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New analytical model for constant-head pumping: Considering rate-dependent factor at well screen
Journal Pre-proofs Research papers New Analytical Model for Constant-Head Pumping: Considering Rate-Dependent Factor at Well Screen Ye-Chen Lin, Hund-Der Yeh PII: DOI: Reference:
S0022-1694(19)31130-8 https://doi.org/10.1016/j.jhydrol.2019.124395 HYDROL 124395
To appear in:
Journal of Hydrology
Received Date: Revised Date: Accepted Date:
24 September 2019 20 November 2019 22 November 2019
Please cite this article as: Lin, Y-C., Yeh, H-D., New Analytical Model for Constant-Head Pumping: Considering Rate-Dependent Factor at Well Screen, Journal of Hydrology (2019), doi: https://doi.org/10.1016/j.jhydrol. 2019.124395
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© 2019 Published by Elsevier B.V.
New Analytical Model for Constant-Head Pumping: Considering Rate-Dependent Factor at Well Screen
Ye-Chen Lin1, Hund-Der Yeh1*
1Institute
of Environmental Engineering, National Chiao Tung University, Taiwan
Resubmitted to Journal of Hydrology as a research article on November 20, 2019 ___________________________________________________________ *Corresponding author. Hund-Der Yeh, Email: [email protected]
1
1
ABSTRACT
2
Non-Darcy flow due to pumping may occur at the well screen and yields some head loss. Unlike the wellbore skin yielding a constant head loss at the well, the studies in petroleum engineering indicate that the head loss due to the non-Darcy flow is proportional to the pumping rate. The head loss due to non-Darcy flow is quantified by a rate-dependent factor at the well screen. To our knowledge, such a head loss is never considered in the area of groundwater hydraulics. In light of this, this study develops a groundwater flow model for the constant-head pumping with a head loss expressed in terms of a rate-dependent factor at a fully penetrating well in a confined aquifer. A skin factor representing the head loss due to the presence of wellbore skin is also incorporated into the model. The consideration of the rate-dependent factor, however, would make the model nonlinear. The model has a steady-state flow equation with a time-dependent finite outer boundary to approximate the transient groundwater flow in the aquifer system. The solutions to the model for the drawdown and wellbore flowrate are developed and compared with the finite difference solutions. With the help of the present solutions, the sensitivity analyses of the aquifer drawdown and wellbore flowrate are made. The effects of the wellbore skin and non-Darcy flow on the drawdown are also investigated. The wellbore flowrate solution is then applied to estimate the aquifer parameters based on the field data from a constant-head injection test in a confined aquifer. The prediction of the present solution gives the best fit to the 3
field measured data as compared with three existing solutions, indicating that the consideration of the rate-dependent factor improves the aquifer parameter estimation. The results of the parameter estimation also reflect the quality of the well installation. Keywords: constant-head pumping, fully penetrating well, skin factor, rate-dependent factor, non-Darcy flow.
1. Introduction The test for constant-head pumping (CHP) is commonly performed to estimate the aquifer parameters (i.e., hydraulic conductivity and specific storage) for lowpermeability aquifers in engineering practices. The method of CHP can avoid overdewatering problems in a shallow aquifer (Hiller and Levy, 1994) or overdrawing the pumping well in low-permeability formations (Jones, 1993; Ciftci, 2018). The CHP is usually carried out by maintaining a constant water level or drawdown in the test well and the wellbore flowrate and the drawdown at the observation well are recorded continuously and simultaneously. The aquifer parameters can then be determined by a type-curve match approach or by an analytical drawdown model coupled with an optimization method to analyze the recorded data. In the past, many studies addressed the issue involved the CHP at the aquifer with various configurations. For instance, Jacob and Lohman (1952) firstly presented a formula in analogy to Smith’s (1937) work for heat conduction to describe the wellbore flowrate due to CHP in a confined aquifer. 4
Hantush (1964) obtained a drawdown solution for the CHP in confined and leaky aquifer systems. Later, Mishra and Guyonnet (1992) employed the Boltzmann transform to obtain approximate solutions for both wellbore flowrate and drawdown in a confined aquifer. A problem similar to the CHP was also explored in the field of electrochemistry, which has the same mathematical formulation expressed in terms of a radial diffusion equation. Fang et al. (2009) proposed an approximate solution for chronoamperometric current at a microcylinder electrode. They solved the model from a steady-state diffusion equation with a finite outer boundary represented by a timedependent function called the diffusion boundary layer. In analogy with Fang et al.’s (2009) work, Yang et al. (2014) proposed a solution for the CHP by coupling two steady-state flow equations to describe the hydraulic head distribution in a confined aquifer. In their model, an averaging hydraulic head along the screened part and zeroflux along the unscreened part are specified as the inner boundary condition and a timedependent function in analogy to Fang et al.’s (2009) work is employed as the outer boundary condition. A region around the wellbore having different hydraulic properties from the aquifer may be presented due to the well drilling, development, and/or completion. The region, known as wellbore skin, would lead to a head loss during the pumping. The wellbore skin sometimes can be depicted as a zone with a finite thickness near the 5
pumping well (Barker and Herbert, 1982; Novakowski, 1989; Yang and Yeh, 2005; Barua and Bora, 2010; Lin et al., 2016; Jia et al., 2017; Wang et al., 2018). The properties of the wellbore skin are commonly lumped as a parameter called “skin factor” and included in the inner boundary condition. Many studies have been devoted to the model incorporating the skin factor for estimating the aquifer parameters subject to the constant rate pumping (CRP) (e.g., van Everdingen and Hurst, 1949; van Everdingen, 1953; Hawkins, 1956; Agarwal et al., 1970; Moench, 1997; Chen and Chang, 2006), slug test (e.g., Moench and Hsien, 1985), and CHP (e.g., Cassiani et al., 1999; Chang and Chen 2003; Lin et al., 2017). van Everdingen and Hurst (1949) indicated that the Darcian model usually leads to an underestimate of the pressure at the well. He therefore proposed a term with a skin factor in the model to reflect such a pressure drop. Hawkins (1956) introduced a skin factor in terms of the hydraulic conductivity ratio and thickness ratio of the formation zone to the skin zone to represent the skin effect. Hurst et al. (1969) derived a wellbore flowrate solution for a flowing well problem by adding a skin factor at the inner boundary in a manner similar to that described by van Everdingen and Hurst (1949). Moench (1997) developed a skin factor model for the CRP at a finite-radius well in an unconfined aquifer. He assumed that the water flux across the pumping well is equal to the difference between the averaged hydraulic head within the wellbore and the averaged head in the skin divided by the skin thickness. 6
Chen and Chang (2006) explored the problem regarding the effect of non-uniform skin zone on the groundwater flow due to CRP in confined aquifers. Chen and Chang (2002) conducted two constant-head injection tests and analyzed the measured cumulative volume data of both tests using a large-time solution developed with a skin factor accounting for the skin effect. In petroleum engineering, the skin factor determined from a build-up test or pressure drawdown test commonly display a linear relation with the pumping rate (e.g., Smith, 1961; Ramey, 1965; Meehan and Schell, 1983; Gunaydin et al., 2007; Gringarten et al., 2011; Beidokhti et al., 2011; Mohamed et al., 2016; Al-Rbeawi, 2016). Such a rate-dependent effect may be attributed to the problem of non-Darcy flow at the well screen (Smith, 1961). To reflect this effect, Ramey (1965) firstly introduced a ratedependent factor expressed as a product of a non-Darcy flow constant and the wellbore flowrate. Further, Ramey (1982) demonstrated that combining van Everdingen and Hurst’s (1949) skin factor solution and the rate-dependent factor has an equivalent form to the well loss equation proposed by Jacob (1946). Beidokhti et al. (2011) derived the rate-dependent factor from the Forchheimer equation. The result shows a similar form to Ramey’s (1965) one. The factor is widely applied to pressure models in petroleum engineering; contrarily, the use of this factor for non-Darcy flow at the well screen is overlooked in the field of groundwater hydraulics. Instead, the Forchheimer and Izbash 7
laws commonly used in the groundwater area are applied along with the conservation of mass to develop the groundwater flow equation (see, e.g., Mathias et al., 2008; Wen et al., 2013; Wang et al., 2014; Liu et al., 2017). Those approaches implying the flow in the entire aquifer system under the non-Darcy flow condition may not be suitable to depict the pump-induced flow because the flow at a farther distance from the pumping well usually complies with the Darcy law. Commonly, the water from the aquifer formation flowing through the gravel pack and wellbore screen is under non-Darcian condition. Shekhar (2006) mentioned that the flow near the well screen has the turbulence nature even when the water is pumped with a rate lower than a critical limit to avoid the turbulent flow. Hence, the non-Darcy flow effect may play a role influencing the groundwater flow near the wellbore. Motivated by this problem, the objective of this article is to develop a groundwater flow model with considering the non-Darcy flow effect at the inner boundary for the CHP problems. The use of rate-dependent factor for the non-Darcy flow effect however would result in a nonlinear model, which cannot be solved by the integral transform method. Notice that the approach proposed by Yang et al. (2014) and Fang et al. (2009) for the time-dependent outer boundary can solve the present model without using the integral transform method. We herein propose a transient flow model for CHP at a fully penetrating well composed of a steady-state 8
flow equation with an inner boundary condition having skin and rate-dependent factors and a time-dependent outer boundary condition under zero-drawdown. The drawdown solution to the model is verified by comparing with the finite difference solutions. Then, the present solution is used to examine the effects of the wellbore skin and non-Darcy flow on the drawdown distribution. Further, we perform the sensitivity analysis to explore the aquifer drawdown and wellbore flowrate behavior in response to the change in each of the hydraulic parameters. Finally, the present solution coupled with the leastsquares approach is used to determine the aquifer parameters when analyzing the recorded data obtained from a constant-head injection test reported in the article of Chen and Chang (2002). The quality of the well installation for these two test wells is also discussed based on the results of the parameter estimation.
2. Method 2.1 Model Description Figure 1 shows a schematic representation for the conceptual model depicting the aquifer drawdown distribution 𝑠(𝑟,𝑡) [L] induced by a CHP at a fully penetrating well in confined aquifers. The variable 𝑟 [L] represents the radial direction and 𝑡 [T] denotes the operating time. The well has a radius 𝑟𝑤 [L] and a constant drawdown 𝑠𝑤 [L]. The origin of the cylindrical coordinates is set at the center of the test well. The aquifer has the transmissivity denoted as 𝑇 [L2/T]. The water extracted by the CHP is 9
assumed instantaneously from the storativity 𝑆 [-] of the aquifer. The groundwater flow equation describing the drawdown distribution in an aquifer can be expressed as 1∂
( ∂𝑠)
∂𝑠
(1)
𝑇𝑟∂𝑟 𝑟∂𝑟 = 𝑆∂𝑡
The initial condition is (2)
𝑠 = 0, 𝑡 = 0
The head loss at the interface between the aquifer and the wellbore may be expressed as ∂𝑠
(3a)
𝑠 ― 𝑟𝑤𝑆𝑒∂𝑟 = 𝑠𝑤, 𝑟 = 𝑟𝑤
where 𝑆𝑒 [-] is called the effective skin factor. The first term on the left-hand side (LHS) of Eq. (3a) represents the aquifer drawdown, the second LHS term denotes as the additional head loss generated around the well screen, and the term on the righthand side is the wellbore drawdown. According to Ramey’s (1965) work, 𝑆𝑒 can be further defined as 𝑆𝑓 +𝐷𝑄 in which 𝑆𝑓 [-] is a skin factor due to the drilling damage adjacent to the wellbore and 𝐷𝑄 represents a rate-dependent factor due to the nonDarcy flow effect at the well screen with a constant 𝐷 [T/L3] and wellbore flowrate 𝑄 [L3/T]. The variable 𝑄 is equal to ―2𝜋𝑟𝑤𝑇∂𝑠/∂𝑟 at 𝑟 = 𝑟𝑤. Thus, Eq. (3a) can be written as
(
)
∂𝑠 ∂𝑠 ∂𝑟
𝑠 ― 𝑟𝑤 𝑆𝑓 ― 2𝜋𝑟𝑤𝑇𝐷∂𝑟
(3b)
= 𝑠𝑤, 𝑟 = 𝑟𝑤 10
If the non-Darcy effect is negligible, i.e., 𝐷 = 0, Eq. (3b) reduces to ∂𝑠
(3c)
𝑠 ― 𝑟𝑤𝑆𝑓∂𝑟 = 𝑠𝑤, 𝑟 = 𝑟𝑤
which is commonly adopted in the existing groundwater models for the CHP problems (e.g., Chen and Chang, 2003, 2006; Lin et al., 2017). The outer boundary condition can be written as (4)
𝑠 = 0, 𝑟→∞ 2.2 Similarity to Well Loss Function
Jacob (1946) proposed a well loss formula for describing the wellbore drawdown for the step-drawdown test; he gave 𝐵𝑄 + 𝐶𝑄2 = 𝑠𝑤, 𝑟 = 𝑟𝑤
(5a)
where 𝐵 [T/L2] and 𝐶 [T2/L5] are linear and nonlinear well-loss coefficients, respectively. Kruseman and de Ridder (1994) defined that 𝐵 is the sum of linear aquifer-loss coefficient 𝐵1 [T/L2] and linear well-loss coefficient 𝐵2 [T/L2]. The coefficient 𝐵1 is to reflect the aquifer head loss due to pumping and 𝐵1𝑄 is the aquifer drawdown s. The coefficient 𝐵2 is to represent the linear head loss due to the flow across the residual mud, gravel pack, and/or the screen entrance; and the coefficient 𝐶 is to denote the nonlinear loss due to the turbulent flow at the well screen. Thus, Eq. (5a) can be arranged as 𝑠 + 𝐵2𝑄 + 𝐶𝑄2 = 𝑠𝑤, 𝑟 = 𝑟𝑤
(5b) 11
Substituting 𝑄 = ―2𝜋𝑟𝑤𝑇∂𝑠/∂𝑟 into Eq. (5b) yields
(
)
∂𝑠 ∂𝑠 ∂𝑟
𝑠 ― 𝑟𝑤 2𝜋𝑇𝐵2 ― 4𝜋2𝑇2𝑟𝑤𝐶∂𝑟
= 𝑠𝑤, 𝑟 = 𝑟𝑤
(5c)
Comparing Eq. (3b) with Eq. (5c), the relations of parameters in both equations can be found as (6a)
𝑆𝑓 = 2𝜋𝑇𝐵2 and
(6b)
𝐷 = 2𝜋𝑇𝐶 These findings have been pointed out by Ramey (1982). 2.3 Dimensionless Equations
The use of the dimensionless parameters can reduce the number of variables in the system for the analysis. Herein, we define the dimensionless parameters as 𝑠𝐷 = 𝑠/𝑠𝑤, 𝑟𝐷 = 𝑟/𝑟𝑤, 𝑟𝑠𝐷 = 𝑟𝑠/𝑟𝑤, 𝑡𝐷 = 𝑇𝑡/𝑆𝑟2𝑤, 𝑞𝐷 = 𝑄/2𝜋𝑠𝑤𝑇, 𝐷 ∗ = 2𝜋𝑠𝑤𝑇𝐷 (7) The dimensionless form for Eq. (1) can be written as 1 ∂ 𝑟𝐷∂𝑟𝐷
(𝑟 ) = ∂𝑠𝐷 𝐷∂𝑟𝐷
∂𝑠𝐷
(8)
∂𝑡𝐷
The dimensionless initial condition is (9)
𝑠𝐷 = 0, 𝑡𝐷 = 0 The dimensionless inner boundary condition, Eqs (3b), is ∂𝑠𝐷 ∂𝑠𝐷
𝑠𝐷 ―(𝑆𝑓 ― 𝐷 ∗ ∂𝑟𝐷)∂𝑟𝐷 = 1, 𝑟𝐷 = 1
(10)
12
The outer boundary condition, Eq. (4), in a dimensionless form is (11)
𝑠𝐷 = 0,𝑟𝐷→∞ 2.4 Steady-State Model with Time-Dependent Outer Boundary
It should be noted that Eq. (10) is nonlinear such that the model cannot be solved by using the method of integral transforms. Alternatively, we adopt the approach of Yang et al. (2014) by defining a steady-state dimensionless flow equation as 1 d 𝑟𝐷d𝑟𝐷
(𝑟 ) = 0 d𝑠𝐷 𝐷d𝑟𝐷
(12)
Furthermore, Eq. (11) is replaced by a transient boundary condition specified at a finite distance from the pumping well as (13)
𝑠𝐷 = 0,𝑟𝐷 = 𝑅𝐷
where 𝑅𝐷 is defined as 1 + 𝜋𝑡𝐷 (Yang et al., 2014). Notice that the initial condition of Eq. (9) becomes redundant in solving Eq. (12). Eq. (12) can be written as d2𝑠𝐷 d𝑟𝐷2
1 d𝑠𝐷
(14)
+ 𝑟𝐷d𝑟𝐷 = 0
It has a general solution denoted as (15)
𝑠𝐷 = 𝐴1 + 𝐴2ln (𝑟𝐷) where 𝐴1 and 𝐴2 are undetermined coefficients and ln ( ∙ ) is the natural
logarithm. With the boundary conditions, Eqs. (10) and (13), the coefficients can be determined as 𝐴1 = ―
(ln (𝑅𝐷) + 𝑆𝑓 ±
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2) 2𝐷 ∗
ln (𝑅𝐷) 13
(16a)
and 𝐴2 =
(ln (𝑅𝐷) + 𝑆𝑓 ±
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2)
(16b)
2𝐷 ∗
Substituting Eqs. (16a) and (16b) into (15), the dimensionless aquifer drawdown solution is then obtained as 𝑠𝐷 = ―
(ln (𝑅𝐷) + 𝑆𝑓 ±
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2) 2𝐷
∗
ln
( ) 𝑅𝐷 𝑟𝐷
(17)
The symbol ± indicates that there are two solutions to the model. To ensure Eq. (17) having a positive value, Eq. (17) should be 𝑠𝐷 = ―
(ln (𝑅𝐷) + 𝑆𝑓 ―
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2) 2𝐷
ln
∗
( ) 𝑅𝐷 𝑟𝐷
(18)
The dimensionless wellbore flowrate is obtained by applying Darcy’s law as 𝑞𝐷 = ―
2 𝑆𝑓 + ln (𝑅𝐷) ― 4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)
(19)
2𝐷 ∗
2.5 Special Cases If the skin factor 𝑆𝑓 is negligible, Eqs. (18) and (19) can be respectively reduced to 𝑠𝐷 = ―
(ln (𝑅𝐷) ―
4𝐷 ∗ + ln (𝑅𝐷)2) 2𝐷
∗
ln
( ) 𝑅𝐷
(20)
𝑟𝐷
and 𝑞𝐷 = ―
ln (𝑅𝐷) ― 4𝐷 ∗ + ln (𝑅𝐷)2
(21)
2𝐷 ∗
If 𝐷 ∗ = 0, Eq. (18) reduces to ln (𝑅𝐷/𝑟𝐷)
(22)
𝑠𝐷 = ln (𝑅𝐷) + 𝑆𝑓 and Eq. (19) becomes 14
―1
(23)
𝑞𝐷 = ln (𝑅𝐷) + 𝑆𝑓
Moreover, if the effective skin factor exerts an insignificant effect on the flow system, Eqs. (18) and (19) may be respectively simplified as 𝑠𝐷 =
ln (𝑅𝐷/𝑟𝐷)
(24)
ln (𝑅𝐷)
and ―1
(25)
𝑞𝐷 = ln (𝑅𝐷)
Note that Eqs. (24) and (25) are the same as the equations given in Yang et al. (2014, Eqs. (35) and (36)).
3. Results and discussion Consider a CHP conducted in a confined aquifer with the following parameter values 𝑟𝑤 = 0.05 m, 𝑠𝑤 = 10 m, 𝑇 = 1 × 10 ―3 m2/min, 𝑆 = 1 × 10 ―5, 𝑆𝑓 = 1, and 𝐷 = 50 min/m3. The following sections are analyzed based on these values. 3.1 Verification of Present Solution To verify the present aquifer drawdown solution, an implicit finite difference solution is developed based on the transient groundwater model composed of Eqs. (8) – (10) using the Mathematica function NDSolve. The remote boundary condition for the finite difference model is set at 𝑟𝐷 = 105. The grid size in radial direction is discretized as unit and the time increment is also unit. Figure 2 displays (a) temporal and (b) spatial dimensionless aquifer drawdown curves predicted by the present solution and finite 15
difference solution when 𝐷 ∗ equals 0.1, 1, and 10 with 𝑆𝑓 = 1. Ramey (1965) mentioned that the non-Darcy flow profoundly affects the flow near the wellbore at the early time; thus, 𝑟𝐷 is set as one (at the screen) in Figure 2(a) and 𝑡𝐷 is 100 (3 sec) in Figure 2(b) to emphasize its effect. Figure 2(a) indicates that the curves predicted by the present solution gives a fairly good match with the finite difference solution within 𝑡𝐷 = 5 × 107 (around 21 h). Figure 2(b) demonstrates that the present solution gives good aquifer drawdown predictions except at the observation point close to the timedependent outer boundary, i.e., 1 + 𝜋𝑡𝐷. Overall, the results indicate that the present solution is well developed. 3.2 Sensitivity Analysis Sensitivity analysis is a useful tool to examine the influence of each aquifer parameter on the aquifer drawdown or wellbore flowrate. The normalized sensitivity defined by Liou and Yeh (1997) gives ∂𝑂
(26)
𝑋𝑘 = 𝑃𝑘∂𝑃𝑘
where 𝑋𝑘 is the normalized sensitivity of the output 𝑂 (the aquifer drawdown or wellbore flowrate) to the 𝑘𝑡ℎ input parameter 𝑃𝑘. Eq. (26) can then be approximated by the forward finite difference formula as 𝑂(𝑃𝑘 + ∆𝑃𝑘) ― 𝑂(𝑃𝑘)
𝑋𝑘 = 𝑃 𝑘
(27)
∆𝑃𝑘
where ∆𝑃𝑘 is set to 10-3 𝑃𝑘 (Yeh and Han, 1989). 16
Figure 3 demonstrates the temporal normalized sensitivity curves for (a) the aquifer drawdown 𝑠 observed at 𝑟 = 𝑟𝑤 and (b) the wellbore flowrate 𝑄. Note that the positive value of the normalized sensitivity indicates the increment of the specific parameter leading to an increase in 𝑠 or 𝑄 at a given time. On the other hand, the negative value of that implies that the increasing parameter value would result in a decrease in 𝑠 or 𝑄. Figure 3 (a) indicates that 𝑆 is the most sensitive parameter before the pumping time at 30 sec according to the absolute value of the normalized sensitivity. After that time, 𝑆𝑓 has the largest influence on the aquifer drawdown, followed by 𝑇 and 𝐷. This is contrary to our expectation that 𝑇 should be the most sensitive parameter. The reason may be that the non-Darcy flow effect, which delays the water flow propagation, reduces the influence of the parameter 𝑇 on the aquifer drawdown. One can observe that 𝑇 and 𝑆 decreasingly affect the aquifer drawdown when operating time elapses; however, 𝑆𝑓 and 𝐷 constantly exert certain effects on the aquifer drawdown with increasing time. Figure 3(b) shows that 𝑇 has the largest sensitivity values fallowed by 𝑆𝑓, 𝑆, and 𝐷 for the case of 𝑄. This result is different from that given in Figure 3 (a) because the wellbore flowrate is mainly dependent on 𝑇 (i.e., 𝑄 = ―2𝜋𝑟𝑤𝑇∂𝑠/∂𝑟). Notably, 𝑆 and 𝐷 are relatively insensitive parameters compared with others but they still produce some effects on the wellbore flowrate. Judged from these results, the parameters appear to have varying degrees of 17
influence on the aquifer drawdown and wellbore flowrate. Therefore, analyzing the data set from the aquifer drawdown or wellbore flowrate would reveal different impacts of the aquifer parameters. 3.3 Effect of Skin Factor and Rate-Dependent Factor In Eq. (10), the head loss due to wellbore skin is described by ― 𝑆𝑓∂𝑠𝐷/∂𝑟𝐷 while that due to the non-Darcy flow effect at the well screen is represented by 𝐷 ∗
(∂𝑠𝐷/∂𝑟𝐷)2. Obviously, the head loss at the well increases as the coefficient 𝑆𝑓 or 𝐷 ∗ increases. Figure 4 shows the temporal dimensionless aquifer drawdown curves for (a) 𝑆𝑓 varying from 0.01 to 100 with 𝐷 ∗ = 1 and (b) 𝐷 ∗ from 0.01 to 100 with 𝑆𝑓 = 1 observed at the rim of wellbore, i.e., 𝑟𝐷 = 1. The figure shows that the aquifer drawdown curves decrease with increasing 𝑆𝑓 and 𝐷 ∗ because more head loss occurs at the well screen. The aquifer drawdown is sensitive to the change in 𝑆𝑓 than in 𝐷 ∗ because the loss value due to 𝐷 ∗ (∂𝑠𝐷/∂𝑟𝐷)2 yields a smaller value than that due to ― 𝑆𝑓∂𝑠𝐷/∂𝑟𝐷. It helps account for the outcome of the sensitivity analysis in the previous section that why the parameter 𝐷 always yields a smaller impact than 𝑆𝑓. Additionally, the aquifer drawdown curves for 𝐷 ∗ = 0.01 and 0.1 almost coincide with each other indicating that 𝐷 ∗ exerts an insignificant effect on the aquifer drawdown when its value is less than 0.1. 3.4 Field Data Analyses 18
Chen and Chang (2002) conducted two constant-head injection tests, one was at well 2 (referred to as the CHIT1 in their study) and the other at well 5 (CHIT2). Both wells have a 0.05 m radius and fully penetrate a silty sand confined aquifer of 10-m thickness at Taoyuan Tableland in the northern part of Taiwan. The test at the CHIT1 ran 58 hours with the 𝑠𝑤 kept at 10.1 m while the CHIT2 ran 67 hours with 𝑠𝑤 maintained at 13.48 m. They measured the cumulative volume in the injection well for parameter estimations. The formula they used to estimate the parameters is 𝑠𝑤𝑡 𝑉𝑤
=
(
)+
0.183 0.83𝑇𝑡 𝑇 log 𝑆𝑟2𝑤
𝑆𝑓
(28)
2𝜋𝑇 𝑡
where 𝑉𝑤 is the cumulative volume defined as ∫0𝑄 (𝑥)𝑑𝑥 which can be easily evaluated by a numerical integral function in Mathematica called NIntegrate (Wolfram Research Inc., 2018). The aquifer parameters were estimated by graphically fitting the measured data in a semi-log plot of 𝑠𝑤𝑡/𝑉𝑤 versus time. Chen and Chang (2002) pointed out that the measured data displayed two different portions in early-time and late-time periods. One can expect that the late-time portion can reflect the feature of the flow behavior in the formation zone and the well losses; therefore, the data points from the CHIT1 and CHIT2 after around 100 min were used to analyze for the aquifer parameters. They firstly determined the value of 𝑇 as 1.68 × 10 ―3 m2/min by fitting the late-time data from the CHIT1 and CHIT2. Later, they determined the value of 𝑆 as 2.52 × 10 ―4 by assuming the absent of the wellbore skin in the CHIT2. With the 19
obtained 𝑇 and 𝑆, the skin factor 𝑆𝑓 was finally determined as 18.57 for the CHIT1. Their results of the parameter estimation are listed in Table 1. Following their work, the 𝑠𝑤𝑡/𝑉𝑤 data of CHIT1 and CHIT2 after 100 min were read from Chen and Chang (2002, Figure 4) by using WebPlotDigitizer (Rohatgi, 2018). We analyze these data using three different approaches. The first approach is to use Jacob and Lohman’s (1952) solution in which both skin factor and rate-dependent factor are not considered. The second one is to employ Hurst et al.’s (1969) solution in which only the skin factor is considered while the third is the present solution. Each of three solutions is coupled with the least-squares approach and solved by the Levenberg Marquardt algorithm in Mathematica routine NonlinearModelFit (Wolfram Research Inc., 2018) for the parameter estimation. Two statistics are chosen to assess the goodness of fit for different solutions with the estimated parameters. They are standard error of estimate (SEE) and mean error (ME) defined in Yeh (1987) and respectively expressed as 𝑆𝐸𝐸 =
1 𝑁 ∑ 𝜈 𝑖=1
|𝑒2𝑖 |
(29a)
and 1
𝑁
𝑀𝐸 = 𝑁∑𝑖 = 1𝑒
(29b)
𝑖
where 𝜈 is the degree of freedom defined as the number of observed data 𝑁 minus the number of unknown parameters 𝑀, i.e., 𝜈 = 𝑁 ― 𝑀, and 𝑒𝑖 is the 𝑖th prediction 20
error between the predicted 𝑠𝑤𝑡/𝑉𝑤 and observed data. The estimated results by three different solutions are also listed in Table 1. According to the estimated results of the CHIT1, the values of 𝑇 are obtained as 1.04 × 10 ―3 m2/min, 1.62 × 10 ―3 m2/min, and 1.61 × 10 ―3 m2/min by Jacob and Lohman’s (1952), Hurst et al.’s (1969), and present solutions, respectively. The estimate of 𝑇 obtained from the first solution has the smallest value compared to the others. That obtained from the second and the present solutions are rather close to that determined by Chen and Chang (2002). The first solution, however, gives the estimated 𝑆 unrealistically small. It may be attributed to the fact that the skin effect was not considered in Jacob and Lohman’s (1952) model. Additionally, the estimated values of 𝑆 by the second and present solutions are larger than that by Chen and Chang (2002) about an order of magnitude. The reason for that is Chen and Chang (2002) determined its value by analyzing the data set from the CHIT2. Based on the SEE and ME values shown in Table 1, we consider that the value of 𝑆 estimated by Chen and Chang (2002) cannot represent the estimated result from the CHIT1. The 𝑆𝑓 values estimated by Hurst et al.’s (1969) and present solutions confirms that the wellbore has a positive skin as indicated by Chen and Chang (2002). Table 1 indicates that the present solution yields the smallest SEE and ME values indicating that the consideration of both skin factor and rate-dependent factor gives the best fit to the data set of CHIT1. 21
The estimated results for data from CHIT2 indicate that those three solutions give the same estimates of 𝑇 (i.e., 1.64 × 10 ―3 m2/min) which are not far from Chen and Chang’s (2002) estimation. The values of 𝑆 estimated by all solutions are on the same order of magnitude. Besides, the estimates of 𝑆𝑓 obtained by Hurst et al.’s (1969) and present solutions and that of 𝐷 obtained by the present solution are smaller than those obtained from the CHIT1. Both effects of wellbore skin and non-Darcy flow are insignificant for data in the CHIT2. It is consistent with Chen and Chang’s (2002) statement that there are little or no wellbore skin existing around the well 5. Figure 5 shows the measured data of (a) CHIT1 and (b) CHIT2 and the temporal curves of 𝑠𝑤𝑡/ 𝑉𝑤 predicted by three solutions with the parameters listed in Table 1. It demonstrates that the curve predicted by the present solution matches well with the measured data while those by the other solutions show some significant departure from the measured data, especially for the CHIT1. It implies that the consideration of the rate-dependent factor improves the parameter estimation if non-Darcy flow effect is profound. 3.5 Performance of Test Wells Kurtulus et al. (2019) proposed a well efficiency criterion by analyzing the data of step-drawdown test from 290 wells coupled with Rorabaugh’s (1953) well loss function. Notice that Rorabaugh’s (1953) well loss function can be viewed as a modified version of Jacob’s (1946) one. That is 22
𝐵𝑄 + 𝐶𝑄𝑘 = 𝑠𝑤, 𝑟 = 𝑟𝑤
(30)
in which 𝑘 is the exponent parameter equal to 2 in this study. Kurtulus et al. (2019, Table 3) summarized the following that the well is in the deteriorating condition when 𝐶 > 1800 s2/m5 or 0.5 min2/m5 for 𝑘 within 1 and 2. Employing the estimated results, we obtain that 𝐶 equals 1163.51 min2/m5 for well 2 and 0.97 min2/m5 for well 5. These values indicate that both test wells were not properly designed or developed. It may be explained why the measured data from the CHIT1 and CHIT2 are merely capable of reflecting the feature of the flow behavior in the aquifer formation after the operating time up to 100 min (1.67 h).
4. Concluding remarks This study presents a new mathematical model with considering both skin factor and rate-dependent factor at the pumping well for describing aquifer drawdown distribution and wellbore flowrate induced by a CHP at a fully penetrating well in confined aquifers. The inner boundary condition is nonlinear due to the consideration of the rate-dependent factor. In light of this, we adopt the work of Yang et al. (2014) to formula the model comprised of a steady-state groundwater flow equation with a timedependent finite outer boundary. The solutions to the model for the wellbore flowrate and the drawdown can then be derived. The comparison of the aquifer drawdown predicted by the present and finite difference solutions indicates that the present 23
solution is well developed. The results of sensitivity analysis indicate that 𝑆 is the most sensitive parameter to the aquifer drawdown before the pumping time of 30 sec while 𝑇 is the most sensitive parameter for the wellbore flowrate. The non-Darcy flow constant 𝐷 exerts relatively small effect on both aquifer drawdown and wellbore flowrate as compared to other aquifer parameters. An increase in the skin factor or ratedependent factor has the effect of decreasing the drawdown and the flowrate values because the head losses are generated. The present solution is used to estimate the aquifer parameters when analyzing a measured data set from a field constant-head injection test. The results show that the present solution used in the parameter estimation gives the best-estimated results compared with those estimated by Jacob and Lohman solution (1952), and Hurst et al. (1969) solution as well as that determined by Chen and Chang (2002). According to these results, the consideration of the ratedependent factor improves the parameter estimation. Also, the results of the parameter estimation indicate that the test wells in Chen and Chang’s (2002) work may be not properly installed or developed. Overall, the present solutions can be used as a convenient tool to estimate the aquifer parameters and predict the aquifer drawdown and flowrate distributions.
Acknowledgments The research leading to this paper has been partially supported by the grant from 24
Taiwan Ministry of Science and Technology under the contract number MOST 1072221-E-009 -019 -MY3. References Agarwal, R. G., Al-Hussainy, R., Ramey Jr, H. J., 1970. An investigation of wellbore storage and skin effect in unsteady liquid flow: I. Analytical treatment. Society of Petroleum Engineers Journal, 10(03), 279-290. https://doi.org/10.2118/2466-PA. Al-Rbeawi, S., 2016. New technique for determining rate dependent skin factor & nonDarcy coefficient. Journal of Natural Gas Science and Engineering, 35, 1044-1058. https://doi.org/10.1016/j.jngse.2016.09.028. Barker, J. A., Herbert, R., 1982. Pumping tests in patchy aquifers, Ground Water, 20(2), 150-155. https://doi.org/10.1111/j.1745-6584.1982.tb02742.x. Barua, G., Bora, S. N., 2010. Hydraulics of a partially penetrating well with skin zone in a confined aquifer, Advances in Water Resources, 33, 1575-1587. https://doi.org/10.1016/j.advwatres.2010.09.008. Beidokhti, M., Arabjamaloei, R., Hashemi, A., Edalatkhah, S., Nabaei, M., Malak ooti, R., Jamshidi, E., 2011. Dealing with the challenges of the rate-dependent skin phenomenon in a gas condensate reservoir: a simulation approach. Petroleu m Science and Technology, 29(21), 2177-2190. https://doi.org/10.1080/1091646 1003610363. 25
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33
Figure captions Figure 1. Schematic representation of drawdown distribution induced by a constanthead pumping at fully penetrating well. Figure 2 (a) Temporal and (b) spatial drawdown curves predicted by the present and finite difference solutions with 𝐷 ∗ = 0.1, 1, and 10. Figure 3 The temporal distribution curves of normalized sensitivity for (a) aquifer drawdown and (b) wellbore flowrate. Figure 4 Temporal dimensionless drawdown distribution with (a) 𝑆𝑓 from 0.01 to 100, 𝐷 ∗ = 1 and (b) 𝐷 ∗ from 0.01 to 100, 𝑆𝑓 = 1 observed at the wellbore 𝑟𝐷 = 1. Figure 5 The measured data and the curves of swt/Vw predicted by the present solution and two existing solutions with estimated parameters for the tests (a) CHIT1 and (b) CHIT2.
34
Nomenclature Parameters and variables (𝑟, 𝑟𝑤, 𝑡)
Radial distance, well radius, and time
(𝑇, 𝑆)
Transmissivity and storativity of the aquifer
(𝑆𝑒, 𝑆𝑓, 𝐷)
Effective factor, skin factor, and non-Darcy flow constant
(𝑠, 𝑠𝑤)
Drawdown in the aquifer and the wellbore
𝑄(𝑡)
Wellbore flowrate
𝑉𝑤
Cumulative volume defined as ∫0𝑄 (𝑥)𝑑𝑥
𝑡
(𝐵,
𝐵1, Linear head loss, linear aquifer-loss, linear well-loss, and nonlinear well-
𝐵2, 𝐶)
𝑒𝑖)
loss coefficients
(𝑋, 𝑃, 𝑂)
Normalized sensitivity value, input parameter, and output
(𝑀, 𝑁, 𝜈,
The number of unknown parameters, the number of observed data, the degree of freedom, the 𝑖th prediction error
35
Table 1 Parameters estimated by Jacob and Lohman’s (1952), Hurst et al.’s (1969), and present solutions as well as that determined by Chen and Chang (2002) with the associated prediction errors Estimated
Jacob and Lohman (1952)
parameters
solution
Hurst et al. (1969) solution
Present solution
Chen and Chang (2002)
CHIT1 𝑇(m2/min)
1.04 × 10 ―3
1.62 × 10 ―3
1.61 × 10 ―3
1.68 × 10 ―3
𝑆(-)
3.93 × 10 ―12
8.37 × 10 ―5
1.40 × 10 ―5
2.52 × 10 ―4
𝑆𝑓(-)
―
16.91
15.90
18.57
𝐷
―
―
11.66
―
(min/m3) SEE
28.91
ME
9.40 × 10
10.45 ―2
4.33 × 10
4.61 ―2
3.16 × 10
11.77 ―3
―9.79
CHIT2 𝑇(m2/min)
1.64 × 10 ―3
1.64 × 10 ―3
1.64 × 10 ―3
1.68 × 10 ―3
𝑆(-)
3.55 × 10 ―4
4.16 × 10 ―4
4.21 × 10 ―4
2.52 × 10 ―4
𝑆𝑓(-)
―
0.07
0.08
0
𝐷
―
―
0.01
―
(min/m3) SEE
2.28
ME
1.25 × 10
2.26 ―3
―1.23 × 10
1.25 ―3
―1.01 × 10
2.91 ―3
1.39
Note. ME = mean error; SEE = standard error of estimate.
Highlights
Rate-dependent factor is adopted in a confined flow model for constant-head pumping This factor is to reflect the effect of non-Darcy flow at the pumping well screen
The model comprises a stationary flow equation with a time-dependent outer boundary
Sensitivity analyses are made to show the effect of aquifer parameters on the flow
The model is used to analyze field drawdown data and well installation quality
Ye-Chen Lin: Conceptualization, Methodology, Software, Formal analysis, 36
Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization. Hund-Der Yeh: Formal analysis, Writing - Review & Editing, Supervision.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
37
38
39
40
41
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
42
Ye-Chen Lin: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization. Hund-Der Yeh: Formal analysis, Writing - Review & Editing, Supervision.
43
Highlights
Rate-dependent factor is adopted in a confined flow model for constant-head pumping This factor is to reflect the effect of non-Darcy flow at the pumping well screen
The model comprises a stationary flow equation with a time-dependent outer boundary
Sensitivity analyses are made to show the effect of aquifer parameters on the flow
The model is used to analyze field drawdown data and well installation quality
44
S0022-1694(19)31130-8 https://doi.org/10.1016/j.jhydrol.2019.124395 HYDROL 124395
To appear in:
Journal of Hydrology
Received Date: Revised Date: Accepted Date:
24 September 2019 20 November 2019 22 November 2019
Please cite this article as: Lin, Y-C., Yeh, H-D., New Analytical Model for Constant-Head Pumping: Considering Rate-Dependent Factor at Well Screen, Journal of Hydrology (2019), doi: https://doi.org/10.1016/j.jhydrol. 2019.124395
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© 2019 Published by Elsevier B.V.
New Analytical Model for Constant-Head Pumping: Considering Rate-Dependent Factor at Well Screen
Ye-Chen Lin1, Hund-Der Yeh1*
1Institute
of Environmental Engineering, National Chiao Tung University, Taiwan
Resubmitted to Journal of Hydrology as a research article on November 20, 2019 ___________________________________________________________ *Corresponding author. Hund-Der Yeh, Email: [email protected]
1
1
ABSTRACT
2
Non-Darcy flow due to pumping may occur at the well screen and yields some head loss. Unlike the wellbore skin yielding a constant head loss at the well, the studies in petroleum engineering indicate that the head loss due to the non-Darcy flow is proportional to the pumping rate. The head loss due to non-Darcy flow is quantified by a rate-dependent factor at the well screen. To our knowledge, such a head loss is never considered in the area of groundwater hydraulics. In light of this, this study develops a groundwater flow model for the constant-head pumping with a head loss expressed in terms of a rate-dependent factor at a fully penetrating well in a confined aquifer. A skin factor representing the head loss due to the presence of wellbore skin is also incorporated into the model. The consideration of the rate-dependent factor, however, would make the model nonlinear. The model has a steady-state flow equation with a time-dependent finite outer boundary to approximate the transient groundwater flow in the aquifer system. The solutions to the model for the drawdown and wellbore flowrate are developed and compared with the finite difference solutions. With the help of the present solutions, the sensitivity analyses of the aquifer drawdown and wellbore flowrate are made. The effects of the wellbore skin and non-Darcy flow on the drawdown are also investigated. The wellbore flowrate solution is then applied to estimate the aquifer parameters based on the field data from a constant-head injection test in a confined aquifer. The prediction of the present solution gives the best fit to the 3
field measured data as compared with three existing solutions, indicating that the consideration of the rate-dependent factor improves the aquifer parameter estimation. The results of the parameter estimation also reflect the quality of the well installation. Keywords: constant-head pumping, fully penetrating well, skin factor, rate-dependent factor, non-Darcy flow.
1. Introduction The test for constant-head pumping (CHP) is commonly performed to estimate the aquifer parameters (i.e., hydraulic conductivity and specific storage) for lowpermeability aquifers in engineering practices. The method of CHP can avoid overdewatering problems in a shallow aquifer (Hiller and Levy, 1994) or overdrawing the pumping well in low-permeability formations (Jones, 1993; Ciftci, 2018). The CHP is usually carried out by maintaining a constant water level or drawdown in the test well and the wellbore flowrate and the drawdown at the observation well are recorded continuously and simultaneously. The aquifer parameters can then be determined by a type-curve match approach or by an analytical drawdown model coupled with an optimization method to analyze the recorded data. In the past, many studies addressed the issue involved the CHP at the aquifer with various configurations. For instance, Jacob and Lohman (1952) firstly presented a formula in analogy to Smith’s (1937) work for heat conduction to describe the wellbore flowrate due to CHP in a confined aquifer. 4
Hantush (1964) obtained a drawdown solution for the CHP in confined and leaky aquifer systems. Later, Mishra and Guyonnet (1992) employed the Boltzmann transform to obtain approximate solutions for both wellbore flowrate and drawdown in a confined aquifer. A problem similar to the CHP was also explored in the field of electrochemistry, which has the same mathematical formulation expressed in terms of a radial diffusion equation. Fang et al. (2009) proposed an approximate solution for chronoamperometric current at a microcylinder electrode. They solved the model from a steady-state diffusion equation with a finite outer boundary represented by a timedependent function called the diffusion boundary layer. In analogy with Fang et al.’s (2009) work, Yang et al. (2014) proposed a solution for the CHP by coupling two steady-state flow equations to describe the hydraulic head distribution in a confined aquifer. In their model, an averaging hydraulic head along the screened part and zeroflux along the unscreened part are specified as the inner boundary condition and a timedependent function in analogy to Fang et al.’s (2009) work is employed as the outer boundary condition. A region around the wellbore having different hydraulic properties from the aquifer may be presented due to the well drilling, development, and/or completion. The region, known as wellbore skin, would lead to a head loss during the pumping. The wellbore skin sometimes can be depicted as a zone with a finite thickness near the 5
pumping well (Barker and Herbert, 1982; Novakowski, 1989; Yang and Yeh, 2005; Barua and Bora, 2010; Lin et al., 2016; Jia et al., 2017; Wang et al., 2018). The properties of the wellbore skin are commonly lumped as a parameter called “skin factor” and included in the inner boundary condition. Many studies have been devoted to the model incorporating the skin factor for estimating the aquifer parameters subject to the constant rate pumping (CRP) (e.g., van Everdingen and Hurst, 1949; van Everdingen, 1953; Hawkins, 1956; Agarwal et al., 1970; Moench, 1997; Chen and Chang, 2006), slug test (e.g., Moench and Hsien, 1985), and CHP (e.g., Cassiani et al., 1999; Chang and Chen 2003; Lin et al., 2017). van Everdingen and Hurst (1949) indicated that the Darcian model usually leads to an underestimate of the pressure at the well. He therefore proposed a term with a skin factor in the model to reflect such a pressure drop. Hawkins (1956) introduced a skin factor in terms of the hydraulic conductivity ratio and thickness ratio of the formation zone to the skin zone to represent the skin effect. Hurst et al. (1969) derived a wellbore flowrate solution for a flowing well problem by adding a skin factor at the inner boundary in a manner similar to that described by van Everdingen and Hurst (1949). Moench (1997) developed a skin factor model for the CRP at a finite-radius well in an unconfined aquifer. He assumed that the water flux across the pumping well is equal to the difference between the averaged hydraulic head within the wellbore and the averaged head in the skin divided by the skin thickness. 6
Chen and Chang (2006) explored the problem regarding the effect of non-uniform skin zone on the groundwater flow due to CRP in confined aquifers. Chen and Chang (2002) conducted two constant-head injection tests and analyzed the measured cumulative volume data of both tests using a large-time solution developed with a skin factor accounting for the skin effect. In petroleum engineering, the skin factor determined from a build-up test or pressure drawdown test commonly display a linear relation with the pumping rate (e.g., Smith, 1961; Ramey, 1965; Meehan and Schell, 1983; Gunaydin et al., 2007; Gringarten et al., 2011; Beidokhti et al., 2011; Mohamed et al., 2016; Al-Rbeawi, 2016). Such a rate-dependent effect may be attributed to the problem of non-Darcy flow at the well screen (Smith, 1961). To reflect this effect, Ramey (1965) firstly introduced a ratedependent factor expressed as a product of a non-Darcy flow constant and the wellbore flowrate. Further, Ramey (1982) demonstrated that combining van Everdingen and Hurst’s (1949) skin factor solution and the rate-dependent factor has an equivalent form to the well loss equation proposed by Jacob (1946). Beidokhti et al. (2011) derived the rate-dependent factor from the Forchheimer equation. The result shows a similar form to Ramey’s (1965) one. The factor is widely applied to pressure models in petroleum engineering; contrarily, the use of this factor for non-Darcy flow at the well screen is overlooked in the field of groundwater hydraulics. Instead, the Forchheimer and Izbash 7
laws commonly used in the groundwater area are applied along with the conservation of mass to develop the groundwater flow equation (see, e.g., Mathias et al., 2008; Wen et al., 2013; Wang et al., 2014; Liu et al., 2017). Those approaches implying the flow in the entire aquifer system under the non-Darcy flow condition may not be suitable to depict the pump-induced flow because the flow at a farther distance from the pumping well usually complies with the Darcy law. Commonly, the water from the aquifer formation flowing through the gravel pack and wellbore screen is under non-Darcian condition. Shekhar (2006) mentioned that the flow near the well screen has the turbulence nature even when the water is pumped with a rate lower than a critical limit to avoid the turbulent flow. Hence, the non-Darcy flow effect may play a role influencing the groundwater flow near the wellbore. Motivated by this problem, the objective of this article is to develop a groundwater flow model with considering the non-Darcy flow effect at the inner boundary for the CHP problems. The use of rate-dependent factor for the non-Darcy flow effect however would result in a nonlinear model, which cannot be solved by the integral transform method. Notice that the approach proposed by Yang et al. (2014) and Fang et al. (2009) for the time-dependent outer boundary can solve the present model without using the integral transform method. We herein propose a transient flow model for CHP at a fully penetrating well composed of a steady-state 8
flow equation with an inner boundary condition having skin and rate-dependent factors and a time-dependent outer boundary condition under zero-drawdown. The drawdown solution to the model is verified by comparing with the finite difference solutions. Then, the present solution is used to examine the effects of the wellbore skin and non-Darcy flow on the drawdown distribution. Further, we perform the sensitivity analysis to explore the aquifer drawdown and wellbore flowrate behavior in response to the change in each of the hydraulic parameters. Finally, the present solution coupled with the leastsquares approach is used to determine the aquifer parameters when analyzing the recorded data obtained from a constant-head injection test reported in the article of Chen and Chang (2002). The quality of the well installation for these two test wells is also discussed based on the results of the parameter estimation.
2. Method 2.1 Model Description Figure 1 shows a schematic representation for the conceptual model depicting the aquifer drawdown distribution 𝑠(𝑟,𝑡) [L] induced by a CHP at a fully penetrating well in confined aquifers. The variable 𝑟 [L] represents the radial direction and 𝑡 [T] denotes the operating time. The well has a radius 𝑟𝑤 [L] and a constant drawdown 𝑠𝑤 [L]. The origin of the cylindrical coordinates is set at the center of the test well. The aquifer has the transmissivity denoted as 𝑇 [L2/T]. The water extracted by the CHP is 9
assumed instantaneously from the storativity 𝑆 [-] of the aquifer. The groundwater flow equation describing the drawdown distribution in an aquifer can be expressed as 1∂
( ∂𝑠)
∂𝑠
(1)
𝑇𝑟∂𝑟 𝑟∂𝑟 = 𝑆∂𝑡
The initial condition is (2)
𝑠 = 0, 𝑡 = 0
The head loss at the interface between the aquifer and the wellbore may be expressed as ∂𝑠
(3a)
𝑠 ― 𝑟𝑤𝑆𝑒∂𝑟 = 𝑠𝑤, 𝑟 = 𝑟𝑤
where 𝑆𝑒 [-] is called the effective skin factor. The first term on the left-hand side (LHS) of Eq. (3a) represents the aquifer drawdown, the second LHS term denotes as the additional head loss generated around the well screen, and the term on the righthand side is the wellbore drawdown. According to Ramey’s (1965) work, 𝑆𝑒 can be further defined as 𝑆𝑓 +𝐷𝑄 in which 𝑆𝑓 [-] is a skin factor due to the drilling damage adjacent to the wellbore and 𝐷𝑄 represents a rate-dependent factor due to the nonDarcy flow effect at the well screen with a constant 𝐷 [T/L3] and wellbore flowrate 𝑄 [L3/T]. The variable 𝑄 is equal to ―2𝜋𝑟𝑤𝑇∂𝑠/∂𝑟 at 𝑟 = 𝑟𝑤. Thus, Eq. (3a) can be written as
(
)
∂𝑠 ∂𝑠 ∂𝑟
𝑠 ― 𝑟𝑤 𝑆𝑓 ― 2𝜋𝑟𝑤𝑇𝐷∂𝑟
(3b)
= 𝑠𝑤, 𝑟 = 𝑟𝑤 10
If the non-Darcy effect is negligible, i.e., 𝐷 = 0, Eq. (3b) reduces to ∂𝑠
(3c)
𝑠 ― 𝑟𝑤𝑆𝑓∂𝑟 = 𝑠𝑤, 𝑟 = 𝑟𝑤
which is commonly adopted in the existing groundwater models for the CHP problems (e.g., Chen and Chang, 2003, 2006; Lin et al., 2017). The outer boundary condition can be written as (4)
𝑠 = 0, 𝑟→∞ 2.2 Similarity to Well Loss Function
Jacob (1946) proposed a well loss formula for describing the wellbore drawdown for the step-drawdown test; he gave 𝐵𝑄 + 𝐶𝑄2 = 𝑠𝑤, 𝑟 = 𝑟𝑤
(5a)
where 𝐵 [T/L2] and 𝐶 [T2/L5] are linear and nonlinear well-loss coefficients, respectively. Kruseman and de Ridder (1994) defined that 𝐵 is the sum of linear aquifer-loss coefficient 𝐵1 [T/L2] and linear well-loss coefficient 𝐵2 [T/L2]. The coefficient 𝐵1 is to reflect the aquifer head loss due to pumping and 𝐵1𝑄 is the aquifer drawdown s. The coefficient 𝐵2 is to represent the linear head loss due to the flow across the residual mud, gravel pack, and/or the screen entrance; and the coefficient 𝐶 is to denote the nonlinear loss due to the turbulent flow at the well screen. Thus, Eq. (5a) can be arranged as 𝑠 + 𝐵2𝑄 + 𝐶𝑄2 = 𝑠𝑤, 𝑟 = 𝑟𝑤
(5b) 11
Substituting 𝑄 = ―2𝜋𝑟𝑤𝑇∂𝑠/∂𝑟 into Eq. (5b) yields
(
)
∂𝑠 ∂𝑠 ∂𝑟
𝑠 ― 𝑟𝑤 2𝜋𝑇𝐵2 ― 4𝜋2𝑇2𝑟𝑤𝐶∂𝑟
= 𝑠𝑤, 𝑟 = 𝑟𝑤
(5c)
Comparing Eq. (3b) with Eq. (5c), the relations of parameters in both equations can be found as (6a)
𝑆𝑓 = 2𝜋𝑇𝐵2 and
(6b)
𝐷 = 2𝜋𝑇𝐶 These findings have been pointed out by Ramey (1982). 2.3 Dimensionless Equations
The use of the dimensionless parameters can reduce the number of variables in the system for the analysis. Herein, we define the dimensionless parameters as 𝑠𝐷 = 𝑠/𝑠𝑤, 𝑟𝐷 = 𝑟/𝑟𝑤, 𝑟𝑠𝐷 = 𝑟𝑠/𝑟𝑤, 𝑡𝐷 = 𝑇𝑡/𝑆𝑟2𝑤, 𝑞𝐷 = 𝑄/2𝜋𝑠𝑤𝑇, 𝐷 ∗ = 2𝜋𝑠𝑤𝑇𝐷 (7) The dimensionless form for Eq. (1) can be written as 1 ∂ 𝑟𝐷∂𝑟𝐷
(𝑟 ) = ∂𝑠𝐷 𝐷∂𝑟𝐷
∂𝑠𝐷
(8)
∂𝑡𝐷
The dimensionless initial condition is (9)
𝑠𝐷 = 0, 𝑡𝐷 = 0 The dimensionless inner boundary condition, Eqs (3b), is ∂𝑠𝐷 ∂𝑠𝐷
𝑠𝐷 ―(𝑆𝑓 ― 𝐷 ∗ ∂𝑟𝐷)∂𝑟𝐷 = 1, 𝑟𝐷 = 1
(10)
12
The outer boundary condition, Eq. (4), in a dimensionless form is (11)
𝑠𝐷 = 0,𝑟𝐷→∞ 2.4 Steady-State Model with Time-Dependent Outer Boundary
It should be noted that Eq. (10) is nonlinear such that the model cannot be solved by using the method of integral transforms. Alternatively, we adopt the approach of Yang et al. (2014) by defining a steady-state dimensionless flow equation as 1 d 𝑟𝐷d𝑟𝐷
(𝑟 ) = 0 d𝑠𝐷 𝐷d𝑟𝐷
(12)
Furthermore, Eq. (11) is replaced by a transient boundary condition specified at a finite distance from the pumping well as (13)
𝑠𝐷 = 0,𝑟𝐷 = 𝑅𝐷
where 𝑅𝐷 is defined as 1 + 𝜋𝑡𝐷 (Yang et al., 2014). Notice that the initial condition of Eq. (9) becomes redundant in solving Eq. (12). Eq. (12) can be written as d2𝑠𝐷 d𝑟𝐷2
1 d𝑠𝐷
(14)
+ 𝑟𝐷d𝑟𝐷 = 0
It has a general solution denoted as (15)
𝑠𝐷 = 𝐴1 + 𝐴2ln (𝑟𝐷) where 𝐴1 and 𝐴2 are undetermined coefficients and ln ( ∙ ) is the natural
logarithm. With the boundary conditions, Eqs. (10) and (13), the coefficients can be determined as 𝐴1 = ―
(ln (𝑅𝐷) + 𝑆𝑓 ±
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2) 2𝐷 ∗
ln (𝑅𝐷) 13
(16a)
and 𝐴2 =
(ln (𝑅𝐷) + 𝑆𝑓 ±
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2)
(16b)
2𝐷 ∗
Substituting Eqs. (16a) and (16b) into (15), the dimensionless aquifer drawdown solution is then obtained as 𝑠𝐷 = ―
(ln (𝑅𝐷) + 𝑆𝑓 ±
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2) 2𝐷
∗
ln
( ) 𝑅𝐷 𝑟𝐷
(17)
The symbol ± indicates that there are two solutions to the model. To ensure Eq. (17) having a positive value, Eq. (17) should be 𝑠𝐷 = ―
(ln (𝑅𝐷) + 𝑆𝑓 ―
4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)2) 2𝐷
ln
∗
( ) 𝑅𝐷 𝑟𝐷
(18)
The dimensionless wellbore flowrate is obtained by applying Darcy’s law as 𝑞𝐷 = ―
2 𝑆𝑓 + ln (𝑅𝐷) ― 4𝐷 ∗ + (ln (𝑅𝐷) + 𝑆𝑓)
(19)
2𝐷 ∗
2.5 Special Cases If the skin factor 𝑆𝑓 is negligible, Eqs. (18) and (19) can be respectively reduced to 𝑠𝐷 = ―
(ln (𝑅𝐷) ―
4𝐷 ∗ + ln (𝑅𝐷)2) 2𝐷
∗
ln
( ) 𝑅𝐷
(20)
𝑟𝐷
and 𝑞𝐷 = ―
ln (𝑅𝐷) ― 4𝐷 ∗ + ln (𝑅𝐷)2
(21)
2𝐷 ∗
If 𝐷 ∗ = 0, Eq. (18) reduces to ln (𝑅𝐷/𝑟𝐷)
(22)
𝑠𝐷 = ln (𝑅𝐷) + 𝑆𝑓 and Eq. (19) becomes 14
―1
(23)
𝑞𝐷 = ln (𝑅𝐷) + 𝑆𝑓
Moreover, if the effective skin factor exerts an insignificant effect on the flow system, Eqs. (18) and (19) may be respectively simplified as 𝑠𝐷 =
ln (𝑅𝐷/𝑟𝐷)
(24)
ln (𝑅𝐷)
and ―1
(25)
𝑞𝐷 = ln (𝑅𝐷)
Note that Eqs. (24) and (25) are the same as the equations given in Yang et al. (2014, Eqs. (35) and (36)).
3. Results and discussion Consider a CHP conducted in a confined aquifer with the following parameter values 𝑟𝑤 = 0.05 m, 𝑠𝑤 = 10 m, 𝑇 = 1 × 10 ―3 m2/min, 𝑆 = 1 × 10 ―5, 𝑆𝑓 = 1, and 𝐷 = 50 min/m3. The following sections are analyzed based on these values. 3.1 Verification of Present Solution To verify the present aquifer drawdown solution, an implicit finite difference solution is developed based on the transient groundwater model composed of Eqs. (8) – (10) using the Mathematica function NDSolve. The remote boundary condition for the finite difference model is set at 𝑟𝐷 = 105. The grid size in radial direction is discretized as unit and the time increment is also unit. Figure 2 displays (a) temporal and (b) spatial dimensionless aquifer drawdown curves predicted by the present solution and finite 15
difference solution when 𝐷 ∗ equals 0.1, 1, and 10 with 𝑆𝑓 = 1. Ramey (1965) mentioned that the non-Darcy flow profoundly affects the flow near the wellbore at the early time; thus, 𝑟𝐷 is set as one (at the screen) in Figure 2(a) and 𝑡𝐷 is 100 (3 sec) in Figure 2(b) to emphasize its effect. Figure 2(a) indicates that the curves predicted by the present solution gives a fairly good match with the finite difference solution within 𝑡𝐷 = 5 × 107 (around 21 h). Figure 2(b) demonstrates that the present solution gives good aquifer drawdown predictions except at the observation point close to the timedependent outer boundary, i.e., 1 + 𝜋𝑡𝐷. Overall, the results indicate that the present solution is well developed. 3.2 Sensitivity Analysis Sensitivity analysis is a useful tool to examine the influence of each aquifer parameter on the aquifer drawdown or wellbore flowrate. The normalized sensitivity defined by Liou and Yeh (1997) gives ∂𝑂
(26)
𝑋𝑘 = 𝑃𝑘∂𝑃𝑘
where 𝑋𝑘 is the normalized sensitivity of the output 𝑂 (the aquifer drawdown or wellbore flowrate) to the 𝑘𝑡ℎ input parameter 𝑃𝑘. Eq. (26) can then be approximated by the forward finite difference formula as 𝑂(𝑃𝑘 + ∆𝑃𝑘) ― 𝑂(𝑃𝑘)
𝑋𝑘 = 𝑃 𝑘
(27)
∆𝑃𝑘
where ∆𝑃𝑘 is set to 10-3 𝑃𝑘 (Yeh and Han, 1989). 16
Figure 3 demonstrates the temporal normalized sensitivity curves for (a) the aquifer drawdown 𝑠 observed at 𝑟 = 𝑟𝑤 and (b) the wellbore flowrate 𝑄. Note that the positive value of the normalized sensitivity indicates the increment of the specific parameter leading to an increase in 𝑠 or 𝑄 at a given time. On the other hand, the negative value of that implies that the increasing parameter value would result in a decrease in 𝑠 or 𝑄. Figure 3 (a) indicates that 𝑆 is the most sensitive parameter before the pumping time at 30 sec according to the absolute value of the normalized sensitivity. After that time, 𝑆𝑓 has the largest influence on the aquifer drawdown, followed by 𝑇 and 𝐷. This is contrary to our expectation that 𝑇 should be the most sensitive parameter. The reason may be that the non-Darcy flow effect, which delays the water flow propagation, reduces the influence of the parameter 𝑇 on the aquifer drawdown. One can observe that 𝑇 and 𝑆 decreasingly affect the aquifer drawdown when operating time elapses; however, 𝑆𝑓 and 𝐷 constantly exert certain effects on the aquifer drawdown with increasing time. Figure 3(b) shows that 𝑇 has the largest sensitivity values fallowed by 𝑆𝑓, 𝑆, and 𝐷 for the case of 𝑄. This result is different from that given in Figure 3 (a) because the wellbore flowrate is mainly dependent on 𝑇 (i.e., 𝑄 = ―2𝜋𝑟𝑤𝑇∂𝑠/∂𝑟). Notably, 𝑆 and 𝐷 are relatively insensitive parameters compared with others but they still produce some effects on the wellbore flowrate. Judged from these results, the parameters appear to have varying degrees of 17
influence on the aquifer drawdown and wellbore flowrate. Therefore, analyzing the data set from the aquifer drawdown or wellbore flowrate would reveal different impacts of the aquifer parameters. 3.3 Effect of Skin Factor and Rate-Dependent Factor In Eq. (10), the head loss due to wellbore skin is described by ― 𝑆𝑓∂𝑠𝐷/∂𝑟𝐷 while that due to the non-Darcy flow effect at the well screen is represented by 𝐷 ∗
(∂𝑠𝐷/∂𝑟𝐷)2. Obviously, the head loss at the well increases as the coefficient 𝑆𝑓 or 𝐷 ∗ increases. Figure 4 shows the temporal dimensionless aquifer drawdown curves for (a) 𝑆𝑓 varying from 0.01 to 100 with 𝐷 ∗ = 1 and (b) 𝐷 ∗ from 0.01 to 100 with 𝑆𝑓 = 1 observed at the rim of wellbore, i.e., 𝑟𝐷 = 1. The figure shows that the aquifer drawdown curves decrease with increasing 𝑆𝑓 and 𝐷 ∗ because more head loss occurs at the well screen. The aquifer drawdown is sensitive to the change in 𝑆𝑓 than in 𝐷 ∗ because the loss value due to 𝐷 ∗ (∂𝑠𝐷/∂𝑟𝐷)2 yields a smaller value than that due to ― 𝑆𝑓∂𝑠𝐷/∂𝑟𝐷. It helps account for the outcome of the sensitivity analysis in the previous section that why the parameter 𝐷 always yields a smaller impact than 𝑆𝑓. Additionally, the aquifer drawdown curves for 𝐷 ∗ = 0.01 and 0.1 almost coincide with each other indicating that 𝐷 ∗ exerts an insignificant effect on the aquifer drawdown when its value is less than 0.1. 3.4 Field Data Analyses 18
Chen and Chang (2002) conducted two constant-head injection tests, one was at well 2 (referred to as the CHIT1 in their study) and the other at well 5 (CHIT2). Both wells have a 0.05 m radius and fully penetrate a silty sand confined aquifer of 10-m thickness at Taoyuan Tableland in the northern part of Taiwan. The test at the CHIT1 ran 58 hours with the 𝑠𝑤 kept at 10.1 m while the CHIT2 ran 67 hours with 𝑠𝑤 maintained at 13.48 m. They measured the cumulative volume in the injection well for parameter estimations. The formula they used to estimate the parameters is 𝑠𝑤𝑡 𝑉𝑤
=
(
)+
0.183 0.83𝑇𝑡 𝑇 log 𝑆𝑟2𝑤
𝑆𝑓
(28)
2𝜋𝑇 𝑡
where 𝑉𝑤 is the cumulative volume defined as ∫0𝑄 (𝑥)𝑑𝑥 which can be easily evaluated by a numerical integral function in Mathematica called NIntegrate (Wolfram Research Inc., 2018). The aquifer parameters were estimated by graphically fitting the measured data in a semi-log plot of 𝑠𝑤𝑡/𝑉𝑤 versus time. Chen and Chang (2002) pointed out that the measured data displayed two different portions in early-time and late-time periods. One can expect that the late-time portion can reflect the feature of the flow behavior in the formation zone and the well losses; therefore, the data points from the CHIT1 and CHIT2 after around 100 min were used to analyze for the aquifer parameters. They firstly determined the value of 𝑇 as 1.68 × 10 ―3 m2/min by fitting the late-time data from the CHIT1 and CHIT2. Later, they determined the value of 𝑆 as 2.52 × 10 ―4 by assuming the absent of the wellbore skin in the CHIT2. With the 19
obtained 𝑇 and 𝑆, the skin factor 𝑆𝑓 was finally determined as 18.57 for the CHIT1. Their results of the parameter estimation are listed in Table 1. Following their work, the 𝑠𝑤𝑡/𝑉𝑤 data of CHIT1 and CHIT2 after 100 min were read from Chen and Chang (2002, Figure 4) by using WebPlotDigitizer (Rohatgi, 2018). We analyze these data using three different approaches. The first approach is to use Jacob and Lohman’s (1952) solution in which both skin factor and rate-dependent factor are not considered. The second one is to employ Hurst et al.’s (1969) solution in which only the skin factor is considered while the third is the present solution. Each of three solutions is coupled with the least-squares approach and solved by the Levenberg Marquardt algorithm in Mathematica routine NonlinearModelFit (Wolfram Research Inc., 2018) for the parameter estimation. Two statistics are chosen to assess the goodness of fit for different solutions with the estimated parameters. They are standard error of estimate (SEE) and mean error (ME) defined in Yeh (1987) and respectively expressed as 𝑆𝐸𝐸 =
1 𝑁 ∑ 𝜈 𝑖=1
|𝑒2𝑖 |
(29a)
and 1
𝑁
𝑀𝐸 = 𝑁∑𝑖 = 1𝑒
(29b)
𝑖
where 𝜈 is the degree of freedom defined as the number of observed data 𝑁 minus the number of unknown parameters 𝑀, i.e., 𝜈 = 𝑁 ― 𝑀, and 𝑒𝑖 is the 𝑖th prediction 20
error between the predicted 𝑠𝑤𝑡/𝑉𝑤 and observed data. The estimated results by three different solutions are also listed in Table 1. According to the estimated results of the CHIT1, the values of 𝑇 are obtained as 1.04 × 10 ―3 m2/min, 1.62 × 10 ―3 m2/min, and 1.61 × 10 ―3 m2/min by Jacob and Lohman’s (1952), Hurst et al.’s (1969), and present solutions, respectively. The estimate of 𝑇 obtained from the first solution has the smallest value compared to the others. That obtained from the second and the present solutions are rather close to that determined by Chen and Chang (2002). The first solution, however, gives the estimated 𝑆 unrealistically small. It may be attributed to the fact that the skin effect was not considered in Jacob and Lohman’s (1952) model. Additionally, the estimated values of 𝑆 by the second and present solutions are larger than that by Chen and Chang (2002) about an order of magnitude. The reason for that is Chen and Chang (2002) determined its value by analyzing the data set from the CHIT2. Based on the SEE and ME values shown in Table 1, we consider that the value of 𝑆 estimated by Chen and Chang (2002) cannot represent the estimated result from the CHIT1. The 𝑆𝑓 values estimated by Hurst et al.’s (1969) and present solutions confirms that the wellbore has a positive skin as indicated by Chen and Chang (2002). Table 1 indicates that the present solution yields the smallest SEE and ME values indicating that the consideration of both skin factor and rate-dependent factor gives the best fit to the data set of CHIT1. 21
The estimated results for data from CHIT2 indicate that those three solutions give the same estimates of 𝑇 (i.e., 1.64 × 10 ―3 m2/min) which are not far from Chen and Chang’s (2002) estimation. The values of 𝑆 estimated by all solutions are on the same order of magnitude. Besides, the estimates of 𝑆𝑓 obtained by Hurst et al.’s (1969) and present solutions and that of 𝐷 obtained by the present solution are smaller than those obtained from the CHIT1. Both effects of wellbore skin and non-Darcy flow are insignificant for data in the CHIT2. It is consistent with Chen and Chang’s (2002) statement that there are little or no wellbore skin existing around the well 5. Figure 5 shows the measured data of (a) CHIT1 and (b) CHIT2 and the temporal curves of 𝑠𝑤𝑡/ 𝑉𝑤 predicted by three solutions with the parameters listed in Table 1. It demonstrates that the curve predicted by the present solution matches well with the measured data while those by the other solutions show some significant departure from the measured data, especially for the CHIT1. It implies that the consideration of the rate-dependent factor improves the parameter estimation if non-Darcy flow effect is profound. 3.5 Performance of Test Wells Kurtulus et al. (2019) proposed a well efficiency criterion by analyzing the data of step-drawdown test from 290 wells coupled with Rorabaugh’s (1953) well loss function. Notice that Rorabaugh’s (1953) well loss function can be viewed as a modified version of Jacob’s (1946) one. That is 22
𝐵𝑄 + 𝐶𝑄𝑘 = 𝑠𝑤, 𝑟 = 𝑟𝑤
(30)
in which 𝑘 is the exponent parameter equal to 2 in this study. Kurtulus et al. (2019, Table 3) summarized the following that the well is in the deteriorating condition when 𝐶 > 1800 s2/m5 or 0.5 min2/m5 for 𝑘 within 1 and 2. Employing the estimated results, we obtain that 𝐶 equals 1163.51 min2/m5 for well 2 and 0.97 min2/m5 for well 5. These values indicate that both test wells were not properly designed or developed. It may be explained why the measured data from the CHIT1 and CHIT2 are merely capable of reflecting the feature of the flow behavior in the aquifer formation after the operating time up to 100 min (1.67 h).
4. Concluding remarks This study presents a new mathematical model with considering both skin factor and rate-dependent factor at the pumping well for describing aquifer drawdown distribution and wellbore flowrate induced by a CHP at a fully penetrating well in confined aquifers. The inner boundary condition is nonlinear due to the consideration of the rate-dependent factor. In light of this, we adopt the work of Yang et al. (2014) to formula the model comprised of a steady-state groundwater flow equation with a timedependent finite outer boundary. The solutions to the model for the wellbore flowrate and the drawdown can then be derived. The comparison of the aquifer drawdown predicted by the present and finite difference solutions indicates that the present 23
solution is well developed. The results of sensitivity analysis indicate that 𝑆 is the most sensitive parameter to the aquifer drawdown before the pumping time of 30 sec while 𝑇 is the most sensitive parameter for the wellbore flowrate. The non-Darcy flow constant 𝐷 exerts relatively small effect on both aquifer drawdown and wellbore flowrate as compared to other aquifer parameters. An increase in the skin factor or ratedependent factor has the effect of decreasing the drawdown and the flowrate values because the head losses are generated. The present solution is used to estimate the aquifer parameters when analyzing a measured data set from a field constant-head injection test. The results show that the present solution used in the parameter estimation gives the best-estimated results compared with those estimated by Jacob and Lohman solution (1952), and Hurst et al. (1969) solution as well as that determined by Chen and Chang (2002). According to these results, the consideration of the ratedependent factor improves the parameter estimation. Also, the results of the parameter estimation indicate that the test wells in Chen and Chang’s (2002) work may be not properly installed or developed. Overall, the present solutions can be used as a convenient tool to estimate the aquifer parameters and predict the aquifer drawdown and flowrate distributions.
Acknowledgments The research leading to this paper has been partially supported by the grant from 24
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33
Figure captions Figure 1. Schematic representation of drawdown distribution induced by a constanthead pumping at fully penetrating well. Figure 2 (a) Temporal and (b) spatial drawdown curves predicted by the present and finite difference solutions with 𝐷 ∗ = 0.1, 1, and 10. Figure 3 The temporal distribution curves of normalized sensitivity for (a) aquifer drawdown and (b) wellbore flowrate. Figure 4 Temporal dimensionless drawdown distribution with (a) 𝑆𝑓 from 0.01 to 100, 𝐷 ∗ = 1 and (b) 𝐷 ∗ from 0.01 to 100, 𝑆𝑓 = 1 observed at the wellbore 𝑟𝐷 = 1. Figure 5 The measured data and the curves of swt/Vw predicted by the present solution and two existing solutions with estimated parameters for the tests (a) CHIT1 and (b) CHIT2.
34
Nomenclature Parameters and variables (𝑟, 𝑟𝑤, 𝑡)
Radial distance, well radius, and time
(𝑇, 𝑆)
Transmissivity and storativity of the aquifer
(𝑆𝑒, 𝑆𝑓, 𝐷)
Effective factor, skin factor, and non-Darcy flow constant
(𝑠, 𝑠𝑤)
Drawdown in the aquifer and the wellbore
𝑄(𝑡)
Wellbore flowrate
𝑉𝑤
Cumulative volume defined as ∫0𝑄 (𝑥)𝑑𝑥
𝑡
(𝐵,
𝐵1, Linear head loss, linear aquifer-loss, linear well-loss, and nonlinear well-
𝐵2, 𝐶)
𝑒𝑖)
loss coefficients
(𝑋, 𝑃, 𝑂)
Normalized sensitivity value, input parameter, and output
(𝑀, 𝑁, 𝜈,
The number of unknown parameters, the number of observed data, the degree of freedom, the 𝑖th prediction error
35
Table 1 Parameters estimated by Jacob and Lohman’s (1952), Hurst et al.’s (1969), and present solutions as well as that determined by Chen and Chang (2002) with the associated prediction errors Estimated
Jacob and Lohman (1952)
parameters
solution
Hurst et al. (1969) solution
Present solution
Chen and Chang (2002)
CHIT1 𝑇(m2/min)
1.04 × 10 ―3
1.62 × 10 ―3
1.61 × 10 ―3
1.68 × 10 ―3
𝑆(-)
3.93 × 10 ―12
8.37 × 10 ―5
1.40 × 10 ―5
2.52 × 10 ―4
𝑆𝑓(-)
―
16.91
15.90
18.57
𝐷
―
―
11.66
―
(min/m3) SEE
28.91
ME
9.40 × 10
10.45 ―2
4.33 × 10
4.61 ―2
3.16 × 10
11.77 ―3
―9.79
CHIT2 𝑇(m2/min)
1.64 × 10 ―3
1.64 × 10 ―3
1.64 × 10 ―3
1.68 × 10 ―3
𝑆(-)
3.55 × 10 ―4
4.16 × 10 ―4
4.21 × 10 ―4
2.52 × 10 ―4
𝑆𝑓(-)
―
0.07
0.08
0
𝐷
―
―
0.01
―
(min/m3) SEE
2.28
ME
1.25 × 10
2.26 ―3
―1.23 × 10
1.25 ―3
―1.01 × 10
2.91 ―3
1.39
Note. ME = mean error; SEE = standard error of estimate.
Highlights
Rate-dependent factor is adopted in a confined flow model for constant-head pumping This factor is to reflect the effect of non-Darcy flow at the pumping well screen
The model comprises a stationary flow equation with a time-dependent outer boundary
Sensitivity analyses are made to show the effect of aquifer parameters on the flow
The model is used to analyze field drawdown data and well installation quality
Ye-Chen Lin: Conceptualization, Methodology, Software, Formal analysis, 36
Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization. Hund-Der Yeh: Formal analysis, Writing - Review & Editing, Supervision.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
37
38
39
40
41
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
42
Ye-Chen Lin: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - Original Draft, Writing - Review & Editing, Visualization. Hund-Der Yeh: Formal analysis, Writing - Review & Editing, Supervision.
43
Highlights
Rate-dependent factor is adopted in a confined flow model for constant-head pumping This factor is to reflect the effect of non-Darcy flow at the pumping well screen
The model comprises a stationary flow equation with a time-dependent outer boundary
Sensitivity analyses are made to show the effect of aquifer parameters on the flow
The model is used to analyze field drawdown data and well installation quality
44