A temporal sampling strategy for hydraulic tomography analysis

A temporal sampling strategy for hydraulic tomography analysis

Ronglin Sun,1 Tian-Chyi Jim Yeh,2 Deqiang Mao,2,3 Menggui Jin,1,4 Wenxi Lu,5 and Yonghong Hao6

Received 22 February 2012; revised 17 May 2013; accepted 26 May 2013; published 2 July 2013.

[1] This paper investigates optimal sampling times of drawdowns for the analysis ofhydraulic tomography (HT) survey. The investigation was carried out by analyzing thespatial and temporal evolution of cross-correlations between the head responses at anobservation well and transmissivity (T) and storage coefficient (S) properties during apumping test in homogeneous and heterogeneous aquifers. The analysis shows that thecross-correlation between the head and S values is limited to the region between theobservation and the pumping well in the aquifers: It reaches the highest value near the earlytime (tm), and decays to zero afterwards. The time tm is approximately equal to the time t0 atwhich the extrapolated drawdown from the first straight line portion of an observeddrawdown-log time plot becomes zero. At early times, the high cross-correlation betweenthe head and T is confined to the region between the observation and the pumping well. Thisregion then evolves into two humps: One on each side of the circular region encompassingthe observation well and the pumping well. The size of the two humps expands and theirvalues reach the maximum as flow reach steady-state. As a consequence, we hypothesizethat pairs of head data at t0 and those at either the steady-state or a late time during an HTsurvey could yield the best estimates of the heterogeneous T and S fields. Results fromnumerical experiments have verified this hypothesis and demonstrated that this samplingstrategy is generally applicable even when the boundary condition is unknown. We,therefore, recommend in principle that (1) carrying out pumping tests of HT surveys forsufficiently long period of time such that drawdown reaches the entire area of interest and(2) using a constant head or zero drawdown for all boundaries during the inverse modelinganalysis.

Citation: Sun, R., T.-C. J. Yeh, D. Mao, M. Jin, W. Lu, and Y. Hao (2013), A temporal sampling strategy for hydraulic tomographyanalysis, Water Resour. Res., 49, 3881–3896, doi:10.1002/wrcr.20337.

1. Introduction

[2] Transmissivity (T) and storage coefficient (S) are twoimportant properties that control groundwater flow in aqui-fers and are of practical importance for water resourcesdevelopment and management as well as protection andremediation of groundwater. After Yeh and Liu [2000] andLiu et al. [2002] sequential pumping tests, multiwell inter-ference tests, or hydraulic tomography (HT) have been the

subject of active theoretical, laboratory, and recently fieldresearch to characterize the spatial distributions of hydraulicparameters at a higher level of detail than traditional meth-ods [e.g., Gottlieb and Dietrich, 1995; Yeh and Liu, 2000;Liu et al., 2002; Bohling et al., 2002, 2007; Brauchler et al.,2003, 2011; Zhu and Yeh, 2005, 2006; Liu et al., 2007; Niand Yeh, 2008; Straface et al., 2007; Xiang et al., 2009; Ill-man et al., 2007, 2008, 2009, 2010, 2011; Fienen et al.,2008; Kuhlman et al., 2008; Castagna and Bellin, 2009;Berg and Illman, 2011; Cardiff et al., 2009, 2012; Cardiffand Barrash, 2011; Huang et al., 2011; Li et al., 2005,2008; Liu and Kitanidis, 2011; Yin and Illman, 2009]. Notethat Cardiff and Barrash (2011) provided a summary of allpeer-reviewed HT studies (1D/2D/3D).

[3] These studies showed that transient HT can identifynot only the pattern of the heterogeneous hydraulic conduc-tivity (K) or transmissivity (T) field, but also the variationof specific storage (Ss) or the storage coefficient (S) [seeZhu and Yeh, 2005, 2006; Liu et al., 2007; Xiang et al.,2009, in particular]. More importantly, they have demon-strated that the hydraulic property fields estimated by HTcan yield much better predictions of flow and solute trans-port processes than other conventional characterizationapproaches [Illman et al., 2011]. Berg and Illman [2011],in particular, substantiated the robustness of HT for a

1School of Environmental Studies, China University of Geosciences,Wuhan, China.

2Department of Hydrology and Water Resources, The University of Ari-zona, Tucson, Arizona, USA.

3School of Water Resources and Environmental science, China Univer-sity of Geosciences, Beijing, China.

4State Key Laboratory of Biogeology and Environmental, China Uni-versity of Geosciences, Wuhan, China.

5College of Environment and Resources, Jilin University, Changchun,China.

6Tianjin Key Laboratory of Water Environment and Resources, TianjinNormal University, Tianjin, China.

Corresponding author: R. Sun, School of Environmental Studies, ChinaUniversity of Geosciences, Wuhan 430074, China. ([email protected])

©2013. American Geophysical Union. All Rights Reserved.0043-1397/13/10.1002/wrcr.20337

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WATER RESOURCES RESEARCH, VOL. 49, 3881–3896, doi:10.1002/wrcr.20337, 2013

highly heterogeneous geological medium with a varianceof log hydraulic conductivity of 5.4 and a vertical correla-tion scale of 0.15 m.

[4] Nonetheless, many practical issues remain, including(1) the design of spatial sampling and pumping locations,(2) the duration and magnitude of the pumping rate duringan HT survey, and (3) the frequency of temporal sampling.Specifically, is it necessary to conduct the pumping test toreach a steady-state? How many time/drawdown data in awell hydrograph should be used to obtain good estimates ofT and S field while minimizing computational effort for theHT analysis? While Yeh and Liu [2000] conducted a pre-liminary investigation of the spatial sampling issues, thetemporal sampling issues are opened for investigation.

[5] The sampling time requirements vary with themethod of interpretation. For instance, Vasco et al. [2000],Vasco and Karasaki [2006], Brauchler et al. [2003, 2011],and He et al. [2006] developed methods based on traveltime and amplitude of drawdown-time curves at differentlocations from the pumping tests conducted in tomographicformat to estimate hydraulic properties. Similar to thetravel time approaches, Li et al. [2005] and Zhu and Yeh[2006] developed temporal moment approaches, which arebased on multidimensional flow model. These approacheseither implicitly or explicitly require a complete wellhydrograph to determine the travel time and the amplitude(i.e., the temporal moments).

[6] On the other hand, Yeh and Liu [2000], Fienen et al.[2008], and Liu and Kitanidis [2011] developed HT inter-pretation methods based on steady-state flow models whichrequire only steady-state heads or drawdowns. Zhu and Yeh[2005] reported that heads are highly correlated in timeduring HT experiments, As a result, Zhu and Yeh [2005],Liu et al. [2007], Straface et al. [2007], Xiang et al. [2009],Illman et al. [2009], Castagna et al. [2011], Berg andIllman [2011], Cardiff and Barrash [2011], and Cardiff etal. [2012] analyzed transient HT data using only a smallnumbers (four or five) of selected heads or drawdowns ofthe drawdown-time well hydrograph. Bohling et al. [2002],however, used 51 drawdown-time data points of each wellhydrograph of 42 sampling ports of 14 pumping tests (atotal of 29,988 drawdown data points) for transient inver-sion for hydraulic conductivity field only, assuming theS field was known exactly. Because of the large amount ofhead data used, the amount of computational resourcerequired was overwhelming. Consequently, they appliedthe concept of steady shape to their analysis of HT data forthe K field. The term ‘‘steady shape’’ is used to designateconditions in an unsteady-state flow regime in which draw-down continues to change with time but the hydraulic gra-dient remains constant. Specifically, they suggested that thehead differences between different observation points(hydraulic gradient) can be analyzed using steady-shapeflow model during the steady-shape regime even thoughthe drawdown remains transient. Hu et al. [2011] similarlyused the steady-shape concept and travel time approach toanalyze the HT survey. Bohling [2009] further claimed thatthe steady-shape approach reduces the influence of uncer-tainty in boundary conditions when compared with theapproach that uses drawdowns in a steady-state flow model.Bohling et al. [2002], Bohling [2009], and Bohling and

Butler [2010] also advocated that neither steady-shape nortransient approach for HT analysis was capable of reveal-ing hydraulic conductivity variations outside of the regionencompassed by the pumping and observation wells.

[7] This paper investigates the time of drawdown meas-urements of well hydrographs that can maximize the reso-lutions of T and S estimates in the analysis of a HT survey.We employed a first-order cross-correlation analysis toinvestigate the temporal and spatial evolutions of cross-correlation between the head at an observation well and Tand S in homogeneous and heterogeneous aquifers during apumping test. Based on the analysis, we explored the headinformation content at different time periods during apumping test on spatial distribution of T and S, then pro-posed a temporal sampling strategy for HT analysis, and atlast tested it using numerical examples with and withoutprior knowledge of the boundary conditions. At the end,the cross-correlation analysis is used to explain the robust-ness of HT on T estimation, even beyond well field, andalso to explain the limitation of HT on S estimation.

2. Cross-Correlation Analysis

[8] The cross-correlation analysis presented in this paperassumes that the groundwater flow in two-dimensional,depth-averaged, saturated, heterogeneous, or homogeneousaquifers can be described by the following equations:

r � T xð ÞrH½ � þ Q xp

� �¼ S xð Þ @H

@tð1Þ

subject to boundary and initial conditions:

H jG1 ¼ H1; T xð ÞrH½ � � njG2 ¼ q; and H jt¼0 ¼ H0 ð2Þ

where H is the total head [L], x is the spatial coordinate(x¼{x, y}, [L]), Q(xp) is the pumping rate (L3/T/L3/) at thelocation xp, T(x) is the transmissivity [L2/T], and S(x) is thestorage coefficient [–]. If the aquifer is conceptualized ashomogenous, values of T(x) and S(x) are independent of x.H1 is the prescribed total head at Dirichlet boundary u1, q isthe specific flux at Neumann boundary u2, n is a unit vectornormal to the union of u1 and u2, and H0 represents the ini-tial total head.

[9] These equations are necessary to derive the cross-correlation between the head or drawdown at an observa-tion well and the transmissivity and storage coefficientfields of homogeneous and heterogeneous aquifers.

2.1. Homogeneous Aquifers

[10] We employ a probabilistic approach in order to ana-lyze the uncertainty of the prediction of transient hydraulichead response to a pumping test in a homogeneous aquiferwith uncertain T and S values, either due to measurementerrors or lack of measurements. This approach assumes thatthe natural log of T (ln T) and that of S (ln S) are randomvariables, and ln T ¼ Y þ y and ln S ¼ Z þ z, where Y andZ are their means and y and z denote the perturbations.Note that logs of the parameters are dimensionless. Like-wise, the uncertainty in the head is represented byH ¼ H þ h, where H is the mean and h is the perturbation.Expanding the transient hydraulic head in equation (1) in a

SUN ET AL.: A TEMPORAL SAMPLING FOR TRANSIENT HYDRAULIC TOMOGRAPHY

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Taylor series about the mean values of parameters, andneglecting second-order and higher order terms, the head per-turbation at location x at a given time t can be expressed as:

h x; tð Þ � @H x; tð Þ@ln T

��Y ;Z

" #yþ @H x; tð Þ

@ln S

��Y ;Z

" #z

¼ Jhy x; tð ÞyþJhz x; tð Þzð3Þ

where Jhy [L] and Jhz [L] are scalars and are the sensitivityof h at location x at a given time t with respect to change inln T and ln S values of the homogeneous aquifer (scalars),respectively. Equation (3) states that the head perturbationat x and t is approximately a weighted sum of the perturba-tions, y and z. The weights are the sensitivity of h to y andthat of h to z at (x, t).

[11] If the perturbations are casted into a probabilisticframework, and if the perturbation of ln T and that of ln S(i.e., y and z, respectively) are mutually independent fromeach other, the cross-covariance of h and y and that of hand z can be given as

�2hy x; tð Þ ¼ Jhy x; tð Þ�2

y and �2hz x; tð Þ ¼ Jhz x; tð Þ�2

z ð4Þ

where �2y and �2

z are the variances of ln T and ln S, whichrepresent the uncertainty associated with the ln T and ln Svalues of the homogeneous aquifer. The correspondinghead variance at (x, t) based on equation (3) is given as:

�2h x; tð Þ ¼ Jhy x; tð Þ

� �2�2

y þ Jhz x; tð Þð Þ2�2z ð5Þ

[12] The cross-covariances, �2hy x; tð Þ [L] and �2

hz x; tð Þ[L],are then normalized by the square root of the product of thevariances of h(x, t) and ln T (i.e., �2

h x; tð Þ and �2y , respec-

tively) or those of h (x, t) and ln S (i.e., �2h x; tð Þ and �2

z ,respectively) to obtain their corresponding cross-correlations �hy and �hz at location x at time t.

�hy x; tð Þ ¼�2

hy x; tð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

h x; tð Þ�2y

q ¼Jhy x; tð Þ�2

yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJhy x; tð Þ� �2

�2y þ Jhz x; tð Þð Þ2�2

z

� ��2

y

r

�hz x; tð Þ ¼ �2hz x; tð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

h x; tð Þ�2z

q ¼ Jhz x; tð Þ�2zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Jhy x; tð Þ� �2

�2y þ Jhz x; tð Þð Þ2�2

z

h i�2

z

r :

ð6Þ

[13] Note that these cross-correlations are dimensionless(ranging from �1 to þ1) and represent the relationshipbetween the uncertainty of a given parameter of the homo-geneous aquifer and the uncertainty in the head at a givenlocation and time due to uncertainty of all parameters. Ineffect, the cross-correlation between the head and a param-eter is a product of a weighted sensitivity and uncertaintyof the parameter.

[14] To analyze the cross-correlation between thehydraulic head at a given location and T and S parametersin unbounded homogeneous and isotropic aquifers, we usethe Theis solution [Theis, 1935]. That is, the drawdown,s r; tð Þ, at time t in an observation well at a radial distance r

from a pumping well, which discharges at a constant rateQ, is given by

s r; tð Þ ¼ Q

4�TW uð Þ; W uð Þ ¼

Z 1u

e-u

udu; and u ¼ r2S

4Ttð7Þ

[15] Equation (7) leads to the following dimensionlesssensitivities,

T@H r; tð ÞQ@ln T

¼ � T@s r; tð ÞQ@ln T

¼ 1

4�w uð Þ � e�u½ � ð8Þ

T@H r; tð ÞQ@ln S

¼ � T@s r; tð ÞQ@ln S

¼ 1

4�e�u ð9Þ

[16] Subsequently, the cross-correlation between h andln T can be expressed as:

�hy ¼@H r;tð Þ@ln T

��Y ;Z

�2yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@H r;tð Þ@ln T

��Y ;Z

� �2�2

y þ@H r;tð Þ@ln S

��Y ;Z

� �2�2

z

� �2

y

s

¼ w uð Þ � e�uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw uð Þ½ �2 � 2w uð Þe�u þ 1þ 1=bð Þe�2u

qð10Þ

[17] The cross-correlation between h and ln S thenbecomes

�hz ¼@H r;tð Þ@ln S

��Y ;Z

�2zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@H r;tð Þ@ln T

��Y ;Z

� �2�2

y þ@H r;tð Þ@ln S

��Y ;Z

� �2�2

z

� �2

z

s

¼ e�uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib w uð Þ½ �2 � 2bw uð Þe�u þ bþ 1ð Þe�2u

qð11Þ

[18] Note that in equations (10) and (11), we assume that�2

y ¼ b�2z , where b is a constant.

[19] To investigate the effects of a straight-line imperme-able or recharge boundary on these cross-correlations, weuse the image well (superposition) approach [Bear, 1972].That is, the drawdown or built up, sa r; tð Þ, at a distance rfrom the pumping or injection well can be formulated usingthe Theis solution as:

sa r; tð Þ ¼ s r; tð Þ6si ri; tð Þ ¼ Q

4�TW uð Þ6 Q

4�TW uið Þ ð12Þ

where ui ¼ r2i S=4Tt ; ri denotes the distance between the

observation well and the image well ; the minus sign inequation (12) is used for the recharge boundary conditionand the plus sign is for the impermeable boundary. Accord-ingly, we obtain the dimensionless sensitivities of the heador drawdown with respect to change in ln T or ln S.

T@H r; tð ÞQ@ln T

¼ �T@sa r; tð ÞQ@ln T

¼ 1

4�w uð Þ � e�u½ �6 w uið Þ � e�ui½ �ð Þ

ð13ÞT@H r; tð Þ

Q@ln S¼ �T@sa r; tð Þ

Q@ln S¼ 1

4�e�u6e�uið Þ ð14Þ


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[20] Subsequently, we have the expressions for their cor-responding cross-correlations:

�hy ¼w uð Þ � e�uð Þ6 w uið Þ � e�uið Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w uð Þ6w uið Þð Þ2 � 2 w uð Þ6 w uið Þð Þ e�u6e�uið Þ þ 1þ 1=bð Þ e�u6 e�uið Þ2q ð15Þ

�hz ¼e�u6 e�uið Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b w uð Þ6w uið Þð Þ2 � 2b w uð Þ6w uið Þð Þ e�u6 e�uið Þ þ 1þ bð Þ e�u6 e�uið Þ2q ð16Þ

[21] The cross-correlations between h and ln T and thatbetween h and ln S (i.e., equations (15) and (16)) are plottedas a function of dimensionless time t� which equals to 1/u inthree homogeneous aquifers in Figure 1a. They are (1) anunbounded aquifer with infinite lateral extents (equations(10) and (11)), (2) an aquifer with a line impermeableboundary (equations (15) and (16) with plus sign), and (3)an aquifer with a line recharge boundary (equations (15) and(16) with minus sign). In all these cases, the value of b is setto one. The observation well is assumed to be on the lineperpendicular to the boundary and connecting the pumpingwell and the boundary. Effects of distances between the ob-servation well and the boundary (i.e., 1r, 5r, and unbound)on the cross-correlations are also plotted in this figure. Thedistance 5r means that the distance between the boundaryand the observation well is five times, the distance betweenthe observation and pumping well, and so on.

[22] As illustrated by the solid black and red lines (forthe unbounded aquifer with infinite lateral extents) in Fig-ure 1a, at early times the head at the observation well is

positively correlated with ln S and negatively correlatedwith ln T. Physically, this means that if the observed headis high (or drawdown is small), the S value is likely to belarge and the T value is small, and vice versa. At intermedi-ate time (t� ¼ 2.2) in the figure when the correlation of hand ln T is zero, the cross-correlation of the h and ln Sreaches maximum. This dimensionless time can be con-verted to real time as:

tm ¼2:2r2S

4T¼ r2S

1:8Tð17Þ

[23] This time is approximately equal to the intercept t0on time axis of a log time-drawdown plot of well hydro-graph at any r, when the straight line portion of the plot isextrapolated to the point where the drawdown is zeroaccording Cooper-Jacob’s approach [Cooper and Jacob,1946]. That is,

t0 ¼r2S

2:25Tð18Þ

Figure 1. Cross-correlation between the head at an observation well and aquifer properties ln T (hy)and ln S (hz) as a function of dimensionless time for homogeneous aquifers with (a) different boundariesand (b) different b values (the ratio of variance in ln T to that of ln S). Unbnd denotes unbounded aqui-fers, Im stands for aquifers with an impermeable boundary, and Re stands for aquifers with a rechargeboundary. r is the distance between the pumping well and observation. Im-5r or Re-5r denotes that thedistance between the impermeable or recharge boundary and the observation is 5 times of r.


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[24] Afterward, the value of �hy continuously grows andapproaches some asymptotic value (close to 1) at largetimes, whereas the �hz value decreases rapidly andapproaches some value (zero if steady-state condition ismet). These results are consistent with the fact that at earlytime the head is controlled by both T and S and is onlyinfluenced by T near steady-state. Therefore, if the head inan observation well is high, the T of the aquifer must behigh.

[25] As expected, the cross-correlations exhibit effects ofeither impermeable or recharge boundary once the drawdownreaches the boundary (see Figure 1a). In comparison with thecross-correlation between head and ln T of the unbounded aq-uifer, the presence of the recharge boundary accelerates therise of the correlation and the impermeable boundary delaysit. On the other hand, the recharge boundary accelerates thereduction of the correlation between head and ln S to zero,while the impermeable boundary postpones it. As shown inthe figure, the dimensionless time where the maximum cross-correlation between head and ln S occurs is independent ofthe boundary location and type.

[26] Effects of the magnitude of b (ratio of �2y to �2

z ) onthe cross-correlations are illustrated in Figure 1b forunbounded media. The larger the b value is, �hy starts at amore negative value, and the earlier it reaches one. On theother hand, the larger the b value is, the smaller the�hzvalue starts, the faster it rises, and the sharper it drops.The value of b does not alter the time to reach maximum�hz. These effects are also observed for aquifers with imper-meable or recharge boundary.

2.2. Heterogeneous Aquifers

[27] To analyze the relationship between the head and lnT and that between the head and ln S values in a heteroge-neous aquifer, we assume that ln T and ln S at every loca-tion of the aquifer are random variables with some spatialcorrelation (i.e., the ln T and ln S fields of the aquifer arecollections of random variables: Stochastic processes orrandom fields). Again, ln T ¼ Y þ y and ln S ¼ Z þ z,where Y and Z are mean values and y and z denote pertur-bations, which represent spatial variability or uncertaintydue to lack of measurements of these parameters. Likewise,the head is represented by H ¼ H þ h, where H is themean and h is the perturbation caused by spatial variabilityor uncertainty of the parameters. The first-order approxima-tion of the head perturbation at location xi at a given timet is then given as:

h xi; tð Þ � @H xi; tð Þ@ln T xj

� � ��Y ;Z

" #y xj

� �þ @H xi; tð Þ

@ln S xj

� � ��Y ;Z

" #z xj

� �¼ Jhy xi; xj; t

� �y xj

� �þJhz xi; xj; t

� �z xj

� � ð19Þ

where y xj

� �and z xj

� �are perturbation of ln T and ln S at

location xj and j¼1, . . . , N, which is the total number ofparameters in the aquifer (i.e., number of elements in a fi-nite element domain); Jhy xi; xj; t

� �and Jhz xi; xj; t

� �are the

sensitivity of h at location xi at a given time t with respectto ln T and ln S perturbation at location xj. Here, the Ein-stein’s summation convention over the repeated suffix isused. In other words, the head perturbation at xi; tð Þ is a

weighted sum of perturbation of parameters ln T and ln Severywhere in the aquifer. The weights are the correspond-ing sensitivity values. Assuming ln T and ln S are mutuallyindependent from each other, the cross-covariance matricesbetween h and y and between h and z are

Rhy xi; xj; t� �

¼ Jhy xi; xj; t� �

Ryy xi; xj

� �i; j ¼ 1; . . . ;N

Rhz xi; xj; t� �

¼ Jhz xi; xj; t� �

Rzz xi; xj

� � ð20Þ

[28] Ryy xi; xj

� �and Rzz xi; xj

� �are covariance matrices of

perturbation of ln T and ln S, which are modeled with thesame exponential function using the same correlationscales Lx, Ly, and Lz in x, y, and z directions, respectively.The corresponding head covariance matrix based on equa-tion (19) is given as:

Rhh xi; xj; t� �

¼ Jhy xi; xj; t� �

Ryy xi; xj

� �Jhy

T xi; xj; t� �

þJhz xi; xj; t� �

Rzz xi; xj

� �Jhz

T xi; xj; t� � ð21Þ

[29] The superscript T denotes the transpose. The com-ponents of Rhh xi; xj; t

� �at xi ¼ xj are the head variances

�2hh xi; tð Þ, which represents the uncertainty in head at loca-

tion xi at a given time t due to the unknown heterogeneityin the aquifer. The cross-covariances, Rhy and Rhz, are thennormalized by the square root of the product of the varian-ces of h at xi; tð Þ and ln T or those of h at xi; tð Þ and ln S toobtain their corresponding cross-correlation �hy and �hz atlocation i and j at time t.

�hy xi; xj; t� �

¼Jhy xi; xj; t� �

Ryy xi; xj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

h xi; tð Þ�2y

q ¼Jhy xi; xj; t� �

Ryy xi; xj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRhh xi; xi; tð ÞRyy xj; xj

� �q�hz xi; xj; t� �

¼Jhz xi; xj; t� �

Rzz xi; xj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2

h xi; tð Þ�2z

q ¼Jhy xi; xj; t� �

Rzz xi; xj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRhh xi; xi; tð ÞRzz xj; xj

� �qð22Þ

where �2h xi; tð Þ is the head variance at xi and t ; �2

y and �2z

are the variances of ln T and ln S, respectively. The cross-correlation (dimensionless) represents how the head pertur-bation at the location xi at a given time t is influenced bythe ln T or ln S perturbation at location xj in a statisticalsense. With a given mean T, S, and a pumping rate, thesecross-covariances are evaluated numerically using the HTinverse model by Zhu and Yeh [2005] which is an extensionof the earlier work by Yeh et al. [1996], Zhang and Yeh[1997], Li and Yeh [1999], Hughson and Yeh [2000], andYeh and Liu [2000].

[30] The above cross-correlation analysis in heterogene-ous aquifers is similar to the sensitivity analysis by Oliver[1993] but it adopts the stochastic or geostatistics concept.In particular, the cross-correlation analysis considers thevariance (spatial variability) of the parameter and its spatialcorrelation structure (covariance function or variogram ofthe parameter) in addition to the most likely flow field,which is considered in the sensitivity analysis. Physically,the correlation structure represents the average dimensionsof aquifer heterogeneity. The cross-correlation in essencerepresents the statistical relationship of spatial variabilityof a given parameter (T or S) at any location and the vari-ability of head at an observation location in the aquifer.


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Note that the cross-correlation is the foundation of cokrig-ing approach [e.g., Kitanidis and Vomvoris, 1983; Hoek-sema and Kitanidis, 1984; Yeh and Zhang, 1996; Li andYeh, 1999; Yeh et al., 1995], nonlinear geostatisticalinverse approach [e.g., Kitanidis, 1995; Yeh et al., 1996;Zhang and Yeh, 1997; Hanna and Yeh, 1998; Hughson andYeh, 2000], and HT inverse model [e.g., Yeh and Liu,2000; Zhu and Yeh, 2005], geostatistical inverse modelingof electrical resistivity tomography [Yeh et al., 2002].

[31] To illustrate the behaviors of equation (22), asquare-shaped synthetic 2-D confined aquifer (200 m �200 m) was discretized into 100 � 100 square elementswith 2 m in length and width. Four sides (i.e., east, west,north, and south) of the aquifer were assumed to have aprescribed hydraulic head of 100 m. An observation wellwas placed at x¼ 80 m, and y¼ 100 m and a pumping wellwas placed at x¼ 120 m, y¼ 100 m, where a constant dis-charge of 0.0006 m3/s was imposed. The geometric meanvalues of T and S of the aquifer are 0.000116 m2/s andS¼ 0.00014, respectively. The variance of ln T is 1.0 and

variance of ln S is 0.2. Covariance functions of T and S arebased on the exponential model with isotropic correlationscales in x and y directions that are 30 m. These propertiesof the aquifer and pumping rate are based on the fieldexperiments reported by Way and McKee [1982].

[32] Contour maps of the cross-correlations between thehead at an observation well and T perturbation everywherein the domain (i.e., �hy) as well as the head field are plottedin Figures 2a and 2c for two early times (i.e., dimensionlesstime t� ¼ 0.4 and 2.2), and Figures 3a and 3c for late timeand steady-state (i.e., t� ¼ 6.2 and 41.4), respectively. Thedimensionless time t� which equals to 1/u was evaluatedusing the mean values of T and S. The corresponding cross-correlation maps for the head and the S field (�hz) areshown in Figures 2b, 2d, 3b, and 3d. Note that the headfields are the means, which were obtained using the mean Tand S values.

[33] At very early time (t� ¼ 0.4, t¼ 200 s), the �hy val-ues are significantly negative over a large area and are withthe highest correlation (i.e., around �0.6) between the

Figure 2. Contour maps of cross-correlation between the head at the observation well (the white circle)and ln T (a and c) and ln S (b and d) everywhere in the aquifer at early dimensionless time 0.4, and 2.2after pumping started at the pumping well (the black circle). The aquifer is surrounded by constant headsof 100 m. White contours are equipotential lines.


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pumping well and observation well (Figure 2a). Mean-while, the cross-correlation �hz ranges from 0.1 to 0.4 (Fig-ure 2b). The high positive values (i.e., around 0.4) areconfined to the small circular area between the pumpingand the observation well. Within the aquifer domain, thenonzero velocities at this very early time are limited to thevicinity of the pumping well and the drawdown has notpropagated to the observation well yet. As a result, the highcross-correlation at this time stage is not useful to relate hy-draulic properties to the head at the observation well andbeyond.

[34] At early time (t� ¼ 2.2, t¼ 1100 s, approximatelyequal to t0 time), the noticeable drawdown have expandedto the observation well, but only to some limited arealextents (Figures 2c and 2d). The �hy values are close tozero everywhere except the small circular area centeredbetween the pumping well and the observation well, wherethe values range from �0.5 to 0 (Figure 2c). The negativecorrelation means that if the head at the observation well ina heterogeneous aquifer is higher than the head calculatedfrom the mean T and S values of the aquifer, the T valuesdownstream of the observation are likely to be lower thanthe mean T value. Meanwhile, the �hz values are rangingfrom 0.1 to 0.4 over the area with the highest correlationare located in a small circular area between the pumpingwell and the observation well (Figure 2d). The positivecross-correlation implies that if the observed head in a het-erogeneous aquifer is high in comparison with the headderived from the mean T and S values of the aquifer, theS values in the area between the pumping well and the ob-servation well are likely to be higher than the mean S value.Note that the area of significant �hz values is much largerthan that of �hy. These high cross-correlation values suggestthat the head at the observation well at this time is highlyinfluenced by the S heterogeneity within the small redcircle with �hz¼ 0.4 enclosing the pumping and the obser-vation wells although the area outside the circle also hassome influence.

[35] At t� ¼ 6.2 (intermediate time, t¼ 3000 s), as shownin Figures 3a and 3b, the edge of the cone of depression hasreached the right-hand side of the constant head boundary.The shapes of �hz remain similar to that at t� ¼ 2.2 but thevalues drop below 0.4 (Figure 3b), indicating that the effectof storage coefficient on head is diminishing and flow ismoving toward a steady-state. At this time, �hy has evolvedto positive values everywhere. The cross-correlation pat-tern forms two kidney-shaped humps: One at the left-hand(upstream) side of the observation well and the other at theright-hand (upstream) side of the pumping well (Figure 3a).The values in the area between the two wells are smaller.

[36] In order to explain the formation of this peculiar dis-tribution, we will examine the cross-correlations along twostreamlines opposite to each other since the flow is radialsymmetrical and close to steady-state. The first streamlineis the one starting from the left constant head boundary andpassing through the observation well to the pumping well.The second one is the streamline starting from the right-hand side constant head boundary to the pumping welldirectly opposite to the first streamline. We will discuss thevariation of �hy along the first streamline first. If the flow tothe pumping well approaches a steady condition, and thehead at the observation well is higher than the head at the

same location that is simulated with the mean T, the aver-age hydraulic gradient along the streamline upstream fromthe observation well is smaller than the mean gradient ofthe homogeneous medium. The T values upstream, there-fore, are most likely to be higher than the mean T, and theT values along the same streamline downstream from theobservation well to the pumping well are likely to besmaller. If the T values between the observation and pump-ing well are small on average, then the contribution to thepumping well from this streamline is likely to be smallerthan that calculated using the mean T. As a result, the con-tribution from the second streamline to the pumping wellmust increase, and the T values, on average, along the sec-ond streamline upstream from the pumping well are in turngreater than that based on the mean T in order to sustain thewell discharge.

[37] For this radial flow field, all streamlines converge tothe pumping well. The cross-correlation distributionbetween head at observation well and ln T along eachstreamline follows the same principle as discussed above.However, such a cross-correlation pattern along otherstreamlines diminishes as the distance between thesestreamlines and the streamline that passes through theobservation well increases. As a result, two kidney-shapedhumps of high correlation values (one near the observationwell and one near the pumping well) form.

[38] When the flow state reaches the steady-state(t� ¼ 41.4 or t¼ 20,000 s), the shape of the spatial distribu-tion of �hy values are similar to that at t� ¼ 6.2 (Figure 3a)but their magnitudes everywhere in the aquifer are elevated(Figure 3c). Meanwhile, the �hz values diminish to zeroeverywhere reflecting the fact that the steady hydraulichead is not related to S (Figure 3d). Note that this twokidney-shaped spatial pattern of the cross-correlationbetween ln T and the head is different from the sensitivitybehavior based on radial symmetric flow model as used byBohling et al. [2002], which indicates that the head is onlysensitive to T values in the region between the observationwell and the pumping well.

[39] As shown in equation (22), the cross-correlationswill be affected by the covariance functions of the parame-ters. When the correlation scales in both directions decreaseto 2 m (one element size) (i.e., the parameter fields becomeuncorrelated random fields), the spatial patterns of thecross-correlation functions are identical to that of the sensi-tivity results by Oliver [1993] and Leven and Dietrich[2006], with the exception that they are normalized. On theother hand, when the correlation scales become muchlarger than the domain size, the temporal behavior of thecross-correlation functions becomes that for homogeneouscase as shown in Figures 1a and 1b. These results in Figure3 are similar to those in Figure 10 of Wu et al. [2005].

[40] These spatial patterns of �hy and �hz provide someinsight to the limitation of HT for S estimation and itsrobustness for T estimation. Specifically, heterogeneities atthe same correlation contour have the same amount of con-tribution to the head responses at the observation well. Theradial symmetrical pattern of �hz (Figures 2b, 2d, 3b, and3d) thus indicates that contributions of an S anomaly in theaquifer will be the same for any pairs of a pumping and anobservation well that are separated by the same distanceand centered at the same location. Hence, it is difficult to


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identify the S anomaly unless data from pairs of wells withdifferent separation distances are available. Furthermore,the high resolution of the S estimates is only observedwithin the high correlation circular area enclosed by thetwo wells.

[41] The spatial patterns of �hy evolve from a symmetricpattern at early times, similar to those of �hz to radial non-symmetric two-hump patterns at late times. The highest �hyvalues are at the two humps at the upstream of the observa-tion and the pumping well. This unique two-hump patternimplies that if the location of one of the two wells (i.e., theflow) is altered and the heterogeneity at the hump associ-ated with the new location is different from that at the pre-vious location, the head at the observation well (regardlessif its position is changed or not) will be different and willcarry new information about heterogeneity. Furthermore,�hy values over entire area of cone of depression are rela-tively high at late time in comparison with other times.Therefore, adding more head data at late time or steady-state from new observation or pumping locations in the

inverse modeling process helps to decipher the spatial dis-tribution of T anomalies, and thus yield a better resolutionof the spatial distribution of heterogeneity. This is the rea-son that joint interpretation of sequential pumping tests ormultiwell pumping tests are superior to inverse modelingusing one pumping test as demonstrated by Huang et al.[2011]. The flow-dependent nature of �hy (or the flow inde-pendence nature of �hz) also explains that HT is a more ro-bust approach for estimating T than S, and HT can depictheterogeneity patterns not only within the well filed butalso far away from the well field although at lowresolutions.

3. Temporal Sampling Strategy for HT analysis

[42] According to temporal and spatial distributions ofthe cross-correlations, and head fields presented in the pre-vious section, we conclude that the head observed at thegiven observation well at t� ¼ 2.2 (or t0) carries the highestlevel of information about S perturbations over a large area

Figure 3. Contour maps of cross-correlation between the head at the observation well (the white circle)and ln T (a and c) and ln S (b and d) everywhere in the aquifer at late dimensionless times 6.2, and 41.4after pumping started at the pumping well (the black circle). The aquifer is surrounded by constant headsof 100 m. White contours are equipotential lines. Black lines with arrows are streamlines.


3888

of influence. The head data at this time is highly desirablefor estimating S distribution. On the other hand, at late timeor steady-state, the observed head at the given observationlocation carries the greatest level of information about Tperturbations over a larger portion of the aquifer in compar-ison with the head at other time periods. Furthermore, the bvalues in equations (15) and (16) are generally greater than10 or above based on our knowledge of T and S values formost aquifers. As indicated in Figure 1b, for these b values,the correlation between the head and S at these values dropsrapidly after t0. There is only a very small time windowwhere the drawdown is more correlated with S than T, and

for most of the time during a pumping test, drawdownin most aquifers is dominated by T values. Hence, it ismost appropriate to choose the head data at steady-stateor late times to estimate the distribution of T, usingHT. Combining these two observations, we hypothesizea temporal sampling strategy for interpretation of HT.That is, pumping tests of HT should last sufficientlylong such that drawdown reaches the entire area of in-terest. HT analysis should use head data at the earlytime (t0) and late time (or steady-state) for the estima-tion of T and S to maximize the power of HT as wellas to reduce computational expenses.

Figure 4. The true T and S fields of the synthetic aquifer are shown in Figures 4a and 4b, respectively.White circles are the nine wells used in HT and dashed white lines delineate the four zones for HT per-formance evaluation. The estimated T and S field from the HT analysis with the correct impermeableboundary on the east side using the pair of head data at t0 and steady-state are illustrated in Figures 4cand 4d. The estimated T and S fields using a wrong boundary condition (constant head) on the east sideare shown in Figures 4e and 4f. The impermeable boundary is manifested as low permeable zones in theestimated T field in Figure 4e.


3889

[43] To test this hypothesis which is built upon the sto-chastic ensemble concept and the first-order analysis, wecreated a synthetic, two-dimensional confined aquifer. Thisconfined aquifer has the same dimension, numerical discre-tization, and spatial statistics describing heterogeneity ofT and S of the aquifers used in section 2.2. However, theright-hand side of this aquifer is an impermeable boundaryand all others three boundaries are constant head of 100 m.One realization of 10,000 pairs of T and S values of the aq-uifer were generated by the spectral method [Gutjahr,1989] using the spatial statistics in section 2.2. The T and Sfields were assumed to be independent. The generated Tand S fields are shown in Figures 4a and 4b, respectively.

[44] Nine wells were placed in the aquifer. The coordi-nates for the nine wells from 1 to 9 (small circles in Figures4a and 4b) are (80 m, 120 m), (100 m, 120 m), (120 m, 120m), (80 m, 100 m), (100 m, 100 m), (120 m, 100 m), (80 m,80 m), (100 m, 80 m), and (120 m, 80 m), respectively. Apumping test was simulated at one of the five wells (wells1, 3, 5, 7, and 9) with a constant pumping rate 0.0006 m3/s(51.84 m3/d) for 70,000 s with a time step of 5 s, and thehead responses at the other eight wells were monitored.This simulation of the pumping test was repeated for theother wells until all five wells were considered. As a result,there are five pumping tests and each test has eightobserved drawdown-time curves (Figure 5). For this hetero-geneous aquifer, the flow field has become steady at nearly30,000 s. Using the eight drawdown-time curves for eachpump test, eight t0 values of Cooper-Jacob straight-linemethod were obtained for each one of the pumping tests.Since the distance between observation and pumping wellsare different and the aquifer is heterogeneous, t0 time foreach drawdown-log time curve is different and they rangearound 160–700 s. The reason we use t0 instead of tm (equa-tion (17)) are the following: (1) the difference between thetwo is small, (2) tm requires the knowledge of the mean Tand S, which are unknowns themselves, and (3) t0 can beobtained conveniently from well hydrographs of each indi-vidual well.

[45] Drawdowns (i.e., heads) at some selected times ofsimulated drawdown-time curves of the HT survey (Figure5) were sampled to test our proposed strategy for estimat-ing 10,000 pairs of T and S values of the aquifer. Theseselected time intervals include t0, 1000 s, 2000 s, 3000 s,5000 s, 8000 s, and steady-state. These head data are noise-free since noise effects on HT interpretations have beeninvestigated extensively by Xiang et al. [2009] and Maoet al. [2013]. The principle of reciprocity [Bruggeman,1972] was not considered in the HT analysis even though itcan reduce the analysis time.

[46] The interpretation or inverse modeling usingselected head data were carried out by employing Simulta-neous Successive Linear Estimator (SimSLE) [Xiang et al.,2009]. Instead of incorporating data sequentially into theestimation as in SSLE [Zhu and Yeh, 2006], SimSLEincludes all selected drawdown data from different pump-ing tests during an HT survey simultaneously to estimatehydraulic properties of aquifers. A detailed description ofthe SimSLE can be found in Xiang et al. [2009].

[47] To evaluate the estimates, scatter plots of the truevs. estimated T and S values for each case are plotted and alinear model is then fitted to each case without forcing the

intercept to zero. The slope and intercept of the fitted linearmodel, the coefficient of determination (R2), the meanabsolute error (L1), and the mean square error (L2) normsare then used for evaluation since a single criterion is notsufficient. The L1 and L2 norms of T and S are computedas:

L1 ¼1

N

Xn

i¼1

jln Pi� � ln Pij and L2 ¼

1

N

Xn

i¼1

ln Pi� � ln Pið Þ2 ð23Þ

where N is the total element of the model, i indicates theelement number, Pi

� is the estimated T or S value for theith element, and Pi is the true T or S value of the ithelement.

[48] In order to further evaluate the influence of boundaryon the estimates, and to examine the resolution of HT esti-mates at different distances away from well field, we dividedthe aquifer into four zones using white dashed lines. As illus-trated in Figure 4a, zone 1 represents the smallest squarearea, which is the area surrounded by the nine wells (wellfield); zone 2 is the next large square; while zone 3 denotesthe area surrounded by the line, which includes zones 1 and2; and zone 4 is the entire aquifer in Figure 4a.

[49] Three cases using head data at the five differentsampling times as discussed previously were examined.Unless otherwise specified, analysis of all cases used theknown geometric means, variances, and correlation scalesof the T and S fields as the inputs to SimSLE. The effectsof uncertainty of these parameters on HT interpretation aresmall as reported by Yeh and Liu [2000]. Their finding hasalso been verified by HT applications to many sandbox andfield experiments over the past decade.

3.1. Case 1[50] Five pairs of T and S fields were estimated using

head data at the five pairs of sampling times: (1) 1000sþ t0, (2) 3000 sþ t0, (3) 5000 sþ t0, (4) 8000 sþ t0, and(5) steady-stateþ t0. Again, t0 time is different for eachobserved well hydrographs during HT.

[51] Evaluation metrics (L1, L2, R2, and slope) of the esti-mated T fields in the four zones (different colors) are plot-ted for the five sampling time pairs (i.e., 1, 2, 3, 4, and 5 onthe horizontal axis) in Figures 6a–6d, respectively. As indi-cated by the four metrics, the pair of t0 and late time dataalways results in a better of T field for all four zones thanthe pair of t0 and the early time. These figures also showthat the estimated T fields in zone 1 (area within the wellfield) using the five different sampling time pairs are con-sistently better than zones 2, 3, and 4. Note that theimprovement of T estimates in zone 1 tails off but improve-ments in outer zones continue although slightly as headdata at later times were used.

[52] Plots of the evaluation metrics (L1, L2, R2, andslope) of the estimated S fields in the four zones (differentcolors) for the five sampling time pairs (i.e., 1, 2, 3, 4, and5 on the horizontal axis) are given in Figures 7a–7d,respectively. According to these figures, the pair of headdata at late time and those at t0 also leads to better S esti-mates for the four zones than the pair of the early time andt0. Overall, these metrics consistently indicate that the Sestimates in zone 1 (area within the well field) are betterthan zone 2, and so on, although there are some deviations


3890

(e.g., Figure 7c). Furthermore, results shown in bothFigures 6 and 7 indicate that the pairs of head data atsteady-state with those at t0 yields best estimates of T and Sfields for all zones.

3.2. Case 2[53] To further confirm the hypothesis that the maximum

cross-correlation between head and S at t0 can lead to betterestimate of S field, head data from three pairs of samplingtimes, that is, (1) steady-state and t0, (2) steady-state and

1000 s, and (3) steady-state and 2000 s, were employed toestimate three sets of S fields. In this case, we first usedsteady-state head data and the steady HT code to estimate Tfield. Then, we treated the estimated T fields as known, andproceeded to estimation of three S fields using head data att0, 1000 s, and 2000 s, respectively, using the transient HTcode. Plots similar to those in Figures 6 and 7 are shown inFigure 8. All evaluation metrics (Figures 8a–8d) confirmthat data from t0 and T estimates from the steady-state yield

Figure 5. Drawdown time curves of five pumping tests. In the legend, the first digit denotes the pump-ing well number and the second digit represents that of the observation well.


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the best estimated S fields for all four zones (zone 1, in par-ticular). The metrics for different zones fluctuate a little fortimes 2 and 3 but they all are less satisfactory in comparisonwith time 1. These results further substantiate the fact that thehigh cross-correlation between head and S at t0 time is indeeduseful for estimating the heterogeneous S field.

[54] The estimated T and S fields using the suggestedsampling strategy (i.e., head data from t0 and steady-state)for HT analysis are illustrated in Figures 4c and 4d, respec-tively. In comparison with the true fields in Figures 4a and4b, these estimated fields are much smoother (i.e., at lowerresolutions) than the true fields. This is attributed to thefact that only nine wells were used for HT and the nature ofthe conditional expectation from SimSLE [Yeh and Liu,2000]. Nevertheless, the general patterns of dominant het-erogeneity in T and S (not limited to the area encompassedwith the wells) have merged in the estimates. In particular,the low permeable regions along the upper-right boundaryand at the left center of the aquifer (left of well #4) as wellas the high permeable zone near well #9 stand out in theestimated T field. Likewise, the low and high S regions onthe left-side and right-side of the aquifer are also depicted.Generally, the S estimates are not as vivid as in the T esti-mates, as explained before using the cross-correlationanalysis.

[55] Contrary to the claims by Bohling et al, [2002],Bohling [2009], and Bohling and Bulter [2010], the aboveresults unequivocally demonstrate that HT can reveal T andS heterogeneity outside the well field, as elucidated by ourcross-correlation analysis, and as supported by the resultsof Huang et al. [2011], Berg and Illman [2011], Liu andKitanidis [2011], Xiang et al. [2009], Liu et al. [2007], andLiu et al [2002]. Note these estimates are the conditionaleffective T and S fields, which are the most likely andunbiased T and S fields that can reproduce observed headsof the HT survey, with the given limited information.

[56] These results support our proposed samplinghypothesis: (1) pumping tests of HT should last sufficientlylong such that drawdown reaches the entire area of interestand (2) HT analysis should use head data at the early time(t0) and late time (or steady-state) for the estimation of T andS to maximize the power of HT as well as to reduce computa-tional expenses. However, these results are based on theassumption that the exact boundary conditions are known.

3.3. Case 3[57] In order to investigate impacts of a wrong boundary

on the estimates based on our suggested sampling strategy,we interpreted the same pair data at steady-state and t0 timeof Case 1 using a constant head boundary condition for thefour sides of the aquifer (i.e., the right-hand side

Figure 6. (a) L1, (b) L2, (c) R2, and (d) slope norms for estimated T field within the four zones, respec-tively, using head data at five pairs of sampling times: 1 (t0þ 1000 s), 2 (t0þ 3000 s), 3 (t0þ 5000 s), 4(t0þ 8000 s), and 5 (t0þ steady). Head data at t0 time paired with head data at later time yield better Testimates for the four zones.


3892

impermeable boundary was replaced by a constant head of100 m, the same as the other three.) The estimated T and Sfields based on this wrong boundary condition are illus-trated in Figures 4e and 4f, respectively. Comparing esti-mates based on incorrect east side boundary with the truefields (Figures 4a and 4b), we see that even though thewrong boundary condition was used, HT using SimSLEstill yield the general patterns of T and S estimates similarto the true ones. The general patterns of T and S estimatesbased on the incorrect boundary are also similar to thoseestimates based on the data set with correct boundary con-ditions (Figures 4c and 4d). However, the high T zones(greenish zones in Figure 4c) in the vicinity of the east-sideimpermeable boundary are replaced by low permeablezones (blue zones in Figure 4e) to reflect the presence ofthe impermeable boundary. The pattern of the estimated Sfield based on the incorrect boundary condition remainssimilar to that based on the correct boundary condition,except the values of S on the right-hand side increasesgreatly due to the introduction of the low permeable zonesin the region.

[58] Based on these results, we may conclude that HTanalysis using the SimSLE, and the drawdown data at t0time and steady-state yields reasonable estimates of thegeneral patterns of T and S fields even without any prior

knowledge of the boundary conditions. Specifically, usingthe assumption of constant head boundaries to interpret HTsurveys in an aquifer with an impermeable boundary, thetrue T field near the impermeable boundary was identifiedas low T zones. On the negative side, the true T field adja-cent to the boundary was incorrectly estimated to reflectthe impermeable boundary. On the other hand, we mayargue that the likely location of the impermeable boundarywas identified. As a matter of fact, this result also supportsthe interpretation by Straface et al. [2007] of an HT analy-sis of a field sequential pumping test experiment. Theyattributed the low permeability zones identified by HT,which surround the modeling domain, to possible pinch-outof the confined aquifer based on the local geologicalsetting.

[59] Overall, results of our cross-correlation analysis andnumerical experiments challenge those reported by Bohlinget al. [2002], Bohling [2009], and Bohling and Butler[2010]. We agree with Bohling and Butler [2010] that aninfinite number of possible T and S fields could match allthe head data from the sequential pumping tests. Weemphasize that nonunique or uncertain solutions could existin both forward and inverse modeling problems unless theproblem is well-defined with necessary conditions [Yehet al., 2011; Mao et al., 2013]. HT is an approach to collect

Figure 7. (a) L1, (b) L2, (c) R2, and (d) slope norms for estimated S field within the four zones, respec-tively, using head data at five pairs of sampling times: 1 (t0þ 1000 s), 2 (t0þ 3000 s), 3 (t0þ 5000 s), 4(t0þ 8000 s), and 5 (t0þ steady). Head data at t0 time paired with head data at later time yield better Sestimates for the four zones.


3893

data from an existing well field to reduce the uncertainty.We attribute the cautionary notes about limitations of HTby Bohling and Butler [2010] and Butler [2008] to theirchoice of forward model (axial symmetric radial flowmodel), and inverse methodology (i.e., pilot pointapproach) as explained by Huang et al. [2011].

[60] At last, while the cross-correlation analysis is builtupon the ensemble concept (describing the general behav-iors) and the first-order analysis (small perturbations),applications of HT to an aquifer with a single realization ofT and S fields with relatively large variability confirm itsusefulness for providing insights to the HT analysis.

4. Summary and Conclusions

[61] Selection of drawdown data at appropriate timeintervals is important to joint interpretation of HT tests.Based on the analysis of the spatial and temporal distribu-tions of the cross-correlation between head observed at awell and ln T and ln S everywhere in the aquifer, and draw-down distributions, we recommend that pumping tests ofHT should last sufficiently long such that drawdownreaches the entire area of interest. Head data at the earlytime (t0) and late time or steady-state should be used for theestimation of T and S during the HT analysis. The t0 time is

the time at which the drawdown extrapolated from firststraight line portion of an observed drawdown-log timeplot becomes zero, according to the Jacob straight-linemethod. At this time, the cross-correlation between h and Sis the highest over a large portion of the aquifer. On theother hand, the head at an observation well is highlycorrelated with ln T heterogeneity over a large portion ofthe aquifer at the late time, steady or near steady-state.Note that longer pumping tests in HT surveys allow detec-tion of heterogeneity at great distances at low resolutionsbut do not improve the estimate within the well field signif-icantly. Ultimately, resolutions of the HT estimates dependhighly on the number of monitoring wells [Yeh and Liu,2000; Liu et al., 2002].

[62] If the flow regime approaches steady-state condi-tions, one should estimate T field first using the steady-flowinverse approach and then S field by the transient flow anal-ysis using drawdowns at t0. We also recommend the use ofconstant head (or zero drawdown) boundaries for the inter-pretation of the HT data. Effects of impermeable bounda-ries manifested in head measurements will be reflected inthe estimated T field using the SimSLE.

[63] The cross-correlation analysis also offers insightsinto the robustness of HT for estimating the T distributionand its limitation for estimating S. Specifically, the radially

Figure 8. (a) L1, (b) L2, (c) R2, and (d) slope norms for estimated S field within the four zones, respec-tively, using head data at three pairs of sampling times:1 (steadyþ t0), 2 (steadyþ 1000 s), and 3(steadyþ 2000 s). Head data at t0 time paired with head data at later time yield better T estimates for thefour zones. Using T estimates from steady head data as known, head data at t0 yield the best S estimatesin comparison with other times.


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symmetric pattern of the cross-correlation of the head at awell and S heterogeneity everywhere in the aquifer duringpumping at the pumping well attributes to the difficulty indelineating the heterogeneity of S even data from morepairs of wells are available. On the other hand, the twokidney-shaped humps near the observation well and thepumping well of the cross-correlation of head and T hetero-geneity suggest that if the location of one of a pair ofpumping and observation wells is changed and the hetero-geneity near the well is not the same, the observed headcontains new information. This explains the scenario-dependent nature of T and S estimates based on singlepumping test experiment discussed by Huang et al. [2011].Furthermore, it explicates the rationale behind sequentialpumping test (HT, or multiwell interference pumping tests),and the reason that joint interpretation of multiwell pump-ing tests are superior to inverse modeling efforts using onepumping test and the same number of observation wells. Atlast, our results appears to support the call for change theway we collect and analyze data for characterizing aquifersby Yeh and Lee [2007] and promote the concept for cat-scanning the groundwater basin using natural stimuli asproposed by Yeh et al. [2008].

[64] Acknowledgments. This research was carried out during the firstauthor’s visit at the Department of Hydrology and Water Resources at theUniversity of Arizona. The first author acknowledges supported by theNational Natural Science Foundation of China (41102155, 41272258),National Basic Research Program of China (2010CB428802), the Funda-mental Research Fund for National University, China University of Geo-sciences (Wuhan) (CUGL090215, CUG120113), open research fund ofState Key Laboratory of Geohazard Prevention and Geoenvironment Pro-tection (Chengdu University of Technology) (GZ2006-07). The secondand third authors acknowledge support of NSF grant EAR-1014594, thesupport of ‘‘China Scholarship Council.’’ Supports from Jilin University,Jilin, China, and a grant from ESTCP subcontract through AMEC are alsoacknowledged. Yonghong Hao acknowledges funding from the NationalNatural Science Foundation of China 40972165. The authors would alsolike to give special thanks to Jeff Gawad and Michael Tso for proofreadingthe manuscript. Finally, the authors are also grateful to the editor, the AE,and the four reviewers for their encouraging, insightful and constructivecomments.

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A temporal sampling strategy for hydraulic tomography analysis