A mathematical model for regional land subsidence due to pumping: 3. Integrated equations for a phreatic aquifer

A mathematical model is developed for the areal distribution of drawdown, land subsidence, and horizontal displacements due to groundwater pumping from a deformable phreatic aquifer. This paper is a continuation of earlier works published by the authors (Bear and Corapcioglu, 1981a, b) on the subject of regional land subsidence due to pumping in confined and leaky aquifers. All the basic assumptions underlying these previous works are also valid here. However, in this case the same methodology is extended to the case of a phreatic aquifer. The main additional feature in the present paper is that due to the decline of the water table, the total stress at points within the flow domain changes continuously. Following the development of a three‐dimensional mathematical model consisting of a mass conservation equation, quasi‐static equilibrium equations, and stress‐strain relations for an assumed perfectly elastic solid matrix, a regional, horizontal two‐dimensional model is derived by averaging the three‐dimensional model over the vertical thickness of the aquifer, taking into account the continuous variation in total stress as a result of water table fluctuations. The separation of an initially coinciding phreatic surface, which serves as the Upper boundary of the aquifer, and a surface consisting of a set of solid particles during the lowering of the water table and the compaction of the aquifer has been incorporated into the averaging procedure. An analytical solution is given for the case of a single pumping well. The solution, which is expressed in terms of the classical well function for a confined aquifer but with a modified storativity, provides estimates of drawdown, vertical subsidence, and horizontal displacement. The displacements for a sample problem are compared with those of a confined aquifer under similar conditions. The results indicate that under the conditions of the studied case of radial flow, vertical and horizontal displacements due to pumping from a phreatic aquifer with the same discharge rate would be less than those of a confined one. Furthermore, the same averaging procedure is extended for a system of phreatic and leaky confined aquifers and for the unsaturated zone above the water table. An additional change in total stress is obtained in terms of moisture content of the zone above the phreatic surface.