PRESSURE PREDICTION BETWEEN TWO WELLS USING THE LAPLACIAN COMPONENT IN THE DIFFUSION EQUATION: ANALYTICAL SOLUTION IN ONE DIMENSION

Transient flow describes the change of hydraulic head (h) with time at a given point in an aquifer. It has been widely studied due to the capacity of estimating aquifer parameters from pumping and slug tests. As well, transient flow conditions are more common in nature than steady-state conditions. It can take several months or years for an aquifer to be in steady-state conditions. The diffusion equation can describe the changes in h with time in confined aquifers, using hydraulic conductivity (K), h and the second spatial derivative of the hydraulic head.

The diffusion equation is a mathematical formulation based on the conservation of mass and can have components for x, y, and z axes. However, by using an integration of the Laplacian component (second spatial derivative of h), we can arrive at a solution in one dimension (distance, x), to predict the changes of h with distance at a given point between two wells. Some assumptions and simplifications are made to arrive at a one-dimensional solution, for example, it has to be assumed that K is spatially homogeneous. Despite the fact that this is a simplified form of predicting head changes, the prediction of h at a given point between the wells is easy to calculate.

Boundary conditions for both wells are necessary to arrive at the solution, which will be related to K and h for both wells. Hence, if both wells have the same K and h along the aquifer, no change in h will be expected. Nevertheless, the changes between the hydraulic properties of both wells and the development of the Laplacian will estimate at a given point the changes in head. With the result in terms of head, we can use the definition of h to express it in terms of pressure head. These types of solutions are simple and useful tools for predicting how hydraulic properties vary in an aquifer in one-dimension and can lead to two-dimensional models integrating other equations like Theis method and the diffusion equation for analysis of change in pressure in transient conditions, not only in space but also in time, along with changes in pumped or injected water volumes.