An Analytical Model With a Generalized Nonlinear Water Transfer Term for the Flow in Dual‐Porosity Media Induced by Constant‐Rate Pumping in a Leaky Fractured Aquifer

Abstract

In the past, many mathematical models based on the dual‐porosity (DP) concept were developed to describe the groundwater flow in fractured aquifer systems. Most of them seemingly have problems in predicting accurate drawdown at the early and/or intermediate times as compared with field measured data. Thus, this study proposes a new analytical model with a generalized transfer term (GTT) to describe the flow induced by pumping in such systems. The new model is nonlinear because the GTT representing the matrix‐to‐fracture flux gives different weights to the fracture and matrix drawdowns. The GTT reduces to the existing first‐order transfer term if the weight equals zero and second‐order term if the weight is one. The present model also includes a leakage term accounting for flow from the overlain or underlain aquitard. The drawdown solution of the model is developed based on the Laplace transform method and integration by parts formula and then verified through the comparison with the finite‐element solution. The effect of different weight values in the GTT on the DP flow is investigated. Additionally, the sensitivity analysis is performed to assess the impact of the change in each of the aquifer parameters on the flow. Furthermore, the present solution is used to analyze two sets of pumping drawdown data from test sites in Canada and India. We found that the drawdown predictions from the present solution fit field measured data very well, suggesting that the present model can adequately describe the real‐world DP flow system.

Analytical study of dual‐porosity coastal aquifer with generalized nonlinear water transfer term

“…According to the results presented by Lin and Yeh (2021), the parameter λ has the effect of decreasing the drawdown in fractures and increasing that in matrix blocks on the pump‐induced DP flow. That means that a larger λ leads to a smaller MTF flux.…”

“…Moreover, without loss of generality, Lin and Yeh (2021) proposed a generalized expression called a GTT for the MFT flux that can be transferred from the first‐order to second‐order terms for the fluxes by assigning a weight, λ , from 0 to 1. The GTT can be expressed as $$$$ \Gamma =\alpha \left(\frac{h_f^2}{\leftthreetimes {h}_m+\left(1-\leftthreetimes \right){h}_f}-{h}_m\right) $$$$ …”

“…The water interaction between these two superimposed continua is formulated by a source term reflecting the matrix‐to‐fracture (MTF) flux. There are two major approaches to express the MTF term—the distributed‐parameter approach (DPA) and lumped‐parameter approach (LPA; Lin & Yeh, 2021). The DPA assumes that the water in a matrix block can flow from its centre to the interface between the matrix block and the fracture.…”

Two‐region flow caused by pumping at a partial penetration well in a leaky confined aquifer

“…Sensitivity analysis is carried out to assess the influence of various controlling parameters on drawdown and wellbore flowrate 14,40–42 . In this study, the normalized sensitivity analysis of drawdown is implemented, and the normalized sensitivity coefficient takes the form 40 : $$$$\begin{equation}{X^{\prime}_{i,j}} = {P_j}\frac{{\partial {O_i}}}{{\partial {P_j}}} = {P_j}\frac{{{O_i}\left( {{P_j} + \Delta {P_j}} \right) - {O_i}\left( {{P_j}} \right)}}{{\Delta {P_j}}},\end{equation}$$$$in which $${X^{\prime}_{i,j}}$$ represents the normalized sensitivity due to the change of the j th parameter ( P j ) at the i th time; O i refers to the response of the dimensionless drawdown and the small increment Δ P j = 10 −2 × P j in this study.…”

“…Sensitivity analysis is carried out to assess the influence of various controlling parameters on drawdown and wellbore flowrate. 14,[40][41][42] In this study, the normalized sensitivity analysis of drawdown is implemented, and the normalized sensitivity coefficient takes the form 40 :…”