Progress in Pumping Test Analysis via Generalisation and Reciprocity

We possess a growing toolbox of models for the analysis of pumping tests: in recent years those tools have taken the form of software packages containing collections of well functions. New hydrogeological situations demand new models but often what is required is closely related to an existing model. It is therefore useful to consider to what extent we can generalise our well functions or adapt them to different circumstances. Most well functions are based on the assumption of cylindrical flow to a well. This two-dimensional analysis can be generalised to any flow dimension, giving, for example, the incomplete gamma function as a generalisation of the Theis function. This 'fractional-dimension' flow model is a special case of a fractal model; both of these are becoming widely applied. Double-porosity models are based on local flow between 'fracture' and 'matrix' components of the aquifer, where the matrix blocks are normally taken to be slabs or spheres. Given improvements in data quality and density, we can begin to consider generalisations of that block geometry and it is possible to formulate well functions for any mixture of block shapes and sizes. A hierarchical porosity model has also been formulated recently. A third type of geometrical generalisation that is possible is in the shape of a pumping well. It is, for example, possible to produce a well function for a slug test in a rectangular shaft. This suggests new field techniques such as constructing short trenches instead of boreholes for pumping tests in very shallow aquifers. In the context of pumping tests, the reciprocity theorem states that if we invert the positions of a pumping well and an observation well the two well functions will be identical. (Strictly, both wells should be represented as points.) This has been proven for Darcian flow in both continuum and discretely fractured aquifers, no matter how heterogeneous. When applied to the problem of observation well storage, the theorem shows that the normal large-diameter well solution is applicable. Another application has been to find the influence of pumping on a wetland, modelled as a circular pond overlying a leaky-aquifer.