Fractional Models Simulating Non‐Fickian Behavior in Four‐Stage Single‐Well Push‐Pull Tests

Abstract

Four‐stage single‐well push‐pull (SWPP) tracer tests, including injection, chasing, resting, and pumping, were conducted in a fractured aquifer at Newark basin. An anomalous transport phenomenon observed in the SWPP tests is the linear decline of breakthrough curves (BTCs) at late time with slope of −1.8 in log‐log plots. A time‐dependent fractional model is developed to interpret the anomalous transport behavior. This model considers a time‐dependent power law memory function and a time‐dependent fractional advection‐dispersion operator. The fractional advection‐dispersion equations (fADE) are solved in a radial coordinate system using the implicit Euler method. A semi‐analytical solution of the first‐order rate‐limited mobile‐immobile model (FORMIM) is derived for comparison. It is found that both the nonlocal transport in time and space can produce the long‐tailed BTC. A smaller time‐fractional or space‐fractional index leads to a lower peak concentration and a larger late‐time slope. The mass distribution of the fractional‐in‐space (FS) model exhibits power law decline at the leading plume edge. Early breakthrough during pumping is not observed because the mobile mass at the start of pumping is nonzero and more concentrated near the wellbore. The capacity ratio is an important factor that affects the peak concentration. A larger capacity ratio leads to greater peak concentration. A smaller time‐fractional index in the injection, chasing, or resting stage will move the BTC downward and the slope of the late time BTC is determined by the space‐fractional index over all stages and the time‐fractional index in the pumping stage. The capability of the existing models to recover the BTC of the SWPP test is discussed and some guidelines for how to choose the appropriate model to interpret the SWPP test data are proposed.

Models of Single‐Well Push‐Pull Test With Mixing Effect in the Wellbore

“…Interestingly, FCC and RCC perform essentially the same when α r is 0. In previous studies, C inj was assumed to be constant in equations and , and C ext was assumed to be equal to in equation , as can be seen in Gelhar and Collins (), Schroth and Istok (), Huang et al (), Jung and Pruess (), and Chen et al (). It is interesting to note that equation reduces to the Danckwerts boundary condition if the concentration continuity at the wellbore‐aquifer interface is maintained at the extraction phase (Danckwerts, ): which has been widely used to describe the boundary condition of solute transport at the wellbore in the extraction phase of SWPP tests, such as Gelhar and Collins (), Schroth and Istok (), Huang et al (), Jung and Pruess (), and Chen et al ().…”

“…The hydraulic diffusion refers to the ratio of hydraulic conductivity over specific storage of a confined aquifer. Although such an assumption has been widely used to describe solute transport in the SWPP test, such as Gelhar and Collins (), Schroth and Istok (), Huang et al (), Jung and Pruess (), and Chen et al (), its validity may be acceptable only under certain circumstances. This is because the transient flow effect is inevitable when switching from the injection phase to the extraction phase.…”

“…Due to the advantages of computational efficiency, easy implementation, and free of numerical errors (like numerical dispersion and numerical oscillation), analytical solutions are preferred by many scientists/engineers to interpret the SWPP test results. As a result, numerous analytical models have been developed to facilitate the parameter estimation in the SWPP test, such as Gelhar and Collins (), Schroth and Istok (), Huang et al (), Jung and Pruess (), and Chen et al (). The analytical solutions of Gelhar and Collins (), Schroth and Istok (), Huang et al (), and Chen et al () were developed for SWPP tests conducted in porous media aquifers, while the solution of Jung and Pruess () was for a fractured aquifer.…”

“…The main difference among these models for the porous media aquifers was the type of wellbore utilized. For instance, the well was assumed to fully penetrate the aquifer in Gelhar and Collins () and Chen et al (), to partially penetrate the aquifer in Huang et al (), and to have a zero penetration length in Schroth and Istok (), which was also named a point source well. It seems that the model of a partially penetrating well is a direct extension of the models of fully penetrating and point source wells.…”

“…In contrast, FCC can keep the mass balance at the wellbore‐aquifer interface, but the concentration continuity at the wellbore‐aquifer is not satisfied (Parker & van Genuchten, ). Gelhar and Collins (), Schroth and Istok (), Jung and Pruess (), and Chen et al () developed the RCC solutions for the SWPP tests, while Huang et al () presented both RCC and FCC solutions for the SWPP tests. Hansen et al () recently conducted a parametric study quantifying the scattering effects of ambient flow, local scale dispersion, and velocity field heterogeneity on BTCs of the SWPP test and compared them to the effect of mobile‐immobile mass transfer (MIMT).…”