a simplified solution using izbash's equation for non
TRANSCRIPT
Methods Note/
A Simplified Solution Using Izbash’s Equation forNon-Darcian Flow in a Constant RatePumping Testby Liang Xiao1,2,3,4, Ming Ye5, Yongxin Xu6,7, and Fuwan Gan1,2,3
AbstractThis paper derives an equivalent of Darcian Theis solution for non-Darcian flow induced by constant rate
pumping of a well in a confined aquifer. The derivation, which is valid at later times only, is original. It utilizesIzbash’s equation. This introduces an additional parameter to Darcian condition, namely, empirical exponent. Thesolution is a non-Drcian equivalent of Jacob straight line method for analyzing pumping tests at late times. It canbe used to determine aquifer parameters: storativity, analogous hydraulic conductivity, and empirical exponent.However, while the Jacob method requires a minimum of only one pumping test with one observation well, theadditional parameter in the present solution means that a minimum of two observation wells in one test or twopumping tests at different rates with one observation well are required. The derived solution is applied to a casestudy at Plomeur in Brittany, France, and is shown to provide a practical and efficient method for analyzingpumping tests where non-Darcian groundwater flow occurs.
IntroductionGroundwater flow mostly obeys Darcy’s law with the
Reynolds numbers ranging from 1 to 10 (Bear 1972).
1College of Civil Engineering and Architecture, GuangxiUniversity, Nanning, 530004, China.
2Guangxi Key Laboratory of Disaster Prevention andEngineering Safety, Guangxi University, Nanning, 530004, China.
3Key Laboratory of Disaster Prevention and Structural Safetyof Ministry of Education, Guangxi University, Nanning, 530004,China.
4Institute of Africa Water Resources and Environment, HebeiUniversity of Engineering, Handan, 056038, China.
5Department of Earth, Ocean, and Atmospheric Science andDepartment of Scientific Computing, Florida State University,Tallahassee, FL, 32306.
6Department of Earth Sciences, University of the WesternCape, Cape Town, South Africa.
7Corresponding author: Institute of Africa Water Resourcesand Environment, Hebei University of Engineering, Handan 056038,China; [email protected]
Article Impact Statement: This study proposes new methodsto characterize late-time drawdown of non-Darcian flow and toanalyze aquifer hydraulic properties.
Received June 2018, accepted March 2019.© 2019, National Ground Water Association.doi: 10.1111/gwat.12886
Outside of this range the flow becomes non-Darcian with anonlinear relationship of specific discharge and hydraulicgradient (e.g., Bordier and Zimmer 2000; Sen 2000; Wu2001, 2002a, 2002b; Moutsopoulos and Tsihrintzis 2005).The non-Darcian flow can occur in low permeabilitymaterials under very low gradients and when the flow islarge through very high permeability media (Hansbo 2001;Xu et al. 2007; Sedghi-Asl et al. 2014; Yang et al. 2017).
For the case of large flow with high permeability,a number of equations have been proposed to quantifythe nonlinear relationship between the specific dischargeand hydraulic gradient. The most commonly methodswere the Forchheimer’s equation (Forchheimer 1901)(a second-order polynomial function) and the Izbash’sequation (Izbash 1931) (a power function). The choicebetween the two equations is depended on field conditions(Bordier and Zimmer 2000; Yamada et al. 2005; Wenet al. 2006; Yeh and Chang 2013; Houben 2015; Menget al. 2015). Based on the two equations, many analyticaland numerical models for the non-Darcian flow inducedby constant rate pumping in different aquifer systemshave been derived (e.g., Sen 1990; Wen et al. 2008a,2008b, 2008c; Wen et al. 2011; Wen et al. 2013; Eck
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et al. 2012; Mijic et al. 2013; Wang et al. 2015; Chenet al. 2015a, 2015b, 2015c; Liu et al. 2016).
An important practical application of the analyticalsolutions to groundwater flow is in the estimation ofaquifer hydraulic properties using drawdown data. Fornon-Darcian flow, Sen (1989) proposed a curve matchingmethod based on the Boltzmann’ solution. However,determining the most suitable matching is subjective andinefficient due to a complicated computation procedure.Le Borgne et al. (2004) developed other fitting methodbased on the Barker’s model (Barker 1988). It wassuccessfully used to estimate the hydraulic properties ofthe Ploemeur aquifer, France. Liu et al. (2016) proposeda generalized non-Darcian radial flow model by which thehydraulic properties could be estimated via a curve-fittingprocedure with the least squares methods.
Analytical solutions for groundwater flow at greatertimes can often be approximated by simpler expressions.This makes it easier to apply them to analyze pumpingtests, especially in field conditions. Analytical solutionsfor late time drawdown are also of interest becausepumping test data can become more accurate with time asconditions in a pumping test stabilize (Wen et al. 2008b;Liu et al. 2016).
A literature review of published non-Darcian flowstudies suggests that there is a lack of research on theanalytical solution on the late-time drawdown for non-Darcian flow in pumping tests in confined aquifers. There-fore, a new simplified analytical solution is derived herefor this situation. The derivation utilizes Izbash’s equationand the Boltzmann transform. The solution is a non-Darcian equivalent of the Jacob straight line method foranalyzing pumping tests at later times. Apart from aquiferstorativity and an analogous hydraulic conductivity, thesolution contains a third aquifer parameter, the empiricalexponent from Izbash’s equation. All three parameters canbe determined by applying the solution in pumping tests.The effectiveness of the solution in practice is evaluatedusing data from pumping tests performed at the Ploemeursite in Brittany, France.
Problem StatementTo derive the analytical solution, the same assump-
tions as those for the Theis equation in the case of Darcianflow are made as: (1) the aquifer is confined, homoge-nous, and horizontally isotropic; (2) the aquifer has con-stant thickness, and is infinite in the horizontal direction;(3) the pumping well and the observation wells fully pen-etrate the aquifer; (4) the pumping rate is constant overtime; and (5) the aquifer is hydrostatic before pump-ing. A schematic diagram of the aquifer is shown inFigure 1.
According to the mass conservation, the governingequation for two-dimensional radial flow during thepumping period (e.g., Wen et al. 2008b) is
∂q
∂r+ q
r= S
b
∂s
∂t(1)
where b is the aquifer thickness [L], r is the radial distancefrom the pumping well [L], t is the pumping time [T],q(r , t) is specific discharge [LT−1], s(r , t) is thedrawdown [L], and S is the dimensionless storativity.
The far-field boundary condition is
s (r → ∞, t) = 0. (2)
The boundary condition representing the fully pene-trating well is
limr→0
2πbrq = −Q (3)
where Q is the pumping rate [L3 T−1].The initial condition is
s (r, 0) = 0 (4)
For modeling purposes for the non-Darcian flow,Basak (1977) and Bordier and Zimmer (2000) pointedout that the Izbash’s equation is likely to be preferred,because it is comparable with the Darcy’s law. Hence, wewill employ the Izbash’s equation in the study. Izbash’sequation for the specific discharge is
q =(
K∂s
∂r
) 1n
(5)
where n and K are empirical constants. The n constant isa dimensionless constant of the non-Darcian flow withinthe range 1 to 2, representing the degree of deviation fromlinearity. The coefficient, K , is analogous to hydraulicconductivity with the unit as [LT−1]n. When n valueequals to one, the pumping flow becomes Darcian andK is hydraulic conductivity.
Linearization MethodBased on Equation 5, the derivative of q with respect
to r is
∂q
∂r= 1
nK
1n
(∂s
∂r
) 1−nn ∂2s
∂r2. (6)
Substituting Equation 6 into Equation 1 yields
∂2s
∂r2+ n
r
∂s
∂r= S
b
n
K1n
(∂s
∂r
) n−1n ∂s
∂t(7)
To linearize such Equation 7, an approximation isgiven as
∂s
∂r= (q)n
K≈ −
(Q
2πrb
)n
K. (8)
Equation 8 implies that the water amount through anyradial face per unit time is approximately equal to Q .The error of this approximation is small at places near to
2 L. Xiao et al. Groundwater NGWA.org
Figure 1. Schematic diagrams of the constant rate pumping test.
the pumping well or at late time (e.g., Odeh and Yang1919; Ikoku and Ramey Jr 1979; Qian et al. 2005; Wenet al. 2008b). After substituting Equation 8 to Equation 7,the linearized governing equation of the non-Darcian flowis obtained as
∂2s
∂r2+ n
r
∂s
∂r= εr1−n ∂s
∂t(9)
where ε = nK
(Q
2πb
)n−1Sb
.Substituting Equation 5 into Equation 3, the bound-
ary condition representing the fully penetrating wellbecomes
limr→0
2πbr
(K
∂s
∂r
) 1n
= −Q. (10)
The far-field boundary condition (Equation 2) andinitial boundary condition (Equation 4) remain thesame.
Boltzmann Transform SolutionEmploying the Boltzmann transform, the solution
of Equation 9 subject to the boundary conditions (2),(4), and (10) gives the drawdown under non-Darcianflow as
s = Q
4Kπb
(Q
2πrb
)n−1 ∫ ∞
nS r24Kbt
(Q
2πrb
)n−1
exp (−y)
ydy.
(11)
The derivation process is documented in the Support-ing Information and is original. It is necessary to highlightthat Equation 11 is only valid near to the well or at latetime because it has been derived using Equation 8, which
is subject to this condition. Assuming dimensionless itemsof p, v , and s′as
p = nS r2
4Kbt
(Q
2πrb
)n−1
(12a)
v =(
Q
2πrK1n b
)n−1
(12b)
s′ = s4K
1n πb
Q. (12c)
Equation 11 can be rewritten as
s′ = W (p) = v
∫ ∞
p
exp (−y)
ydy (13)
where W (p) is the well function for the non-Darcian flow.If n = 1, the value of v is equal to 1 and, thus, the W (p) isreduced to W
(Sr2
4Kbt
), the well function for Darcian flow.
Approximate Analytical Solution for Late-TimeDrawdown
Similar to the derivation of Jacob’s method for Dar-cian flow (Kruseman and Ridderna 1991), the wellfunction of the non-Darcian flow (Equation 13) can bewritten as
W(p) = v
[−0.5772 − ln p + p − (p)2
2.2!+ (p)3
3.3!− . . .
].
(14)
According to Equation 12a, the value of p becomessmall if the radius, r , is small or the pumping time, t ,
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is large. When p < 0.01, the p and its higher order termsin Equation 14 can be neglected (Kruseman and Ridderna1991). Hence, the well function at the late time can berewritten as
W(p) = v[−0.5772 − ln p
]. (15)
After changing from natural to 10-base logarithm inEquation 15 and subjecting it into Equation 13, the late-time solution is given as
s = 2.3Q
4πK1n b
v log 0.5621
p. (16)
It is indicated that the late-time drawdown curve canbe presented as a straight line for variable of log 0.562 1
p
with a constant slope of 2.3Q
4πK1n b
v. Equation 16 is a valid
solution for non-Darcian flow but not a solution for smallt and/or big r .
Interpretation of Drawdown for Non-DarcianFlow
The assessments of the hydraulic properties (K , S ,and n) of the pumped aquifer can be satisfied byobservations at least two different constant rate pumpingtests (Figure 2). Applying similar derivation to the Jacobmethod as in Kruseman and Ridderna (1991), the ratio ofthe two slopes is given by using the late-time drawdownsolution (Equation 16) as
(Q
Q′
)n (r ′
r
)n−1
=s2−s1
log t2−log t1
s′2−s
′1
log t′2−log t
′1
. (17)
The value of the constant n can be obtained asthe solution of Equation 17. Based on the constant n ,the analysis proceeds as in the Jacob method to get Kand S as
K = 2.3Qn (log t2 − log t1)
4πbn (s2 − s1)
(1
2πr
)n−1
(18)
S = K1n
v
2.25bt0
nr2. (19)
Case StudyThe Ploemeur aquifer is the main source of drinking
water in the south coast of Brittany, France. To charac-terize the Ploemeur aquifer after groundwater explorationsince 1990, two constant rate pumping tests were carriedout as documented by Le Borgne et al. (2004) and Liuet al. (2016). They were a long-term test (LT ) in June1991 and a short-term test (ST ) in September 1995. Allwells fully penetrated the aquifer. The former studies (LeBorgne et al. 2004; Liu et al. 2016) indicated that flow in
Figure 2. Schematic diagrams of the drawdown-time curvesof non-Darcian flow with different pumping rates in theobservation wells with different radial distances.
Table 1Data for the Pumping Tests (after Liu et al. 2016)
Test Short-Term Long-Term
Pumping rate (m3/h) 34 80Duration (day) 5 88Distance from pumping well
and observation wells (m)46, 61 330
the Ploemeur aquifer was non-Darcian. Hence, the datafrom these tests can be used to evaluate the analyticalsolution developed in the paper.
To meet the requirement of the use of the developedlate-time solution, that is, p < 0.01, the drawdown fromthe observation well with small radial distance or fromthe sufficiently LT is preferred. As a result, the drawdowndata from the STs with the radial distances of 46 m and61 m are chosen to determine the hydraulic properties ofthe Ploemeur aquifer and the LT with a raial distanceof 330 m is adopted to verify the proposed solution fordrawdown simulation of the non-Darcian flow (Figure 4aand 4b) in the case. The variable values associated withthe pumping tests are given in Table 1. Accordingly, thelocations of the pumping well and the observation wellsfor the pumping tests used in the case study are given inFigure 3.
With the drawdown-time curve, the procedures forparameter assessments are highlighted as following.Firstly, the slopes of the late-time drawdown curves fromthe ST s with the radial distances of 46 m and 61 m can bedirectly obtained in Figure 4a as 1.15 and 1.1. Secondly,the n value is estimated as 1.13 by using Equation 17,indicating a moderate non-Darcian flow. And then, basedon the n value, the analogous hydraulic conductivitycoefficient, K , is obtained as 2.08 × 10−6 [m/s]n by usingEquation 18. Finally, the S value can be obtained by usingEquation 19 as 3.03 × 10−3.
A comparison between our results of parameterestimation and those from the several previous studies is
4 L. Xiao et al. Groundwater NGWA.org
Figure 3. Locations of the pumping well and observationwells of concern in the case study (after Liu et al. 2016).
Figure 4. Drawdown-time series for (a) the STs (b) the LTin the semilog plot (after Liu et al. 2016).
Figure 5. Drawdown simulations for LT with radial distanceof 330 m.
shown in Table 2. It is suggested the K values are similarfor the three results with n = 1.13, and for the two resultswith n = 1. Similarly, the S values seem remarkablyconsistent in all cases. Assuming that the Ploemeur aquiferis under the Darcian condition, the results from the Theissolution are also given in Table 2. It is indicated that theK and S values of the aquifer under the Darcian conditionare greater than that under the non-Darcian condition.
With the estimated hydraulic properties, thedrawdown-time curve of the LT can be simulated by ouranalytical solution of Equation 11 and the former studiesof Sen (1990) and Wen et al. (2008b) via software,namely, MATLAB. It is clear to see that the drawdowncurve by the proposed solution is well matched with thatfrom the field work and the former studies (Figure 5) atlate time. Otherwise, a good match at most data suggeststhat the error due to larger p is not very significant inthis case, possibly because the n value is not very muchgreater than 1. It proves that conditions in the tests atPlomeur approximate the assumptions of the proposedsolution.
ConclusionIn this paper, a simplified analytical solution using
Izbash’s equation is developed for the non-Darcian flowinduced by pumping of a well in a confined aquifer. Thesolution is derived by the Boltzmann transform for late-time drawdown. The result indicates that the late-time
Table 2Hydraulic Properties of the Ploemeur Aquifer (after Liu et al. 2016)
TestSolution ofThis Study
Liu et al.(2016)
Wen et al.(2008b)
Le Borgneet al. (2004) Sen (1990) Theis (1935)
n 1.13 1.25 1.13 1 1.13 1K ([m/s]n) 2.03 × 10−6 4.6 × 10−7 1.95 × 10−6 1.89 × 10−5 1.84 × 10−6 1.71 × 10−5
S 3.03 × 10−3 3.10 × 10−3 3.10 × 10−3 2.92 × 10−3 2.95 × 10−3 3.95 × 10−2
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drawdown-time curve is approximated as a straight linein the semilog plot.
For data interpretation, the analytical solution can beused for estimation of the power law index, analogoushydraulic conductivity, and storativity of the pumpedaquifer by observations at least two different constant ratepumping tests. The case of the constant rate pumpingtests performed during 1990-1995 at the Ploemeur sitein Brittany, France, is studied to investigate the practicalapplication of the proposed model. Being comparedwith those of previous studies, the estimated hydraulicproperties by our analytical solution can well characterizethe hydraulic properties in the aquifer conditions onthe field scale after the groundwater exploitation, andthe drawdown can be well predicted at late time in astraightforward way.
AcknowledgmentsThis research was partly supported by the National
Natural Science Foundation of China (Grant Num-bers 2017YFC0405900, 41807197 and 51469002),and the Natural Science Foundation of GuangxiZhuang Autonomous Region (Grant Numbers2017GXNSFBA198087 and GuiKeAB17195073).We would like to thank Prof. Zhang Wen from ChinaUniversity of Geosciences for constructive comments,and the Editors and two anonymous reviewers who helpus improve the quality of the manuscript.
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