Exact Analytical Solutions for Elliptical Flow Toward Extended Wells in Fractured Confined Aquifers: Application to Groundwater-Head Interpretation in Shale-Gas Development Areas

This study develops exact analytical solutions for transient elliptical groundwater flow toward an extended well in an anisotropic fractured confined aquifer and then discusses how the resulting hydraulic response can support groundwater-head interpretation in shale-gas development areas. The environmental connection is made at the aquifer-protection scale: the model is not a shale-gas reservoir production model, and it does not solve contaminant transport directly. Instead, it provides a hydraulic interpretation framework for estimating anisotropy, equivalent fracture length, wellbore-storage effects, and the preferential direction of head propagation around possible leakage points, old wells, fractures, or monitoring wells. Based on Mathieu-function theory and the separation-of-variables method, constant-rate and constant-head solutions are derived in Laplace space and inverted to the time domain with the Stehfest algorithm. The analytical results are validated against COMSOL5.2 finite-element simulations, and the effects of anisotropy coefficient and wellbore storage are analyzed through drawdown and flow-rate type curves. A synthetic but field-style water-head example is included to demonstrate how monitoring records can be converted to drawdown, fitted to the elliptical-flow solution, and used to delineate a preliminary hydraulic response zone. The results show that anisotropy mainly controls early-to-middle time response, whereas wellbore storage may obscure early head changes and delay the recognition of fracture connectivity. Therefore, the solution is best regarded as a rapid hydraulic-screening and monitoring-design tool that can precede, but not replace, site-specific contaminant-transport modeling in shale-gas groundwater-protection studies. The relevant technical issues are possible head disturbances and preferential groundwater pathways associated with surface spills, flowback-water handling, old wells, faults, and fracture-connected water-bearing zones. Because verified local field-monitoring records were not available for us, the application example is explicitly described as a synthetic field-style demonstration; it is used to show the workflow and its limitations, not to claim site-specific prediction of contaminant concentration.