Analyzing non-Darcian flow in a confined aquifer toward a well with a linearization method

In this study, we have developed a new method to analyze non-Darcian flow toward a well in a confined aquifer with and without wellbore storage. The power law has been used to describe the relationship of the specific discharge and hydraulic gradient for non-Darcian flow. This new method is based on a combination of the linearization approximation of the non-Darcian flow equation and the Laplace transform. Approximate analytical solutions of steady-state and late-time drawdowns are also obtained. The drawdowns at any distance and time are computed by using the Stehfest numerical inverse Laplace transform with MATLAB programs. The results of this study agree perfectly with previous Theis solution for an infinitesimal well and with the Papadopulos and Cooper's solution for a finite-diameter well under the special case of Darcian flow. The Boltzmann transform, which is commonly employed for solving non-Darcian flow problems before, is problematic for studying radial non-Darcian flow. Comparison of drawdowns obtained by our proposed method and the Boltzmann transform method suggests that the Boltzmann transform method differs from the linearization method at early and moderate times, and it yields similar results as the linearization method at late times. The drawdowns decrease at late times as the power index n or the quasi hydraulic conductivity k increases, regardless of the wellbore storage. It has also been found when n is larger, flow approaches steady state earlier. The approximate analytical solutions indicate that the drawdown at steady state is approximately proportional to r over a power of (1-n), where r is the radial distance from the pumping well; the late time drawdown is a superposition of the steady- state solution and a negative time-dependent term that is proportional to time t over a power of (1-n)/(3-n).