Journal of Hydrology 407 (2011) 115–128

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Journal of Hydrology

journal homepage: www.elsevier .com/ locate / jhydrol

Joint inversion of steady-state hydrologic and self-potential data for 3Dhydraulic conductivity distribution at the Boise Hydrogeophysical Research Site

Salvatore Straface a,b,⇑, Francesco Chidichimo a, Enzo Rizzo b, Monica Riva c, Warren Barrash d, André Revil e,f,Michael Cardiff g,1, Alberto Guadagnini c

a Dipartimento Difesa del Suolo, Università della Calabria, Rende (CS), Italyb Hydrogeosite Laboratory, CNR-IMAA, Tito - Marsico Nuovo (PZ), Italyc Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Politecnico di Milano, Milano, Italyd Center for Geophysical Investigation of the Shallow Subsurface, Boise State University, Boise, ID, USAe Department of Geophysics;Colorado School of Mines, Golden, CO, USAf ISTerre, CNRS, UMR 5559, Université de Savoie, Equipe Volcan, Le Bourget du Lac, Franceg Department of Civil and Environmental Engineering, Stanford University, Stanford, CA, USA

a r t i c l e i n f o

Article history:Received 7 June 2010Received in revised form 3 June 2011Accepted 13 July 2011Available online 23 July 2011This manuscript was handled by P. Baveye,Editor-in-Chief

Keywords:Inverse problemMultiple Indicator KrigingSelf potential

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.07.013

⇑ Corresponding author at: Dipartimento DifesaCalabria, Rende (CS), Italy.

E-mail address: straface@unical.it (S. Straface).1 Present address: Center for Geophysical Inv

Subsurface;Boise State University, Boise, ID, USA.

s u m m a r y

We combine sedimentological, hydraulic and geophysical information to characterize the 3D distributionof transport properties of an heterogeneous aquifer. We focus on the joint inversion of hydraulic head andself-potential measurements collected during an extensive experimental campaign performed at theBoise Hydrogeophysical Research Site (BHRS), Boise, Idaho, and involving a series of dipole tests. Whilehydraulic head data obtained from piezometric readings in open wells represent a depth-averaged value,self-potential signals provide an estimate of the water table location. The aquifer is conceptualized as amultiple-continuum, where the volumetric fraction of a geo-material within a cell of the numerical flowmodel is calculated by Multiple Indicator Kriging. The latter is implemented on the basis of availablesedimentological information. The functional format of the indicator variograms and associated param-eters are estimated on the basis of formal model identification criteria. Self-potential and hydraulic headdata have been embedded jointly within a three-dimensional inverse model of groundwater flow at thesite. Each identified geo-material (category) is assumed to be characterized by a constant hydraulicconductivity. The latter constitute the set of model parameters. The hydraulic conductivity associatedwith a numerical block is then calculated as a weighted average of the conductivities of the geo-materialswhich are collocated in the block by means of Multiple Indicator Kriging. Model parameters are estimatedby a Maximum Likelihood fit between measured and modeled state variables, resulting in a spatiallyheterogeneous distribution of hydraulic conductivity. The latter is effectively constrained on thesedimentological data and conditioned on both self-potential and borehole hydraulic head readings.Minimization of the Maximum Likelihood objective function allows estimating the relative weight ofmeasurement errors associated with self-potential and borehole-based head data. The procedure adoptedallowed a reconstruction of the heterogeneity of the site with a level of details, which was not obtained inprevious studies and with relatively modest computational efforts. Further validation against dipole testswhich were not used in the inversion procedure supports the robustness of the results.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

A key-problem in aquifers characterization is the requirementof a large number of direct and high resolution measurements ofsystem parameters and state variables. Measurements of state

ll rights reserved.

del Suolo;Università della

estigation of the Shallow

variables typically include the drawdown induced in observationboreholes by pumping tests. This method is intrusive andconsequently the hydrological system is perturbed by drillings.Geophysical approaches instead aim at using non-intrusive tech-niques to obtain a large amount of information on subsurfaceand with moderate costs. In the last few decades hydrologists haveincreasingly resorted to hydrogeophysical information to estimategroundwater flow parameters (Purvance and Andricevic, 2000;Slater and Lesmes, 2002; Binley et al., 2005; Yeh et al., 2008; Hördtet al.,2009; Koestel et al., 2009).

116 S. Straface et al. / Journal of Hydrology 407 (2011) 115–128

A key innovation in this field is based on the passive measure-ments of the self-potential generated by (1) ground water flowthrough a so-called electrokinetic coupling (Fournier, 1989; Birch,1993; Birch, 1998; Aubert and Atangana, 1996; Jardani and Revil,2009) and (2) electro-chemical processes associated with gradientsof the chemical potentials of charge carriers (ionic species andelectrons) in the pore water (e.g., Naudet et al., 2004; Revil et al.,2005). A general theory of all these effects has been developedby Revil and Linde (2006), and Revil et al. (2009, 2010). In theself-potential approach, the measured response is a function ofunknown source of electrical currents and resistivity structure.Therefore, there is an inherent ambiguity when interpreting theself-potential data when the earth resistivity is unknown. Thisdifficulty can be solved upon merging different kinds of measure-ments obtained from other non-intrusive techniques, DirectCurrent (DC) and Electromagnetic (EM)-based electrical resistivitytomography, seismic method and geothermal measurements(Jardani and Revil, 2009) or borehole data (Straface et al., 2010).

Interpretation schemes that could model the self-potential fieldrecorded during pumping tests have been developed (Revil et al.,2003; Rizzo et al., 2004; Titov et al., 2002; Titov et al., 2005; Jardaniet al., 2009; Malama et al., 2009a,b). Data from several pumpingtests can be combined and used in a tomographic fashion to char-acterize heterogeneity in the aquifer (for recent field examples see,Bohling et al., 2007; Li et al., 2007; Straface et al., 2007b; Cardiffet al., 2009). Recently, Straface et al. (2007a) used experimentalhydraulic heads and self potential signals associated with a pump-ing test in an inverse model based on the Successive Linear Estima-tion (SLE) (Yeh et al., 1996), to estimate the transmissivitydistribution of a small-scale aquifer. Bianchi Jannetti et al. (2010)extended the moment equations-based inverse method of Hernan-dez et al. (2003, 2006) to quasi-steady state flow conditions andpresented its first application by using hydraulic heads and selfpotential signals collected during a pumping test at the MontaltoUffugo research site. In these cited works the authors used self-potential signals in a two-dimensional inverse modeling. Never-theless, the self-potential method is able to locate the spatialdistribution of electrical sources in the earth generated by the cou-pling electrokinetic mechanism. In other words, it provides a 3Destimate of the free surface location unlike borehole readingswhich provide a depth-averaged hydraulic head. In fact, eventhough a pumping test generates a velocity field in the aquifer,the pressure distribution within an observation well, located atsome distance from a pumping well, is basically hydrostatic afterpseudo-steady state conditions has been attained. Therefore,hydraulic head does not depend on the transducer vertical posi-tion, but attains a constant value representing, in that location, adepth-averaged quantity. On the other hand, the self-potentialsource inversion problem is highly non-unique. There are severalpossible distributions of sources that can fit the data equally. Thisdilemma is common to nearly all potential field inverse problems,and is exacerbated by the fact that measurement locations areoften restricted to relatively few locations on the ground surface.Moreover, as with all geophysical techniques, data errors degradeour ability to interpret the measured signal. Common sources ofself-potential measurement error can be associated with thedegradation or drift of the measuring electrodes, poor contactbetween the electrode and soil, and cultural noise (Corwin, 1973;Perrier et al., 1997; Clerc et al., 1998; Petiau, 2000). As a conse-quence, joint inversion of self-potential and borehole based headdata requires estimating the relative weight of the different datatypes adopted (Bianchi Jannetti et al., 2010).

In this context, a relevant question is how these two types ofmeasurements can be combined within a three-dimensionalinverse modeling approach. It is important to underline that manyresearchers are making a great effort to overcome the

technological and interpretative limits to locate groundwater freesurface via the self-potential method (Minsley et al., 2007). Analternative is represented by the approach proposed by Jardaniand Revil (2009). In this work, the authors develop a joint inversionalgorithm for Self Potential (SP) and temperature data in order toestimate hydraulic conductivity. They apply their methodologyon a two-dimensional large scale real case, with 10 different zones(i.e., homogenous and anisotropic), conditioning the hydraulicconductivity on SP and Temperature. The application of theirmethodology to the Boise experiments should be possible butnot straightforward. In fact, the difficulty is to apply this inversionalgorithm to a 3D model, with the porous medium modeled asmultiple continua (fully heterogeneous) and conditioning hydrau-lic conductivity only on SP (without any hydraulic head measure-ments) that is more highly unstable on this small scale. In thispaper we adopt a joint inversion of hydraulic head and self poten-tial measurements collected during an extensive experimentalcampaign performed at the Boise Hydrogeophysical Research Site(BHRS), Boise, Idaho, in June 2007. A previous study (Cardiff etal., 2009) estimated two-dimensional depth-averaged hydraulicconductivity variations at the BHRS using only the hydrologic datafrom the above-mentioned experimental field campaign. In thisstudy, we estimate the three-dimensional distribution of thehydraulic conductivity K throughout the site by jointly invertinghydrologic and self-potential data while incorporating prior infor-mation from porosity and grain-size logs. We conceptualize theaquifer as a multiple-continuum, where the volumetric fractionof a geo-material within a cell of the numerical flow model is cal-culated by an application of Multiple Indicator Kriging (MIK). Thelatter is implemented on the basis of available stratigraphic infor-mation obtained from 18 wells and GPR reflection surveys at thesite (Barrash and Reboulet, 2004). The functional format of theindicator variograms and associated parameters are estimated onthe basis of formal model identification criteria (ICs). Each identi-fied geo-material is assumed to be characterized by a constanthydraulic conductivity. The latter constitute the set of modelparameters. The hydraulic conductivity associated with a numeri-cal grid block is then calculated as a weighted average of the con-ductivities of the geo-materials which are found in the blockaccording to the results of MIK. Model parameters are estimatedby a Maximum Likelihood fit between measured and modeledstate variables resulting in a spatially heterogeneous distributionof hydraulic conductivity, which is constrained on the sedimento-logical data and is conditioned on both self-potential and boreholehydraulic head readings.

2. The Boise Hydrogeophysical experiment

In June 2007, during the Summer field school on ‘‘Hydrogeophys-ics: Theory, Methods and Modeling’’, a combined hydrogeophysicalstudy including self-potential, electrical resistivity, and hydrologicdata collection was performed at the Boise HydrogeophysicalResearch Site. The primary aquifer stimulation during this studywas a series of dipole pumping tests performed using the site’s fullypenetrating wells. Detailed descriptions of the site, testing design,and field techniques can be found in Jardani et al. (2009), and Cardiffet al. (2009). Here, we briefly summarize the key elements which aredirectly relevant to our work.

The BHRS is located on a gravel bar adjacent to the Boise River,15 km from downtown Boise, Idaho (Fig. 1). The aquifer at theBHRS consists of unconsolidated and unaltered Pleistocene toHolocene coarse fluvial deposits underlain by a red clay formation(Barrash and Reboulet, 2004). The thickness of the fluvial depositsis in the range of 18–20 m, and saturated thickness of the aquifer isgenerally about 16–17 m. The well field consists of 13 wells in the

Fig. 1. Site layout of BHRS showing experimental field: white line represents theboundary on which fixed head boundary conditions are applied and red line the60 m � 60 m numerical model domain where mixed boundary conditions areapplied. Inner domain: small dot correspond to Petiau electrodes used for the self-potential monitoring and circles in the central area of the BHRS indicate the wellslocation.

Table 1Hydraulic pump test data available.

Date Test number Extraction well Injection well Rate (L/s)

June 18 2007 1 B6 B3 3.9June 19 2007 2 B1 B4 4.2June 20 2007 3 B5 B2 4.1June 21 2007 4 C5 C2 4.1June 22 2007 5 C6 C3 4.0June 25 2007 6 C6 C5 1.7June 26 2007 7 B6 Monopole 2.3June 27 2007 8 C1 C4 4.2June 27 2007 9 C4 C1 4.1June 28 2007 10 C4 C3 4.3June 29 2007 11 C5 C1 4.3

S. Straface et al. / Journal of Hydrology 407 (2011) 115–128 117

central area (�20 m diameter) and five boundary wells about10–35 m from the central area. The design and the method of wellconstruction support a wide variety of single-well, cross-hole, andmultiple-well hydrologic and geophysical tests for thorough three-dimensional characterization to determine the distributions ofhydrodynamic and hydrodispersive properties of naturally hetero-geneous aquifers. Further details about the geology of the site andavailable instrumentation are provided by Barrash and Clemo(2002).

The following paragraphs define some important details aboutthe type of data available for site-characterization.

2.1. Hydraulic measurements

Ten dipole tests, in which water was pumped from one well andinjected into another, and one traditional pumping test were per-formed. A high pumping rate of about 4 L/s was chosen for thishighly conductive aquifer (an average hydraulic conductivity,Kave � 7.5 � 10�4 m/s, has been estimated, see Fox, 2006 or Barrashet al., 2006) in order to produce head changes that were well dis-cernible from instrument error. Preliminary modeling efforts sug-gested that a reasonable approximation to steady state would bereached after about 2–3 h of pumping, allowing a full set of draw-down and recovery data to be collected in a one-day for each test.The number and choice of specific well pairs for the dipole testswere determined using a time constraint of 10–12 days withinwhich completion of the experimental campaign could be achievedand preliminary analytical modeling, on the basis of an homoge-neous hydraulic conductivity, K � 5 � 10�4 m/s (Cardiff et al.,2009). Hydraulic data were collected throughout pumping andrecovery phases using a combination of instruments. Fifteenvented pressure transducers were placed down-well to measuretotal head change for all wells, with the exception of X1, X3, andX5 (the wells located furthest from the central area (Fig. 1)). Theresponses of X1 and X5 were expected to be relatively slow to de-velop and were measured manually using electric tapes. Well X3was outfitted with an in-well battery-powered logger for the dura-tion of the experiment to allow continuous collection of both test-related water level changes (on weekdays) in addition to dailyevapotranspiration (ET) signals (overnight and during the week-ends). Additionally, well A1 was set up with a multilevel packerand port setup (absolute pressure sensor spacing 2 m) in order tocollect some depth-dependent pressure change information, inpseudo-logarithmically spaced time increments. Boise River stageand river edge positions were measured manually to establishboundary condition control in conjunction with a pressure trans-ducer placed in the river to monitor for possible stage changes dur-ing the testing period. The tests chosen included seven cross-sitetests (B1–B4, B2–B5, B3–B6, C1–C4, C4–C1, C2–C5, and C3–C6) inaddition to four other tests using non-opposite pairs of C wells.The pumping tests performed are summarized in Table 1. Overall,the majority of instruments worked as planned. As an example ofthe type of data collected, Fig. 2 reports histories of drawdownand raising of the water table during dipole test C5–C1.

2.2. Self-potential measurements

Self-potential signals were measured at the ground surface in theset of 89 electrodes depicted in Fig. 1 and in seven additionalelectrodes placed in boreholes. A reference electrode was placed90 m from the region of investigation. Each in-ground nonpolarizingelectrode was placed in boreholes. Each in-ground nonpolarizingelectrode was placed inside a hole 10 cm deep and filled with a

Fig. 2. Plot of water level change during pumping and injection phases in dipoletest number 11 (Table 1).

Fig. 3. Porosity logs from water table to base of fluvial deposits.

Table 2Elevations used for the basal contact picks for each porosity unit. Note absence of Unit5 in four wells on the eastern side of the BHRS.

Well Aquiferbottomelevation(m)

Unit1topelevation(m)

Unit2topelevation(m)

Unit3topelevation(m)

Unit4topelevation(m)

Unit5top*

elevation(m)

A1 832.1 834.0 840.1 843.0 847.2 847.6B1 832.4 834.5 841.2 843.8 847.9 –B2 830.8 833.4 839.0 843.5 847.4 –B3 831.7 834.6 838.6 841.5 847.5 –B4 832.4 835.0 838.4 841.9 846.7 847.6B5 830.8 834.0 838.8 843.7 846.2 847.4B6 831.2 833.9 840.0 842.0 846.6 847.6C1 831.4 833.8 838.0 842.7 847.6 –C2 831.0 834.0 838.4 842.6 847.6 –C3 831.0 834.3 840.3 843.7 847.1 847.3C4 830.9 834.4 840.9 844.0 845.8 847.6

118 S. Straface et al. / Journal of Hydrology 407 (2011) 115–128

moistened bentonite and gypsum mixture to ensure good contactbetween the electrode and the ground, and stones were placedabove the electrodes. Measurements of the self-potential signalswere carried out with a Keithley 2701 multichannel voltmeter,and we used nonpolarizing Pb/PbCl2 (Petiau) electrodes (Perrieret al., 1998). The voltmeter was connected to a laptop computerwhere the data were stored. All the electrodes were scanned duringa period of 30 s. To account for drift effects associated with temper-ature changes, an additional reference electrode was located 50 mfrom the middle of the investigated region, and temperature of thepacking medium around the reference electrode was measured peri-odically. According to SDEC (the company manufacturing the Petiauelectrodes used in this study), the temperature dependence of theseelectrodes is 0.210 mV/�C. During the day, we measured variationsof temperature ranging from a few degrees Celsius to 15 �C. A differ-ence in temperature of 10 �C is responsible for a drift of the self-po-tential measurement of 2 mV, and therefore this effect cannot beneglected. Following Rizzo et al. (2004), a filtering operation hasbeen used to improve the signal-to-noise ratio of these data and tocorrect the temperature drift.

C5 831.6 832.7 840.6 842.9 845.7 847.4C6 831.3 834.2 839.4 842.1 846.7 847.3

* Unit 5 top is limited by position of water table for porosity log measurements.

Table 3Lithological categories classified in BHRS (Barrash and Reboulet, 2004).

Lithological categories

5 Sand (average porosity = 0.429)4 Pebble- and cobble dominated units (average porosity = 0.232)3 (average porosity = 0.172)2 (average porosity = 0.243)1 (average porosity = 0.182)0 Basalt

2.3. Stratigraphic information

Stratigraphic information from wells drilling are included in adatabase containing location and local thickness of identifiedgeo-materials within each column. Barrash and Clemo (2002)derived porosity values for each well by means of a neutron logsperformed in each well at the BHRS at regular intervals (spacing0.03 m) from the bottom of each well to the water table. Theneutron-derived porosity logs at the BHRS indicate the occurrenceof five hydrostratigraphic units across the central area of the wellfield. These subsurface units are also consistent with GPR reflectionprofiles (Peretti et al., 1999). Four of these units are cobble-dominated deposits and can be traced across the central portionof the site. The fifth unit is a channel sand that is continuous inthe western part of the site and that pinches out from the riverto the central portion of the well field (Barrash and Clemo, 2002).Porosity from Units 1 and 3 have similar means, variances, andkey vertical geostatistical descriptors and are considered to besimilar (if not the same type of) hydrostratigraphic units basedon the trends and statistics detected within porosity logs (Fig. 3).Units 2 and 4 also display similar porosity statistics and may beconsidered as similar hydrostratigraphic units, although they havedissimilar vertical geostatistical interpretations. Unit 5 is typically

sand. Table 2 provides the elevations used as the basal contacts foreach of the porosity units at the BHRS.

Six different lithological categories, derived from core samplescollected during wells drilling, have been finally classified in theBHRS (Barrash and Clemo, 2002; Barrash and Reboulet, 2004)and are reported in Table 3. These data constitute our base of ref-erence for the reconstruction of the internal architecture of the sys-tem on the basis of a multiple continua approach.

S. Straface et al. / Journal of Hydrology 407 (2011) 115–128 119

3. Theoretical background

3.1. Aquifer characterization as an overlapping continuum

A variety of studies suggest the importance of honoring geolog-ical features and information in hydrogeologic modeling (e.g.,Webb, 1995; Scheibe and Freyberg, 1995; Webb and Anderson,1996; Ritzi et al., 1994, 1995, 1996; Boggs and Adams, 1992; Boggset al., 1992; Rehfeldt et al., 1992; Lee et al., 2007 and referencestherein). Increasingly powerful aquifer characterization techniquesare available at various stages of development (Jardani and Revil,2009), so that the approximate boundaries between regions wheredifferent geo-materials occur can sometimes be reliably character-ized by geophysical surveying techniques. Errors associated withboundary locations can be derived (in principle) through geostatis-tics. Techniques which are available to describe the spatial vari-ability of the properties and attributes of the host porous matrixfrom geologic observations and local measurements include con-tinuous geostatistical models, discontinuous facies models (e.g.,Indicator or Gaussian-Threshold models, Markov chain model), orgenetic models. De Marsily et al. (2005) offer an insightful reviewon these techniques, to which we refer the interested reader. A re-cent review focusing on (stochastic) composite models to charac-terize heterogeneous porous media in terms of disjoint units (orlitho-facies) within which hydraulic attributes (typically porosityand hydraulic conductivity) can be spatially variable is presentedby Winter et al. (2003).

Sometimes block geometries cannot be characterized becauseadequate hydrogeological and geophysical data are not availableor because there is not sufficient resolution to determine bound-aries. In this case, models that can be adopted to reflect thecomposite nature of the system include multiple-continuaapproaches. These models have typically been used to quantifythe exchange of mass and momentum between fractured andmatrix phases in fractured media. The idea is conceptually similarto that imbued in stationary models with multi-modal distribu-tions. It relies on the adoption of a random indicator function,Ii(x), to designate the membership of a point x in material Mi. Mate-rials (continua) Mi are allowed to overlap in multiple-continuamodels. The possibility of such a co-existence of various materialsat the same point x relies on the idea that a point in the continuumdescription of porous media represents a volume that can becomprised of several materials Mi.

Here, we characterize the distribution of materials at the site inthe spirit of multiple- (overlapping) continua concepts. We use thestratigraphic information presented in Section 2.3 and adoptsuccessive applications of Indicator Kriging to provide an estimateof the probability of occurrence of each geo-material within theblocks of the numerical grid adopted to solve the groundwater flowproblem. The procedure we adopt is detailed in the following.

1. Stratigraphic and sedimentological information are analyzedand grouped into n main classes. Each one of these identifiesa material type. Here, we adopt the data and results of Barrashand Clemo (2002) and Barrash and Reboulet (2004) and rely onthe identification of n = 6 categories, i.e., 6 geo-materials,identified as Mi (i = 1, 2, . . . , 6).

2. We start upon assigning a value of the indicator, I = 1, tosamples where material M1 is observed, while assigning I = 0to locations where other samples are available.

3. Three-dimensional indicator variography is then performed.Resulting sample variograms are interpreted and modeled bydifferent theoretical variograms and key geostatistical parame-ters (sill, range, nugget, anisotropy pattern) are identified foreach class.

4. Indicator Kriging is performed. Kriged values of the indicatorcoincide with the estimated probability of finding M1 withineach block.

5. The procedure is repeated upon assigning a value of the indica-tor, I = 1, to samples associated with material M2, while settingI = 0 to locations where other samples are available. Kriged val-ues of the indicator coincide with the estimated probability ofoccurrence of M2 within each block.

6. This is repeated in sequence to identify block-volumetric frac-tions of (n � 1) materials. The volumetric fraction of materialn is finally calculated as the complement to unity for each gridblock.

The hydraulic conductivity value of a grid block (Kj) is then cal-culated as a weighted average of the conductivities of materialsoccurring in the block (ki) according to the following expression:

Kj ¼Xn

i¼1

Iji � ki j ¼ 1; . . . ;Nj ð1Þ

where I is the estimated percentage of the materials in a block, jindicates the position of a specific grid block and Nj is the numberof grid blocks. For simplicity, we assume that each material isassociated with a uniform hydraulic conductivity. In our applica-tion, these result in six hydraulic conductivity values which arecalibrated in the inverse modeling procedure presented in Section3.2.

3.2. Inverse modeling with multiple data sources

We consider steady-state groundwater flow in a randomly het-erogeneous domain. The flux vector q(x) and the hydraulic headh(x) obey the continuity equation and Darcy’s Law subject to givenforcing terms (sources and boundary conditions) where x is a vec-tor of spatial coordinates. We write:

h�Pj ¼ hPj þ e�Pj j ¼ 1; . . . ;NP ð2Þh�Ei ¼ hEi þ e�Ei i ¼ 1; . . . ;NE ð3Þ

here, hPj is the unknown true value of the vertically average hydrau-lic head, hP, at measurement point xj, h�Pj its measurement value ob-tained by borehole reading, e�Pj the corresponding zero-meanmeasurement error; hEi is the unknown true value of the locationof the water table, hE, at measurement point xi, h�Ei its measurementvalue, as obtained by self-potential interpretations, e�Ei the corre-sponding zero-mean measurement error; NP and NE are the numberof available measurements of head and self-potential, respectively.Following the work of Carrera and Neuman (1986) we assume (a)lack of space correlation between measurement errors of hP andhE (b) e�Pj and e�Ei being multivariate Gaussian; (c) the covariance ma-trix of measurements errors, ChP and ChE

ChP ¼ r2hPVhP ChE ¼ r2

hEVhE ð4Þ

to be known up to the positive statistical parameters r2hP , and r2

hE,which are typically unknown and need to be estimated duringinversion; and (d) lack of spatial cross-correlation between mea-surement errors of hP and hE. The latter assumption renders theknown symmetric positive-definite matrices VhE = VhP = I (Identitymatrix)

In the absence of direct measurements of hydraulic conductiv-ity at the scale of interest, the Maximum Likelihood estimate ofthe values of hydraulic conductivity associated with each materialare obtained by minimizing the negative log likelihood criterion(Carrera and Neuman, 1986; Medina and Carrera, 2003)

Table 4Variograms models and related parameters. The Nugget is zero for all the variograms.

Model Sill (m2) Range (m)

Category 5 Horizontal Exponential 0.051 10.628Vertical Spherical 0.070 5.304

Category 4 Horizontal Exponential 0.083 8.303Vertical Spherical 0.173 5.089

Category 3 Horizontal Exponential 0.094 25.057Vertical Spherical 0.280 6.494

Category 2 Horizontal Exponential 0.094 25.057Vertical Spherical 0.302 7.151

Category 1 Horizontal Exponential 0.051 2.313Vertical Exponential 0.080 3.323

120 S. Straface et al. / Journal of Hydrology 407 (2011) 115–128

NLL ¼ RhP

r2hP

þ RhE

r2hE

þ ln jVhPj þ ln jVhEj þ NhP ln r2hP þ NhE ln r2

hE

þ ðNhP þ NhEÞ ln 2p ð5Þ

here, RhP and RhE are defined

Rhk ¼ ðh�hk � hmhkÞ

T V�1hk ðh

�hk � hm

hkÞ k ¼ P; E ð6Þ

h�hk is the vector of measurements of type k, hmhk is a vector of mod-

eled state variables of type k. If the statistical parameters r2hP and

r2hE are known, minimizing Eq. (5) is equivalent to minimizing the

following general least squares criterion

R ¼ RhE þ lRhP ð7Þ

where l is the relative weight between measurement errors

l ¼ r2hE

r2hP

ð8Þ

In the common case where these statistical parameters are un-known, they could in principle be estimated jointly with the esti-mates of materials’ conductivities by minimizing Eq. (5). However,such estimation is likely to be unstable (Carrera and Neuman,1986). We therefore follow recent works of Hernandez et al.(2006) and Bianchi Jannetti et al. (2010) by minimizing Eq. (7) withrespect to the vector of hydraulic parameters for selected values ofr2

hE and r2hP . We then improve upon these values as described below

and repeat the process iteratively till all parameters converge towithin predetermined tolerance levels.

We solve the three-dimensional flow problem with the widelyused and thoroughly tested finite difference MODFLOW software(Hill et al., 2000). Modeling details are provided in Section 6. Forgiven r2

hE and r2YE we minimize Eq. (7) using the iterative Leven-

berg–Marquardt algorithm PEST implemented in the public do-main (Doherty, 2002).

The minimization algorithm computes an updated parameterestimate KH of the (unknown) true vector of hydraulic conductivi-ties, K, and a Cramer-Rao lower bound approximation for thecovariance matrix, Q, of the corresponding estimation errors (Car-rera and Neuman, 1986); the latter is evaluated according to:

Q ¼ r2hE½J

ThEV�1

hE JhE þ lJThPV�1

hP JhP��1 ð9Þ

where Jhk (k = E, P) is a Jacobian matrix of derivatives of the statevariable (water level or vertically integrated heads) with respectto K, evaluated at KH (see also Medina and Carrera, 2003). Next,KH is projected onto the finite difference grid of MODFLOW viathe algorithm presented in Section 3.1. The iterative process contin-ues until R in Eq. (7) stabilizes to within a desired tolerance.

In essence, our inverse process iterates between the estimationof hydraulic and statistical parameters. Each iteration starts byadopting the most recent estimates of the statistical parameters,r2

hE and r2hP . It then proceeds by estimating hydraulic parameters

through the minimization of (7) and adopting the Cramer-Rao low-er bound as an estimate of its covariance matrix Q. Next, we obtainan estimate of the weight l by minimizing (with respect to l) thatbetween the following functions which defines it most sharply:

1. NLL in (5)2. The Kashyap (1982) Bayesian model selection criterion

KIC ¼ NLLþ NhP lnNhP þ NhE

2p

� �� ln jQ j: ð10Þ

We follow this by ML estimation of r2hE according to

r2hE ¼

Rmin

NhE þ NhPð11Þ

and of r2hP based on the definition of l,

r2hP ¼

r2hE

lð12Þ

4. Geostatistical analysis of lithostratigraphic information

Spatial variability of the material types was analyzed by three-dimensional indicator variography, along the lines of Section 3.1.Sample directional three-dimensional variograms of the indicatorshave been reconstructed. The results are summarized in Table 4.The analysis of the data allowed interpreting a spatial variabilitypattern associated with sample horizontal variograms exhibitingno clear evidence of directional anisotropy while the vertical rangeresulted significantly smaller than its horizontal counterpart. Table4 also indicates that there is a directional dependence of the vari-ograms sill. This might be associated with (a) occurrence of a zonalanisotropy in the system (Isaaks and Srivastava, 1989) or, (b) direc-tional undersampling of the complete range of variability. Selec-tion of the variogram models was performed as follows: (1) keyparameters associated with a given variogram model were cali-brated on the basis of a maximum likelihood fit between sampleand model variograms; (2) the Kashyap’s selection criterion, KIC,was then adopted for model selection. Figs. 4–8 illustrate our re-sults by depicting the calculated experimental variograms andthe adopted interpretative models. The Kriging algorithm is imple-mented by means of the widely used suite GSLIB (Deutsch andJournel, 1998), upon considering the occurrence of the detected zo-nal anisotropy. Kriged values are associated with the center of ele-ments of the computational grid described in Section 6.

5. Reconstruction of water table by means of self-potentialmeasurements

Under steady-state conditions, Jardani et al. (2009) demon-strated, inverting the Poisson equation in a probabilistic (Bayesian)framework and using also the available potentiometric data in theinverse problem, that self potential signals exhibit a linear relation-ship with the saturated thickness of the aquifer, according to

uðrÞ ffi u0 � C 0ðe� hðrÞÞ ¼ C 0ðhðrÞ � h0Þ ð13Þ

here, C0 is the electrokinetic coupling coefficient (V/m) associatedwith hydraulic head variations, /(r) is the electrical potential (V)at vector location r, and h0 (m) is the hydraulic head where the ref-erence electrode is placed. The assumption of a linear relationshipbetween the saturated thickness of the aquifer and the measuredself potential signals allows performing an estimation of the watertable depth by means of an application of Kriging with external drift(Delhomme, 1979; Galli and Meunier, 1987; Ahmed and de Marsily,1987; Troisi et al., 2000). In this sense, the Kriging with externaldrift may be considered as an extension of Kriging with a trendmodel in which self-potential variable is assumed to reflect thespatial trends of the hydraulic head variability up to a linear rescal-

Fig. 4. Category 1 – (a) vertical and (b) horizontal variogram (m2) vs. lag distance (m).

Fig. 5. Category 2 – (a) vertical and (b) horizontal variogram (m2) vs. lag distance (m).

Fig. 6. Category 3 – (a) vertical and (b) horizontal variogram (m2) vs. lag distance (m).

S. Straface et al. / Journal of Hydrology 407 (2011) 115–128 121

ing of units, or in a way more general Kriging with external drift canbe considered a special case of the Bayesian interpolation withoutconstraints (Linde et al., 2007). As pointed out by Straface et al.(2010), in the Kriging with external drift method secondaryvariable (self-potential) is assumed to reflect the spatial trends of

the hydraulic head (primary variable) distribution up to a linearrescaling of units within small search neighborhoods. In this sensethe value of coupling coefficient may be considered locally constantwithin the search neighborhoods. In order to implement Krigingwith external drift on h(x), values of u(x) should be available at

Fig. 7. Category 4 – (a) vertical and (b) horizontal variogram (m2) vs. lag distance (m).

Fig. 8. Category 5 – (a) vertical and (b) horizontal variogram (m2) vs. lag distance (m).

Fig. 9. Omnidirectional sample variogram of self potential signals and the relatedtheoretical variogram that models the spatial distribution of u.

122 S. Straface et al. / Journal of Hydrology 407 (2011) 115–128

each point where h(x) should be estimated together with values ofu at all points where h data are available. If these values of u(x) arenot directly available at all required locations, they may be deter-mined in advance by an ordinary Kriging implementation. A vario-gram model for u(x) is inferred from the available data and cross-validated according to a standard-check procedure (Isaaks and Sri-vastava, 1989; Deutsch and Journel, 1998). It is important to under-line that, while previous methods proposed like the semiempirical,the tomographic and the simplex algorithms (Revil et al., 2003; Riz-zo et al., 2004), need to estimate, by means of laboratory tests, thevalue of the electrokinetic coupling coefficient C0, application of Kri-ging with external drift does not require prior knowledge of thisterm because it is estimated directly by means of Kriging systemequations (Straface et al., 2010).

The reconstruction of the water table was performed using theGSLIB library (Deutsch and Journel, 1998). An isotropic exponentialvariogram model with a sill equal to 1.6 mV2 and a range value of9.0 m was inferred for measured self-potential signals in the hori-zontal plane. Fig. 9 shows the experimental and modeled vario-gram inferred for u. Ordinary Kriging was then performed forself-potential over a regular two-dimensional grid, with a size of0.5 m, covering the site. At this point, the self-potential variogrammodel was cross-validated by implementing the ordinary Krigingto re-estimate self-potential values at points where experimentalvalues of u were available and so calculating the estimation error.

The residual variogram, cR(h), was then inferred from pairs of h-values that are unaffected (or only slightly affected) by the trend,i.e., from data pairs such that u(x) = u(x + h), and kriging withexternal drift was run for h over the same 2D grid. Fig. 10 depicts

Fig. 10. Two-dimensional water level changes (cm) obtained by means of a Kriging with external drift for Dipole Test C5–C1, negative values (sea colors) representgroundwater drawdown while positive values (land colors) represent the raising of water table. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

S. Straface et al. / Journal of Hydrology 407 (2011) 115–128 123

the distribution of the location of the water table obtained with thedescribed procedure.

6. Inverse flow model at Boise Hydrogephysical Research Site

6.1. Description of the numerical model

Three-dimensional modeling of the BHRS has been performedunder steady state flow conditions. The horizontal dimensions ofthe heterogeneous model domain are 60 m � 60 m while the asso-ciated depth is 21 m. The model is vertically discretized by meansof 60 layers with uniform thickness of 0.35 m. Each layer is discret-ized by a regular grid of 14,400 square cells having a side of 0.5 m.Simulations have been performed by imposing a mixed boundarycondition along the boundary of the 60 m � 60 m computationaldomain. The mixed boundary condition has been obtained byextending the size of the simulation region from 60 m up to500 m from the well-field center (at this distance it had been ver-ified experimentally that head changes were negligible) and is ofthe form:

Krh ¼ N0 þ Cðhb � hÞ ð14Þ

where the flux N0 is equal to zero, C is the average conductance ofthe region comprised between the 60 m � 60 m domain and theextended one (Fig. 1), hb is the constant hydraulic head value im-posed on the boundary of the enlarged domain and h is the hydrau-lic head value on the 60 m � 60 m numerical model domain. Thehead distribution within the enlarged domain has been estimatedby projecting a planar trend surface obtained from water levelsmeasured during the two weeks before the dipole tests and hasbeen adopted to estimate the values of hb in the mixed boundary

condition. The conductance value C has been calculated by assum-ing the aquifer comprised within the two domains (Fig. 1) to behomogeneous with an average hydraulic conductivity correspond-ing to that reported by Fox (2006) and Barrash et al. (2006).

SP signals acquired during our dipole test, provide informationon the late-time part of the experiment, when pseudo-steady stateregime is reached. It is well known that, during this regime,hydraulic heads are subjected to space–time variations while headgradients are time independent. This implies that one can repre-sent drawdown increments (i.e., Dh(x, x0) = h(x0) � h(x)) by meansof a steady state flow equation. In our case, we compute drawdownincrements, Dh(x, x0), relative to that recorded at an observationelectrode positioned at location x0, which is farthest from thepumping and injection wells and outside of the investigated area.These quantities constitute our state variable measurements in (6).

6.2. Results of the Inversion

Here, we present the results of the inversion of the test termedC5(Extraction)/C1(Injection) in Table 1. Data collected during thedipole tests are obtained in two different ways: direct samplingwithin observation wells, and derived measures from self potentialacquisition. With reference to self-potential measures, since the lin-ear relationship between electrical and hydraulic information iseffective under the hypothesis of small water table variations (Stra-face et al., 2010), estimates of water table are affected by a generallyunquantifiable error in the proximity of the pumping and injectionwells. For all these reasons, the strength of measurement errors isestimated according to the procedure explained in Section 3.2.From an operational standpoint, we select five values of l(l = 0.1, 1, 3, 5, 10) for our inversion. These represent a wide range

Fig. 11. Dependence of KIC and NLL on l during the inversion procedure.

Table 5Estimated hydraulic conductivity values.

Lithological categories Hydraulic conductivities Uncertainties

5 1.33 � 10�3 m/s ±2.52 � 10�4 m/s4 2.42 � 10�4 m/s ±1.67 � 10�4 m/s3 1.94 � 10�4 m/s ±1.27 � 10�4 m/s2 2.18 � 10�4 m/s ±2.16 � 10�4 m/s1 5.59 � 10�5 m/s ±2.08 � 10�5 m/s0 4.00 � 10�12 m/s ±8.97 � 10�10 m/s

124 S. Straface et al. / Journal of Hydrology 407 (2011) 115–128

of variability of the relative importance of the measurement errorsassociated with the two types of data available. For each value of lthe steady state flow inversion is performed and the optimum valueof l is identified by applying the methodology described in Section3.2. Fig. 11 depicts the values of KIC and NLL associated with the se-lected values of l. Both KIC and NLL displays a minimum for the sce-nario associated with l = 3. The observed ability of KIC to identifythe statistical parameters included in (4) is consistent with previ-ous observations of Hernandez et al. (2006), Riva et al. (2009),and Bianchi Jannetti et al. (2010), in the context of the inversion

Fig. 12. Two vertical transects crossing the wells B1–A1–A4 and C5–A1–C2 showing the etransect, is showed a stratigraphical comparison between the inverse modeling result a

of mean groundwater flow equations. The optimization process re-turns a standard deviation of ±4.6 cm for the electrodes and ±2.7 cmfor piezometric readings at wells, which are consistent, given thethree-dimensional model configuration, the available lithologicalinformation and the degree error associated to water table elabora-tion (Section 5).

The optimum set of hydraulic parameters estimated at the va-lue of l associated with the minimum value of KIC is reported inTable 5. A qualitative analysis of the consistency of inversion re-sults against previously published geological observations can thenbe performed. Two vertical transects crossing the wells C5–A1–C2and B1–A1–A4 have been carried out in which hydraulic conduc-tivity value of a model grid block has been calculated as a weightedaverage of the estimated hydraulic conductivity values of Table 5.Fig. 12 shows the estimated vertical distribution of heterogeneityat BHRS after the calibration of data from Dipole Test C5–C1. Be-cause of the linear combination of the estimated hydraulic conduc-tivities, the resulting maps present a different permeability rangewith respect to the one obtained in Table 5. In the transect C5–A1–C2, the boundaries between the five stratigraphic units derivedfrom porosity logs and obtained on the basis of Table 2 are avail-able up to a depth of about 15 m, the juxtaposition to the recon-structed conductivity field, indicates a good agreement betweenthe two types of reconstructions (Fig. 12).

We then compare the spatial distribution of depth-average con-ductivities against that obtained by Cardiff et al. (2009) by meansof a two-dimensional inversion on the basis of hydraulic tomogra-phy data. Fig. 13 shows that the depth averaged log-conductivitydistribution obtained in our work displays some key featureswhich are consistent with the results obtained by Cardiff et al.(2009). In both cases, the aquifer is characterized by lower perme-ability areas on the Western region of the field (left side of Fig. 13aand b). The system then tend to display a sequence of more perme-able geomaterials moving towards the Eastern side. It is interestingto note that the distribution of depth-averaged conductivitiesobtained on the basis of a three-dimensional flow model is associ-ated with less pronounced horizontal variations than those result-ing from the inversion of a two-dimensional flow model. Arelatively large conductivity variability is instead observable alongvertical slices (Fig. 12), consistently with pronounced stratificationobserved at the site.

stimated vertical distribution of hydraulic conductivity (m/s) at BHRS. In the secondnd the one derived from the neutron porosity logs at the BHRS.

Fig. 13. Estimate of depth averaged log-conductivity field from (a) 3D inverse modeling of BHRS field using hydraulic and geophysical data, and (b) 2D inverse modeling ofBHRS field using hydraulic data (Cardiff et al., 2009).

S. Straface et al. / Journal of Hydrology 407 (2011) 115–128 125

Finally, we validate our inverse modeling results upon simulat-ing all the remaining dipole tests on the calibrated three-dimen-sional conductivity field. As an example of the quality of theresults obtained, Fig. 14 depicts the observed and modeled depthaveraged head values obtained at steady-state for dipole testsC4(Extraction)/C3(Injection) and C1(Extraction)/C4(Injection),which represent the best and worst validation results, respectively.As a complement to these results, Fig. 14 shows also a scatterplotof the observed and modeled (steady-state) drawdown for thecomplete set of dipole tests. The simulation of the experimental

tests not included in the inversion procedure, returned hydraulichead values close to the observed one (the correlation coefficientwith the unitary slope factor line is about 0,97)

7. Conclusions

We combined the use of sedimentological, hydraulic and hydro-geophysical information to characterize a three-dimensionalheterogeneous aquifer. We focused on the joint inversion of

Fig. 14. Right side: Worst validation result – Test C1(Extraction)/C4(Injection); center: scatterplot of observed drawdown vs calculated one for all dipole tests; Left side: Bestvalidation result – test C4(Extraction)/C3(Injection).

126 S. Straface et al. / Journal of Hydrology 407 (2011) 115–128

sedimentological (porosity), hydraulic head and self-potentialobservations collected at the Boise Hydrogeophysical Research Site(BHRS), Boise, Idaho, during an extensive series of tests. The aquiferwas conceptualized as a multiple-continuum, where the volumet-ric fraction of a geo-material within a cell of the numerical flowmodel has been calculated by Multiple Indicator Kriging (MIK).The latter was implemented on the basis of available sedimento-logical information obtained from porosity measurements per-formed by means of neutron probe acquisition. Self potential andhydraulic head data have been embedded jointly within a three-dimensional (pseudo-steady state) inverse model of groundwaterflow at the site. Each identified geo-material was assumed to becharacterized by a constant hydraulic conductivity. The hydraulicconductivity associated with a numerical block was then calcu-lated as a weighted average of the conductivities of the geo-mate-rials which were found in the block according to the results of MIK.Model parameters were estimated by a Maximum Likelihood fitbetween measured and modeled state variables resulting in a spa-tially heterogeneous distribution of hydraulic conductivity, whichwas constrained on the sedimentological data and conditionedon both free surface location derived from self-potential signalsand depth-averaged hydraulic head obtained from borehole read-ings. Minimization of the ML objective function allowed estimatingthe relative weight of errors associated with self-potential andborehole-based data.

The procedure adopted allowed a three-dimensional recon-struction of the heterogeneity of the site with a level of detailwhich was not obtained in previous studies and relatively modestcomputational efforts. Further validation against dipole testswhich were not used in the inversion procedure supports therobustness of the results.

It is important to underline that the technological and interpre-tative limits associated with the identification of the subsurfacelithology by means of neutron probe acquisition and location ofthe free surface via self-potential are surmountable, e.g., by jointly

using resistivity information as described by Jardani and Revil(2009). Consequently, in spite of these limitations, the procedureadopted in this paper, can represent a useful instrument for realheterogeneous aquifers characterization and for predictive analysisof their behavior.

Acknowledgements

The field experiment was supported by PRIN2006 – Project‘‘Statistical estimation of heterogeneity in complex randomly het-erogeneous geologic media’’, EPA grants X-96004601-0 and X-96004601-1 and by NSF grant DMS-0934680. M. Cardiff was alsosupported during his PhD program at Stanford by an NSF GraduateResearch Fellowship. Significant help during the field experimentswas provided by participants in the Boise State and Calabria Uni-versities ‘‘Hydrogeophysics: Theory, Methods, and Modeling’’ sum-mer program: Steve Berg, Francesco Chidichimo, Agnès Crespy,Carlyle Miller, Harry Liu, Timothy Johnson, Domenico Sorbo, LeahSteinbrunn, and Jianwei Xiang. We thank Frédéric Perrier for theuse of his non-polarizing electrodes.

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