Groundwater flow near a recirculation well with mixed-type boundary condition along wellbore in confined, unconfined, leaky, or reservoir-covered aquifers

Comment on “Tu, K., Q. Wu, N. Zhang, and X. Liu. 2023. An analytical model for a single‐well circulation system in a confined aquifer with a finite thickness well skin. International Journal for Numerical and Analytical Methods in Geomechanics 47(06) https://doi.org/10.1002/nag.3514. First published 17 February 2023″

“…On the other hand, if only one screen segment is used, the summation for individual screen segments, 8 (Equation 56) reduces to a single term and the solution represents a model with uniform radial inflow along the screen. [7][8][9] The same uniform radial inflow solution can be obtained by treating the integrand in the well face boundary condition as constant. 8 Perina, 7 (Figure 2) used a superposition of the Hantush 10 solutions for an extraction and injection well to represent a recirculation well with two screens to show that the general well function reduces to the model with uniform flux along the well screen when one segment is used for each well screen.…”

“…[7][8][9] The same uniform radial inflow solution can be obtained by treating the integrand in the well face boundary condition as constant. 8 Perina, 7 (Figure 2) used a superposition of the Hantush 10 solutions for an extraction and injection well to represent a recirculation well with two screens to show that the general well function reduces to the model with uniform flux along the well screen when one segment is used for each well screen.…”

“…Perina 7 summarized the derivation of the general well function for four types of aquifers. The top boundary condition is different for each aquifer type and requires a specific type of finite integral transform (or equivalent separation of variables).…”

“…To arrive at the Laplace domain solution of Tu et al, 2 specify zero casing radius to eliminate wellbore storage, 7 (Equation 44), use the same length for the extraction and injection screen intervals, and assign just two screen segments (j = 1 for injection screen, j = 2 for extraction screen) in Equation (62) of Perina 7 to determine the radial fluxes per unit screen length to be q 1 = −Q/(bη 1 ) and q 2 = Q/(bη 2 ) (from the last two of the set of equations), where bη 1 = bη 2 is the length of screen interval and Q is the pumping rate (this relation for Q holds in both the Laplace and the time domains). Then substitute for q 1 and q 2 into the expression for summation coefficient B n , 7 (Equation 34) and Equations ( 35) and (36) 7 provide Laplace domain drawdown at a point in the aquifer and in the skin zone, respectively. The summation has to be modified for Fourier cosine transform to be applicable to a confined aquifer, 7 (Equation 21).…”

“…3 Kabala 4 showed that Tu et al 3 did not present a new model but repeated both Ni et al 5 and Kabala. 6 As shown below, the Tu et al 2 solution is a special, simpler case of a more general problem solved by Perina 7 for a recirculation well installed in a confined, unconfined, leaky, or reservoir-covered aquifer that is surrounded by a finite thickness skin zone and may have an arbitrary number of extraction and injection screens of arbitrary length. The Perina 7 model determines the distribution of inflows and outflows along the well screens as part of the solution by discretizing the screen intervals into small segments.…”