The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests

We present a distributed-order fractional diffusion-wave equation (dofDWE) to describe radial groundwater flow to or from a well, and three sets of solutions of the dofDWE for flow from a well for aquifer tests: one for pumping tests, and two for slug tests. The dofDWE is featured by two temporal orders of fractional derivatives, b1 and b2, which characterise small and large pores, respectively. By fitting the approximate solutions of the dofDWE to data from slug tests in the field, we determined the effective saturated hydraulic conductivity, Ke, transmissivity, Tf, and the order of fractional derivatives, b2 in one test and b2 and b1 in the second test. We found that the patterns of groundwater flow from a well during the slug tests at this site belong to the class of sub-diffusion with b2 < 1 and b1 < 1 using both the short-time and large-time solutions. We introduce the concept of the critical time to link Ke as a function of b2 and b1. The importance of the orders of fractional derivatives is obvious in the approximate solutions: for short time slug tests only the parameter b2 for flow in large pores is present while for long time slug tests the parameters b2 and b1 are present indicating both large and small pores are functioning.