
Citation: | Shi PY, Liu JJ, Zong YJ, et al. 2023. Analytical solution for Non-Darcian effect on transient confined-unconfined flow in a confined aquifer. Journal of Groundwater Science and Engineering, 11(4): 365-378 doi: 10.26599/JGSE.2023.9280029 |
Coal resources account for 94.22% of total primary energy resources in China, leading to significant development in coal mining, particularly in deep coal mining. Deep coal mining operations are typically situated beneath confined aquifer systems and are prone to water inrushes, which have been proven one of the important factors affecting the mining safety (Xiao et al. 2022; Zhao et al. 2021). In order to prevent these water inrushes and ensure mining safety, mine drainage has proven to be a highly effective method for reducing hydraulic pressure in the confined aquifer. However, this drainage process can have detrimental effects on the groundwater resources within the mining area. Consequently, it is crucial to accurately assess the decline in water levels caused by drainage in the confined aquifer to safeguard mine safety and protect groundwater resources in the mining area.
The constant-rate pumping has been widely acknowledged an effective approach for drainage design (Moench et al. 2001; Zong et al. 2022). During the pumping process, if the pumping rate is sufficiently high or the duration of drainage is long enough, the hydraulic head in the confined aquifer can drop below the bottom of the overlying aquitard. This phenomenon leads to the development of a transient unconfined flow near the pumping well. Since 1970s, many analytical and numerical solutions have been proposed to investigate the flow mechanism associated with the transient conversion from confined to unconfined conditions. Moench and Prickett (1972) introduced the MP model and derived an analytical solution with the assumption that the aquifer transmissivity is constant. In the same vein, Elango and Swaminathan (1980) developed a numerical method to study the characteristics of the transient confined-unconfined flow within a confined region. Li et al. (2003) analyzed the mechanism of confined-unconfined flow using both analytical and numerical approaches in an initially dry aquifer. Hu and Chen (2008) presented an analytical solution based on Theis' equation. Wang et al. (2009) extended the MP model and Chen's model to develop a solution for transient conversion flow in a pumping well. Considering the variations in hydraulic properties (such as transmissivity, storativity and diffusivity) between confined and unconfined regions, Xiao et al. (2018) developed an analytical solution. They employed the Boltzmann transform under Darcian conditions. Subsequently, Xiao et al. (2022) utilized a semi-analytical method to investigate the delayed response of drawdown.
Based on a comprehensive literature review, it is noted that previous studies have predominantly assumed Darcian conditions for pumping-induced flow. However, fieldwork and actual experiments have provided evidences of non-Darcian flow occurring in a wide range of porous and fractured media surrounding pumping wells, regardless of the flow rate (e.g. Basak, 1976; Soni et al. 1978; Sen, 1987, 1989, 1990; Bordier and Zimmer, 2000; Wu, 2001, 2002a, 2002b; Moutsopoulos and Tsihrintzis, 2005; Wen et al. 2006, 2011; Houben, 2015; Feng and Wen, 2016; El-Hames, 2020; Jiong et al. 2021; Hao et al. 2021). In term of the non-Darcian flow induced by high pumping rates, two common functions are used to describe the nonlinear specific discharge: The Forchheimer equation and the Izbash's equation. The Forchheimer equation expresses the specific discharge as a second-order polynomial function of hydraulic gradient (e.g. Sen, 1987, 1990; Wu, 2002a; Moutsopoulos and Tsihrintzis, 2005; Mathias and Wen, 2015; Mathias and Moutsopoulos, 2016; Liu et al. 2017), while the Izbash's equation describes the specific discharge as exponentially related to the hydraulic gradient. The Forchheimer equation considers viscous and inertial forces of water flow, and its polynomial form allows flexible representation of flow velocities, regardless of their magnitude (Wen et al. 2008c; Moutsopoulos and Tsihrintzis, 2005). The Izbash's equation is suitable for modelling post-linear non-Darcian flow (Wen et al. 2009; Chen et al. 2003; Qian et al. 2005), and can be more easily linearized in comparison with the Forchheimer equation. Over the past two decades, the validity of these two functions has been verified through their application in various types of aquifer hydraulic tests. These tests include slug test (Wang et al. 2015; Ji and Koh, 2015), and pumping tests in different aquifer systems such as aquifer-aquitard system (Wen et al. 2008a), fractured aquifers (Wen et al. 2006), leaky aquifers (e.g. Wen et al. 2011; Wen and Wang, 2013; Wang et al. 2015), unconfined aquifers (Bordier and Zimmer, 2000; Mathias and Wen, 2015; Moutsopoulos, 2007, 2009), and confined aquifers (e.g. Wen et al. 2008b, 2008c, 2013).
The authors have observed that a high discharge rate during well pumping can cause a temporary transition from confined to an unconfined state in a confined aquifer (e.g. Chen et al. 2006; Wang et al. 2009; Mawlood and Mustafa, 2016; Xiao et al. 2018, 2020, 2023). Additionally, it has also been reported that a large pumping rate can result in non-Darcian flow within the aquifer with a high Reynolds number (Rec>10). Consequently, the drawdown in a confined aquifer induced by a high pumping rate is expected to be influenced by both the non-Darcian flow and transient confined-unconfined conversion. However, to date, there has been a lack of research on groundwater modelling on the transient confined-unconfined flow under non-Darcian conditions.
The paper presents an analytical solution for modelling the transient confined-unconfined flow under non-Darcian conditions. The flow in a confined region is described by a two-dimensional differential equation that represents the seepage system, while the flow in an unconfined region follows the Boussinesq equation with distinct hydraulic parameters. The boundary conditions in the conversion interface are expressed by the flow continuity. To capture the nonlinear relationship between specific discharge and hydraulic gradient, the Izbash's equation is employed for modelling purposes. The analytical solution is obtained by using the Boltzmann transform, which enables the development of a practical approach for assessing the dynamic development of the unconfined region in real-world scenarios. The time-drawdown curves are used to quantify the effect of the non-Darcian index and other hydraulic parameters, and a normalized sensitivity analysis is conducted to evaluate the response of drawdown to the different hydraulic parameters.
Fig. 1 illustrates a schematic diagram of pumping test resulting from the transient confined-unconfined flow. The modelling assumptions are as follows: (1) The confined aquifer is isotropic and horizontal infinitely; (2) Both pumping and observation wells fully penetrate the confined aquifer; (3) The pumped rate,
At the beginning of pumping, the flow from the well is fully confined, and the pumping operation causes the hydraulic head to continuously decrease over time. Once hydraulic head in the pumping well (
The transient non-Darcian flow in the unconfined region can be described by the Boussinesq equation as follows:
∂q∂r+qr=−Syhm∂h1∂t |
Where:
The boundary condition representing the fully penetrating well is described as:
limr→02πrh1q=−Q |
Where:
In the region of
∂q∂r+qr=−Sb∂h2∂t |
Where:
h2(r→∞,t)=h0 |
The initial condition is given by:
h1(r,0)=h2(r,0)=h0 | (3) |
At the conversion interface between the confined and unconfined regions (
∂h1(R,t)∂r=∂h2(R,t)∂r |
h1(R,t)=h2(R,t)=b |
Using the Izbash's equation, the nonlinear specific discharge is depicted as:
q=(−Kr∂h∂r)1n | (5) |
Where:
By substituting Equation (5) into Equation (2a), we obtain:
nr(∂h2∂r)+(∂2h2∂r2)=nKr1nSb∂h2∂t(∂h2∂r)n−1n | (6) |
Similarly, the governing equation of the unconfined flow can be re-expressed by substituting Equation (5) into Equation (1a) as:
nr(∂h1∂r)+(∂2h1∂r2)=nKr1nSyhm∂h1∂t(∂h1∂r)n−1n | (7) |
Assuming that the flow rate in the confined region is equal to
∂h2∂r=(q)nKr≈(Q2πrb)nKr | (8) |
Combining Equation (8) with Equation (6) yields:
nr(∂h2∂r)+∂2h2∂r2=ε2∂h2∂tr1−n | (9) |
Where:
Similar to Equation (8), a linearization approach for Equation (7) is implemented by defining:
∂h1∂r=(q)nKr≈(Q2πrhm)nKr | (10) |
With Equation (10), Equation (7) and (1b) can be re-written as:
nr(∂h1∂r)+∂2h1∂r2=ε1∂h1∂tr1−n |
limr→02πrh1(−Kr∂h1∂r)=−Q |
Where:
In the section, the solution of the mathematical models for transient confined-unconfined flow is derived using the Boltzmann transform. As shown in Supporting Information, the analytical solution for the transient flow in the unconfined region is given by:
h1(r,t)=b−(Q4πKrhm)(Q2πrhm)n−1{W[nSyr24tKrhm(Q2πrhm)n−1]−W[nSyR24tKrhm(Q2πrhm)n−1]} | (12) |
Where:
The hydraulic head in the confined region is expressed as:
h2(r,t)=h0−(Q2πhm)n2Krrn−1exp[−nSyR24Krthm(Q2πRhm)n−1]exp[−nSR24Krtb(Q2πRb)n−1]W[−nSr24Krtb(Q2πRb)n−1] | (13) |
Using Equations (12) and (13), we have developed an approach to simulate drawdown for transient confined-unconfined flow under non-Darcian conditions in practice. The parameters required for drawdown simulation, including constant pumping rate (
By subjecting the boundary condition Equation (4b) into Equation (13), an expression at the conversion interface can be expressed as:
b=h0−(Q2πhm)n2KrRn−1exp[−nSyR24Krthm(Q2πRhm)n−1]exp[−nSR24Krtb(Q2πRb)n−1]W[−nSR24Krtb(Q2πRb)n−1] | (14) |
As shown in Fig. 2, Hu and Chen (2008) considered that the total amount of groundwater drained to the pumping well is equal to the changes of groundwater storage of both the confined and unconfined regions. This can be expressed as:
Qt=V1′+V2′ |
Where:
V1′=Sy×V1, |
V1=∫R02πr[b−h1(r,t)]dr=∫R02πr(12Krrn−1(Q2πhm)n{W[nSyr24tKrhm(Q2πrhm)n−1]−W[nSyR24tKrhm(Q2πrhm)n−1]})dr |
Similarly, the volume of groundwater pumped from the confined region is given by:
V2′=S×V2, |
V2=πR2(h0−b)+∫∞R2πr[h0−h2(r,t)]dr=πR2(h0−b)+∫∞R2πr{(Q2πhm)n2KrRn−1exp[−nSyR24Krthm(Q2πRhm)n−1]exp[−nSR24Krtb(Q2πRb)n−1]W[−nSR24Krtb(Q2πRb)n−1]}dr |
Substituting Equations (15b) and (15c) into Equation (15a) yields:
Qt=Sy∫R02πr(12Krrn−1(Q2πhm)n{W[nSyr24tKrhm(Q2πrhm)n−1]−W[nSyR24tKrhm(Q2πrhm)n−1]})dr+{πR2(h0−b)+∫∞R2πr((Q2πhm)n2KrRn−1exp[−nSyR24Krthm(Q2πRhm)n−1]exp[−nSR24Krtb(Q2πRb)n−1]W[−nSR24Krtb(Q2πRb)n−1])dr} | (16) |
Using Equations (14) and (16), the unknown values of
The pumping test has been widely acknowledged as an effective method for assessing aquifer hydraulic parameters. By analyzing drawdown data obtained from fieldwork, an inversed analytical approach has been developed to determine the dynamic behavior of the unconfined region, as well as the diffusivity and specific yield of unconfined region under confined-unconfined conditions. This approach relies on the assumption that the parameters
As shown in Fig. 2, the unconfined region is radially expanded as pumping continues. The dynamic development of the unconfined region is typically characterized by the radial distance (
During the early pumping stage, the piezometric surface is generally located above the top of the aquifer (
h′=h0−(Q2πhm)n2Krr1n−1exp[−nSyR24Krthm(Q2πRhm)n−1]exp[−nSR24Krtb(Q2πRb)n−1]W[−nSr124Krtb(Q2πRb)n−1] | (17) |
Considering the flow continuity and the Equation (4a), the expression at the transient interface can be obtained using Equation (13) as:
b=h0−(Q2πhm)n2KrRn−1exp[−nSyR24Krthm(Q2πRhm)n−1]exp[−nSR24Krtb(Q2πRb)n−1]W[−nSR24Krtb(Q2πRb)n−1] | (18) |
By combining Equations (17) and (18) and taking the ratio, it can be obtained:
h0−h′h0−b=Rn−1r1n−1W[−nSr124Krtb(Q2πRb)n−1]W[−nSR24Krtb(Q2πRb)n−1] | (19) |
Based on Equation (19), there are only two unknown parameters:
As the pumping continues, there comes a point where the water level in the observation well falls below the top of the confined aquifer (
h′=b−12Krr1n−1(Q2πhm)n{W[nSyr124tKrhm(Q2πr1hm)n−1]−W[nSyR24tKrhm(Q2πr1hm)n−1]} | (20) |
By using Equations (16), (18) and (20), the values of
This section aims to evaluate the validity of the proposed solution by comparing it with the work conducted by Xiao et al. (2018). Additionally, the effects of hydraulic parameters, namely
Since the proposed analytical solution is intended for non-Darcian flow, it can also be employed to analyze Darcian flow under confined-unconfined conditions by considering a special case when
h1(r,t)=b−Q4πKrhm{W(nSyr24tKrhm)−W(nSyR24tKrhm)} | (21) |
h2(r,t)=h0−Q4πKrhmexp(−nSyR24Krthm)exp(−nSR24Krtb)W(−nSr24Krtb) | (22) |
To verify the accuracy of the proposed solution, the results of drawdown simulation using the proposed solution when
Fig. 4 illustrates the drawdown curves obtained from a hypothetical study considering different
Fig. 5 depicts the R-time curves of transient confined-unconfined flow under non-Darcian conditions. The curves are generated using the following parameters values:
The time-drawdown curves for the same hypothetical case but with different specific storages are compared in Fig. 6. The differences between three drawdown curves, corresponding to different
Furthermore, a decreasing slope is observed in the time-drawdown curve from the confined region to the unconfined region. This can be attributed to fact that, at the early pumping stage, the flow predominantly stems from the artesian storage of
Fig. 7 shows the R-value and drawdown at various time points of interest for the hypothetical scenario of
The objective of the sensitivity analysis is to assess the impact of different parameters on the hydrogeological model and streamline the parameter calibration process. Among the various methods available, local sensitivity analysis has been selected as the calculation approach to evaluate the influence of individual parameters on the analytical solution. In this case, the hydraulic parameters are assumed to be independent of each other. In comparison with Theis' solution, the modeling for transient confined-unconfined flow under non-Darcian conditions involves four key parameters: Quasi-hydraulic conductivity
The sensitivity is defined as a rate of change in one factor with respect to a change in another factor. Based on the work by Huang and Yeh (2007), the normalized sensitivity parameter is defined as:
X′i,j=Pj∂Oi∂Pj | (23) |
Where:
X'i,j=Pj∂Oi(Pj+ΔPj)−∂Oi(Pj)ΔPj | (24) |
Where:
Fig. 8 displays the temporal variation of normalized sensitivity with respect to
Greater values of
A larger specific storage implies a greater release of water from the elastic storage of the aquifer. This initially has a negative impact on the drawdown and the development of the unconfined region during the early stage of pumping. However, as pumping continues, these effects diminish.
The drawdown is particularly sensitive to the power index
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[6] | Hong-bo HAO, Jie LV, Yan-mei CHEN, Chuan-zi WANG, Xiao-rui HUANG, 2021: Research advances in non-Darcy flow in low permeability media, Journal of Groundwater Science and Engineering, 9, 83-92. doi: 10.19637/j.cnki.2305-7068.2021.01.008 |
[7] | ZHANG Han-xiong, HU Xiao-nong, 2018: Simulation and analysis of Chloride concentration in Zhoushan reclamation area, Journal of Groundwater Science and Engineering, 6, 150-160. doi: 10.19637/j.cnki.2305-7068.2018.02.008 |
[8] | ZHU Wei, TANG Wen, LIU Qiang, ZHANG Mei-gui, 2017: Analysis on variation characteristics of geothermal response in Liaoning Province, Journal of Groundwater Science and Engineering, 5, 336-342. |
[9] | WU Jian-qiang, WU Xia-yi, 2016: Geological environment impact analysis of a landfill by the Yangtze River, Journal of Groundwater Science and Engineering, 4, 96-102. |
[10] | ZHOU Xun, WANG Xiao-cui, CAO Qin, LONG Mi, ZHENG Yu-hui, GUO Juan, SHEN Xiao-wei, ZHANG Yu-qi, TA Ming-ming, CUI Xiang-fei, 2016: A discussion of up-flow springs, Journal of Groundwater Science and Engineering, 4, 279-283. |
[11] | DAI Wen-Bin, ZHANG Wei-Jun, COWEN Taha, 2015: An analysis of River Derwent pollution and its impacts, Journal of Groundwater Science and Engineering, 3, 39-44. |
[12] | YANG Yun, WU Jian-feng, LIU De-peng, 2015: Numerical modeling of water yield of mine in Yangzhuang Iron Mine, Anhui Province of China, Journal of Groundwater Science and Engineering, 3, 352-362. |
[13] | CUI Qiu-ping, LIU Zhi-gang, LIU Li-jun, ZHANG Shao-cai, CHEN Yun-qian, 2014: Emergency water supply capacity analysis of major cities in Hebei, Journal of Groundwater Science and Engineering, 2, 76-86. |
[14] | HE Hong, GUO Hong-bin, LIU Hong-yun, 2014: Analysis of effect of water construction in different phases on groundwater environment, Journal of Groundwater Science and Engineering, 2, 54-59. |
[15] | SHI Jian-sheng, LIU Chang-li, DONG Hua, YAN Zhen-peng, WANG Yan-jun, LIU Xin-hao, GUO Xiu-yan, JIAO Hong-jun, YIN Mi-ying, HOU Huai-ren, 2014: Stability assessment and risk analysis of aboveground river in lower Yellow River, Journal of Groundwater Science and Engineering, 2, 1-18. |
[16] | GU Ming-xu, LIU Yu, HAN Chong, SHANG Lin-qun, JIANG Xian-qiao, WANG Lin-ying, 2014: Analysis of impact of outfalls on surrounding soil and groundwater environment, Journal of Groundwater Science and Engineering, 2, 54-60. |
[17] | BAI Xi-qing, LIU Yan, 2014: Feasibility Analysis on Resuming Flow of Large Karst Spring in Heilongdong, Journal of Groundwater Science and Engineering, 2, 80-87. |
[18] | , 2013: Analysis of Groundwater Environmental Conditions and Influencing Factors in Typical City in Northwest China, Journal of Groundwater Science and Engineering, 1, 60-73. |
[19] | Cheng Yanpei, Ma Renhui, 2013: Analysis of Water Resource Demands: Based on the Hydrological Unit, Journal of Groundwater Science and Engineering, 1, 48-59. |
[20] | Zong-jun Gao, Yong-gui Liu, 2013: Groundwater Flow Driven by Heat, Journal of Groundwater Science and Engineering, 1, 22-27. |
1. | Xiao, L., Chen, B., Shi, P. et al. An analytical solution for non-Darcian flow induced by variable-rate pumping in a radially heterogeneous confined aquifer | [径向非均质承压含水层中变速抽水引起的非达西流的解析解]. Hydrogeology Journal, 2024, 32(7): 1873-1886. doi:10.1007/s10040-024-02841-8 | |
2. | Shi, P., Xiao, L., Mei, G. et al. Investigation of the chloride ion transport mechanism in unsaturated concrete considering the nonlinear seepage effect. Construction and Building Materials, 2024. doi:10.1016/j.conbuildmat.2024.135383 |
JGSE-ScholarOne Manuscript Launched on June 1, 2024.