with known values along the boundaries. Hence, the moments of the impulse response function may be computed exactly by constructing (relatively simple) steady models. The exact impulse response function may then be approximated by predefined analytical functions through moment matching. We propose an approximate impulse response function that has four parameters, so we need to determine four moments at a point to compute the approximate impulse response function there. This means that we have to build four steady models for each transient stress. At any point in space we can then compute four moments, and thus the four parameters of the approximate impulse response function. It will be shown that this approach gives accurate transient results for a number of cases, while it requires the solution of a small number of steady models only.