Solute transport in divergent radial flow with multistep pumping

An efficient approach is developed to analytically evaluate solute transport in a horizontal, divergent radial flow field with a multistep injection flow rate and an arbitrary input concentration history. By assuming a piecewise steady state flow and transforming the time domain to the cumulative injected flow domain, the concentration distribution is found to be completely determined by the total volume of injected flow and independent of specific flow rates. Thus, on the cumulative flow domain, the transport problem with a temporally varying velocity field can be transformed into a steady state flow problem. Linear convolution can then be applied on the cumulative injected flow domain to evaluate the solution for an arbitrarily time-dependent input concentration. Solutions on the regular time domain can be conveniently obtained by mapping the solution on the cumulative injected flow domain to the time domain. Furthermore, we theoretically examine the conditions for the assumption of piecewise steady state flow to be valid. On the basis of the critical time scale of the "pseudosteady state condition," defined as when velocity changes accomplish 99% of their steady state differences, and the relative error in the mean travel time of plume front, we obtain conditions for neglecting the transitional period between two pumping steps. Such conditions include the following: (1) the duration of a pumping step, t _p, must be longer than the critical time scale, t _c, i.e., t _p ≥ t _c = 25r ²S/T, where r is the radial distance, S is the storage coefficient, and T is the transmissivity, or similarly, a maximum problem domain needs to be defined for a given pumping strategy. (2) the maximum well pumping rate, q _max, should satisfy q _max ≤ πθT/25S, where θ is the effective porosity. When both conditions are satisfied, transitional periods may be neglected.