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A mathematical model is presented for a constant-head test performed in a partially penetrating well with a finite-thickness skin. The model uses a no-flow boundary condition for the casing and a constant-head boundary condition for the screen to represent the partially penetrating well. The Laplace-domain solutions for the dimensionless flow rate at the wellbore and the hydraulic heads in the skin and formation zones are derived using the Laplace and finite Fourier cosine transforms. The solutions of hydraulic heads have been shown to satisfy the governing equations, related boundary conditions, and continuity requirements for the pressure head and flow rate at the interface of the skin zone and undisturbed formation. In addition, an efficient algorithm for evaluating those solutions is also presented. The dimensionless flow rates obtained from new solutions have been shown to be better than those of Novakowski's solutions, especially when the penetration ratio is large.

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SUMMARYA mathematical model describing the constant pumping is developed for a partially penetrating well in a heterogeneous aquifer system. The Laplace-domain solution for the model is derived by applying the Laplace transforms with respect to time and the finite Fourier cosine transforms with respect to vertical co-ordinates. This solution is used to produce the curves of dimensionless drawdown versus dimensionless time to investigate the influences of the patch zone and well partial penetration on the drawdown distributions. The results show that the dimensionless drawdown depends on the hydraulic properties of the patch and formation zones. The effect of a partially penetrating well on the drawdown with a negative patch zone is larger than that with a positive patch zone. For a single-zone aquifer case, neglecting the effect of a well radius will give significant error in estimating dimensionless drawdown, especially when dimensionless distance is small. The dimensionless drawdown curves for cases with and without considering the well radius approach the Hantush equation (Advances in Hydroscience. Academic Press: New York, 1964) at large time and/or large distance away from a test well.

[1] An analytical model for the constant flux pumping test is developed in a radial confined aquifer system with a partially penetrating well. The Laplace domain solution is derived by the application of the Laplace transforms with respect to time and the finite Fourier cosine transforms with respect to the vertical coordinates. A time domain solution is obtained using the inverse Laplace transforms, convolution theorem, and Bromwich integral method. The effect of partial penetration is apparent if the test well is completed with a short screen. An aquifer thickness 100 times larger than the screen length of the well can be considered as infinite. This solution can be used to investigate the effects of screen length and location on the drawdown distribution in a radial confined aquifer system and to produce type curves for the estimation of aquifer parameters with field pumping drawdown data.

Abstract:A mathematical model is developed for predicting the temperature distribution in an aquifer thermal energy storage (ATES) system, which consists of a confined aquifer bounded from above and below by the rocks of different geological properties. The main transfer processes of heat include the conduction and advection in the aquifer and the conduction in the rocks. The semi-analytical solution in dimensionless form for the model is developed by Laplace transforms and its corresponding time-domain solution is evaluated by the modified Crump method. Field geothermal property data are used to simulate the temperature distribution in an ATES system. The results show that the heat transfer in the aquifer is fast and has a vast effect on the vicinity of the wellbore. However, the aquifer temperature decreases with increasing radial and vertical distances. The temperature in the aquifer may be overestimated when ignoring the effect of thermal conductivity. The temperature distribution in an ATES system depends on the vertical thermal conduction in the rocks and the horizontal advection and thermal conduction in the aquifer. The present solution is useful in designing and simulating the heat injection facility in the ATES systems.

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This study develops a mathematical model for simulating the hydraulic head distribution in response to pumping in a sloping fault zone aquifer under a water table boundary condition. A twodimensional equation with a sink term representing the pumping is used for describing the head distribution in the aquifer. In addition, a first-order free surface equation is adopted to represent the change in water table at the outcrop. The analytical solution of the model, derived by the Laplace and finite Fourier cosine transforms, is expressed in terms of a double series. A finite difference solution within a deformable grid framework is developed to assess the solution obtained by specifying the free surface equation at the outcrop. Based on the analytical solution, we have found that the model's prediction tends to overestimate drawdown in a late pumping period. The temporal head distribution is independent of the aquifer slope if the water table change is small, and exhibits a double-humped shape due to the effect of the free surface. The temporal drawdown predicted from the analytical solution is further compared with those measured from a pumping test conducted in northern Portugal.