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Forward modelling and inversion of streaming potential for the interpretation of hydraulic conditions.. Sheffer, Megan Rae 2007-12-20

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Forward modelling and inversion of streamingpotential for the interpretation of hydraulicconditions from self-potential databyMegan Rae ShefierB.Sc., Queen’s University, 1995M.A.Sc., University of British Columbia, 2002A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinThe Faculty of Graduate Studies(Geological Engineering)The University Of British ColumbiaDecember, 2007c Megan Rae Shefier 2007AbstractThe self-potential method responds to the electrokinetic phenomenon of streaming potentialand has been applied in hydrogeologic and engineering investigations to aid in the evalua-tion of subsurface hydraulic conditions. Of speciflc interest is the application of the methodto embankment dam seepage monitoring and detection. This demands a quantitative inter-pretation of seepage conditions from the geophysical data.To enable the study of variably saturated  ow problems of complicated geometry, athree-dimensional flnite volume algorithm is developed to evaluate the self-potential dis-tribution resulting from subsurface  uid  ow. The algorithm explicitly calculates the dis-tribution of streaming current sources and solves for the self-potential given a model ofhydraulic head and prescribed distributions of the streaming current cross-coupling con-ductivity and electrical resistivity. A new laboratory apparatus is developed to measurethe streaming potential coupling coe–cient and resistivity in unconsolidated soil samples.Measuring both of these parameters on the same sample under the same conditions enablesus to properly characterize the streaming current cross-coupling conductivity coe–cient. Ipresent the results of a laboratory investigation to study the in uence of soil and  uid pa-rameters on the cross-coupling coe–cient, and characterize this property for representativewell-graded embankment soils. The streaming potential signals associated with preferen-tial seepage through the core of a synthetic embankment dam model are studied using theforward modelling algorithm and measured electrical properties to assess the sensitivity ofthe self-potential method in detecting internal erosion. Maximum self-potential anomaliesare shown to be linked to large localized hydraulic gradients that develop in response topiping, prior to any detectable increase in seepage  ow through the dam. A linear inversionalgorithm is developed to evaluate the three-dimensional distribution of hydraulic headfrom self-potential data, given a known distribution of the cross-coupling coe–cient andiiAbstractelectrical resistivity. The inverse problem is solved by minimizing an objective function,which consists of a data misflt that accounts for measurement error and a model objectivefunction that incorporates a priori information. The algorithm is suitable for saturated  owproblems or where the position of the phreatic surface is known.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviStatement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quantitative interpretation of SP data . . . . . . . . . . . . . . . . . . . . 61.2 Thesis objective and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3-D forward modelling of streaming potential . . . . . . . . . . . . . . . . 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.1 Primary  ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.2 Coupled  ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Forward modelling methodology . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Finite volume solution for self-potential . . . . . . . . . . . . . . . . 31ivTable of Contents2.3.2 Example: Injection well in a homogeneous halfspace . . . . . . . . . 332.3.2.1 Numerical solution . . . . . . . . . . . . . . . . . . . . . . 342.3.2.2 Analytical solution . . . . . . . . . . . . . . . . . . . . . . 352.4 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.1 The cross-coupling conductivity coe–cient . . . . . . . . . . . . . . 372.4.2 Deflning  and L property distributions . . . . . . . . . . . . . . . . 372.4.3 Example: Homogeneous lab-scale embankment . . . . . . . . . . . . 392.5 Sources of charge contributing to the self-potential . . . . . . . . . . . . . . 432.5.1 Example: Pumping well in a heterogeneous halfspace . . . . . . . . 452.6 Field example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Apparatus for streaming potential and resistivity testing . . . . . . . . 593.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.2 Test cell and loading assembly . . . . . . . . . . . . . . . . . . . . . 633.2.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2.4 Fluid  ow system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.5 Data acquisition and control . . . . . . . . . . . . . . . . . . . . . . 673.2.6 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3 Streaming potential measurements . . . . . . . . . . . . . . . . . . . . . . . 683.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.2 Calibration and test design . . . . . . . . . . . . . . . . . . . . . . . 713.3.3 Comparison of unidirectional and oscillatory  ow test methods . . . 723.4 Resistivity measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4.2 Calibration and test design . . . . . . . . . . . . . . . . . . . . . . . 743.4.3 Comparison of 2- and 4-electrode methods . . . . . . . . . . . . . . 773.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79vTable of Contents3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804 Laboratory testing in well-graded soils . . . . . . . . . . . . . . . . . . . . 824.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.1 Streaming potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2 Electrical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.2 Streaming potential measurements . . . . . . . . . . . . . . . . . . . 904.3.3 Resistivity measurements . . . . . . . . . . . . . . . . . . . . . . . . 914.3.4 Sample properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.5 Sample equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.1 In uence of density . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4.2 In uence of gradation . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4.3 In uence of  uid conductivity . . . . . . . . . . . . . . . . . . . . . 1074.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155 Sensitivity of the self-potential method to detect internal erosion . . . 1215.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2 The streaming potential phenomenon . . . . . . . . . . . . . . . . . . . . . 1235.3 Forward modelling of streaming potential . . . . . . . . . . . . . . . . . . . 1255.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3.2 Assessing the in uence of unsaturated  ow . . . . . . . . . . . . . . 1265.4 Two-zone embankment model . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4.1 Model geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4.2 Seepage analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.4.3 SP forward modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 132viTable of Contents5.4.4 Residual SP response at surface . . . . . . . . . . . . . . . . . . . . 1365.4.5 Residual SP response at depth . . . . . . . . . . . . . . . . . . . . . 1385.4.6 In uence of the electrical resistivity distribution . . . . . . . . . . . 1415.5 Practical SP anomaly detection limits . . . . . . . . . . . . . . . . . . . . . 1435.6 Comparison of SP and hydraulic response to defects . . . . . . . . . . . . . 1445.7 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546 3-D inversion of self-potential data . . . . . . . . . . . . . . . . . . . . . . . 1576.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.2 Coupled  ow model of streaming potential . . . . . . . . . . . . . . . . . . 1606.3 Inversion Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.3.1 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.3.3 Physical property models . . . . . . . . . . . . . . . . . . . . . . . . 1646.3.4 Active cell approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.4 Flow under a cut-ofi wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.4.1 Laboratory measurements . . . . . . . . . . . . . . . . . . . . . . . 1676.4.2 Discrete model development . . . . . . . . . . . . . . . . . . . . . . 1696.4.3 Case 1: Homogeneous subsurface approximation . . . . . . . . . . . 1716.4.4 Case 2: Heterogeneous subsurface approximation . . . . . . . . . . 1766.4.5 Case 3: Inactive wall approximation . . . . . . . . . . . . . . . . . . 1806.4.6 Case 4: Incorporating known head values . . . . . . . . . . . . . . . 1846.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1947.1 Practical contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.1.1 Survey design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.1.2 SP data interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 1967.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198viiTable of Contents7.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200AppendicesA Discrete approximation using the flnite-volume method . . . . . . . . . 201A.1 Discrete approximation of equation (A.5) . . . . . . . . . . . . . . . . . . . 202A.2 Discrete approximation of equation (A.6) . . . . . . . . . . . . . . . . . . . 204viiiList of Tables1.1 Primary and coupled  ow phenomena. . . . . . . . . . . . . . . . . . . . . . 24.1 Sample data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.3 Comparison of measured results with published data. . . . . . . . . . . . . . 1135.1 Defect cross-sectional geometry. . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2 Physical properties of 2-zone embankment model. . . . . . . . . . . . . . . . 1305.3 Predicted volumetric seepage  ow rate increase. . . . . . . . . . . . . . . . . 1336.1 Characteristic inversion output parameters. . . . . . . . . . . . . . . . . . . 170ixList of Figures1.1 Conceptual model of streaming potential. . . . . . . . . . . . . . . . . . . . 41.2 Quantitative interpretation of hydraulic conditions from SP data. . . . . . . 122.1 Schematic of the spatial  ow domain. . . . . . . . . . . . . . . . . . . . . . 312.2 Discrete grid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Schematic of an injection well model. . . . . . . . . . . . . . . . . . . . . . . 342.4 Injection well model: hydraulic head distribution. . . . . . . . . . . . . . . . 352.5 Injection well model: self-potential distribution. . . . . . . . . . . . . . . . . 362.6 Schematic of a homogeneous embankment. . . . . . . . . . . . . . . . . . . . 402.7 Embankment with 18cm reservoir: potential distributions. . . . . . . . . . . 412.8 Embankment with 22.5cm reservoir: potential distributions. . . . . . . . . . 412.9 Embankment with 22.5cm reservoir:  ux patterns. . . . . . . . . . . . . . . 422.10 Schematic of a pumping well model. . . . . . . . . . . . . . . . . . . . . . . 462.11 Pumping well model: hydraulic head distribution. . . . . . . . . . . . . . . . 472.12 Pumping well model: streaming current source density. . . . . . . . . . . . . 482.13 Pumping well model: plan map of self-potential distribution. . . . . . . . . 492.14 Pumping well model: proflle plots of self-potential. . . . . . . . . . . . . . . 502.15 Plan map of an embankment dam site. . . . . . . . . . . . . . . . . . . . . . 522.16 Model of the embankment dam and foundation. . . . . . . . . . . . . . . . . 532.17 Modelled and predicted SP at an embankment dam site. . . . . . . . . . . . 543.1 Schematic of apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2 Cross-sectional view of the test cell. . . . . . . . . . . . . . . . . . . . . . . 643.3 Unidirectional  ow test time series data. . . . . . . . . . . . . . . . . . . . . 69xList of Figures3.4 Coupling coe–cient measurement: unidirectional  ow test. . . . . . . . . . . 703.5 Oscillatory  ow test time series data. . . . . . . . . . . . . . . . . . . . . . . 713.6 Coupling coe–cient measurement: oscillatory  ow test. . . . . . . . . . . . 723.7 Comparison of streaming potential measurement methods. . . . . . . . . . . 733.8 Frequency dependence of sample resistance. . . . . . . . . . . . . . . . . . . 753.9 Fluid resistivity derived from 2-electrode measurements. . . . . . . . . . . . 763.10 Fluid resistivity derived from 4-electrode measurements. . . . . . . . . . . . 773.11 Comparison of 2- and 4-electrode measurements. . . . . . . . . . . . . . . . 784.1 Schematic of laboratory apparatus. . . . . . . . . . . . . . . . . . . . . . . . 894.2 Comparison of streaming potential measurement methods. . . . . . . . . . . 924.3 Grain-size distribution of glass bead samples. . . . . . . . . . . . . . . . . . 934.4 Grain-size distribution of embankment soil samples. . . . . . . . . . . . . . 944.5 Sample equilibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.6 In uence of density on the electrical properties in glass bead samples. . . . 994.7 In uence of density on the electrical properties of embankment soil samples. 1014.8 In uence of gradation on the electrical properties in glass bead samples. . . 1054.9 In uence of gradation on the electrical properties of embankment soils. . . . 1064.10 In uence of  uid resistivity on the electrical properties of glass beads. . . . 1084.11 Comparison of sample resistivities versus  uid resistivity. . . . . . . . . . . 1094.12 Comparison of formation factors versus saturating  uid resistivity. . . . . . 1104.13 Comparison of C and L versus sample resistivity. . . . . . . . . . . . . . . . 1115.1 Conceptual model of streaming potential. . . . . . . . . . . . . . . . . . . . 1245.2 Electrical property distribution. . . . . . . . . . . . . . . . . . . . . . . . . . 1265.3 Model dam and defect geometry. . . . . . . . . . . . . . . . . . . . . . . . . 1295.4 Transverse cross-section of head in full defect models. . . . . . . . . . . . . 1315.5 Transverse cross-section of head in half defect models. . . . . . . . . . . . . 1325.6 Padded electrical resistivity model. . . . . . . . . . . . . . . . . . . . . . . . 1345.7 Transverse surface SP proflle: intact embankment. . . . . . . . . . . . . . . 1355.8 Longitudinal surface SP proflle: intact embankment. . . . . . . . . . . . . . 135xiList of Figures5.9 Surface SP residual proflles: full defect models. . . . . . . . . . . . . . . . . 1365.10 Surface SP residual proflles: half defect models. . . . . . . . . . . . . . . . . 1375.11 Transverse cross-section of SP residual: full pipe defect models. . . . . . . . 1395.12 Transverse cross-section of SP residual: half pipe defect models. . . . . . . . 1405.13 Proflles of SP residual for full pipe defect at 45 m depth. . . . . . . . . . . . 1425.14 Proflles of SP residual for half pipe defect at 45 m depth. . . . . . . . . . . 1425.15 Transverse cross-section of head: progression of pipe defect at 45 m depth. . 1475.16 Vertical proflles of SP residual: progression of pipe defect at 45 m depth. . 1485.17 Comparison of SP and seepage: progression of pipe defect at 45 m depth. . 1495.18 Vertical proflles of SP and head residual intersecting centre of defect. . . . . 1505.19 Vertical proflles of SP and head residual 10 m from centre of defect. . . . . 1516.1 Schematic of the spatial  ow domain. . . . . . . . . . . . . . . . . . . . . . 1656.2 Schematic of the laboratory cut-ofi wall model. . . . . . . . . . . . . . . . . 1686.3 Cut-ofi-wall model input: homogeneous subsurface. . . . . . . . . . . . . . . 1716.4 Cut-ofi wall model results: homogeneous subsurface - surface data. . . . . . 1736.5 Cut-ofi wall model results: homogeneous subsurface - subsurface data. . . . 1746.6 Cut-ofi wall model results: homogeneous subsurface - all data. . . . . . . . 1756.7 Cut-ofi-wall model input: heterogeneous subsurface. . . . . . . . . . . . . . 1766.8 Cut-ofi wall model results: heterogeneous subsurface - surface data. . . . . . 1776.9 Cut-ofi wall model results: heterogeneous subsurface - subsurface data. . . . 1786.10 Cut-ofi wall model results: heterogeneous subsurface - all data. . . . . . . . 1796.11 Cut-ofi-wall model input: inactive wall. . . . . . . . . . . . . . . . . . . . . 1806.12 Cut-ofi wall model results: inactive wall - surface data. . . . . . . . . . . . . 1816.13 Cut-ofi wall model results: inactive wall - subsurface data. . . . . . . . . . . 1826.14 Cut-ofi wall model results: inactive wall - all data. . . . . . . . . . . . . . . 1836.15 Cut-ofi-wall model input: reference head values. . . . . . . . . . . . . . . . . 1856.16 Cut-ofi wall model results: reference head values - surface data. . . . . . . . 1866.17 Cut-ofi wall model results: reference head values - subsurface data. . . . . . 1876.18 Cut-ofi wall model results: reference head values - all data. . . . . . . . . . 188xiiList of Figures7.1 Quantitative interpretation of hydraulic conditions from SP data. . . . . . . 195A.1 Discrete grid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202A.2 Discrete volume for gradient operation. . . . . . . . . . . . . . . . . . . . . . 205xiiiAcknowledgementsThe research work presented in this thesis was supported by the Natural Sciences and Engi-neering Research Council of Canada and the Canadian Electricity Association TechnologiesInc. Dam Safety Interest Group. Their funding is greatly appreciated.I extend thanks to BC Hydro and Powertech Labs for releasing the fleld and laboratorydata sets for use in the forward modelling study. I am grateful to Eldad Haber for his pearlsof wisdom, and thank both him and Roman Shekhtman for their signiflcant contributionsto the forward modelling code. I also thank Ross Penner for willingly conducting the cut-ofiwall experiment as his undergraduate research project.The laboratory investigation could not have been completed without the expert guidanceof Phil Reppert, the continuous support of John Howie, the tireless efiorts of Bill Leung infabricating the test cell, the help of Scott Jackson with the electronics, and the assistanceof Roger Middleton. I extend thanks to the geotechnical group in the Department of CivilEngineering for their support during this project.My involvement with BC Hydro and the Dam Safety Interest Group has fostered manyprofessional relationships that have turned into friendships. I extend gratitude to Ken Lumfor facilitating the research sponsored by the Dam Safety Interest Group, and for his supportand respect. Many thanks to Bob Corwin and Rich Markiewicz for our fruitful discussionson SP, and to Sam Johansson, Torleif Dahlin and Pontus Sj˜odahl for graciously hostingvisits to Sweden.I extend immense gratitude to Doug Oldenburg for his support and encouragement,and for creating a friendly and dynamic research group environment. I am also grateful toRoger Beckie for his support and open door policy. Special thanks to Garry Clarke andJohn Howie for their helpful contributions as members of my supervisory committee. Itwould have been a long and painful road without the help and company of fellow UBC-GIFxivAcknowledgementsmembers, EOS-East residents and Rock Dogs, both past and present. In particular, I wouldlike to thank Len and Peter for patiently flelding and answering my many questions.I am eternally grateful to Mike for his patience, encouragement, love and un aggingsupport, and to Anna for allowing me to complete my thesis before her flrst birthday.xvDedicationIn memory of Bob CorwinxviStatement of Co-AuthorshipI was responsible for writing all flve manuscripts, which were reviewed by my co-authors.Prof. DouglasOldenburgsupervisedthenumericalmodellingresearchpresentedinChapters2, 5 and 6. I acted as principal investigator for the laboratory research presented in Chapters3 and 4. The scope of work was deflned with input from Prof. John Howie, who served ina supervisory role. Dr. Philip Reppert collaborated on the design and development of theapparatus.xviiChapter 1IntroductionTheself-potential(SP)methodisapassivegeophysicaltechniquethatrespondstonaturally-occurring potentials in the subsurface. Since its inception in the early 19th century [Fox,1830], the method has been applied to mineral exploration, oil well logging, geothermalexploration and more recently hydrogeologic, environmental and engineering investigations.Self-potential data are voltage difierences measured using a pair or an array of non-polarizing electrodes and a high input-impedance measurement device. Despite the relativesimplicity of the method, its use has been hindered in part by a lack of understanding of theunderlying mechanisms that give rise to an SP response. All source mechanisms generatean accumulation of electrical charge in the subsurface. A physical understanding of thesesource mechanisms is required to enable an efiective interpretation of self-potential data torecover practical information about the subsurface.Mineralization potentials are the signal of interest in mineral exploration. These areassociated primarily with sulphide ore bodies and have been explained using an electro-chemical mechanism [Sato and Mooney, 1960] or related to oxidation potential [Corry,1985]. The amplitude of anomalies encountered over mineralized targets can range in thehundreds of milliVolts, such that other naturally-occurring potentials are typically ignoredand considered noise. A number of these \background" potentials are the signals of interestin most other applications of the method. These source mechanisms may be described usingcoupled  ow theory.Coupled  ow phenomena describe the  ux of some quantity induced by a driving forceother than the primary force typically associated with the  ow. An example is thermaldifiusion, where a temperature gradient induces the  ow of ions in a liquid. When  ows aresmall (viscous laminar) and forces vary slowly, linear  ow laws may be used to describe a ux as the product of a potential gradient and a conductivity term. Table 1.1 lists primary1Chapter 1. Introductionand coupled  ow phenomena and illustrates the interdependence of  uid, temperature,electrical and ionic systems. Primary  ow laws are listed as diagonal terms in the table, andcoupled  ow phenomena are listed as ofi-diagonal terms. Chapter 2 presents a mathematicaldescription of the electrokinetic streaming potential source mechanism. More detail on thegeneral theory can be found in de Groot [1951], Onsager [1931] and Mitchell [1991].GradientFlow Hydraulic Temperature Electrical ChemicalPotential Potential ConcentrationFluid Darcy’s Law Thermo- Electro- Chemical(hydraulic osmosis osmosis osmosisconduction)Heat Isothermal Fourier’s Law Peltier Efiect Dufour Efiectheat transfer (thermalconduction)Electrical Streaming current Seebeck Efiect Ohm’s Law Difiusion andcurrent (electrokinetic (thermo- (electrical membranestreaming electric) conduction) potentialspotential) (electrochemical)Ionic Streaming current Soret Efiect Electrophoresis Fick’s Law(chemical) (thermal (chemicaldifiusion) difiusion)Table 1.1: Primary and coupled  ow phenomena. Primary  ows are listed as diagonalterms, and coupled  ows are listed as ofi-diagonal terms. (adapted from Mitchell [1991]).Thermoelectric, electrochemical and electrokinetic source mechanisms all give rise toelectrical conduction current  ow that can be characterized using the self-potential method.Geothermal exploration and the study of volcanic systems relies on both thermoelectric andelectrokinetic sources, where  uid  ow is driven both by advection and thermal convection[Corwin and Hoover, 1979; Di Maio and Patella, 1994; Revil et al., 1999b]. Environmental2Chapter 1. Introductionapplications of the SP method include the study of electrochemical signals associated withthe migration of concentration fronts [Maineult et al., 2005]. An emerging area is themapping of reduction-oxidation (redox) potential for the delineation of contaminant plumesand monitoring of remediation strategies. The source mechanism in this application is acurrent topic of investigation [Naudet et al., 2003, 2004; Naudet and Revil, 2005; Linde andRevil, 2007].The electrokinetic phenomenon of streaming potential is the main mechanism of interestfor the study of  uid  ow in porous media. Figure 1.1 presents a conceptual pore-scalemodel of streaming potential. An electrical double layer forms at the solid- uid interface,which consists of an adsorbed layer of tightly bound positive ions and a more loosely bounddifiuseouterlayer. Understaticconditions, thesaturatedmediumiselectricallyneutralwithadsorbed ions from the  uid completely balancing the negative surface charge of the solidparticles. The onset of  uid  ow pulls positive ions from the difiuse layer in the direction of ow, which generates a streaming current. A charge imbalance results from the movementof ions in the direction of  ow, which induces an opposing conduction current. Although thestreaming current is strictly limited to the saturated pore channels, the conduction currentpermeates the entire medium and may be quantifled with self-potential measurements.Chapter 4 gives some insight on the properties of the electrical double layer. A more detailedtreatment with reference to electrokinetic phenomena is given in Ishido and Mizutani [1981],Morgan et al. [1989], Revil et al. [1999a] and Shefier [2002].The self-potential method is the only geophysical technique that responds directly to uid  ow. Consequently, the method has been applied to a wide range of problems in anefiort to interpret characteristics of the hydraulic regime from the measured electrical po-tential distribution. Examples include earthquake prediction from  uid  ow into a dilatantzone preceding an earthquake [Mizutani et al., 1976; Corwin and Morrison, 1977; Jouni-aux and Pozzi, 1995], hydraulic fracturing in rocks for hydrocarbon recovery [Moore andGlaser, 2007], and the study of subglacial  ow systems [Blake and Clarke, 1999; Kulessaet al., 2003].A signiflcant body of work exists in the application of SP to the study of groundwater.SP fleld data have been analysed in the study of  ow into pumping wells [Bogoslovsky and3Chapter 1. Introduction------ ----- -----++++++++++++++++++++++FluidflowJcSOIL GRAINSOIL GRAIN++JsFigure 1.1: Conceptual model of streaming potential, showing streaming current JS andconduction current JC  ow paths in the pore space of a saturated soil.Ogilvy, 1973; Rizzo et al., 2004; Titov et al., 2005b],  uid  ow in the vadose zone [Sailhacet al., 2004; Darnet and Marquis, 2004], brine-contamination in an aquifer [Titov et al.,2005a], and spring  ow [Schiavone and Quarto, 1984]. Other studies involve delineating andmonitoring  uctuations in the position of the water table [Revil et al., 2004; Suski et al.,2006], locating sinkholes [Jardani et al., 2006a, b] and subsurface cavities [Jardani et al.,2006c] from vertical in ow in karstic terrain, assessing the performance of granular drainagesystems [Bogoslovsky and Ogilvy, 1973], and delineating buried paleochannels [Revil et al.,2005].One key application of the SP technique is the monitoring of seepage  ow throughearth dams. This has prompted research involving SP measurements on experimental lab-scale [Armbruster et al., 1989] and fleld-scale [Merkler et al., 1989; Wurmstich et al., 1991]embankments, as well as on existing dams [e.g. Ogilvy et al., 1969; Bogoslovsky and Ogilvy,1970; Johansson et al., 2005].Embankment dams are constructed of permeable materials and are designed to allowlimited amounts of seepage  ow from the impounded reservoir. Internal erosion, or the lossof flne-grained material due to seepage forces, is one of the most prevalent failure modes4Chapter 1. Introductionin embankment dams [ICOLD, 1995; Sherard et al., 1963]. Deviations from expected  owrates through an embankment may be indicative of \piping" or preferential seepage zonesthrough the core or foundation material, which can severely compromise the stability of thestructure.Dam safety surveillance typically consists of visual inspections supported with instru-mentation readings. Standard instrumentation is used to monitor seepage  ow from thedam drainage system, displacements, and pore water pressure within the dam body at dis-crete measurement points. These systems may be installed at the time of construction,but their existence and e–ciency is dependent on the design and age of the dam. Internalproblems can become quite advanced before instrumentation alerts to a problem or visualsigns of damage appear. This has sparked interest in the use of geophysical techniques toaugment standard monitoring methods.The SP method has been used to successfully delineate anomalous zones that correspondwith areas of preferential seepage in embankments [Ogilvy et al., 1969; Bogoslovsky andOgilvy, 1970; Black and Corwin, 1984; Butler et al., 1989; Corwin, 1991]. Practical dataacquisition techniques have been developed for one-time surveys and simple monitoringarrays [Corwin, 1990, 2005]. However, the current state of practice in SP data interpretationfor dam seepage studies is limited to qualitative and simple quantitative techniques thatestimate the depth and location of possible seepage paths using point source modelling [e.g.Panthulu et al., 2001; Rozycki et al., 2006; Corwin, 2007]. As discussed in Section 1.1, thesemethods do not provide an adequate physical representation of the underlying hydraulicregime to enable a quantitative interpretation of hydraulic parameters that are of interest.The research presented in this thesis is aimed at advancing the current state of practicein self-potential data interpretation, with a focus on the application to seepage monitoringin embankment dams. The following sections discuss past developments in the quantitativeinterpretation of SP data, and outline the current research in the context of this work.5Chapter 1. Introduction1.1 Quantitative interpretation of SP dataSelf-potential data may be interpreted using geometric source techniques, imaging meth-ods, forward modelling and inversion. Although these techniques vary in complexity, pastdevelopments can be grouped according to the level of knowledge of the underlying sourcemechanism.When the source mechanism is unknown or poorly understood, we can employ geometricsource, source imaging and inversion methods to infer the probable locations of currentsources that give rise to the SP data. Data interpretation consists of relating the calculateddistribution of current sources to possible geologic features or target locations. We cannotinterpret any additional information about the physical parameters that characterize theunderlying source mechanism since none is assumed.Geometric source techniques were originally developed to interpret SP anomalies associ-ated with distinct ore bodies. The general assumption is that the mineralized body may beapproximated by a polarized model of simple geometry, such as a sphere, inflnite cylinderor sheet, which are independent of the polarization mechanism. Curve-matching [de Witte,1948; Y˜ung˜ul, 1950; Meiser, 1962], parametric curve [Paul, 1965; Bhattacharya and Roy,1981; Atchuta Rao and Ram Babu, 1983; Ram Babu and Atchuta Rao, 1988], and least-squares inversion methods [Abdelrahman and Sharafeldin, 1997; Abdelrahman et al., 2003]have been used to estimate source parameters such as surface location, dip or polarizationangle, and depth of the body. Fitterman and Corwin [1982] used a least-squares inversiontechnique to recover the depth, geometry and source intensity of a polarized sheet modelof a geothermal fault zone. In cases where the source geometry is poorly constrained, aderivative analysis method [Abdelrahman et al., 1998, 2003] has been applied to estimatethe depth and a shape factor of the source.A common requirement of geometric source techniques is that the SP data must exhibita discrete anomalous feature that can be attributed to a single polarized body or zone.Regional efiects and signals resulting from other sources or source mechanisms must beremoved from the data or assumed to have minimal in uence. As mentioned above, thistype of approach has been applied to interpret discrete anomalous features observed in SPdata at embankment sites. The calculated source locations are inferred to correspond to6Chapter 1. Introductionthe location of preferential seepage paths. However, the analysis requires an assumption ofsource geometry and does not enable an interpretation of seepage-related properties.Source imaging methods have been used to reconstruct a tomographic map of the dis-tribution of electrical source strength from which the probable location of discrete sourcescan be estimated. These methods assume that measured SP data at the surface resultfrom the superposition of sources in the subsurface, and consequently must employ someform of normalization to enable a comparison of sources located at difierent depths. DiMaio and Patella [1994] performed a 2-D image reconstruction by cross-correlating a singlehorizontal component of the measured electric fleld Ex at surface with a scanning function,which is an analytical representation of Ex generated by a point source of unit strength.They applied their method to interpret SP data collected over a volcanic fracture system.Patella [1997] addressed the in uence of heterogeneous subsurface resistivity distributionon the imaged source distribution and extended the point source analysis to 3-D. Jardaniet al. [2006c] used a similar approach to image the depth to the roof of a subsurface cavity.H˜ammann et al. [1997] built on the work of Di Maio and Patella [1994] by cross-correlatingthe measured electrical fleld with a scanner function, which corresponds to a unit line sourceof charge that extends inflnitely in the direction perpendicular to the surface data proflle.They found that they could only resolve distinct source locations separated by distancesflve times greater than their depth. Their method was applied to the interpretation of SPproflle data collected across a complex hydrogeologic system, which was thought to exhibitelectrokinetic and electrochemical behaviour.The inversion of SP data using an inverse source formulation provides a more rigorousmeans of interpreting a subsurface distribution of current sources. Shi [1998] developeda regularized inversion routine to evaluate a 3-D current source distribution. Her methodrequires signiflcant a priori information to constrain the depth and amplitude of the sources.Given the inherent non-uniqueness of the problem, Minsley et al. [2007a] further developedthis routine by introducing sensitivity scaling (similar to depth weighting) and constraintsto encourage source compactness. They performed an inversion of SP data to delineate theextents of a contaminant plume where redox processes are occurring [Minsley et al., 2007b].Source imaging and inverse source methods are beneflcial in problems where we have7Chapter 1. Introductionlimited prior knowledge of the source mechanism, or where multiple source mechanisms areat work. However, these approaches do not enable a direct interpretation of subsurfaceparameters that are linked to a speciflc source mechanism. This requires the incorporationof a source model into the numerical analysis.When the source mechanism is known, we can employ forward modelling, imaging andinversion methods to interpret relevant characteristics of the underlying source mechanismfrom the SP data. The interpretation of hydraulic conditions from self-potential data relieson the assumption of an electrokinetic source mechanism.Discrete forward modelling methods have been developed to predict the self-potentialresponse to a primary  uid  ow fleld. These methods are based on a coupled  ow model thatdescribes the electrokinetic source mechanism. Consequently, they require some knowledgeof the relevant physical properties to efiectively calculate the distribution of sources andresulting electrical potential in the subsurface. A total potential approach to the coupled ow problem [Nourbehecht, 1963] was used to forward model electrokinetic behaviour pre-ceding earthquakes [Fitterman, 1978], and thermoelectric efiects associated with geothermalareas [Fitterman, 1983]. However, this approach is impractical for the study of streamingpotential, as suggested by Sill [1983].The convection current approach to the coupled  ow problem proposed by Sill [1983]has been widely accepted and implemented in most forward modelling studies of stream-ing potential performed to date. Sill and Killpack [1982] developed the flrst algorithm touse this approach using a 2.5-D solution, which was adapted by Wilt and Butler [1990]and applied to evaluate the streaming potential signal at an embankment site [Wilt andCorwin, 1989]. Ishido and Pritchett [1999] studied electrokinetic behaviour in geothermalreservoir simulations using a flnite difierence post-processor. Revil et al. [1999b] consid-ered both advective and convective  uid  ow in a flnite element study of a geothermalsystem. Wurmstich et al. [1991] used a 2-D flnite element model to evaluate the SP re-sponse to seepage in an embankment. They compared their numerical result to measureddata obtained over the surface of a fleld-scale laboratory embankment model. Berube [2004]developed a flnite element code to study streaming potential signals in dams. Wurmstichand Morgan [1994] developed a 3-D flnite difierence algorithm to study streaming potential8Chapter 1. Introductionsignals generated by radial  ow to an oil well in a  uid-saturated halfspace. Saunders et al.[2006] performed 3-D flnite element modelling of the SP response to oil-well pumping in atwo-phase (oil-water) system. Titov et al. [2002, 2005b] developed and applied a 2-D flnitedifierence algorithm to evaluate streaming potential signals in groundwater  ow problems.Jardani et al. [2006a] performed 3-D flnite element modelling to study the SP responseto vertical inflltration in a karstic environment. Darnet and Marquis [2004] performed asynthetic 1-D modelling study of streaming potentials in the vadose zone. Most recently,Linde et al. [2007] performed 1-D modelling of the SP response to unsaturated  ow in acontrolled drainage experiment using a formulation where streaming current is expressed asa function of pore  uid velocity, porosity and excess charge [Revil and Linde, 2006].Although signiflcant work has been done in developing forward models of streamingpotential, there is a lack of three-dimensional forward modelling methods that are capa-ble of solving variably saturated  ow problems. This type of model is required to solveembankment seepage problems of complicated geometry.Imaging and parametric inversion methods have been developed to extract hydraulicparameters from SP data. However, these methods are limited to speciflc groundwater  owproblems. Examples include work presented by Sailhac and Marquis [2001], who used a 2-Dwavelet-based method to invert for electrokinetic source parameters assuming a homoge-neous electrical conductivity distribution. Darnet et al. [2003] used a weighted least-squaresapproach to invert for the hydraulic conductivity, cross-coupling conductivity and thicknessof a homogeneous aquifer subject to  ow from a pumping well. Electrokinetic sources aredirectly related to the extraction rate of  uid from the well, and the electrical conductivitycontrast between the aquifer and well casing is accounted for in their solution.Other work has focused on determining the position of the water table. Examples in-clude an imaging method presented by Birch [1998], who employed a similar approach asH˜ammann et al. [1997] using line sources of charge. The basis for his approach was flrst pro-posed by Fournier [1989], who developed an equation for electrical potential at the earth’ssurface using a total potential formulation of the coupled  ow problem [Nourbehecht, 1963]for speciflc application to unconflned groundwater problems. Fournier [1989] stated thatall electrokinetic sources in a homogeneous medium can be represented by a dipolar sheet9Chapter 1. Introductionof charge at the position of the water table. This model is assumed in the formulation ofthe source strength term, which consists of a streaming potential coupling coe–cient (C)multiplied by the height of the water table. The position of the water table is interpretedfrom a zone of constant C on the tomographic image of C. Revil et al. [2003] improved themethod proposed by Birch [1998] by normalizing the function to consider only a verticalorientation of dipoles at the phreatic surface. Revil et al. [2003] developed a 2-D weightedleast-squares inversion technique based on the formulation presented by Fournier [1989] torecover the depth to the water table. Revil et al. [2004] improved this routine by incorpo-rating the efiect of an electrical conductivity contrast between saturated and unsaturatedzones. This routine has been applied in a number of studies to recover the shape of thewater table [e.g., Jardani et al., 2006b; Linde and Revil, 2007].Although the imaging and parametric inversion methods listed above are useful in delin-eating the location of the water table, they consider that all electrokinetic charge accumu-lates at this boundary. This precludes the investigation of other sources that may arise dueto heterogeneities in the saturated zone. For embankment seepage monitoring applications,we are interested in studying zones of preferential  ow that develop beneath the phreaticsurface. These zones may not have a signiflcant in uence on the position of the phreaticsurface.We require a numerical modelling method based on a distributed set of parameters tostudy complex  ow problems. Discrete forward modelling methods can be used to predictthe self-potential distribution that results from a given hydraulic regime. However, aninverse methodology facilitates the practical interpretation of measured SP data to extractpertinent hydraulic information. To date, the under-determined problem of inverting self-potential data has only been solved using an inverse source formulation.Successful application of discrete numerical modelling methods is limited by our knowl-edge of the physical properties of the subsurface. Realistic estimates of the relevant proper-ties are required to efiectively model streaming potential in a discretized framework. Whilesigniflcant efiort has been devoted to studying and characterizing hydraulic and electricalproperties for difierent materials, a similar collection of data on coupled  ow properties hasnot been compiled.10Chapter 1. IntroductionPrevious laboratory investigations of streaming potential have focused on measuring thestreaming potential coupling coe–cient C to examine properties of the soil- uid interface[Ishido and Mizutani, 1981; Morgan et al., 1989; Lorne et al., 1999; Revil et al., 1999b;Reppert and Morgan, 2003], or to examine the link between C and hydraulic propertiessuch as permeability and saturation [Jouniaux and Pozzi, 1995; Pengra et al., 1999; Guichetet al., 2003; Revil and Cerepi, 2004]. There is a lack of experimental data reported forsoils typical of embankments, particularly those that exhibit a range of grain sizes. Mostexperimental investigations report measured values of C along with the saturating  uidproperties, but do not include values of electrical resistivity ‰ measured under the samesample conditions. These two properties are required to calculate the streaming currentcross-coupling coe–cient L = ¡C=‰ and are fundamental to any quantitative analysis ofstreaming potential.1.2 Thesis objective and outlineThe main objective of the research presented in this thesis is to identify what useful in-formation about hydraulic parameters and embankment integrity can be garnered fromself-potential data, when streaming potential is the dominant mechanism. I seek to answerthis question through the development and application of forward and inverse modellingtechniques, and a laboratory investigation of the physical properties that govern the phe-nomenon.Figure 1.2 identifles the three key elements of the thesis research and illustrates howthey contribute to a quantitative interpretation of hydraulic conditions from self-potentialdata. A 3-D forward model enables us to predict the self-potential response to a givenseepage regime and enables us to study the streaming potential phenomenon in complexproblems. The inverse problem makes use of the forward model and enables a direct in-terpretation of hydraulic head from self-potential data. Realistic estimates of the relevantelectrical properties are required as input to the numerical methods to efiectively interpretthe measured self-potential.The main body of the thesis consists of flve chapters, each of which has been prepared11Chapter 1. IntroductionRealistic estimates of electrical coupled flow properties3-D forward model of streaming potentialInterpretation of hydraulic conditions from SP data3-D inversion for hydraulic headFigure 1.2: Key elements contributing to the quantitative interpretation of hydraulic con-ditions from SP data.as a separate manuscript.Chapter 2 presents a 3-D flnite volume algorithm for calculating the self-potential dis-tribution resulting from  uid  ow in a porous medium. The algorithm was developed toenable the study of variably saturated  ow problems of complex geometry for applicationto engineering and hydrogeological investigations. This work builds on research flrst pre-sented in [Shefier, 2002]. The chapter presents a coupled  ow model of streaming potential,the governing equations that describe the hydraulic and electrical  ow problems, and thesolution of these equations using the flnite volume method. A series of examples comparespredicted and measured self-potential data, shows the importance of proper representationof heterogeneous physical property distributions, and illustrates the need for 3-D models totackle complex  ow problems.Chapter 3 introduces a laboratory apparatus designed to measure the streaming poten-tial coupling coe–cient C and electrical resistivity ‰ of saturated soil specimens. Theseproperties are measured on the same sample in order to characterize the streaming cur-rent cross-coupling coe–cient L, which is a fundamental parameter required for numericalmodelling. This chapter details the construction and calibration of the apparatus, whichwas designed to study well-graded (poorly sorted) soil samples typical of embankments.It is the flrst of its kind to implement the oscillatory  ow method for streaming potentialmeasurements on well-graded samples, and is unique in its design to measure both C and‰ on the same soil specimen. The chapter includes a description of streaming potential andresistivity measurement techniques and a comparison of data acquired using the difierent12Chapter 1. Introductionmethods.Chapter 4 presents the results of laboratory measurements performed using the appa-ratus described in Chapter 3. The objectives of the study were two-fold: to investigate thein uence of soil and  uid parameters on the streaming current cross-coupling coe–cient, andto characterize this property for representative embankment soils. This chapter presentsexperimental data acquired on a series of glass bead and embankment soil samples, anddiscusses the in uence of sample density, sample gradation and  uid conductivity on themeasured electrical properties. The analysis of glass bead samples includes a comparisonwith theoretical estimates of the electrical properties and illustrates the major in uence ofsample equilibration on the measured electrical properties. The analysis of embankment soildata includes a comparison with published values of the streaming current cross-couplingcoe–cient.Evaluating the sensitivity of the SP method to detect the onset of internal erosion in thecore of a dam is of key importance in determining the suitability of the method for seepagemonitoring investigations. Chapter 5 applies the forward modelling algorithm described inChapter 2 using the electrical properties reported in Chapter 4 to study the SP responseto seepage in a synthetic 2-zone fleld-scale embankment. A series of models are examinedto evaluate the residual SP distributions that result from preferential  ow through defectswithin the core. Detection of the predicted SP anomalies is discussed in terms of practicaldetection limits for the method, and predicted variations in hydraulic head and  ow rate inresponse to the defects.Chapter 6 presents an inversion algorithm that recovers a 3-D distribution of hydraulichead from self-potential data where streaming potential is the dominant mechanism. Theforward model described in Chapter 2 provides a basis for the inverse problem. This chapterdescribes the formulation and solution of the inverse problem, and is the flrst time such anapproach has been documented. SP data collected in a laboratory-scale model of seepageunder a cut-ofi wall are inverted to recover a hydraulic head distribution that is comparedwith observed head measurements. This example also serves to demonstrate how priorinformation about the hydraulic and electrical  ow systems can be used to help constrainthe solution.13Chapter 1. IntroductionChapter 7 summarizes the key flndings of the research, discusses the practical contribu-tions, and comments on areas of future work. A derivation of the discrete approximationof the coupled electrical  ow equation presented in Chapter 2 is included as Appendix A.14Chapter 1. Introduction1.3 ReferencesAbdelrahman, E. M., and S. M. Sharafeldin, A least-squares approach to depth determina-tion from self-potential anomalies caused by horizontal cylinders and spheres, Geophysics,62(1), 44{48, 1997.Abdelrahman, E. M., A. A. B. Ammar, H. I. Hassanein, and M. A. Hafez, Derivativeanalysis of SP anomalies, Geophysics, 63(3), 890{897, 1998.Abdelrahman, E. M., H. M. El-Araby, A. G. Hassaneen, and M. A. Hafez, New methodsfor shape and depth determinations from SP data, Geophysics, 68(4), 1202{1210, 2003.Armbruster, H., J. Brauns, W. Mazur, and G. P. Merkler, Efiect of leaks in dams and trialsto detect leakages by geophysical means, in Detection of Subsurface Flow Phenomena,Lecture Notes in Earth Sciences, vol. 27, edited by G. P. Merkler, H. Militzer, H. H˜otzl,H. Armbruster, and J. Brauns, pp. 3{18, Springer-Verlag, Berlin, 1989.Atchuta Rao, D., and H. V. Ram Babu, Quantitative interpretation of self-potential anom-alies due to two-dimensional sheet-like bodies, Geophysics, 48(12), 1659{1664, 1983.Berube, A. P., Investigating the streaming potential phenomenon using electric measure-ments and numerical modelling with special reference to seepage monitoring in embank-ment dams, Ph.D. thesis, Lule”a University of Technology, Sweden, 2004.Bhattacharya, B. B., and N. Roy, A note on the use of a nomogram for self-potentialanomalies, Geophysical Prospecting, 29, 102{107, 1981.Birch, F. S., Imaging the water table by flltering self-potential proflles, Ground Water,36(5), 779{782, 1998.Black, W. E., and R. F. Corwin, Application of self-potential measurements to the delin-eation of groundwater seepage in earth-flll embankments, Society of Exploration Geophysi-cists Technical Program Expanded Abstracts, 3, 162{164, doi:10.1190/1.1894185, 1984.Blake, E., and G. Clarke, Subglacial electrical phenomena, Journal of Geophysical Re-search, 104(B4), 7481{7495, 1999.15Chapter 1. IntroductionBogoslovsky, V. A., and A. A. Ogilvy, Application of geophysical methods for studyingthe technical status of earth dams, Geophysical Prospecting, 18, 758{773, 1970.Bogoslovsky, V. V., and A. A. Ogilvy, Deformations of natural electric flelds near drainagestructures, Geophysical Prospecting, 21, 716{723, 1973.Butler, D. K., J. L. Llopis, and C. M. Deaver, Comprehensive geophysical investigation ofan existing dam foundation, The Leading Edge, 8(8), 10{18, 1989.Corry, C. E., Spontaneous polarization associated with porphyry sulflde mineralization,Geophysics, 50(6), 1020{1034, 1985.Corwin, R. F., The self-potential method for environmental and engineering applications,in Geotechnical and Environmental Geophysics, vol. 1, edited by S. H. Ward, pp. 127{145,Society of Exploration Geophysicists, Tulsa, 1990.Corwin, R. F., Evaluation of efiects of cutofi wall construction on seepage  ow using self-potential data: East Embankment Wells Dam, Tech. rep., Douglas County Public UtilityDistrict no. 1, Washington, 1991.Corwin, R. F., Investigation of geophysical methods for assessing seepage and internalerosion in embankment dams: Self-potential fleld data acquisition manual, Tech. rep.,Canadian Electricity Association Technologies Inc. (CEATI), Report T992700-0205B/1,Montreal, 2005.Corwin, R. F., Investigation of geophysical methods for assessing seepage and internalerosion in embankment dams: Interpretation of self-potential data for dam seepage inves-tigations, Tech. rep., Canadian Electricity Association Technologies Inc. (CEATI), ReportT992700-0205B/3, Montreal, 2007.Corwin, R. F., and D. B. Hoover, The self-potential method in geothermal exploration,Geophysics, 44(2), 226{245, 1979.Corwin, R. F., and H. F. Morrison, Self-potential variations preceding earthquakes incentral California, Geophysical Research Letters, 4(4), 171{174, 1977.16Chapter 1. IntroductionDarnet, M., and G. Marquis, Modelling streaming potential (SP) signals induced by watermovement in the vadose zone, Journal of Hydrology, 285, 114{124, 2004.Darnet, M., G. Marquis, and P. Sailhac, Estimating aquifer hydraulic properties from theinversion of surface Streaming Potential (SP) anomalies, Geophysical Research Letters,30(13), 1679, doi:10.1029/2003GL017631, 2003.de Groot, S. R., Thermodynamics of irreversible processes, Selected Topics in ModernPhysics, vol. 3, North Holland Publishing Company, Amsterdam, 1951.de Witte, L., A new method of interpretation of self-potential fleld data, Geophysics, 13(4),600{608, 1948.Di Maio, R., and D. Patella, Self-potential anomaly in volcanic areas: The Mt. Etna casehistory, Acta Vulcanologica, 4, 119{124, 1994.Fitterman, D. V., Electrokinetic and magnetic anomalies associated with dilatant regionsin a layered earth, Journal of Geophysical Research, 83(B12), 5923{5928, 1978.Fitterman, D. V., Modeling of self-potential anomalies near vertical dikes, Geophysics,48(2), 171{180, 1983.Fitterman, D. V., and R. F. Corwin, Inversion of self-potential data from the Cerro Prietogeothermal fleld, Mexico, Geophysics, 47(6), 938{945, 1982.Fournier, C., Spontaneous potentials and resistivity surveys applied to hydrogeology in avolcanic area: Case history of the Cha^‡ne des Puys (Puy-de-D^ome, France), GeophysicalProspecting, 37, 647{668, 1989.Fox, R. W., On the electro-magnetic properties of metalliferous veins in the mines ofCornwall, Philosophical Transactions of the Royal Society of London, 120, 399{414, 1830.Guichet, X., L. Jouniaux, and J. P. Pozzi, Streaming potential of a sand columnin partial saturation conditions, Journal of Geophysical Research, 108(B3), 2141, doi:10.1029/2001JB001517, 2003.17Chapter 1. IntroductionH˜ammann, M., H. R. Maurer, A. G. Green, and H. Horstmeyer, Self-potential imagereconstruction: Capabilities and limitations, Journal of Environmental and EngineeringGeophysics, 2(1), 21{35, 1997.ICOLD, Dam failures: Statistical analysis, in Bulletin no. 99, International Commissionon Large Dams, 1995.Ishido, T., and H. Mizutani, Experimental and theoretical basis of electrokinetic phe-nomena in rock-water systems and its application to geophysics, Journal of GeophysicalResearch, 86(B3), 1763{1775, 1981.Ishido, T., and J. W. Pritchett, Numerical simulation of electrokinetic potentials associatedwith subsurface  uid  ow, Journal of Geophysical Research, 104(B7), 15,247{15,259, 1999.Jardani, A., J. P. Dupont, and A. Revil, Self-potential signals associated with preferentialgroundwater  ow pathways in sinkholes, Journal of Geophysical Research, 111, B09204,doi:10.1029/2005JB004231, 2006a.Jardani, A., J. P. Dupont, and A. Revil, Least-squares inversion of self-potential (SP) dataand application to the shallow  ow of ground water in sinkholes, Geophysical ResearchLetters, 33, L19306, doi:10.1029/2006GL027458, 2006b.Jardani, A., A. Revil, and J. P. Dupont, Self-potential tomography applied to the determi-nation of cavities, Geophysical Research Letters, 33, L13401, doi:10.1029/2006GL026028,2006c.Johansson, S., J. Friborg, T. Dahlin, and P.Sj˜odahl, Long term resistivityand self potentialmonitoringofembankmentdams: ExperiencesfromH˜allbyandS˜advadams, Sweden, Tech.rep., Elforsk/Canadian Electricity Association Dam Safety Interest Group, Report 05:15,Stockholm, 2005.Jouniaux, L., andJ.P.Pozzi, Streamingpotentialandpermeabilityofsaturatedsandstonesunder triaxial stress: Consequences for electrotelluric anomalies prior to earthquakes, Jour-nal of Geophysical Research, 100(B6), 10,197{10,209, 1995.18Chapter 1. IntroductionKulessa, B., B. Hubbard, and G. H. Brown, Cross-coupled  ow modeling of coincidentstreaming and electrochemical potentials and application to sub-glacial self-potential data,Journal of Geophysical Research, 108(B8), 2381, doi:10.1029/2001JB001167, 2003.Linde, N., and A. Revil, Inverting self-potential data for redox potentials of contaminantplumes, Geophysical Research Letters, 34, L14302, doi:10.1029/2007GL030084, 2007.Linde, N., D. Jougnot, A. Revil, S. K. Matth˜ai, T. Arora, and D. Renard, Streamingcurrent generation in two-phase  ow conditions, Geophysical Research Letters, 34, L03306,doi:10.1029/2006GL028878, 2007.Lorne, B., F. Perrier, and J. P. Avouac, Streaming potential measurements 1. Propertiesof the electrical double layer from crushed rock samples, Journal of Geophysical Research,104(B8), 17,857{17,877, 1999.Maineult, A., Y. Bernab¶e, and P. Ackerer, Detection of advected concentration and pHfronts from self-potential measurements, Journal of Geophysical Research, 110, B11205,doi:10.1029/2005JB003824, 2005.Meiser, P., A method of quantitative interpretation of self-potential measurements, Geo-physical Prospecting, 10, 203{218, 1962.Merkler, G. P., H. Armbruster, H. H˜otzl, P. Marschall, A. Kassel, and E. Ungar, Modellingof streaming potentials and thermoelectrical measurements at a big laboratory channel, inDetection of Subsurface Flow Phenomena, Lecture Notes in Earth Sciences, vol. 27, editedby G. P. Merkler, H. Militzer, H. H˜otzl, H. Armbruster, and J. Brauns, pp. 223{249,Springer-Verlag, Berlin, 1989.Minsley, B. J., J. Sogade, and F. D. Morgan, Three-dimensional source inversion of self-potential data, Journal of Geophysical Research, 112, B02202, doi:10.1029/2006JB004262,2007a.Minsley, B. J., J. Sogade, and F. D. Morgan, Three-dimensional self-potential inversionfor subsurface DNAPL contaminant detection at the Savannah River Site, South Carolina,Water Resources Research, 43, W04429, doi:10.1029/2005WR003996, 2007b.19Chapter 1. IntroductionMitchell, J. K., Conduction phenomena: From theory to geotechnical practice, Geotech-nique, 43(3), 299{340, 1991.Mizutani, H., T. Ishido, T. Yokokura, and S. Ohnishi, Electrokinetic phenomena associatedwith earthquakes, Geophysical Research Letters, 3, 365{368, 1976.Moore, J. R., and S. D. Glaser, Self-potential observations during hydraulic fracturing,Journal of Geophysical Research, 112, B02204, doi:10.1029/2006JB004373, 2007.Morgan, F. D., E. R. Williams, and T. R. Madden, Streaming potential properties of West-erly granite with applications, Journal of Geophysical Research, 94(B9), 12,449{12,461,1989.Naudet, V., and A. Revil, A sandbox experiment to investigate bacteria-mediated redoxprocesses on self-potential signals, Geophysical Research Letters, 32, L11405, doi:10.1029/2005GL022735, 2005.Naudet, V., A. Revil, and J. Y. Bottero, Relationship between self-potential (SP) sig-nals and redox conditions in contaminated groundwater, Geophysical Research Letters, 30,21,2091, doi:10.1029/2003GL018096, 2003.Naudet, V., A. Revil, E. Rizzo, J. Y. Bottero, and P. B¶egassat, Groundwater redox condi-tions and conductivity in a contaminant plume from geoelectrical investigations, Hydrologyand Earth System Sciences, 8(1), 8{22, 2004.Nourbehecht, B., Irreversible thermodynamic efiects in inhomogeneous media and theirapplications in certain geoelectric problems, Ph.D. thesis, Massachusetts Institute of Tech-nology, Cambridge, 1963.Ogilvy, A. A., M. A. Ayed, and V. A. Bogoslovsky, Geophysical studies of water leakagesfrom reservoirs, Geophysical Prospecting, 17, 36{62, 1969.Onsager, L., Reciprocal relations in irreversible processes, I, Physical Review, 37, 405{426,1931.20Chapter 1. IntroductionPanthulu, T. V., C. Krishnaiah, and J. M. Shirke, Detection of seepage paths in earth damsusing self-potential and electrical resistivity methods, Engineering Geology, 59, 281{295,2001.Patella, D., Introduction to ground surface self-potential tomography, GeophysicalProspecting, 45(4), 653{681, 1997.Paul, M. K., Direct interpretation of self-potential anomalies caused by inclined sheets ofinflnite horizontal extensions, Geophysics, 30(3), 418{423, 1965.Pengra, D. B., S. X. Li, and P. Wong, Determination of rock properties by low-frequencyAC electrokinetics, Journal of Geophysical Research, 104(B12), 29,485{29,508, 1999.Ram Babu, H. V., and D. Atchuta Rao, Short note: A rapid graphical method for theinterpretation of the self-potential anomaly over a two-dimensional inclined sheet of flnitedepth extent, Geophysics, 53(8), 1126{1128, 1988.Reppert, P. M., and F. D. Morgan, Temperature-dependent streaming potentials: 2. Labo-ratory, Journal of Geophysical Research, 108(B11), 2547, doi:10.1029/2002JB001755, 2003.Revil, A., and A. Cerepi, Streaming potentials in two-phase  ow conditions, GeophysicalResearch Letters, 31, L11605, doi:10.1029/2004GL020140, 2004.Revil, A., and N. Linde, Chemico-electromechanical coupling in microporous media, Jour-nal of Colloid and Interface Science, 302, 682{694, 2006.Revil, A., P. A. Pezard, and P. W. J. Glover, Streaming potential in porous media: 1.Theory of the zeta potential, Journal of Geophysical Research, 104(B9), 20,021{20,031,1999a.Revil, A., H. Schwaeger, L. M. Cathles III, and P. D. Manhardt, Streaming potential inporous media: 2. Theory and application to geothermal systems, Journal of GeophysicalResearch, 104(B9), 20,033{20,048, 1999b.Revil, A., V. Naudet, J. Nouzaret, and M. Pessel, Principles of electrography appliedto self-potential electrokinetic sources and hydrogeological applications, Water ResourcesResearch, 39(5), 1114, doi:10.1029/2001WR000916, 2003.21Chapter 1. IntroductionRevil, A., V. Naudet, and J. D. Meunier, The hydroelectric problem of porous rocks:Inversion of the position of the water table from self-potential data, Geophysical JournalInternational, 159, 435{444, 2004.Revil, A., L. Cary, Q. Fan, A. Finizola, and F. Trolard, Self-potential signals associatedwith preferential ground water  ow pathways in a buried paleo-channel, Geophysical Re-search Letters, 32, L07401, 2005.Rizzo, E., B. Suski, A. Revil, S. Straface, and S. Troisi, Self-potential signals associatedwith pumping tests experiments, Journal of Geophysical Research, 109, B10203, doi:10.1029/2004JB003049, 2004.Rozycki, A., J. M. Ruiz Fonticiella, and A. Cuadra, Detection and evaluation of horizontalfractures in earth dams using the self-potential method, Engineering Geology, 82, 145{153,doi:10.1016/j.enggeo.2005.09.013, 2006.Sailhac, P., and G. Marquis, Analytic potentials for the forward and inverse modeling ofSP anomalies caused by subsurface  ow, Geophysical Research Letters, 28(9), 1851{1854,2001.Sailhac, P., M. Darnet, and G. Marquis, Electrical streaming potential measured at theground surface: Forward modeling and inversion issues for monitoring inflltration andcharacterizing the vadose zone, Vadose Zone Journal, 3, 1200{1206, 2004.Sato, M., and H. M. Mooney, The electrochemical mechanism of sulflde self-potentials,Geophysics, 25(1), 226{249, 1960.Saunders, J. H., M. D. Jackson, and C. C. Pain, A new numerical model of electrokineticpotential response during hydrocarbon recovery, Geophysical Research Letters, 33, L15316,doi:10.1029/2006GL026835, 2006.Schiavone, D., and R. Quarto, Self-potential prospecting in the study of water movements,Geoexploration, 22, 47{58, 1984.Shefier, M. R., Response of the self-potential method to changing seepage conditions inembankment dams, M.A.Sc. thesis, University of British Columbia, Vancouver, 2002.22Chapter 1. IntroductionSherard, J. L., R. J. Woodward, S. F. Gizienski, and W. A. Clevenger, Earth and earth-rockdams, John Wiley and Sons, New York, 1963.Shi, W., Advanced modeling and inversion techniques for three-dimensional geoelectricalsurveys, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, 1998.Sill, W. R., Self-potential modeling from primary  ows, Geophysics, 48(1), 76{86, 1983.Sill, W. R., and T. J. Killpack, SPXCPL - Two-dimensional modeling program of self-potential efiects from cross-coupled  uid and heat  ow: User’s guide and documentation forVersion 1.0, Tech. rep., Earth Science Laboratory, University of Utah Research Institute,Salt Lake City, 1982.Suski, B., A. Revil, K. Titov, P. Konosavsky, M. Voltz, C. Dag es, and O. Huttel, Mon-itoring of an inflltration experiment using the self-potential method, Water ResourcesResearch, 42, W08418, doi:10.1029/2005WR004840, 2006.Titov, K., Y. Ilyin, P. Konosavski, and A. Levitski, Electrokinetic spontaneous polarizationin porous media: Petrophysics and numerical modelling, Journal of Hydrology, 267, 207{216, 2002.Titov, K., A. Levitski, P. K. Konosavski, A. V. Tarasov, Y. T. Ilyin, and M. A. Bues,Combined application of surface geoelectrical methods for groundwater- ow modeling: Acase history, Geophysics, 70(5), H21{H31, 2005a.Titov, K., A. Revil, P. Konosavsky, S. Straface, and S. Troisi, Numerical modelling ofself-potential signals associated with a pumping test experiment, Geophysical Journal In-ternational, 162, 641{650, 2005b.Wilt, M. J., and D. K. Butler, Numerical modeling of SP anomalies: Documentationof program SPPC and applications, Report 4, in Geotechnical Applications of the Self-Potential Method, Technical Report REMR-GT-6, Department of the Army, U.S. ArmyCorps of Engineers, Washington, 1990.Wilt, M. J., and R. F. Corwin, Numerical modeling of self-potential anomalies due toleaky dams: Model and fleld examples, in Detection of Subsurface Flow Phenomena, Lec-23Chapter 1. Introductionture Notes in Earth Sciences, vol. 27, edited by G. P. Merkler, H. Militzer, H. H˜otzl,H. Armbruster, and J. Brauns, pp. 73{89, Springer-Verlag, Berlin, 1989.Wurmstich, B., and F. D. Morgan, Modeling of streaming potential responses caused byoil well pumping, Geophysics, 59(1), 46{56, 1994.Wurmstich, B., F. D. Morgan, G. P. Merkler, and R. L. Lytton, Finite-element modeling ofstreaming potentials due to seepage: Study of a dam, Society of Exploration GeophysicistsTechnical Program Expanded Abstracts, 10, 542{544, 1991.Y˜ung˜ul, S., Interpretation of spontaneous polarization anomalies caused by spheroidalorebodies, Geophysics, 15(2), 237{246, 1950.24Chapter 2Three-dimensional forwardmodelling of streaming potential 12.1 IntroductionFluid  ow through porous media generates electrical current  ow, which can be charac-terized through a series of electrode measurements to produce a map of potential. Thiselectrokinetic phenomenon of streaming potential is one of several coupled  ow mecha-nisms, such as thermoelectric and electrochemical efiects, that can be measured using theself-potential (SP) method.Streaming potential is the common if not dominant driving mechanism for the SP signalin geothermal exploration [Corwin and Hoover, 1979; Ishido and Pritchett, 1999], earth-quake prediction [Mizutani et al., 1976; Corwin and Morrison, 1977; Fitterman, 1978],groundwater studies [Titov et al., 2005a], and engineering and environmental investigations[Bogoslovsky and Ogilvy, 1973; Corwin, 1990].Given the analogous behaviour of hydraulic and electrical  ow systems and the principlesof coupled  ow, the self-potential distribution can be studied to evaluate characteristics ofthe hydraulic regime. A coupled  ow model is required to explain the interaction of thehydraulic and electrical  ow systems, and the convection current approach proposed bySill [1983] has been the elemental theory implemented in most numerical investigations ofstreaming potential. Wurmstich et al. [1991] used a 2-D flnite element code to analyzeseepage and the streaming potential response in a model embankment. Wurmstich andMorgan [1994] developed a 3-D flnite difierence algorithm to study streaming potential1A version of this chapter has been published. Shefier, M.R. and Oldenburg, D.W. (2007) Three-dimensional modelling of streaming potential, Geophysical Journal International, 169:839-84825Chapter 2. 3-D forward modelling of streaming potentialsignals caused by oil well pumping in a  uid-saturated halfspace. Ishido and Pritchett[1999] studied the streaming potential response in geothermal reservoir simulations using aflnite difierence post-processor. Revil et al. [1999] used a flnite element model to evaluatethe streaming potential response over a geothermal fleld driven by both advective andconvective  uid  ow. Titov et al. [2002, 2005b] developed a 2-D flnite difierence programto evaluate streaming potential responses in groundwater  ow problems.In this paper we present a 3-D flnite volume algorithm for calculating the self-potentialdistribution resulting from  uid  ow in a porous medium. The algorithm was developed toenable the study of variably saturated  ow problems of complex geometry for application toengineering and hydrogeological investigations. The equations that describe the hydraulicand electrical  ow problems are developed in Section 2.2. Section 2.3 describes the forwardmodelling methodology and the flnite volume solution for self-potential. The numericalsolution of a simple  ow problem is compared with an analytical solution as a means ofverifying the algorithm.A quantitative study of streaming potential relies on realistic estimates of the relevantphysical properties. Section 2.4 discusses the assignment of streaming current cross-couplingconductivity and electrical conductivity distributions for difierent hydraulic systems. Theunconflned  ow problem of seepage through a homogeneous embankment is studied as anexample. Section 2.5 develops an equation that describes primary and secondary sourcesof charge that contribute to the self-potential response. An example of a pumping well isused to illustrate how heterogeneous physical property distributions give rise to secondarysources of charge, which can contribute signiflcantly to the SP response.Finally, a fleld data example of seepage through an earth embankment and foundationis described in Section 2.6. This example clearly illustrates the need for 3-D modellingcapability to correctly capture the  ow regime in a problem of complex geometry.26Chapter 2. 3-D forward modelling of streaming potential2.2 Theory2.2.1 Primary  owFluid  ow through a fully-saturated porous medium is described using the mass conservationequation stated in terms of hydraulic head h [m]:r¢ q = ¡Ss@h@t +Qv –(r¡rs) ; (2.1)where q is the volumetric  ux [m3 s¡1 m¡2 ], Qv is an additional term used to represent anyexternal point sources of  uid  ow imposed on the volume [m3 s¡1 m¡3 ], and rs denotesthe source location. The speciflc storage of the volume, Ss [m3 m¡1 m¡3 ] takes the formSs = ‰fg(fi+nfl) ; (2.2)where‰f is uiddensity[kgm¡3 ], fiisthecompressibilityofthesolidmatrix[m3 Pa¡1 m¡3 ],n is porosity [m3 m¡3 ], fl is the  uid compressibility [m3 Pa¡1 m¡3 ], and g is gravitationalacceleration.Phenomenological laws have been shown empirically to describe irreversible  ow as alinear relation between  ux and the gradient of a scalar potential. Fluid  ux under laminar ow conditions may be described using a form of Darcy’s law:q = ¡Krh; (2.3)where K is the hydraulic conductivity [ms¡1 ] of the volume under study.The conservation of charge equation is used to describe electrical current  ow:r¢(‰ev) = ¡@‰e@t ; (2.4)wherev is the volumetric  ux [m3 s¡1 m¡2 ] and ‰e is the volumetric charge density [Cm¡3 ].The charge  ux term ‰ev may be written instead as a volumetric current densityJ [Am¡2 ],by considering the deflnition of electric current. Equation (2.4) now takes the formr¢J = ¡@‰e@t +Iv –(r¡rs) ; (2.5)where Iv [Am¡3 ] is the current  ow of any external source imposed on the volume. Con-duction current density is described using Ohm’s law:J = ¡ r` ; (2.6)27Chapter 2. 3-D forward modelling of streaming potentialwhere ` [V] is the electrical potential and  [Sm¡1 ] is the electrical conductivity.2.2.2 Coupled  owThe interdependence of  uid, temperature, electrical and chemical  ow systems can bedeflned using the theory of coupled  ow, which describes the  ux of a given quantity interms of the sum of primary and secondary gradients. A generalized constitutive relationthat includes both primary and coupled  ow terms is¡i = ¡nXj=iLijr'j ; (2.7)where ¡i is the  ux of the quantity i under study, 'j is the scalar potential, and Lij is theconductivity term, which is an averaged macroscopic property representative of the volumeunder study.The importance of each term on the total  ux is dependent on the relative magnitude ofthe gradient and the corresponding conductivity coe–cient such that, for su–ciently slowprocesses,  ow may result from not only the primary gradient, but also from the secondarygradients. In the absence of signiflcant chemical or temperature gradients the coupled  owequations that describe electrokinetic phenomena are¡f = ¡Lffr'f ¡Lfer'e (2.8a)and¡e = ¡Leer'e ¡Lefr'f : (2.8b)The cross-coupling conductivity coe–cients link the secondary gradients to the pri-mary  ow terms such that Lfer'e describes  uid  ow due to electro-osmosis and Lefr'fdescribes streaming current  ow. These phenomena are connected through the Onsagerreciprocal relations that state Lfe = Lef, provided the  ux and gradient terms are formu-lated properly, and that  ow can be adequately represented by linear constitutive equationsin a given system [Onsager, 1931].The gradient terms are considered thermodynamic forces and must be formulated touphold the principles of nonequilibrium thermodynamics [de Groot, 1951]. Consequently,28Chapter 2. 3-D forward modelling of streaming potentialpotential must be expressed in terms of energy per unit quantity i. We deflne the hydraulicpotential 'f [Jkg¡1 or m2 s¡2 ], which may be described using Bernoulli’s equation andrelated to hydraulic head through the relation 'f = gh. Accordingly, ¡f = ‰fqT representsthe total mass  ux, and the primary conductivity term can be deflned as Lff = ‰fK=g.Substituting these relations into (2.8a) gives‰fqT = ¡‰fKg rgh¡Lfer` : (2.9)In a similar fashion, 'e = ` is the electric potential [JC¡1 or V], ¡e = JT representsthe total charge  ux [Cs¡1 m¡2 or Am¡2 ], and Lee =  is the electrical conductivity.Equation (2.8b) now becomesJT = ¡ r`¡Lefrgh : (2.10)The cross-coupling coe–cients Lfe and Lef in (2.9) and (2.10) are equal and have unitsof [As2 m¡3 ]. Re-arranging terms simplifles the equations toqT = ¡Krh¡ker` (2.11a)andJT = ¡ r`¡Lrh ; (2.11b)where ke = Lfe=‰f is the coe–cient of electro-osmotic permeability [m2 s¡1 V¡1 ] and L =gLef is the streaming current cross-coupling conductivity coe–cient [Am¡2 ].2.2.3 Governing equationsThe constitutive relations deflned in (2.11) are now combined with the continuity equationsto deflne the hydraulic and electrical  ow problems.In the study of streaming potentials, the hydraulic gradient is the driving force and thesecond term in (2.11a) can be neglected. This is supported by Mitchell [1991], who reportednegligible contribution of the electro-osmotic  ow term in materials with K > 10¡9 m=s.29Chapter 2. 3-D forward modelling of streaming potentialWe recognize this uncoupled  ow equation as a form of Darcy’s law, which is combined with(2.1) to deflne the saturated hydraulic  ow equation:r¢ Krh = Ss@h@t ¡Qv –(r¡rs) : (2.12)Variably saturated  ow problems require the use of both (2.12) and an unsaturated  owequation, such asr¢ K(Sw)rh = n@Sw@t ¡Qv –(r¡rs) ; (2.13)where K is now a function of  uid saturation Sw. Steady-state or transient forms of the uid  ow equation may be used to solve the hydraulic problem, depending on the temporalnature of the boundary conditions.Once we have solved the hydraulic problem for the distribution of hydraulic head h, thesubsequent problem is to solve for the self-potential distribution. Since hydraulic relaxationtimes are orders of magnitude larger than electrical relaxation times, electrical conductioncurrent  ow is treated as a steady-state process regardless of the nature of the hydraulicprocess. This suggests that \snapshots" of the  uid  ow regime over time may be analysedas a DC problem. Substitution of (2.11b) into the steady-state form of (2.5) results in thecoupled electrical  ow equation:r¢  r` = ¡r¢ Lrh ; (2.14)when there are no imposed external sources of current  ow.2.3 Forward modelling methodologyThe  uid and electrical  ow equations are similar in form and hence one could envisageusing the same algorithm [e.g., Wurmstich and Morgan, 1994]. However, we choose touse established programs to solve the hydraulic problem to facilitate both mixed boundaryconditions and model calibration, as well as the solution of variably saturated  ow problems.Fluid  ow modelling is performed using the 3-D flnite difierence codes MODFLOW,developed by the US Geological Survey [Harbaugh et al., 2000], for saturated  ow andMODFLOW-Surfact [HydroGeoLogic Inc., 1996] for variably saturated  ow problems.30Chapter 2. 3-D forward modelling of streaming potentialThe electrical equation (2.14) is solved using a flnite volume approach and our code isindependent of the solution of the hydraulic problem. All that is required is to be ableto import values of the hydraulic head. In the following sections we outline the numericaldetails that pertain to our solution.2.3.1 Finite volume solution for self-potentialThe  ow domain constitutes a three-dimensional volume, which consists of a homogeneousor heterogeneous earth model and in some cases surface water, as indicated in Figure 2.1.We recognize electrical property distributions to be discontinuous at interfaces betweendifierent materials in the subsurface (e.g. between units 1 and 2 in Figure 2.1), as well asat the earth’s surface, where a large conductivity contrast exists. Consequently, we do notwish to evaluate the derivative of  and instead restate (2.14) as two flrst-order equationswith unknown variables J and `:r¢ J = f (2.15a)and ¡1J ¡r` = 0 ; (2.15b)where J is the conduction current density. The source term f describes sources of streamingcurrent:f = ¡r¢ Lrh : (2.15c)waterearthairh=zunit1unit2unit3Figure 2.1: Schematic of the spatial  ow domain.31Chapter 2. 3-D forward modelling of streaming potentialThe difierential equations (2.15a) and (2.15b) are expressed in their weak forms anddiscretized using a flnite volume method [Haber et al., 2000]. The chosen arrangement of(2.15b) is a consequence of the fact that while J is a continuous function,  and r` arediscontinuous across an interface separating regions of difierent conductivity. Integratingthese discontinuous variables separately adds accuracy to the solution for `.The study region is represented by a rectilinear mesh of grid cells. The discrete equationsare solved on a staggered grid, with normal components of J located at cell faces and valuesof potential located at cell centres, as shown in Figure 2.2. Each grid cell is assigned anelectrical conductivity  and a cross-coupling conductivity L. These properties are treatedas a constant for each cell, but are considered to be piecewise constant in the domain andcan vary signiflcantly from one cell to the next.Jxi+1=2;j;kJyi;j+1=2;kJzi;j;k+1=2Ái;j;k¾i;j;k¢yj¢zk¢xiFigure 2.2: Discrete grid cell showing cell dimensions, location of components of currentdensity J at cell faces, and location of potential ` and conductivity  at the cell centre.The equations in (2.15) must be supplied with boundary conditions. In our implemen-tation we deflne the outer surface of the domain @› as a Neumann no- ow boundary:J ¢ ^nj@› = 0 : (2.16)This condition requires that the outer edge of the domain be positioned far from convectivesources to remove the in uence of the boundary on the solution in the region of interest.To achieve this, the mesh is expanded beyond the physical extents of the study region bysuccessively increasing cell dimensions in each direction.32Chapter 2. 3-D forward modelling of streaming potentialThe electrical conductivity and cross-coupling conductivity models are deflned on theexpanded mesh, which is generally much larger than the volume studied in hydraulic mod-elling. The cross-coupling coe–cients in this expanded region are set to zero, but a seriousefiort should be made to estimate reasonable values of electrical conductivity. A moredetailed treatment of electrical property models is given in Section 2.4.Applying the flnite volume technique to the equations in (2.15) yields the discrete sys-tem:DSG` = ¡DLGh; (2.17)where matrix D is the divergence operator, G is the gradient operator and the physicalproperty matrices S and L respectively contain the harmonic averages of electrical conduc-tivity  and cross-coupling coe–cient L. Vectors ` and h respectively contain values ofelectrical potential and hydraulic head. The no- ow boundary condition in (2.16) is im-posed by the divergence matrix D. Equation 2.17 represents a non-unique solution for `.This non-uniqueness is removed by specifying a constant value of ` at a single corner of thegrid.The matrix system in (2.17) is solved using the biconjugate gradient stabilized method(BiCGSTAB) [Van der Vorst, 1992] with preconditioning [Barrett et al., 1994]. For pre-conditioning we use either symmetric successive over relaxation (SSOR) or incomplete LU(ILU).Once (2.17) has been solved, potential or potential difierence data are evaluated as:d = Q`; (2.18)where matrix Q performs a tri-linear interpolation of ` using the eight nearest cell values.In a practical SP survey conflguration, data are potential difierences that are acquired withrespect to a reference electrode. The reference electrode can be located at an arbitrarycoordinate in the grid.2.3.2 Example: Injection well in a homogeneous halfspaceA single point source of  uid  ow in a homogeneous, saturated halfspace is a straightforward ow problem that is useful to illustrate the SP response to a primary source of streaming33Chapter 2. 3-D forward modelling of streaming potentialcurrent, and to verify the numerical algorithm against an analytical solution.The hydraulic problem represents an injection well that is screened over a flnite portionof its length, as illustrated in Figure 2.3. The semi-inflnite homogeneous halfspace wasmodelled as a single isotropic geologic unit 500m-thick extending 10 km in both x andy directions. Halfspace properties are Ko = 1 £ 10¡4 m=s, Lo = 1 £ 10¡5 A=m2, and o = 1£10¡3 S=m . The well screen is centred at a depth of 25 m, and  uid is injected ata constant  ow rate of 10,000 m3=day.r-rsxobservationlocationxz25minjectionwellK=ooc11510m/s10A/m10S/m-4-5 2-3L ==oh=hc h=hcFigure 2.3: Homogeneous halfspace model with injection well showing hydraulic boundaryconditions and physical properties of aquifer.2.3.2.1 Numerical solutionHydraulic head was resolved on a 190 £ 190 £ 35 non-uniform mesh. Cell lengths in the xand y directions ranged from 3.3 m at the centre of the mesh to 100 m at the edges. Cellheights ranged from 3.3 m at the well screen to 20 m at depth. A constant head boundaryof 500 m was deflned at the x and y limits of the mesh. This discretization was chosento facilitate an accurate solution of head in the vicinity of the well. The self-potentialdistribution was resolved on a 212 £ 212 £ 52 mesh in which cell size was progressivelyexpandedbyafactorof1.3. Potentialdataarereferencedtoasurfacepointattheedgeofthestudy region, 5 km from the well. Figure 2.4 shows the predicted hydraulic head distributionin the xz plane of the well. Results are presented as a surface proflle in Figure 2.4(a), and in34Chapter 2. 3-D forward modelling of streaming potentialcross-sectioninFigure2.4(b). Similarly, thepredictedself-potentialdistributionispresentedin Figure 2.5. Although they are of opposite polarity, a strong similarity is evident betweenthe SP and head equipotential distributions.4500 5000 5500500502504506508510Hydraulic head (m)Distance in x (m)Height above datum (m)4500 5000 55000100200300400505510515520525analyticalnumericalhead (m)b)a)Figure 2.4: Hydraulic head distribution resulting from a point injection well in a homoge-neous halfspace: a) analytical and numerical solutions at surface; b) predicted head contours(1 m interval).2.3.2.2 Analytical solutionTo facilitate an analytical solution the equations governing hydraulic and electrical  oware stated as Poisson’s equations. The hydraulic equation that deflnes this problem is thesteady-state form of (2.12):r2h = ¡ 1K Qv –(r¡rs): (2.19)Similarly, substitution of (2.19) into (2.14) reduces the electrical  ow equation to:r2` = L K Qv –(r¡rs): (2.20)35Chapter 2. 3-D forward modelling of streaming potential4500 5000 5500−80−60−40−200SP (mV)Distance in x (m)Height above datum (m)4500 5000 55000100200300400−300−250−200−150−100−50analyticalnumericalSP (mV)b)a)Figure 2.5: Self-potential distribution resulting from a point injection well in a homogeneoushalfspace: a) analytical and numerical solutions at surface; b) predicted SP contours (5 mVinterval).The method of images is used to solve the problem analytically, to account for perturba-tions in the potential flelds caused by the presence of the ground surface. When observationsare made at surface, the solutions to (2.19) and (2.20) reduce to:h(x;y;500) = ¡ Q(rs)2…Kjr¡rsj(2.21)and`(x;y;500) = LQ(rs)2… Kjr¡rsj; (2.22)respectively, where Q(rs) is the volumetric  ow rate of the source [m3 s¡1 ] and jr ¡ rsjdenotes the distance between the source and observation location. The analytical solutionsfor h and ` at the ground surface show excellent agreement with the numerical results, asillustrated in the proflle plots of Figures 2.4 and 2.5.36Chapter 2. 3-D forward modelling of streaming potential2.4 Physical properties2.4.1 The cross-coupling conductivity coe–cientBy applying the divergence theorem, the coupled electrical  ow equation (2.14) can beexpressed as Zs r`¢ds = ¡ZsLrh¢ds : (2.23)In a homogeneous medium, Zsr`¢ ^n ds = ¡LZsrh¢ ^n ds ; (2.24)where L and  are macroscopic properties. If we consider a 1-D  ow experiment in a porousmedium of length ¢x and cross-sectional area A, in which  ow occurs along the length ofthe medium, (2.24) is evaluated to give ¢`¢xA = ¡L¢h¢xA : (2.25)Equation (2.25) may be further simplifled and rearranged to deflne the cross-coupling con-ductivity:L = ¡ C ; (2.26)whereC = ¢`¢h: (2.27)The streaming potential coupling coe–cient C, which is often reported in terms of  uidpressure, may be characterized through laboratory measurements and is typically a negativequantity [e.g., Ishido and Mizutani, 1981; Morgan et al., 1989; Jouniaux and Pozzi, 1995;Pengra et al., 1999]. The above deflnition of C is consistent with our formulation of thecoupled  ow equations in terms of hydraulic head, as shown in (2.11).2.4.2 Deflning  and L property distributionsRepresentative values of the cross-coupling conductivity and electrical conductivity must beprescribed for all porous media in the model domain, and can be assigned using measuredor theoretical estimates. Electrical conductivity can be measured in the fleld or laboratory37Chapter 2. 3-D forward modelling of streaming potentialusing geophysical techniques such as the DC resistivity method, or estimated in certain casesusing an empirical relation such as Archie’s Law [Archie, 1942], which can be adjustedto account for varying degrees of saturation [e.g., Bear, 1972]. The streaming potentialcoupling coe–cient can be measured in the laboratory as noted in the section above, orestimated using a form of the Helmholtz-Smoluchowski equation [Overbeek, 1952], whichshows that C is a function of  uid conductivity. Since both  and C vary with the electricalconductivity of the saturating  uid, appropriate corresponding values must be used tocalculate L using (2.26).As illustrated in Figure 2.1, the  ow domain comprises an earth model and may alsoinclude surface water. The earth model can consist of a number of distinct units, whereeach unit is characterized by representative values of L and  . Since streaming current  owis limited only to porous media, air and water bodies are assigned a null cross-couplingconductivity to impose a no- ow boundary at the physical limits of the porous medium.However, surface water is assigned representative values of  uid conductivity  f to permitconduction current  ow. Air is typically assigned a null electrical conductivity.Inaconflned uid owproblem, theearthmodelunderstudyisinacompletely saturatedstate. Consequently, saturated material properties are used to describe each unit, i.e. L =Lsat and  =  sat. In an unconflned  uid  ow problem, a phreatic surface delineatesthe boundary between saturated and unsaturated zones. The position of this boundary isdeflned where pore  uid pressure is atmospheric (i.e., h = z), as indicated in Figure 2.1. Ifthe phreatic surface intersects a given earth model unit, representative properties must beassigned to both saturated and unsaturated zones within the unit.There are two approaches to determining a hydraulic solution to an unconflned  owproblem. In a saturated or free-surface approach,  uid  ow is assumed to dominate belowthe phreatic surface and is not considered in the unsaturated zone. The position of thephreatic surface, and consequently the size of the saturated model domain, is not knowna-priori and is determined through the hydraulic  ow analysis. In a variably saturatedapproach,  uid  ow is evaluated within both the saturated and unsaturated zones. Thislatter approach generates a more realistic hydraulic solution in problems with steeply-dipping interfaces, and is necessary for transient simulations [Freeze, 1971]. The delineation38Chapter 2. 3-D forward modelling of streaming potentialbetween saturated and unsaturated zones is determined by the condition h = z.The behaviour of streaming current  ow in the unsaturated zone is not addressed here.We consider streaming current  ow to be limited to the saturated zone, such that thephreatic surface acts as a no- ow boundary. This condition is imposed by specifyingLunsat = 0. However, the conduction current permeates both saturated and unsaturatedzones, and a representative  unsat is chosen to re ect the unsaturated or partially saturatedconditions of the material. For simplicity, we assume a constant value of  unsat for each unit,which does not vary with the decline in saturation above the phreatic surface. The nextexample illustrates the assignment of physical properties in an unconflned  ow problem.2.4.3 Example: Homogeneous lab-scale embankmentThe SP response to steady-state seepage through a laboratory-scale embankment is exam-ined as an example of an unconflned  ow problem, and provides a means of verifying theSP algorithm with measured data.The homogeneous embankment under study was constructed of Ottawa sand in anacrylic tank and measures 140 cm in length, 10 cm in width and 31 cm in height. The damwas subjected to steady-state  ow from constant reservoir heights of 18 cm and 22.5 cm. Thehydraulic conductivity and electrical properties of the sand were derived from laboratorytesting and are represented in the model as: K = 4:5 £ 10¡4 m=s, Lsat = 2:5 £ 10¡4 A=m2, sat = 2:5 £ 10¡3 S=m and  unsat = 1:5 £ 10¡3 S=m. The electrical conductivity of thereservoir water  f was measured as 2:8 £ 10¡3 S=m. Figure 2.6 illustrates a schematic ofthe electrical property distribution in the study region.The embankment was modelled using a 70 £ 5 £ 16 mesh with a uniform cell dimensionof 2 cm. The self-potential distribution was resolved on a padded grid, in which paddingcells were assigned an electrical conductivity of 1£10¡8 S=m to represent air. The predictedhydraulic head distribution and SP response at the surface of the embankment are shownfor each reservoir level in Figures 2.7 and 2.8. The SP data are referenced to a pointcentred at the surface of the crest. Predicted and measured SP data at each reservoir levelare compared in Figures 2.7(a) and 2.8(a) and show good agreement. The increase in SPamplitude with reservoir level is a direct consequence of the increase in hydraulic gradient39Chapter 2. 3-D forward modelling of streaming potentialL=0=c115 c115unsatL=0=c115fc115L=0=c115 c115airL=L=c115satsatc115h=zzFigure 2.6: Schematic cross-section of a homogeneous earth embankment, illustrating thedistribution of electrical properties. Hydraulic head equipotential lines are indicated in theflgure, with dotted lines representing hydraulic head in the unsaturated zone. Superscriptssat and unsat refer to representative properties of the saturated and unsaturated zones,respectively. The value  f represents the electrical conductivity of the impounded water.The value  air is typically chosen as 1£10¡8 S/m.across the embankment.The chosen L and  property distributions in uence how electrical current  ows inthe subsurface. Figure 2.9 displays  uid, streaming current and conduction current  uxpatterns for the 22.5 cm reservoir model described in Figure 2.8. Figure 2.9(a) showsvectors of  uid  ux q within the saturated zone below the phreatic surface. Figure 2.9(b)displays streaming current  ow, which is limited to the saturated zone since the electricalproblem is solved using a saturated  ow assumption (Lunsat = 0). Streaming current and uid  ow vector flelds appear very similar since both processes are driven by hydraulicgradients in the saturated zone. Figure 2.9(c) illustrates vectors of conduction current  owJ, which is not conflned to the saturated zone and is governed by the electrical conductivitydistribution.40Chapter 2. 3-D forward modelling of streaming potential0 0.2 0.4 0.6 0.8 1 1.2 1.4-505101520Distance in x (m)SP (mV)measuredpredicteda)b)Height (m)0.10.20.3Figure 2.7: Homogeneous embankment subject to steady-state seepage from an 18 cmreservoir: a) Measured and predicted SP data at surface; b) predicted hydraulic headdistribution (m).0 0.2 0.4 0.6 0.8 1 1.2 1.4-505101520Distance in x (m)SP (mV)measuredpredicteda)b)Height (m)0.10.20.3Figure 2.8: Homogeneous embankment subject to steady-state seepage from a 22.5 cmreservoir: a) Measured and predicted SP data at surface; b) predicted hydraulic headdistribution (m).41Chapter 2. 3-D forward modelling of streaming potentialFigure 2.9: Homogeneous embankment subject to steady-state seepage from a 22.5 cmreservoir: a)  uid  ux q; b) streaming current; c) conduction current J.42Chapter 2. 3-D forward modelling of streaming potential2.5 Sources of charge contributing to the self-potentialConduction current  ow and the corresponding self-potential fleld is governed by the dis-tribution of streaming current sources and electrical conductivity in the subsurface. Toenable an intuitive understanding of the potential response, (2.14) is expressed as a Poissonequation and restated in terms of charge density. This equation clearly deflnes the physicalconditions that dictate the magnitude and sign of current sources in the subsurface. Allproperties are assumed isotropic for the sake of discussion.The source function f deflned in (2.15) describes all sources of streaming current inthe study region. The nature of these convective sources may be examined by expanding(2.15a) to:r¢ J = ¡Lr2h¡rL¢rh : (2.28)The steady-state form of (2.12) is expanded and re-arranged to give:r2h = ¡ 1K Qv –(r¡rs)¡ 1K rK ¢rh : (2.29)Substitution of (2.29) into (2.28) results in:r¢ J = LK Qv –(r¡rs)+ LK rK ¢rh¡rL¢rh : (2.30)Here we see an explicit deflnition of streaming current sources, which may be groupedinto two distinct types: 1) \primary" sources due to the injection or withdrawal of  uidfrom the system, as described by the  uid  ow source term Qv; and 2) \secondary" sourcesthat are generated by gradients in the physical properties K and L in the presence of ahydraulic gradient.Heterogeneities in  will perturb the potential fleld generated by the streaming cur-rent source distribution, such that boundaries between regions of difierent  may also beconsidered as secondary sources. Invoking Ohm’s law, (2.30) is expanded and re-stated asPoisson’s equation for potential `:r2` = L K Qv –(r¡rs)+ L K rK ¢rh¡ 1 rL¢rh¡ 1 r ¢r` ; (2.31)where the right hand side is the source function that describes all sources that contributeto `. Finally, (2.31) is expressed in terms of volumetric charge density ‰e [Cm¡3 ] using43Chapter 2. 3-D forward modelling of streaming potentialGauss’ law:‰e = †o‰¡ L K Qv –(r¡rs)¡ L K rK ¢rh+ 1 rL¢rh+ 1 r ¢r` : (2.32)The self-potential ` is the sum of all potential flelds generated by primary and secondarysources of charge within the system, according to the superposition principle. The potentialdecays with the inverse radial distance jr¡rsj from each source of charge deflned by ‰e(rs),which may be evaluated through the integral solution:`(r) = 14…†oZV‰e(rs)jr¡rsj dV : (2.33)Equations (2.32) and (2.33) provide a basis for an intuitive understanding of the dis-tribution of electrical charge in the subsurface, and the resulting potential response. Theflrst term on the right hand side of (2.32) describes primary sources of charge. Primarysources typically manifest as injection or pumping wells in an aquifer. As was illustrated inthe example shown in Section 2.3.2, a positive injection of  uid into the system results in aconcentration of negative charge and consequently a negative SP response. The magnitudeof this primary source is controlled by the  uid  ow rate and conductivity values at thesource.The remaining three terms on the right hand side of (2.32) describe the secondarysources generated between regions of difierent conductivity. These are the only sources thatcontribute to the self-potential in systems where no primary  ow sources exist, such as theearth dam example of Section 2.4.3. In a discontinuous conductivity model of the earth,secondary sources appear at an interface between difierent materials or between difierentsaturation states within a given material. The sign of the accumulated charge depends onthe sign of the conductivity and potential gradients at the interface.Transient hydraulic conditions can give rise to a variable charge distribution, which canbe evaluated through (2.32) using time-dependent variables. The release of water fromstorage in a transient saturated hydraulic analysis manifests as changes in the hydraulichead fleld with time. Since the conduction current responds instantaneously to hydraulicconditions, as discussed in Section 2.2.3, the charge distribution and corresponding self-potential re ects the head distribution at a given point in time. Similarly, transient  uidsaturation resulting from a variably saturated  ow analysis can give rise to sources of charge,44Chapter 2. 3-D forward modelling of streaming potentialwhich manifest as changes in the hydraulic head and physical property distributions withtime.In any analysis, the magnitude of secondary charge can be signiflcant and can causedi–culty when the SP interpretation is done in terms of a primary source only. The followingexample illustrates this.2.5.1 Example: Pumping well in a heterogeneous halfspaceA pumping well model is used to illustrate the efiect of heterogeneous physical propertydistributions on the SP response to a primary  ow source. This example serves to showthat a heterogeneous hydraulic conductivity distribution can greatly in uence the characterof the self-potential distribution, whose magnitude is in uenced by the electrical propertiesof the subsurface.Figure 2.10 displays a schematic of the heterogeneous halfspace model, which representsa pervious sand lens buried within a silty sand deposit. The chosen hydraulic conductivityand electrical properties of the sand aquifer are K = 1 £ 10¡4 m=s, Lsat = 3 £ 10¡5 A=m2and  sat = 2 £ 10¡3 S=m. The silty sand halfspace properties are K = 1 £ 10¡6 m=s,Lsat = 1 £ 10¡5 A=m2 and  sat = 5 £ 10¡3 S=m. In this example we assume  unsat =  satfor simplicity. The well penetrates the sand aquifer and is pumped at a constant rate of 500m3=day. Steady-state hydraulic conditions are simulated and the water table is assumed tobe at ground surface a distance from the well. The study region was modelled using a non-uniform 99£99£22 cell mesh, extending 100 m in depth and 1000 m in x and y directions.The pumping rate is averaged over the total volume of the two cells used to represent thewell screen, such that the pumping rate per unit volume is Qv = ¡1:3£10¡5 m3s¡1m¡3.Figure 2.11 illustrates the hydraulic head contours in a vertical xz plane centred on thewell. Since the conflned sand aquifer is more permeable to  uid  ow relative to the siltysand halfspace, water is drawn into the aquifer from the surrounding medium. Pumpingcauses a drawdown in the water table to a height of 78 m at the well, as shown in Figure2.11(a). Figure 2.11(b) shows the hydraulic head contours for the same pumping well in ahomogeneous aquifer (K = 1£10¡4 m/s) for comparison. Although the well is pumped atthe same rate, there is no drawdown in the water table.45Chapter 2. 3-D forward modelling of streaming potentialxz25m40m10m100mpumpingwella)b)xy200mK=c1151 10m/s1 10A/m5 10S/mxxx-6-5 2-3L==K=c1151 10m/s3 10A/m2 10S/mxxx-4-5 2-3L==h=hc h=hcFigure 2.10: Heterogeneous halfspace with pumping well: a) model in plan view, show-ing outline of subsurface aquifer; b) model in cross-section, showing hydraulic boundaryconditions and physical properties of aquifer and halfspace. Well screen length is 10 m.The chosen pumping rate and physical property distributions dictate the sign and mag-nitude of streaming current sources in the subsurface. Figure 2.12(a) shows the primarysource associated with the pumping well in a homogeneous halfspace model of the aquifer.The pumping well equates to a primary source of positive charge. The total magnitude ofthis primary current source is equal to 1:74 £ 10¡3 A (3:9 £ 10¡6 A=m3 in each grid cellrepresenting the well screen), which is consistent with that predicted using the flrst term of(2.30) and the pumping rate noted above. The corresponding self-potential distribution atsurface is shown in Figure 2.13(a), which shows a positive SP anomaly centred on the well.46Chapter 2. 3-D forward modelling of streaming potentialDistance in x (m)Height above datum (m)250 300 350 400 450 500 550 600 650 700 750050100708090(m)Distance in x (m)Height above datum (m)250 300 350 400 450 500 550 600 650 700 75005010098.59999.5(m)a)b)Figure 2.11: Vertical cross-section of hydraulic head in the plane of the pumping well:a) heterogeneous halfspace model; b) homogeneous halfspace (K = 1 £ 10¡4 m/s) forcomparison.The relative in uence of each secondary source term is examined by progressively in-creasing the level of heterogeneity in the model. The chosen heterogeneous hydraulic con-ductivity distribution causes positive charge to accumulate at the interface between theaquifer and surrounding halfspace, as shown in Figure 2.12(b). This model assumes thatK varies but that L and  of the halfspace are equal to that of the aquifer. The secondarysources are of smaller magnitude than the primary source, but collectively they contributesigniflcantly to the self-potential response at the ground surface, as shown in Figure 2.13(b).Here we see that the SP contour pattern roughly delineates the shape of the aquifer. Intro-ducing a heterogeneous L distribution in addition to the heterogeneous K model decreasesthe magnitude of the secondary streaming current sources, as indicated in Figure 2.12(c),since the chosen L values result in the accumulation of negative charge at the interface.However, the net SP response remains positive, as shown in Figure 2.13(c). The chosenheterogeneous distribution of  results in a further decrease in the magnitude of secondarysources at the aquifer interface, as illustrated by the surface SP pattern shown in Figure2.13(d). Figure 2.14 illustrates the surface SP data in proflle form, along a line at y = 500m. The secondary sources caused by heterogeneous K, L and  distributions result in a47Chapter 2. 3-D forward modelling of streaming potential100 200 300 400 500 600 700 800 900050100Distance in x (m)Height (m)100 200 300 400 500 600 700 800 900050100Distance in x (m)Height (m)100 200 300 400 500 600 700 800 900050100Distance in x (m)Height (m) −101234567x 10−7(A/m3)c)b)a)Figure 2.12: Vertical cross-section of streaming current source density in the plane of thepumping well: a) homogeneous halfspace; b) heterogeneous K halfspace; c) heterogeneousK and L halfspace. The total magnitude of primary streaming current sources at the wellscreen is 7:8£10¡6 A=m3.SP signature of much difierent amplitude and shape to that predicted using a homogeneoushalfspace model.48Chapter 2. 3-D forward modelling of streaming potentialDistance in x (m)Distance in y (m)200 400 600 800200400600800Distance in x (m)Distance in y (m)200 400 600 800200400600800Distance in x (m)Distance in y (m)200 400 600 800200400600800Distance in x (m)Distance in y (m)200 400 600 80020040060080020406080100120140(mV)c) d)b)a)Figure 2.13: Surface plan map of self-potential: a) homogeneous halfspace; b) heterogeneousK halfspace (dashed outline indicates location of subsurface aquifer); c) heterogeneous Kand L halfspace; d) heterogeneous K, L and  halfspace.49Chapter 2. 3-D forward modelling of streaming potential0 200 400 600 800 1000020406080100120140160Distance in x (m)SP (mV)a)b)c)d)Figure 2.14: Surface proflles of self-potential at y = 500 m: a) homogeneous halfspace; b)heterogeneous K halfspace; c) heterogeneous K and L halfspace; d) heterogeneous K, Land  halfspace.50Chapter 2. 3-D forward modelling of streaming potential2.6 Field exampleOne of the principal motivations for undertaking this research is the interpretation of self-potential data collected during embankment dam seepage investigations. Flow throughan embankment and foundation is a problem of complicated geometry that requires athree-dimensional model to properly characterize the hydraulic head and correspondingself-potential distributions across the structure.Self-potential surveys were performed at a dam site in British Columbia to aid in theassessment of overall seepage conditions, and to investigate the integrity of certain compo-nents of the site, which are discussed below. The surveys took place in May and August2001 to respectively capture representative high and low pool conditions. Data were col-lected over the surface of the embankment and reservoir using CuSO4 electrodes with aflxed-base electrode conflguration, and were corrected for telluric variations. Preliminarymodelling was undertaken to aid in the interpretation of self-potential data collected at thesite. Our goal was to generate a preliminary 3-D simulation of the self-potential distributionusing best estimates of the physical properties K, L and  .The key features of the site are highlighted in the plan map shown in Figure 2.15and a conductivity model shown in Figure 2.16. The main dam is a zoned earthflll dam,which is underlain by alluvial deposits and situated within a granite bedrock valley. Apre-existing diversion dam is located at the upstream toe of this main dam. A clay blanketlines the upstream face of the main dam and keys into the central impervious core of thediversion dam. The foundation materials consist of pervious upper and lower aquifers,which are separated by a thick clay layer. A sheet-pile cut-ofi wall coincides with the crestof the diversion dam and extends vertically through the upper aquifer into the clay layerto prevent seepage through this zone. A deep grout cut-ofi is located further upstream andcontrols seepage through the lower aquifer. The clay blanket, diversion dam core, sheet-pilecut-ofi and grout curtain form a continuous impervious barrier to  ow through the damand foundation.Fourteen distinct soil and geologic units are deflned in the model of the site. Values ofK, L and  were assigned to each zone in the model. Hydraulic conductivity values werederived from published estimates [Terzaghi and Lacroix, 1964]. Values of cross-coupling51Chapter 2. 3-D forward modelling of streaming potentialINSTRUMENT ATIONHOUSE750 700650600600600625650650650650650750740 730 720710690680 670660640630620610610610610620690680660640640630630630640660620630640660630 640760740730720710700630620diversiondamgroutcurt ainmaindamcrestoriginalriverchannel0 100mflowdirectionspillwayA A’B’BFigure 2.15: Plan map of the embankment dam site, indicating key structural features andsurvey lines A-A’ and B-B’.conductivity and electrical conductivity were assigned based on published and theoreticalestimates for typical soil and rock types found at site, using average reservoir water con-ductivity values measured during the course of SP surveys performed at the site. Thesephysical property distributions were assigned as a flrst approximation, and all units wereeach represented by uniform property distributions.Figure 2.17 compares measured and predicted data taken along two perpendicular linesat the surface of the reservoir corresponding to the high pool survey. Figure 2.17(a) showsdata along the diversion dam crest, and Figure 2.17(b) shows data in a N-S direction acrossthe grout curtain. Measured and predicted data compare well, and show general agreementof trends and range of amplitude. This gives us confldence in both the modelling results andour initial estimates of the physical property distributions. The data clearly show the three-dimensional nature of the self-potential distribution at the site. Efiective interpretation willrequire further work and the development of an inverse methodology.52Chapter 2. 3-D forward modelling of streaming potentialgrout curtaindiversion dam main damclay layerFigure 2.16: Model of the embankment dam and foundation.53Chapter 2. 3-D forward modelling of streaming potential-20-15-10-50510152025300 50 100 150 200 250 300Distance north along line (m)SP (mV)measuredmodelled-50510152025300 50 100 150 200 250 300 350Distance east along line (m)SP (mV)measuredmodelleda)b)B B’A A’Figure 2.17: Embankment subject to seepage from the reservoir at high pool: a) Measuredand predicted SP data at the surface of the reservoir above the diversion dam (survey lineA-A’); b) Measured and predicted SP data at the surface of the reservoir in a N-S directionacross the grout curtain (survey line B-B’).54Chapter 2. 3-D forward modelling of streaming potential2.7 ConclusionWe have developed a 3-D forward modelling algorithm that calculates the self-potential fleldinduced by  uid  ow in the subsurface, based on the theory of coupled  ow. The algorithmwas developed using a flnite volume discretization on a staggered grid, and explicitly cal-culates all streaming current sources based on known distributions of hydraulic head andphysical properties, namely the streaming current cross-coupling conductivity and electricalconductivity.The algorithm was used to successfully reproduce an analytical point source solutionand approximate the measured SP response to seepage through a homogeneous earth dam.A synthetic pumping well example illustrated that heterogeneous physical property distrib-utions result in an accumulation of charge at physical boundaries. The sign and magnitudeof this charge is determined by the physical property and potential gradients at the bound-ary, and can contribute signiflcantly to the self-potential signal resulting from a primarysource. Preliminary modelling of the SP response to seepage at an embankment dam siteillustrated the need for a three-dimensional model to characterize the self-potential distrib-ution at a complicated site. These four examples clearly demonstrate the link between theself-potential and hydraulic head distributions. This suggests that information about thehydraulic system may be inferred from the self-potential data, given some knowledge of theelectrical property distribution in the subsurface.55Chapter 2. 3-D forward modelling of streaming potential2.8 ReferencesArchie, G. E., The electrical resistivity log as an aid in determining some reservoir char-acteristics, Transactions of the Society of Petroleum Engineers of the American Instituteof Mining, Metallurgical and Petroleum Engineers (AIME), 146, 54{67, 1942.Barrett, R., et al., Templates for the Solution of Linear Systems: Building Blocks forIterative Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia,1994.Bear, J., Dynamics of Fluids in Porous Media, Dover Publications Inc., New York, 1972.Bogoslovsky, V. V., and A. A. Ogilvy, Deformations of natural electric flelds near drainagestructures, Geophysical Prospecting, 21, 716{723, 1973.Corwin, R. F., The self-potential method for environmental and engineering applications,in Geotechnical and Environmental Geophysics, vol. 1, edited by S. H. Ward, pp. 127{145,Society of Exploration Geophysicists, Tulsa, 1990.Corwin, R. F., and D. B. Hoover, The self-potential method in geothermal exploration,Geophysics, 44(2), 226{245, 1979.Corwin, R. F., and H. F. Morrison, Self-potential variations preceding earthquakes incentral California, Geophysical Research Letters, 4(4), 171{174, 1977.de Groot, S. R., Thermodynamics of irreversible processes, Selected Topics in ModernPhysics, vol. 3, North Holland Publishing Company, Amsterdam, 1951.Fitterman, D. V., Electrokinetic and magnetic anomalies associated with dilatant regionsin a layered earth, Journal of Geophysical Research, 83(B12), 5923{5928, 1978.Freeze, R. A., In uence of the unsaturated  ow domain on seepage through earth dams,Water Resources Research, 7(4), 929{941, 1971.Haber, E., U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, Fast simulation of 3D elec-tromagnetic problems using potentials, Journal of Computational Physics, 163(1), 150{171, 2000.56Chapter 2. 3-D forward modelling of streaming potentialHarbaugh, A. W., E. R. Banta, M. C. Hill, and M. G. McDonald, Mod ow-2000, the U.S.GeologicalSurveyModularGround-WaterModel-UserGuidetoModularizationConceptsand the Ground-Water Flow Process, Tech. rep., United States Geological Survey, Open-File Report 00-92, Washington, 2000.HydroGeoLogicInc., MODFLOW-SURFACT Version 2.2 Documentation, HydroGeoLogicInc., Herndon, 1996.Ishido, T., and H. Mizutani, Experimental and theoretical basis of electrokinetic phe-nomena in rock-water systems and its application to geophysics, Journal of GeophysicalResearch, 86(B3), 1763{1775, 1981.Ishido, T., and J. W. Pritchett, Numerical simulation of electrokinetic potentials associatedwith subsurface  uid  ow, Journal of Geophysical Research, 104(B7), 15,247{15,259, 1999.Jouniaux, L., andJ.P.Pozzi, Streamingpotentialandpermeabilityofsaturatedsandstonesunder triaxial stress: Consequences for electrotelluric anomalies prior to earthquakes, Jour-nal of Geophysical Research, 100(B6), 10,197{10,209, 1995.Mitchell, J. K., Conduction phenomena: From theory to geotechnical practice, Geotech-nique, 43(3), 299{340, 1991.Mizutani, H., T. Ishido, T. Yokokura, and S. Ohnishi, Electrokinetic phenomena associatedwith earthquakes, Geophysical Research Letters, 3, 365{368, 1976.Morgan, F. D., E. R. Williams, and T. R. Madden, Streaming potential properties of West-erly granite with applications, Journal of Geophysical Research, 94(B9), 12,449{12,461,1989.Onsager, L., Reciprocal relations in irreversible processes, I, Physical Review, 37, 405{426,1931.Overbeek, J. T. G., Electrokinetic phenomena, in Colloid Science, Irreversible Systems,vol. 1, edited by H. R. Kruyt, Elsevier Publishing Company, Amsterdam, 1952.Pengra, D. B., S. X. Li, and P. Wong, Determination of rock properties by low-frequencyAC electrokinetics, Journal of Geophysical Research, 104(B12), 29,485{29,508, 1999.57Chapter 2. 3-D forward modelling of streaming potentialRevil, A., H. Schwaeger, L. M. Cathles III, and P. D. Manhardt, Streaming potential inporous media: 2. Theory and application to geothermal systems, Journal of GeophysicalResearch, 104(B9), 20,033{20,048, 1999.Sill, W. R., Self-potential modeling from primary  ows, Geophysics, 48(1), 76{86, 1983.Terzaghi, K., and Y. Lacroix, Mission dam: An earth and rockflll dam on a highly com-pressible foundation, Geotechnique, 14(1), 14{50, 1964.Titov, K., Y. Ilyin, P. Konosavski, and A. Levitski, Electrokinetic spontaneous polarizationin porous media: Petrophysics and numerical modelling, Journal of Hydrology, 267, 207{216, 2002.Titov, K., A. Levitski, P. K. Konosavski, A. V. Tarasov, Y. T. Ilyin, and M. A. Bues,Combined application of surface geoelectrical methods for groundwater- ow modeling: Acase history, Geophysics, 70(5), H21{H31, 2005a.Titov, K., A. Revil, P. Konosavsky, S. Straface, and S. Troisi, Numerical modelling ofself-potential signals associated with a pumping test experiment, Geophysical Journal In-ternational, 162, 641{650, 2005b.Van der Vorst, H. A., Bicgstab: A fast and smoothly converging variant of the Bi-CG forthe solution of nonsymmetric linear systems, Journal on Scientiflc and Statistical Com-puting, 13, 631{644, 1992.Wurmstich, B., and F. D. Morgan, Modeling of streaming potential responses caused byoil well pumping, Geophysics, 59(1), 46{56, 1994.Wurmstich, B., F. D. Morgan, G. P. Merkler, and R. L. Lytton, Finite-element modeling ofstreaming potentials due to seepage: Study of a dam, Society of Exploration GeophysicistsTechnical Program Expanded Abstracts, 10, 542{544, 1991.58Chapter 3A laboratory apparatus forstreaming potential and resistivitymeasurements on soil samples 23.1 IntroductionThe self-potential (SP) method is a geophysical technique that is used to measure the elec-trokinetic process of streaming potential, in which electrical current  ow is generated bythe  ow of  uid through a porous medium [Overbeek, 1952]. Streaming potential measure-ments have been performed in engineering and groundwater [Corwin, 1990; Ogilvy et al.,1969], oil reservoir [Wurmstich and Morgan, 1994], and geothermal investigations [Ishidoand Mizutani, 1981; Revil et al., 1999] to help infer characteristics of the hydraulic regime.Practical numerical analysis and interpretation of streaming potential survey data re-quires estimates of the physical properties that describe the electrical response to  uid  owthrough a porous medium. Streaming potential measurements have been collected on rockand uniform sand samples [e.g., Morgan et al., 1989; Pengra et al., 1999; Ahmad, 1964],but very few data exist for natural unconsolidated sediments. The present investigation isfocused on the development of an apparatus to measure the relevant electrical and sampleproperties of soils over a range of particle sizes.Streaming potential is a function of the properties of the solid- uid interface. An elec-trical double layer forms at the surface of a solid in contact with an electrolyte. This double2A version of this chapter has been published. Shefier, M.R., Reppert, P.M. and Howie, J.A. (2007)A laboratory apparatus for streaming potential and resistivity measurements on soil samples, Review ofScientiflc Instruments, 78:09450259Chapter 3. Apparatus for streaming potential and resistivity testinglayer consists of a tightly adsorbed inner layer and a weakly-bound difiuse outer layer ofions, which act together to balance the surface charge of the solid. A streaming current isgenerated within the pore space by the  uid  ow, which drags ions from the difiuse layerin the direction of  ow. The streaming current may be described using a linear  ow law[de Groot, 1951]:JS = ¡Lrh; (3.1)where L is the streaming current cross-coupling coe–cient and h is hydraulic head. Themovement of ions causes a charge imbalance within the pore space in the direction of  ow,which drives an opposing conduction current JC. This conduction current permeates theporous medium and is described using Ohm’s law:JC = ¡1‰r`; (3.2)where ‰ is the electrical resistivity of the saturated porous medium and ` is the electricalpotential. If no temperature or chemical/ionic concentration gradients exist in the system,the total current J is described using coupled  ow theory as the sum of streaming andconduction currents:J = JC +JS: (3.3)If we consider one-dimensional  ow in a homogeneous porous medium of length l andcross-sectional area A, Equations 3.1 to 3.3 can be shown to reduce to1‰¢`l A = ¡L¢hl A ; (3.4)which may be further simplifled and rearranged to deflne the streaming current cross-coupling conductivity:L = ¡C‰ ; (3.5)where the streaming potential coupling coe–cientC = ¢`¢h: (3.6)Laboratory measurements of hydraulic head may be reported in terms of pressure, suchthatC = ¢`¢P ; (3.7)60Chapter 3. Apparatus for streaming potential and resistivity testingwhere P =  fh and  f is the product of  uid density and gravitational acceleration.The purpose of the present study was to develop an apparatus to measure C and ‰ fora given material. Of particular importance was the ability to measure these parameters onthe same unconsolidated soil specimen in order to characterize L.The streaming potential coupling coe–cient C is typically measured using either theunidirectional  ow method or the oscillatory  ow method [Reppert and Morgan, 2001b].Although oscillatory  ow methods have been implemented in previous studies of streamingpotential on glass capillaries and rock samples [Packard, 1953; Sears and Groves, 1978;Pengra et al., 1999; Reppert and Morgan, 2001a], the oscillatory method had previouslynot been tested on well-graded (poorly sorted) soil samples. Consequently, we adopted aparallel design approach to enable a comparison of the two methods in application to soilstesting. Laboratory measurements of DC resistivity are typically performed using 2- or4-electrode array conflgurations. The apparatus was designed to measure resistivity usingthe 2-electrode technique, and the 4-electrode conflguration was tested for comparison.The streaming potential measurements performed in this study consist of a low voltagemeasurement (0.1 to 1 milliVolt range) across a saturated soil column, which acts as a highimpedance current source (typically 1 to 30 k›). As a result, the collection of streamingpotential data is considered a low voltage-high impedance electrical measurement, andrequires unique attention to sources of error in the electronics and environment that aretypically ignored in most other laboratory measurement systems.This article explains the design of the apparatus, and the development of calibrationand measurement techniques. We compare the results from unidirectional and oscillatory ow measurements of the streaming potential coupling coe–cient, and 2- and 4-electrodemeasurements of resistivity.3.2 Experimental apparatus3.2.1 OverviewA schematic of the experimental apparatus is shown in Figure 3.1. The test cell houses asoil specimen, a pair of non-polarizing electrodes, a pair of stainless steel electrodes and a61Chapter 3. Apparatus for streaming potential and resistivity testingpair of gauge pressure transducers to enable measurements of the electrical potential and uid pressure difierences across the sample. An external loading assembly is used to applya vertical seating load to the top of the soil sample.soilsample V1~V2fluid reservoirfluid reservoirloadunidirectionalflow inlet oscillatoryflow membranecamP1P2“infinite” outflow reservoirinflow tankFigure 3.1: Schematic of apparatus.An \inflnite" reservoir imposes a constant hydraulic head boundary to the top of the soilspecimen for the purposes of both unidirectional and oscillatory  ow testing. In the case ofunidirectional  ow testing, a constant head reservoir is connected to the base of the test cellto impose a hydraulic gradient that drives  uid  ow upwards through the sample. A sealedrubber membrane is connected to the base of the test cell for the purposes of oscillatory ow testing. An oscillating pressure gradient is induced by de ecting the membrane with amechanical actuator, which generates  ow through the sample.Streaming potential measurements (V1 in Figure 3.1) are made using the non-polarizingelectrodes positioned within  uid reservoirs immediately above and below the sample. Pres-sure measurements (P1 and P2 in Figure 3.1) are made at the same position. Pervious plate62Chapter 3. Apparatus for streaming potential and resistivity testingelectrodes in contact with both ends of the soil specimen (V2 in Figure 3.1) are used toapply current and measure the resultant voltage drop across the sample for resistivity mea-surements, which are performed as a separate experiment. An automated data acquisitionand control system is used to perform all measurements related to streaming potential andresistivity testing. Speciflc details about each component of the apparatus are outlined inthe sections that follow.The electrical potential measured across the soil sample must be purely a function ofthe applied hydraulic gradient to accurately characterize the streaming potential couplingcoe–cient. Any secondary currents induced by temperature or ionic gradients in the sample,or stray currents from the instrumentation can result in an inaccurate measurement of thestreaming potential. Consequently, insulating materials were used to construct the cell,and grounded shielding surrounded the entire apparatus to reduce the efiects of 60 Hznoise from ambient sources. The experiments were conducted at constant temperature anddifiusion potential efiects were avoided by allowing time for the soil sample to reach chemicalequilibrium with the saturating  uid.3.2.2 Test cell and loading assemblyThe test cell consists of a sample holder and two cell reservoirs, which are flxed above andbelow the soil specimen, as indicated in Figure 3.1. Figure 3.2 shows a detailed drawing ofthe test cell in cross-section.The sample holder was oriented vertically to allow preparation of fully-saturated, ho-mogeneous soil samples. The cylindrical sample holder was designed as a separate piece toaccommodate difierent sample lengths as required in testing. The sample height had to beshort enough to ensure homogeneity, and long enough to apply small hydraulic gradients(• 0:1 m of H2O/m) as well as measure su–cient streaming potential signal across thesample. All experiments discussed in this paper were performed using a 10 cm-long sampleholder.To enable testing of well-graded samples, the diameter of the sample had to be largeenough to accommodate a wide range of grain sizes. The inner diameter of the sampleholder is 10 cm, which enables testing of samples exhibiting a maximum particle size of63Chapter 3. Apparatus for streaming potential and resistivity testing stainlesssteelwirestainlesssteelplates(endplateelectrodes)pressuretransducerport potentialelectrodeportpotentialelectrodeportloadcellpressuretransducerportstainlesssteelwireperviousendplatesuppercellreservoirtopcaplowercellreservoirbasecapfluidflowoutletfluidflowinletpneumaticcylinderFigure 3.2: Cross-sectional view of the test cell.9.5 mm [ASTM, 1991]. The upper and lower cell reservoir assemblies have the same innerdiameter as the sample, and house the pressure transducers and SP electrodes in side portsaway from the primary  uid  ow path [Morgan, 1989].The cell reservoirs are separated from the sample by pervious top and base plates. Thebase plate is flxed as part of the lower cell reservoir assembly. The top plate is a separatepiece fltted with an acrylic loading ram, to enable the application of a vertical seating loadto the top of the prepared sample. This plate and ram assembly was designed to withstanda nominal load of 25 kPa, to prevent the migration of flner soil particles and instabilities inthe soil matrix caused by upward  uid  ow through the sample. The test cell and loadingapparatus were designed as a freestanding unit, such that the cell assembly itself acts as a64Chapter 3. Apparatus for streaming potential and resistivity testingloading frame. The vertical seating load is applied to the sample using a pneumatic pistonfltted with a ball in contact with the loading ram.Filter screens are used to retain the soil sample between the top and base plates. Thecell reservoirs are enclosed with top and base caps, and the entire cell and loading assemblysecured with three stainless steel rods. All cell components in contact with the sampleand  uid are acrylic, with the exception of the corrosion-resistant stainless steel currentelectrodes and fllter screens.3.2.3 InstrumentationTo characterize the streaming potential coupling coe–cient using Equation (3.6), the ap-paratus had to allow testing of samples subject to laminar  ow conditions, with equivalentgeometry of  uid (streaming current) and conduction current  ow path geometry [Mor-gan, 1989]. Consequently, a unique design approach was implemented in the placementof the streaming potential electrodes and pressure transducers. Measurement points werepositioned immediately above and below the soil sample, instead of along the side of thesample. This design enables the average  ow across the entire cross-sectional area of thesample to be characterized, and implies that the  uid and current  ow path lengths areequal to the physical length of the soil specimen. This same principle was applied in de-signing the electrode conflguration for resistivity measurements. Current electrodes wereplaced in contact with the entire cross-sectional area of the ends of the sample to ensurethat current was applied uniformly through the entire soil volume.Non-polarizing silver-silver chloride electrodes were installed within the upper and lowercell reservoirs for streaming potential measurements. These electrodes are positioned in sideports removed from the  uid  ow path to ensure that electromotive efiects, attributed tothe  ow of water across the tip of the electrode, would not in uence the streaming potentialmeasurements. Silver-silver chloride electrodes were selected for their particular suitabilityin laboratory testing and to avoid polarization efiects, particularly in unidirectional  owtesting [Morgan, 1989].Gauge pressure transducers were installed within the upper and lower cell reservoirs, atthe same vertical position as the streaming potential electrodes, to monitor the pressure65Chapter 3. Apparatus for streaming potential and resistivity testingdifierence across the sample. Independent gauge transducers were chosen over a difierentialtransducer to avoid a parallel low-impedance connection bridging the ends of the sample.The transducers are constructed with a plastic housing and a silicon membrane and arepowered by isolated 10V DC power supplies.Two stainless steel plate electrodes were installed directly onto the pervious top and baseplates used to support the soil specimen for the purposes of resistivity testing. These currentelectrodes were machined to coincide with the drilled surface of the end plates in order tohave maximum surface area in contact with the ends of the soil specimen, without impeding uid  ow through the sample. All components of the current electrodes were designed ofstainless steel exhibiting a high corrosion resistance, to avoid the risk of spurious currentsarising from corrosion or contact between difierent types of metals within the test cell.A regulator is used to control the pressure applied to the top of the sample from thepneumatic piston, and the applied force is measured with a load cell. The length of the soilspecimen is calculated from the referenced height of the loading ram, which is measuredusing a dial gauge.3.2.4 Fluid  ow systemExternal  uid reservoirs include a constant head in ow tank used in unidirectional  owtesting, and an \inflnite" constant head out ow reservoir of large surface area.The cell was designed to initiate  ow from the base of the sample. The in ow constanthead reservoir is connected to a port opening in the base cell reservoir via a plastic ballvalve. The height of this reservoir may be adjusted to impose a range of hydraulic gradientsacross the sample for unidirectional  ow testing. Typical hydraulic gradients used in testingranged from 0.02 up to 2.5 m of H2O/m for pervious samples. The  uid level in the in owreservoir is maintained using an over ow tube and gravity-driven  ow from an isolatedplastic supply tank.A second port and ball valve connects the base cell reservoir to a flxed inlet tube sealedwith a rubber membrane. A piston powered by a motorized cam is used to de ect themembrane, which induces an oscillatory pressure pulse at the base of the sample for thepurposes of oscillatory  ow testing. The cell was designed for upward  ow in part to66Chapter 3. Apparatus for streaming potential and resistivity testingfacilitate the rebound de ection of this membrane.The inflnite reservoir is connected directly to a port opening in the top cap of the uppercell reservoir and is used to apply a constant hydraulic head to the top of the sample. The uid level is maintained during unidirectional  ow testing using an over ow tube. The largesurface area of the reservoir is required to ensure that a constant pressure is maintained atthe top of the sample during oscillatory  ow testing.3.2.5 Data acquisition and controlThe digital data acquisition and control (DAC) system consists of a 200 kHz 16-bit A/Dcard with difierential inputs and 100 G› input impedance. The high input impedance ofthe device is necessary to ensure that the streaming potential is accurately measured acrossresistive samples. Four input channels are conflgured to sample data from the Ag-AgCl elec-trode pair, the upper pressure transducer, the lower pressure transducer, and the load cell,respectively. A grounded channel is sampled prior to each streaming potential measurementto avoid channel cross-talk caused by parasitic capacitance in the multiplexer. One outputchannel is conflgured to supply a voltage waveform via a voltage-to-current converter tothe current electrode pair for resistivity testing. The resulting potential difierences acrossthe Ag-AgCl electrode pair and stainless steel electrode pair are sampled using two inputchannels.Sample length and  uid properties are recorded manually. Sample length is calculatedfrom dial gauge measurements of the height of the loading ram with respect to the base ofthe cell. Fluid properties (electrical conductivity, temperature and pH) are measured in theexternal reservoirs using portable conductivity and pH meters once the  uid and sampleare at equilibrium.Software programs were developed to collect and process self-potential, pressure andload cell data from unidirectional and oscillatory  ow testing, as well as electrical potentialand current data from resistivity testing. Details of data acquisition and processing methodsare discussed in Sections 3.3 and 3.4.67Chapter 3. Apparatus for streaming potential and resistivity testing3.2.6 Sample preparationThe apparatus was designed to enable cell assembly and sample preparation without trap-ping air in the system, which could adversely afiect both streaming potential and resistivitymeasurements. Cell assembly involves the use of a saturation bath and removable top-capover ow reservoir to enable air-free assembly of the system. Sample preparation methodswere chosen to ensure fully saturated conditions and homogeneous samples. The waterpluviation technique is used for uniformly graded samples. A modifled slurry depositiontechnique is used for well-graded samples [Kuerbis and Vaid, 1988].The soil and saturating  uid must be allowed to reach a state of ionic equilibrium tore ect natural conditions and to ensure that difiusion potential efiects do not contribute tothe measured streaming potential. Ionic transfer between the  uid and solid matrix givesrise to changes in  uid chemistry that can in uence the measured electrical properties. Astate of sample equilibrium is marked by stable measured parameters that are repeatableover time, and synchronized streaming potential and pressure responses to the onset of  uid ow in streaming potential measurements.3.3 Streaming potential measurements3.3.1 MethodologyThe unidirectional  uid  ow measurement technique involves measuring the streaming po-tential response to an applied pressure difierence across the soil sample for a range ofhydraulic gradients under steady-state  ow conditions. Figure 3.3 shows streaming poten-tial and pressure time series data acquired using the unidirectional  ow method on a sampleof glass beads. The static DC ofiset is removed from the measured potential difierence andthe hydrostatic pressure is removed from the measured pressure difierence prior to plottingthese data versus each other, as shown in Figure 3.4. The streaming potential couplingcoe–cient is calculated from the linear flt between the measured potential and pressuredifierences. The corrected potential and pressure difierence data respectively exhibited av-erage measurement uncertainties of §0.1 mV and §0.04 kPa in unidirectional  ow testingperformed using the apparatus.68Chapter 3. Apparatus for streaming potential and resistivity testing0 250 500 750 1000 1250 1500−100−75−50−250time (s)∆ V (mV)0 250 500 750 1000 1250 150012345time (s)∆ P (kPa)a)b)Figure 3.3: Time series data acquired using the unidirectional  ow test method: a) stream-ing potential, b) pressure difierence (¢P = P1¡P2).Although this technique is seemingly straightforward, the  uid entering the system mustbe in equilibrium with the soil and pore  uid in the cell to ensure that no ionic gradients existacross the sample. This is one of the main practical limitations of this method, as signiflcanttime and efiort is required to equilibrate the large amount of  uid. Other limitations includethe possibility of electrode drift in the measurements and electrode polarization, as well aslimited sensitivity in the streaming potential and pressure data channels due to ambientnoise. Recording static streaming potential readings in between applied gradients enablesdrift corrections, and electrode polarization efiects can be eliminated through the use ofnon-polarizing electrodes. Noise levels, particularly from 60 Hz sources, are minimizedthrough electrical shielding of the apparatus and data flltering.The oscillatory  ow measurement technique does not necessitate equilibrium with exter-nal  uid sources. It is also less time-consuming to execute for a given sample as a single testrecords the response to a sinusoidally-varying imposed gradient. Figure 3.5 shows streaming69Chapter 3. Apparatus for streaming potential and resistivity testing0 0.5 1 1.5 2 2.5 3 3.5−100−90−80−70−60−50−40−30−20−100∆ P (kPa)∆ V (mV)C = ∆ V/ ∆ P = −28.2 mV/kPaFigure 3.4: Calculation of the streaming potential coupling coe–cient (C) from unidirec-tional  ow test data. Measurement error is within the limits of the marker used to identifydiscrete data points.potential and pressure difierence time series data acquired on a sample of glass beads usingthe oscillatory  ow method. The streaming potential data shown in Figure 3.5 have beencorrected for a static ofiset of 2 mV, and pressure data are plotted as a pressure difierencewith the hydrostatic pressure removed.The oscillatory  ow technique requires somewhat more advanced data collection andprocessing tools. Several methods exist to calculate the streaming potential coupling co-e–cient using this technique [Reppert and Morgan, 2001b]. The approach used here is aspectral analysis of the measured signals. The coupling coe–cient is determined by stack-ing the amplitude spectra of the SP and pressure signals, and calculating their ratio at thepulsation frequency:C =Xni=1¢Vrmsi–Xni=1¢Prmsi : (3.8)This averaging process minimizes random noise in the data and increases the signal to noiseratio. Figure 3.6 illustrates the analysis procedure. Oscillatory  ow test data collected ata pulsation frequency of 0.1 Hz respectively exhibited average uncertainties of §0.001 mV70Chapter 3. Apparatus for streaming potential and resistivity testing0 25 50 75 100−2−1.5−1−0.500.511.52time (s)∆ V (mV) ∆ P (kPa)Figure 3.5: Streaming potential and pressure time series data acquired using the oscillatory ow test method.and §0.002 kPa for this apparatus.3.3.2 Calibration and test designStreaming potential measurements were carried out using the test cell fllled entirely withwater (no solid matrix) to ensure that no streaming potentials were generated across thepervious fllter screens or end plates. The frequency spectrum of the measured backgroundstreaming potential under static  ow conditions was evaluated to determine appropriatesampling rates and digital fllter criteria, and to conflrm no signiflcant noise source in therange from zero (DC) to 0.5 Hz, which is the frequency range of interest for all  ow testingperformed using the apparatus.The optimum frequency for oscillatory  ow testing was determined to be 0.1 Hz basedon the results of preliminary testing. The frequency of oscillation must be low enough toprevent inertial  ow efiects. The amplitude of the pressure pulse must be small enough toprevent liquefaction of the sample, but large enough to generate a discernible response inthe streaming potential and pressure.71Chapter 3. Apparatus for streaming potential and resistivity testingappliedsignal membranedeflectionat base of celltmeasuredreal time signalP2 tP1 t∆V tmeasuredfrequency spectrumfreq.freq.a0freq.a0∆P = P1 –P2pulsationfrequency, a0P2P1∆VFigure 3.6: Spectral analysis of streaming potential and pressure data acquired using theoscillatory  ow test method.All  ow test data were acquired using a digital low-pass fllter with a cut-ofi frequencyof 20Hz. Unidirectional  ow test data were sampled at an average rate of 256 samplesper second and frame size of 256 samples. Each frame of data was flltered and averagedto produce one point of data per second. Oscillatory  ow test data were acquired at anaverage scan rate of 328 samples per second and frame size of 32,768 samples to resolvefrequencies down to 0.01 Hz. Several frames of data were averaged to improve the signal tonoise ratio.3.3.3 Comparison of unidirectional and oscillatory  ow test methodsThe validity of the oscillatory measurement technique was conflrmed through a compari-son of C values derived from unidirectional and oscillatory  ow testing performed on glassbead and soil samples. Figure 3.7 illustrates the results of unidirectional and subsequentoscillatory  ow tests performed on each sample. The oscillatory  ow tests were performedimmediately following each unidirectional  ow test in an attempt to capture similar sampleconditions, which in most cases did not correspond to complete equilibrium conditions. Ev-72Chapter 3. Apparatus for streaming potential and resistivity testingident in Figure 3.7 is a strong agreement between C values derived from corresponding  owtests. Three data points corresponding to samples gb a, gb b and gbnc a deviate notablyfrom this trend. These samples exhibited a very rapid change in the measured propertiesupon termination of steady unidirectional  uid  ow. Consequently, the results of unidirec-tional and oscillatory  ow tests represent difierent sample conditions prior to equilibrium.This is conflrmed by measured resistivity values, which dropped respectively by 7%, 10%and 13% in each sample between unidirectional and oscillatory  ow tests. The lower sampleresistivity translates to a smaller magnitude of C, which is evident in the data shown inFigure 3.7.-30-25-20-15-10-50-30 -25 -20 -15 -10 -5 0C from unidirectional flow test (mV/kPa)C from oscillatory flow test (mV/kPa) gbnf_agbnf_bgbnf_cgbnf_dgbnc_agbnc_bgbnc_cgb_agb_bsg_cgt_bFigure 3.7: Comparison of streaming potential coupling coe–cient (C) values measuredusing unidirectional and oscillatory  ow test methods. Soil samples are labelled accordingto material type (gb: glass beads, s: sand and gravel c: glacial till), changes to the grada-tion (nf: flnes removed, nc: coarse fraction removed) and the sample identifler (e.g., a).Measurement error exceeding marker size is indicated by error bars.73Chapter 3. Apparatus for streaming potential and resistivity testing3.4 Resistivity measurements3.4.1 MethodologyThe DC resistivity method consists of measuring the electrical potential response to current ow generated by a DC or low frequency AC source. An AC current source is typicallyused to prevent error introduced by polarization of the current electrodes. At low sourcefrequencies, the measured resistivity is equivalent to the DC limit [Telford et al., 1990]. Theresistance R (ratio of the measured voltage and current) is converted to resistivity usingthe relation:‰ = kR; (3.9)where k is a geometric factor speciflc to the experiment.Field surveys typically employ a 4-electrode conflguration, where one pair of electrodesis used to inject current and a second pair of electrodes is used to measure the potentialdifierence. The subsurface resistivity may be derived using a factor speciflc to the electrodeconflguration and geometry. The conflned geometry of a laboratory experiment makes itpossible to use either a 2- or 4-electrode conflguration. Current is applied across the sampleby means of two plate or ring electrodes, and the resulting potential difierence is measuredusing the same or a separate pair of electrodes.The apparatus consists of two stainless steel end-cap plate electrodes in contact withthe top and bottom cross-sectional area of the soil specimen. These are used to apply alow frequency AC source current to the sample. The apparatus was designed to measureresistivity using the 2-electrode method, but potential difierence measurements were alsomade using the Ag-AgCl electrode pair (4-electrode method) for comparison.3.4.2 Calibration and test designContact resistance measurements were made to calculate the amplitude of the source voltagewaveform required to apply su–cient current across the sample. A peak source amplitude of0.3 V was chosen to accommodate large sample resistances (»30 k›) and the 10 V outputlimit of the voltage-to-current converter. The in uence of the source waveform type onthe measured resistance was assessed over a range of 50 Hz to 1000 Hz, and a sinusoidal74Chapter 3. Apparatus for streaming potential and resistivity testingwaveform was chosen as the optimal signal type. The source voltage was generated by theDAC board and was converted to a 0.117 mA rms current source.Measurements were made across a range of frequencies to determine a suitable sourcefrequency. Figure 3.8 shows the efiect of source frequency on the measured resistance using2- and 4-electrode techniques on two samples of glass beads. The 2-electrode measurementsare efiectively constant at frequencies above 50 Hz, but deviate signiflcantly at lower fre-quencies. This is attributed to polarization efiects at the current electrodes. The 4-electrodemeasurements fall ofi from the DC resistivity limit at frequencies above 100 Hz. This be-haviour is attributed to the complex impedance of the sample [Olhoeft, 1985]. These datasuggest an optimum source frequency of 100 Hz to enable a comparison of the two methods.100 101 102 103 1047000720074007600780080002−electrode measured resistance (Ohm)frequency (Hz)2000250030003500400045004−electrode measured resistance (Ohm)2−electrode4−electrodeFigure 3.8: Frequency dependence of the resistance of glass bead samples measured using2- and 4-electrode DC resistivity methods. Note that 2- and 4-electrode data series werecollected at difierent stages of sample equilibration. Measurement error is within the limitsof the marker used to identify discrete data points.The conflned sample geometry and chosen electrode conflguration enable us to calculatethe sample resistivity using a geometric factor A=l:‰ = Al R; (3.10)75Chapter 3. Apparatus for streaming potential and resistivity testingwhere A is the cross-sectional area of the sample and l is the sample length. To conflrm thisrelation, resistivity measurements were performed on the test cell fllled with  uids of knownconductivity for current electrode separations ranging from 7 cm to a maximum samplelength of 10 cm. Fluid conductivity was controlled by varying the concentration of sodiumsulphate in de-aired distilled water, and was measured using a portable conductivity meter.Figure 3.9 compares the resistivity derived using (3.10) from 2-electrode measurements tothe resistivity of the  uid contained in the cell. The near unity match conflrms the validityof (3.10) to calculate sample resistivity.0 200 400 600 800 1000120014000200400600800100012001400fluid resistivity (Ohm.m)2−electrode measured resistivity (Ohm.m)y = 0.99xFigure 3.9: Comparison of resistivity measured using the 2-electrode method to the knownresistivity of  uid fllling the test cell. Resistivity is calculated from the measured resistanceusing a geometric factor k, which is calculated as a function of sample volume deflned in(3.10). Measurement error is within the limits of the marker used to identify discrete datapoints.DC resistivity data were collected at a similar scan rate and frame size as the oscillatory ow test data, but no digital flltering was applied due to the high amplitude of the measuredsignals.76Chapter 3. Apparatus for streaming potential and resistivity testing3.4.3 Comparison of 2- and 4-electrode methodsThe potential difierence measured using the Ag-AgCl electrode pair is representative ofthe true potential difierence across the sample if the sample resistance (Rs) is signiflcantlygreater than the resistance of the  uid column between the electrodes and the ends of thesample (Rf). This assumption does not hold in the case of a  uid-fllled cell, as shown bythe data in Figure 3.10. The measured 4-electrode resistance derived using (3.10) is 34%smaller than the known resistivity of the  uid, which is in contrast to the near unity matchfor the 2-electrode case shown in Figure 3.9. This suggests that there is net potential dropbetween the ends of the sample and the Ag-AgCl electrodes, which results in a smallerpotential difierence measured using the 4-electrode technique.0 500 1000 1500 2000 250005001000150020002500fluid resistivity (Ohm.m)4−electrode measured resistivity (Ohm.m)y = 0.66xFigure 3.10: Comparison of resistivity measured using the 4-electrode method to the knownresistivity of  uid fllling the test cell. Resistivity is calculated from the measured resistanceusing a geometric factor k, which is calculated as a function of sample volume deflned in(3.10). Measurement error is within the limits of the marker used to identify discrete datapoints.The potential drop is less signiflcant when a soil sample is present due to the largersample resistance. For a given sample length, the ratio Rs/Rf is on average flve times77Chapter 3. Apparatus for streaming potential and resistivity testinggreater when the test cell houses a soil sample than when the cell is fllled entirely with uid. Figure 3.11 illustrates the difierence between 4-electrode and 2-electrode measuredpotential from resistivity tests performed on a range of soil samples. The linear relationindicates that the potential measured using the Ag-AgCl electrodes underestimates the\true" potential measured at the plate electrodes. Consequently, a 6% correction should beapplied to the 4-electrode data if they are to be used to calculate sample resistivity.0 500 10001500200025003000350005001000150020002500300035002−electrode measured potential (mV)4−electrode measured potential (mV)y = 0.9424xFigure 3.11: Comparison of 4-electrode and 2-electrode potentials measured during resis-tivity experiments on soil samples (0.117 mA rms current source). Measurement error iswithin the limits of the marker used to identify discrete data points.The small discrepancy between 2- and 4-electrode measurements could also be attributedin part to the measurement frequency used to compare the two methods. As discussed inSection 3.4.2, the 4-electrode measured resistivity at 100 Hz may underestimate the true DCresistivity. In oscillatory  ow testing, streaming potentials are generated across a soil sampleat much lower frequencies (0.1 Hz) and result from a difierent current source mechanism.Consequently, no correction is applied to streaming potential measurements obtained usingthe Ag-AgCl electrodes.78Chapter 3. Apparatus for streaming potential and resistivity testing3.5 ConclusionWe have developed a laboratory apparatus that allows us to measure the streaming po-tential coupling coe–cient and electrical resistivity on the same saturated soil specimen.These parameters are required to properly characterize the streaming current cross-couplingconductivity of a given material for use in fleld survey data interpretation. The apparatusenables testing of well-graded soil samples up to a maximum particle size of 9.5 mm, andmeasurement of relevant sample properties including dry density, porosity and hydraulicconductivity.The apparatus was designed to measure the streaming potential coupling coe–cientusing both unidirectional and oscillatory  ow methods, in order to validate use of thelatter method on unconsolidated samples. The oscillatory  ow method was proven to bea valid test method for soils and is the preferred method to achieve sample equilibriummost efiectively. Sample resistivity was characterized using a 2-electrode measurementconflguration, which was shown to give accurate measurements through a calibration with uids of known resistivity. A comparison of measured sample resistivity using 2- and 4-electrodeconflgurationsatasourcefrequencyof100Hzrevealedthatthe4-electrodemethodunderestimates the 2-electrode value by 6%.79Chapter 3. Apparatus for streaming potential and resistivity testing3.6 ReferencesAhmad, M. U., A laboratory study of streaming potentials, Geophysical Prospecting, 12,49{64, 1964.ASTM, D2434-68: Standard test method for permeability of granular soils (constant head),in Annual Book of ASTM Standards, vol. 4, American Society of Testing and Materials,1991.Corwin, R. F., The self-potential method for environmental and engineering applications,in Geotechnical and Environmental Geophysics, vol. 1, edited by S. H. Ward, pp. 127{145,Society of Exploration Geophysicists, Tulsa, 1990.de Groot, S. R., Thermodynamics of irreversible processes, Selected Topics in ModernPhysics, vol. 3, North Holland Publishing Company, Amsterdam, 1951.Ishido, T., and H. Mizutani, Experimental and theoretical basis of electrokinetic phe-nomena in rock-water systems and its application to geophysics, Journal of GeophysicalResearch, 86(B3), 1763{1775, 1981.Kuerbis, R., and Y. P. Vaid, Sand sample preparation: The slurry deposition method,Soils and Foundations, 28(4), 107{118, 1988.Morgan, F. D., Fundamentals of streaming potentials in geophysics: Laboratory methods,in Detection of Subsurface Flow Phenomena, Lecture Notes in Earth Sciences, vol. 27,edited by G. P. Merkler, H. Militzer, H. H˜otzl, H. Armbruster, and J. Brauns, pp. 133{144, Springer-Verlag, Berlin, 1989.Morgan, F. D., E. R. Williams, and T. R. Madden, Streaming potential properties of West-erly granite with applications, Journal of Geophysical Research, 94(B9), 12,449{12,461,1989.Ogilvy, A. A., M. A. Ayed, and V. A. Bogoslovsky, Geophysical studies of water leakagesfrom reservoirs, Geophysical Prospecting, 17, 36{62, 1969.Olhoeft, G. R., Low-frequency electrical properties, Geophysics, 50(12), 2492{2503, 1985.80Chapter 3. Apparatus for streaming potential and resistivity testingOverbeek, J. T. G., Electrokinetic phenomena, in Colloid Science, Irreversible Systems,vol. 1, edited by H. R. Kruyt, Elsevier Publishing Company, Amsterdam, 1952.Packard, R. G., Streaming potentialsacross glass capillaries for sinusoidalpressure, Journalof Chemical Physics, 21(2), 303{307, 1953.Pengra, D. B., S. X. Li, and P. Wong, Determination of rock properties by low-frequencyAC electrokinetics, Journal of Geophysical Research, 104(B12), 29,485{29,508, 1999.Reppert, P. M., and F. D. Morgan, Frequency-dependent streaming potentials, Journal ofColloid and Interface Science, 234, 194{203, 2001a.Reppert, P. M., and F. D. Morgan, Streaming potential collection and data processingtechniques, Journal of Colloid and Interface Science, 233, 348{355, 2001b.Revil, A., H. Schwaeger, L. M. Cathles III, and P. D. Manhardt, Streaming potential inporous media: 2. Theory and application to geothermal systems, Journal of GeophysicalResearch, 104(B9), 20,033{20,048, 1999.Sears, A. R., and J. N. Groves, The use of oscillating laminar  ow streaming potentialmeasurements to determine the zeta potential of a capillary surface, Journal of Colloidand Interface Science, 65(3), 479{482, 1978.Telford, W. M., L. P. Geldart, and R. E. Sherifi, Applied Geophysics, Cambridge UniversityPress, Cambridge, 1990.Wurmstich, B., and F. D. Morgan, Modeling of streaming potential responses caused byoil well pumping, Geophysics, 59(1), 46{56, 1994.81Chapter 4Laboratory measurements ofstreaming potential and resistivityin well-graded soils 34.1 IntroductionThe self-potential method has been used to measure the electrokinetic phenomenon ofstreaming potential in hydrogeological [Revil et al., 2004; Titov et al., 2005; Suski et al.,2006], earth dam seepage [Ogilvy et al., 1969; Wurmstich et al., 1991], earthquake prediction[Mizutani et al., 1976; Corwin and Morrison, 1977], geothermal [Corwin and Hoover, 1979;Revil et al., 1999b], and subglacial  ow investigations [Blake and Clarke, 1999; Kulessaet al., 2003] to garner information about  uid  ow in the subsurface. Quantitative interpre-tation of self-potential data using forward and inverse modelling techniques requires someknowledge of the electrical properties that govern streaming potential to recover informationabout the hydraulic system.Electrokinetic processes involve the relative movement between an electrolyte and acharged solid surface. An electrical double layer, which consists of a tightly bound innerlayer and a more difiuse outer layer of ions, forms at the solid- uid interface in a saturatedporous medium [Adamson, 1990]. The zeta potential is the average electric potential on thesurface of shear within the difiuse layer. This parameter is fundamental in describing thesolid- uid interface, as it gives an approximation of the typically negative surface potentialof the solid.3A version of this chapter will be submitted for publication. Shefier, M.R., Reppert, P.M. and Howie,J.A. (2007) Laboratory measurements of streaming potential and resistivity in well-graded soils.82Chapter 4. Laboratory testing in well-graded soilsMuch of the laboratory work on streaming potentials has focused on measuring thestreaming potential coupling coe–cient C to study the in uence of electrolyte chemistryand temperature on the zeta potential in difierent materials [e.g., Ishido and Mizutani,1981; Morgan et al., 1989; Lorne et al., 1999; Revil et al., 1999b; Reppert and Morgan,2003]. Other work has focused on examining the link between C and hydraulic permeability[Jouniaux and Pozzi, 1995; Pengra et al., 1999], studying the in uence of  uid saturation(two-phase  ow) on the measured coupling coe–cient [Guichet et al., 2003; Revil and Cerepi,2004], and characterizing C for use in SP data interpretation at a speciflc site [e.g., Revilet al., 2002, 2005]. Although most of these studies report measured values of C alongwith the saturating  uid properties, measured values of the electrical conductivity of thesaturated sample  are not often included. Both of these properties are fundamental toany mathematical treatment of streaming potential using the theory of coupled  ow, andare required to calculate the streaming current cross-coupling conductivity L for a givenmaterial.In this paper we present the results of a laboratory testing program designed to study thein uence of soil and  uid parameters on the streaming current cross-coupling conductivityin saturated well-graded soil samples. We are interested in characterizing L to facilitate thequantitative interpretation of streaming potentials using numerical modelling techniques.Section 4.2.1 describes the streaming potential phenomenon using coupled  ow theoryand deflnes L and C for 1-D  ow conditions typical of laboratory experiments. Section4.2.2 describes the properties C,  and L using empirical and theoretical formulations. Thecross-coupling conductivity may be measured directly or derived from the product of C and . In the current study we characterize L by measuring C and the electrical resistivity‰ = 1= on the same saturated soil sample using an apparatus designed speciflcally for thispurpose [Shefier et al., 2007]. The oscillatory  ow method was used to perform streamingpotential measurements on well-graded soil samples for the flrst time using this apparatus.Sections 4.3.1 to 4.3.3 summarize the details of the laboratory apparatus and the testingmethods.Most laboratory studies of geologic materials have focused on uniform samples, such asquartz sand [e.g., Ahmad, 1964; Guichet et al., 2003], or crushed rock [e.g., Morgan et al.,83Chapter 4. Laboratory testing in well-graded soils1989; Lorne et al., 1999] and intact rock samples [Jouniaux and Pozzi, 1995]. Consequently,there is a lack of published streaming potential data for soils, particularly those that exhibita range of grain sizes. In the present study, we are interested in characterizing well-gradedsoils for application to hydrogeologic and earth dam seepage investigations. Section 4.3.4describes the soil samples tested in the current investigation.Previous laboratory studies report varying timescales of interaction between the sampleand the saturating  uid. Section 4.3.5 discusses the importance of sample equilibration andillustrates how  uid properties can vary signiflcantly during the course of ionic exchangewith the solid matrix, particularly when resistive  uids are used. This has implications forthe use of theoretical estimates that rely on  uid conductivity, as discussed in Section 4.4.3.Section 4.4 presents the results of tests performed on samples of glass beads, glacial tillembankment core material and sand and gravel embankment shell material to study the in- uence of density, gradation and  uid conductivity on the streaming current cross-couplingconductivity. These results are compared with theoretical estimates of the electrical proper-ties for glass bead samples. Section 4.5 compares the measured values of L in embankmentsoil samples with published data.4.2 Theory4.2.1 Streaming potentialThe total  ux of electric charge, or current densityJ [Am¡2 ], in a saturated porous mediummay be expressed in terms of the sum of primary and secondary potential gradients [On-sager, 1931]. In the absence of signiflcant thermal or ionic concentration gradients, electricalcurrent  ow is described byJ = JC +JS; (4.1)where JC and JS are respectively the conduction current and streaming current densities.The streaming current is described using a linear  ow law [de Groot, 1951; Shefier andOldenburg, 2007]:JS = ¡Lrh; (4.2)84Chapter 4. Laboratory testing in well-graded soilswhereL [Am¡2 ] is the streaming current cross-coupling conductivity andh[m] is hydraulichead. The conduction current is described using Ohm’s law:JC = ¡ r`; (4.3)where  [Sm¡1 ] is the electrical conductivity and ` is the electrical potential [V].In the absence of any imposed current sources, r¢ J = 0 such that (4.1) reduces tor¢  r` = ¡r¢ Lrh: (4.4)By applying the divergence theorem, (4.4) can be expressed as Zsr`¢ ^n ds = ¡LZsrh¢ ^n ds (4.5)in a homogeneous medium where L and  are macroscopic properties. For 1-D  ow in aporous medium of length l and cross-sectional area A, in which  ow occurs along the lengthof the medium, (4.5) is evaluated to give ¢`l A = ¡L¢hl A : (4.6)Equation (4.6) may be further simplifled and rearranged to deflne the cross-coupling con-ductivity:L = ¡C  ; (4.7)whereC = ¢`¢h: (4.8)The streaming potential coupling coe–cient C is typically measured in the laboratory andreported in terms of pressure, such thatC = ¢`¢P ; (4.9)where P =  fh and  f is the product of  uid density and gravitational acceleration.4.2.2 Electrical propertiesIn a saturated and insulating porous matrix, electrolytic conduction dominates and con-duction current  ows within the interconnected pore space via free ions in solution. Surface85Chapter 4. Laboratory testing in well-graded soilsconduction at the solid- uid interface can contribute signiflcantly to the electrical conduc-tivity of the saturated medium if the solid matrix contains conductive minerals such as clay,or if the saturating  uid exhibits low salinity. [Ishido and Mizutani, 1981; Revil and Glover,1998].Surface conductivity is a function of ionic conductance within the electrical double layerand is attributed to a much smaller extent to proton transfer at the mineral surface [Reviland Glover, 1998]. Saturation with a low-conductivity pore  uid increases the thickness ofthe double layer, such that it becomes a more signiflcant fraction of the total pore diameter[Hunter, 1981]. This efiect can be quantifled by calculating the Debye length ⁄D:⁄D = "kBT2NAe2cI¶1=2(4.10)where " is the electric permittivity of the  uid [Fm¡1 ], kB is Boltzmann’s constant [JK¡1 ],T is temperature [K], NA is Avogadro’s number [mol¡1 ] and ec is the elementary charge[C]. The ionic strength of the  uid I is given byI = 12Xiciz2i ; (4.11)where ci is the ionic concentration [molm¡3 ] and zi is the valency of each ion i in solution.The Debye length is referred to as the thickness of the double layer [Hunter, 1981]. However,free ions in solution may be considered to exist > 2⁄D from the mineral surface [Revil et al.,1999a]. Surface conductivity efiects may be considered to be signiflcant when the pore radiusr < 10⁄D [Morgan et al., 1989].The electrical conductivity of the  uid  f [Sm¡1 ] may be calculated as a function ofionic concentration through the relation f = FXicimi; (4.12)where F is Faraday’s constant [Ceq¡1 ], ci is the ionic concentration stated in terms ofequivalents [eqm¡3 ] and mi is the mobility [m2 s¡1 V¡1 ] of each ion i in solution [Kellerand Frischknecht, 1966].A theoretical description of C is given by the Helmholtz-Smoluchowski equation, whichwas derived for a capillary subject to laminar  uid  ow [Overbeek, 1952]:C = "‡· f; (4.13)86Chapter 4. Laboratory testing in well-graded soilswhere ‡ is the zeta potential [V] and · is  uid viscosity [Pa s]. This equation is validwhen the thickness of the electrical double layer is much smaller than the pore radius, andelectrical conductivity of the saturated matrix is governed solely by the conductivity of thepore  uid.In cases where the surface conductivity of the solid cannot be ignored,  f in (4.13) canbe replaced with an efiective  uid conductivity  efi that accounts for the in uence of surfaceconductivity within the pore:C = "‡· efi: (4.14)The efiective conductivity of the pore channel may be thought of as the sum of  uid andsurface conductivity: efi =  f + 2‚s⁄p; (4.15)where ‚s is the speciflc surface conductance [S], and ⁄p=2 is a characteristic length scale ofthe pore [m], which approximates the ratio of pore volume to pore surface area [Gu¶eguenand Palciauskas, 1994]. Some estimates of surface conductance are given in Ishido andMizutani [1981], Jouniaux and Pozzi [1995] and Revil and Glover [1998]. In a cylindricalpore channel, ⁄p is equivalent to the pore radius [e.g., Overbeek, 1952]. Revil and CathlesIII [1999] derived an expression for ⁄p based on grain size diameter d in uniform sands:⁄p = d2m(F ¡1); (4.16)where Archie’s cementation exponent m and formation factor F are described below.The efiect of surface conductivity can also be estimated by multiplying (4.13) by a ratioof measured formation factors [Jouniaux and Pozzi, 1995; Lorne et al., 1999]:C = "‡· fFFo ; (4.17)where F =  f= is measured under the same conditions as for C when surface conductivityis present. Formation factor Fo represents the high conductivity limit when surface con-ductivity is negligible. The formation factor is a function of pore space topology and maybe described empirically using Archie’s law [Archie, 1942]: f = n¡m; (4.18)87Chapter 4. Laboratory testing in well-graded soilswhere n is porosity. This equation is valid in cohesionless materials saturated with a salinepore  uid, such that surface conductivity efiects can be neglected. When flnite surfaceconductivity exists but the conductivity of the pore  uid still dominates conduction, thebulk conductivity of the saturated medium may be described by =  efiF ; (4.19)where  efi is as deflned in (4.15).A theoretical description of the cross-coupling conductivity can be derived using (4.7),where C and  are deflned in (4.14) and (4.19), respectively:L = ¡ "‡·F : (4.20)This expression illustrates that L is a function of the formation factor, or the ratio of theelectrical conductivity of the saturated sample and pore space.4.3 Experimental methodsThe laboratory apparatus was designed to enable measurements of the streaming potentialcoupling coe–cient and resistivity on the same saturated soil specimen, as well as relevantsample properties including density, porosity and hydraulic conductivity. A complete de-scription of the design and calibration of the apparatus and the experimental methods usedto perform all tests presented in this paper can be found in Shefier et al. [2007].4.3.1 ApparatusA schematic of the experimental apparatus is shown in Figure 4.1. The test cell consists ofa sample holder and two  uid reservoirs which are flxed above and below the soil specimen.The sample holder is oriented vertically to enable the preparation of fully saturated, homo-geneous soil samples. A 10 cm-long sample holder was used for all experiments presentedin this paper. The inner diameter of the sample holder is 10 cm, which enables testingof samples exhibiting a maximum particle size of 9.5 mm [ASTM, 1991a]. The sample iscontained between two pervious acrylic plates, and an external loading assembly is used toapply a nominal vertical seating load of 25 kPa to the top of the sample.88Chapter 4. Laboratory testing in well-graded soilssoilsample V1~V2fluid reservoirfluid reservoirloadunidirectionalflow inlet oscillatoryflow membranecamP1P2“infinite” outflow reservoirinflow tankFigure 4.1: Schematic of laboratory apparatus.The apparatus was designed to enable the use of both unidirectional and oscillatory ow test methods to measure the streaming potential coupling coe–cient. These methodsare described in Section 4.3.2. An \inflnite" reservoir imposes a constant hydraulic headboundary to the top of the soil specimen for the purposes of both testing methods. Aconstant head reservoir is connected to the base of the test cell to drive  uid  ow upwardsthrough the sample for unidirectional  ow testing. A sealed rubber membrane is connectedto the base of the test cell for the purposes of oscillatory  ow testing. De ection of thismembrane using a mechanical actuator generates an oscillating pressure gradient that drives ow through the sample.Streaming potential measurements (V1 in Figure 4.1) are made using a pair of non-polarizing Ag-AgCl electrodes positioned immediately above and below the sample withinthe test cell  uid reservoirs. These electrodes are recessed away from the primary  ow pathto avoid electromotive efiects [Morgan, 1989]. Pressure measurements (P1 and P2 in Figure89Chapter 4. Laboratory testing in well-graded soils4.1) are made at the same vertical position using a pair of gauge pressure transducers. Re-sistivity measurements are performed using a pair of pervious stainless steel disk electrodesthat are a–xed to the acrylic plates in direct contact with both ends of the soil specimen(V2 in Figure 4.1). These electrodes are used to apply current and measure the resultantvoltage drop across the sample to characterize the resistivity as a separate experiment. Anautomated data acquisition and control system with a 100 G› input impedance is usedto perform all streaming potential and resistivity measurements. Insulating materials wereused in the construction of the cell, and grounded shielding surrounds the entire apparatusto reduce the efiects of 60 Hz noise from ambient sources.4.3.2 Streaming potential measurementsStreaming potential measurements were conducted using unidirectional and oscillatory  uid ow techniques to characterize the streaming potential coupling coe–cient C. These meth-ods are described in Reppert and Morgan [2001b], and Shefier et al. [2007] gives detailsspeciflc to the apparatus used in this study. Although the oscillatory  ow method has beenimplemented in previous studies of streaming potential on glass capillaries and rock samples[Packard, 1953; Sears and Groves, 1978; Pengra et al., 1999; Reppert and Morgan, 2001a],it had previously not been tested on well-graded soils. Consequently, we adopted a paralleldesign approach to enable a comparison of the two methods for application to soil samples.Unidirectional  ow testing was performed by measuring the streaming potential responsetoaseriesofhydraulicgradientsundersteady-state owconditions. Thestreamingpotentialcoupling coe–cient was then calculated from the linear flt between the measured electricalpotential and hydraulic head difierences for a range of hydraulic gradients. Oscillatory  owtesting was carried out by measuring the streaming potential and hydraulic response toa sinusoidal 0.1 Hz pressure pulse applied at the base of the sample. This method takesadvantage of the fact that the temporal response of streaming potential and difierential headto an applied pressure pulse are identical under laminar  ow conditions [Chandler, 1981].The coupling coe–cient C was determined through a spectral analysis of the measuredsignals by calculating the ratio of averaged potential and head amplitudes at the pulsationfrequency.90Chapter 4. Laboratory testing in well-graded soilsThe oscillatory  ow method is a more e–cient measurement technique as it subjectsthe sample to a range of hydraulic gradients in a single test. The method facilitates ionicequilibration between the sample and the pore  uid, an issue discussed in detail in Section4.3.5, since external  uid sources are not required to induce  ow through the sample.The validity of the oscillatory measurement technique for use on soil samples was con-flrmed through a comparison of C values derived from unidirectional and oscillatory  owtesting. Figure 4.2 compares the streaming potential coupling coe–cient obtained usingeach method on the same glass bead or soil sample. Evident in the flgure is a strong agree-ment between C values derived from corresponding  ow tests, with the exception of threedata points corresponding to samples gb a, gb b and gbnc a. The relatively high percentageof flner grains and resistive  uid (»500 ›m) used to saturate these samples led to a rapidchange in sample conditions upon termination of steady unidirectional  uid  ow. This isconflrmed by measured resistivity values, which dropped respectively by 7%, 10% and 13%in these samples between  ow tests. The lower sample resistivity translates to a smallermagnitude of C, which is evident in the data shown in Figure 4.2.4.3.3 Resistivity measurementsResistivity measurements were conducted using a 2-electrode conflguration, where a singlepair of stainless steel plate electrodes was used to inject current and measure the potentialdifierence across the length of the soil sample. A 100 Hz AC source current was chosento characterize the DC limit of resistivity while preventing polarization of the electrodes.Sample resistivity was calculated by multiplying the measured resistance by the ratio of thecross-sectional area and length of the sample.4.3.4 Sample propertiesSample density and porosity were calculated using measurements of sample volume andvalues of speciflc gravity for each material tested. Sample length was measured using a dialgauge. The hydraulic conductivity of each sample was determined using Darcy’s law:Q = ¡K@h@zA; (4.21)91Chapter 4. Laboratory testing in well-graded soils-30-25-20-15-10-50-30 -25 -20 -15 -10 -5 0C from unidirectional flow test (mV/kPa)C from oscillatory flow test (mV/kPa)gbnf_agbnf_bgbnf_cgbnf_dgbnc_agbnc_bgbnc_cgb_agb_bsg_cgt_bFigure 4.2: Comparison of streaming potential coupling coe–cient (C) values measuredusing unidirectional and oscillatory  ow test methods. Soil samples are labelled accordingto material type (gb: glass beads, sg: sand and gravel gt: glacial till), variations on thegradation (nf: flnes removed, nc: coarse fraction removed) and the sample identifler (e.g.a).where A is the cross-sectional area of the sample and Q [m3 s¡1 ] is the volumetric  ow ratemeasured during steady-state unidirectional  ow from an applied hydraulic gradient.Figure 4.3 shows the grain size distribution curves for the glass bead samples tested.The base gradation gb represents a well-graded material of uniform mineralogy. Gradationsgbnf and gbnc represent the same gradation but with flne and coarse fractions removed,respectively. These two gradations serve as bounds on the material for the purposes ofstudying the in uence of gradation on the electrical properties. The average speciflc gravityfor the glass particles is 2.475.Figure 4.4 shows the grain size distribution curves for soil obtained from the core andinner shell zones of an embankment site in British Columbia. The core material gt is awell-graded glacial till with an average clay content of 7%, an average fleld dry density of2.11 g/cm3 and a mean speciflc gravity of 2.54 [BC Hydro, 2002]. Mineralogical analyses92Chapter 4. Laboratory testing in well-graded soils10−1 100 1010102030405060708090100Particle size (mm)% finer thangbgbnfgbncFigure 4.3: Grain-size distribution of glass bead samples.performed on the flne fraction of the core revealed weight percentages of 30% quartz and over41% muscovite [Shefier, 2005], where flnes are deflned as particles < 0:075 mm in diameter.The sand and gravel shell material sg consists primarily of sub-angular to sub-roundedparticles of quartzite and limestone with approximately 17% mica schist and gneiss, has anaverage fleld dry density of 2.34 g/cm3, and a mean speciflc gravity of 2.78 [BC Hydro, 2002].The core and shell material was processed using the wet-sieve method [ASTM, 1993a] andsamples were created to match the average gradations of material placed below a particlesize of 9.4 mm, to conform to the capacity of the test cell. The 3% flne fraction was notincluded in any of the reconstituted shell samples. Gradation gt represents core materialwith the full 40% flne fraction, gt25 contains 25% flnes and gt0 contains none of the flnefraction, as indicated in Figure 4.4.All soil samples were saturated under vacuum and reconstituted in the cell using a mod-ifled slurry deposition technique [Kuerbis and Vaid, 1988]. Sample density was increased bytapping on the base of the test cell during sample deposition, for the purposes of studyingthe in uence of density on the electrical properties. The electrical conductivity of the satu-rating  uid was controlled by adding Na2SO4 in solution to de-aired tap water exhibiting a93Chapter 4. Laboratory testing in well-graded soils10−3 10−2 10−1 100 1010102030405060708090100Sieve opening size (mm)% passingsggtgt25gt0Figure 4.4: Grain-size distribution of embankment soil samples (sg: sand and gravel shell,gt: glacial till core).nominal  uid conductivity of 2£10¡3 S/m. Fluid conductivity, temperature and pH weremeasured in the external reservoirs using portable electrical conductivity and pH meters.Saturating  uid properties are listed along with the sample data in Table 4.1. Embankmentsoil samples were saturated with  uid exhibiting a nominal resistivity of 70 ›m to representobserved  uid properties at site. All samples were subjected to a vertical seating load of 25kPa following preparation.4.3.5 Sample equilibrationOne main advantage of oscillatory  ow testing over unidirectional  ow testing is that itallows the soil and pore  uid to reach a state of ionic equilibrium more e–ciently since thepore  uid is not completely  ushed during testing. This process of equilibration is importantto ensure that measured C and ‰ values are more representative of the saturated systemin its natural state, and to prevent the in uence of difiusion potentials on the measuredstreaming potential.Figure 4.5 illustrates the in uence of soil- uid equilibration on the measured electrical94Chapter 4. Laboratory testing in well-graded soilsproperties for glass bead sample gb a. This sample was saturated with a 540 ›m  uid.Evident in the flgure is the rapid change in early time, as the saturated sample startsto equilibrate upon termination of a unidirectional  ow test. All subsequent streamingpotential measurements were performed using the oscillatory  ow method such that noexternal  uid was introduced to the sample. Resistivity measurements were conductedin between streaming potential measurements. The electrical properties trend towardsconstant values as the sample approaches equilibrium. Of particular note is the relativechange in the amplitude of each parameter with time. Both C and ‰ values decrease upto one order of magnitude, whereas L varies by less than half its original value. A  uidresistivity of 327 ›m was measured within the test cell reservoir following the experiment,andrepresentsa39%decreaseinresistivityoftheoriginalsaturating uid. Thisisattributedto ionic exchange between the solid matrix and saturating  uid, which has been observedby other researchers [Jouniaux and Pozzi, 1995; Guichet et al., 2003].0 5 10 15 20 25 30 35 40 45−30−20−100 C  (mV/kPa)0 5 10 15 20 25 30 35 40 450100020003000 ρ  (Ohm m)0 5 10 15 20 25 30 35 40 450.511.5x 10−4Time (hours) L  (A/m2 ) c)a)b)Figure 4.5: In uence of sample equilibration on the electrical properties measured on asample of glass beads (gb a).95Chapter 4. Laboratory testing in well-graded soilsA state of sample equilibrium is marked by measured parameters that are stable overtime, and synchronized streaming potential and pressure responses to the onset of  uid ow. A time lag between these responses is indicative of difiusion potential efiects causedby ionic imbalances within the system. Gradients in  uid chemistry generate an electricalpotential due to the difiusion of ions, which is measured along with the streaming potential.Unidirectional  ow measurements of streaming potential in glass bead samples gavesteady, repeatable results if the pore  uid was completely  ushed prior to testing. Thelower hydraulic conductivity of embankment material precluded reliable unidirectional  owmeasurements in these samples. In oscillatory  ow measurements, difiusion-induced lag wasfound to be most prevalent in samples of embankment material. Oscillatory  ow testingperformed in early time revealed up to a 180o phase shift in the streaming potential response.Mineralogical analyses conflrmed that this apparent reversal in polarity was not caused bya positive surface potential on the solid grains. However, a long-term experiment wasconducted on a sand and gravel sample (sg c) to conflrm that difiusion potential efiectswere indeed the source of this lag. Fluid was re-circulated through the test cell and externalreservoir system over a period of eightweeks to achieve equilibrium conditions, at whichtimeunidirectional owmeasurementsgaveconsistentresultswithoscillatory owmeasurements,as shown in Figure 4.2. Fluid resistivity decreased from 82 ›m at the start of the experimentto 22 ›m at completion due to ionic exchange with the soil. Any lag evident in oscillatory ow measurements performed at the start of an experiment was found to dissipate as thesample reached equilibrium.96Chapter 4. Laboratory testing in well-graded soilsSample n Bulk density Cu  f (‰f) Fluid pHg/cm3 „S/m (›m)gb a 0.26 2.08 5.7 18.5 (540) @ 23.1oC 6.40gb b 0.26 2.08 5.7 20.4 (489) @ 22.8oC 6.40gbnf a 0.32 2.00 2.7 19.7 (508) @ 22.5oC -gbnf b 0.33 1.99 2.7 20.2 (495) @ 22.0oC -gbnf c 0.30 2.03 2.7 23.0 (435) @ 23.5oC 6.40gbnf d 0.32 2.00 2.7 100 (100) @ 23.1oC 6.40gbnc a 0.32 2.00 3.2 23.4 (428) @ 22.9oC 6.60gbnc b 0.32 2.00 3.2 101 (99) @ 23.3oC 6.60gbnc c 0.31 2.02 3.2 232 (43) @ 22.8oC 6.80sg a 0.22 2.20 15 145 (69) @ 22.2oC 6.64sg b 0.22 2.20 15 156 (64) @ 22.1oC 6.38sg c 0.14 2.33 15 122 (82) @ 21.7oC 7.46gt a 0.28 2.29 417 149 (67) @ 22.8oC 6.36gt b 0.29 2.26 417 141 (71) @ 21.5oC 6.46gt0 0.29 2.27 8 147 (68) @ 21.5oC 6.44gt25 0.24 2.36 180 154 (65) @ 22.7oC 6.52Table 4.1: Summary of sample data for tests conducted on glass beads and embankment flllmaterial. Soil samples are labelled according to material type (gb: glass beads, sg: sand andgravel gt: glacial till), variations on the gradation (nf: flnes removed, nc: coarse fractionremoved, #: % flnes) and the sample identifler (e.g. a). Absolute  uid conductivity (resis-tivity), temperature and pH characterize the saturating  uid properties and were measuredduring oscillatory  ow testing in the inflnite reservoir connected to the apparatus. Thesevalues do not necessarily re ect the properties of the pore  uid at ionic equilibrium withthe sample.97Chapter 4. Laboratory testing in well-graded soils4.4 ResultsTable 4.2 summarizes the experimental results for all glass bead and embankment soilsamples studied. The reported electrical properties ‰, C and L represent the saturatedsample following a period of soil- uid equilibration. Measurements of sample resistivity areabsolute values with respect to temperature, with all experiments conducted at a nominalambient temperature of 22:5oC. The quoted uncertainties in the measured properties re ectinstrument sensitivity.In the sections that follow we evaluate the in uence of sample density, sample gradationand saturating  uid conductivity on the electrical properties. Data collected for glass beadsamples are compared to theoretical estimates of C, ‰ and L, which are calculated using(4.14), (4.19) and (4.7), respectively. The average pore radius of each glass bead sample iscalculated using (4.16). The mean grain diameter d is approximated using d50, the medianparticle size for a given gradation. The formation factor is calculated from Archie’s lawF = n¡m, using the measured porosity for each sample and m = 1:3. To calculate Cusing (4.14), we assume constant  uid property values of " = 7:08 £ 10¡10 Fm¡1 and· = 8:9£10¡4 Pas. We calculate  efi using (4.15), where  f is the measured conductivityof the  uid used to saturate the samples in each experiment. Values of surface conductanceand zeta potential are taken from Gu and Li [2000] who reported ‚s = 2 £ 10¡7 S and‡ = ¡25 mV for a glass channel saturated with a 2:4 £ 10¡4 molL¡1 NaCl electrolyte,which approximates the dominant ions and nominal  uid resistivity of the natural  uidused in this study. These values lie within the range of those reported for Fontainebleausandstone, a rock containing 99% quartz, saturated with 10¡3 to 10¡4 M NaCl solutions[Jouniaux and Pozzi, 1995; Reppert and Morgan, 2003].4.4.1 In uence of densitySamples of glass beads, shell and core material were prepared at difierent densities toevaluate the in uence of this parameter on the electrical properties. The bulk density % ofa fully-saturated sample is deflned as% = Gs +e1+e %f; (4.22)98Chapter 4. Laboratory testing in well-graded soilswhere Gs is the speciflc gravity of the material, e = n=(1¡n) is the void ratio and %f is uid density.Three glass bead samples of the same gbnf gradation (gbnf a, gbnf b and gbnf c) weresaturated with  uid exhibiting a nominal resistivity of 500 ›m and prepared at densitiesranging from 1.99 to 2.03 g/cm3, as listed in Table 4.1. These samples exhibit porositiesthat lie within the maximum range of 0.29 to 0.34 measured for the material [ASTM,1993b, 1991b]. Figure 4.6 compares measured and calculated values of C, ‰ and L for eachsample.1.98 1.99 2 2.01 2.02 2.03 2.04−6−4−2 C  (mV/kPa)1.98 1.99 2 2.01 2.02 2.03 2.0405001000 ρ  (Ohm m)1.98 1.99 2 2.01 2.02 2.03 2.0400.511.52 x 10−4 L  (A/m2 )Bulk density (g/cm3)measured theoreticala)c)b)Figure 4.6: In uence of density on the electrical properties in glass bead samples: a) stream-ing potential coupling coe–cient; b) sample resistivity; c) cross-coupling conductivity. Mea-sureddatarepresentsamplesgbnf a, gbnf bandgbnf c. Opensymbolsrepresentcalculatedtheoretical behaviour using parameters described in text. Dashed line is a linear flt to thetheoretical data points.99Chapter 4. Laboratory testing in well-graded soilsAn increase in sample density might be expected to decrease the porosity and averagepore radius of a soil. Given the relation shown in (4.15), a smaller pore radius would causean increase in the efiective conductivity of the pore channels due to the larger in uenceof surface conductivity. The higher efiective conductivity would translate to a decrease inthe magnitude of both C and sample resistivity, according to (4.14) and (4.19). Figures4.6(a) and (b) show that both measured and predicted values of the streaming potentialcoupling coe–cient and resistivity exhibit a slight decrease in magnitude with increasingdensity. However, the relative variation is very small and within the range of error formeasured values of C such that no distinct trend may be inferred conclusively. Calculatedresistivity values are up to two times larger than measured results, which suggests that thetheoretical estimate may not adequately represent true conditions. This issue is discussedfurther in Section 4.4.3. Figure 4.6(c) shows a slight decrease in measured and predictedvalues of L. However, the net efiect of changing density on the cross-coupling conductivityis insigniflcant, and suggests an average measured value of L = 1:1§0:2£10¡4A=m2.Two samples of glacial till core material (gt a and gt b) were saturated with  uid ex-hibiting a nominal resistivity of 70 ›m and prepared at densities of 2.29 and 2.26 g/cm3,respectively. Figure 4.7 reveals that the change in density has no efiect on the measuredelectrical properties for the conductive core material. Three samples of sand and gravel shellmaterial (sg a, sg b and sg c) were also prepared for the purposes of studying the efiect ofchanging density. The density of sample sg b did not increase following preparation usingthe method described in Section 4.3.4. Consequently, sample sg c was prepared in layersand densifled by tamping the sample with a glass rod to achieve a porosity of 0.14. Figure4.7 shows a decrease in the magnitude of C, ‰ and L with an increase in density. How-ever, conclusive interpretation of these results is complicated by difierences in experimentalprocedure and the much longer timescale of the sg c experiment, as discussed in Section4.3.5.100Chapter 4. Laboratory testing in well-graded soils2.15 2.2 2.25 2.3 2.35 2.4−2−10 C  (mV/kPa)2.15 2.2 2.25 2.3 2.35 2.40200400 ρ  (Ohm m)2.15 2.2 2.25 2.3 2.35 2.40123 x 10−5 L  (A/m2 )Bulk density (g/cm3)shellcorec)a)b)Figure 4.7: In uence of density on the electrical properties of embankment soil samples: a)streamingpotentialcouplingcoe–cient; b)sampleresistivity; c)cross-couplingconductivity.Measured shell data represent samples sg a, sg b and sg c; measured core data representsamples gt a and gt b.101Chapter 4. Laboratory testing in well-graded soilsSample K ‰ C Lm/s ›m mV/kPa A/m2gb a 1:3£10¡3 552§8 ¡4:07§0:03 7:3(§0:1)£10¡5gb b 1:1£10¡3 443§7 ¡3:00§0:02 6:8(§0:1)£10¡5gbnf a - 422§6 ¡5:0§0:3 1:14(§0:09)£10¡4gbnf b 8:2£10¡3 541§8 ¡5:5§0:3 1:07(§0:07)£10¡4gbnf c 7:0£10¡3 480§7 ¡4:9§0:2 9:9(§0:6)£10¡5gbnf d 7:0£10¡3 312§5 ¡2:2§0:1 7:1(§0:4)£10¡5gbnc a 1:1£10¡3 502§8 ¡4:10§0:02 8:1(§0:2)£10¡5gbnc b 1:1£10¡3 278§4 ¡2:20§0:01 7:8(§0:2)£10¡5gbnc c 9:2£10¡4 144§2 ¡1:06§0:01 7:2(§0:1)£10¡5sg a 3:2£10¡3 296§4 ¡0:76§0:02 2:5(§0:1)£10¡5sg b 1:1£10¡3 297§4 ¡0:62§0:01 2:12(§0:08)£10¡5sg c 5:5£10¡5 225§3 ¡0:21§0:01 9:3(§0:1)£10¡6gt a - 96§1 ¡0:17§0:01 1:70(§0:07)£10¡5gt b 3:2£10¡7 94§1 ¡0:16§0:01 1:62(§0:03£10¡5gt0 3:4£10¡4 228§3 ¡0:49§0:01 2:10(§0:05)£10¡5gt25 - 133§2 ¡0:14§0:01 1:03(§0:06)£10¡5Table 4.2: Summary of experimental results for tests conducted on glass beads and embank-ment flll material. Soil samples are labelled according to material type (gb: glass beads,sg: sand and gravel gt: glacial till), variations on the gradation (nf: flnes removed, nc:coarse fraction removed, #: % flnes) and the sample identifler (e.g. a). Values of ‰, C andL characterize the sample following a period of ionic equilibration with the pore  uid. Alltesting was performed at a nominal ambient temperature of 22:5oC. Uncertainties in theelectrical properties were calculated based on instrument sensitivities.102Chapter 4. Laboratory testing in well-graded soils4.4.2 In uence of gradationGlass bead samples exhibiting difierent grain size distributions were studied to evaluatethe in uence of gradation on the electrical properties in a material of uniform mineralogy.Similarly, glacial till samples with varying flnes content were studied to examine the efiect ofgradation accompanied by a change in mineralogy. The change in gradation was quantifledusing the coe–cient of uniformity Cu, which is deflned asCu = d60d10; (4.23)where d60 and d10 are representative particle sizes for a given grain size distribution. Thesubscript represents the percentage of soil grains flner than the stated particle size. A largevalue of Cu denotes a sample with a wide range of grain sizes, whereas a smaller valuedenotes a more uniformly graded sample.Two series of glass bead samples were studied for the purposes of evaluating the efiectof gradation. Samples gb a, gb b, gbnf a and gbnc a were saturated with  uid exhibitinga nominal resistivity of 500 ›m. Samples gbnf d and gbnc b were saturated with  uidexhibiting a nominal resistivity of 100 ›m. Figure 4.8 presents the measured and theoreticalvalues of C, ‰ and L for each set of samples. Although gbnf and gbnc samples containdramatically difierent amounts of flnes, as shown in Figure 4.3, the similar shape of thegrain size curves results in similar values of Cu, as listed in Table 4.1.The porosity can be expected to decrease with a wider range of grain sizes, as evidencedby the lower measured porosity of gb samples as compared to gbnf and gbnc samples shownin Table 4.1. Consequently, with larger values of Cu we might expect a decrease in poreradius and an increased in uence of surface conductivity on the efiective conductivity. Anincrease in  efi would decrease the amplitude of C and ‰. Figure 4.8(a) shows a slightdecrease in measured values of C in samples saturated with 500 ›m  uid, with no cleartrend visible in the theoretical values. Figure 4.8(b) shows no distinct trend in measured orpredicted resistivity values for these samples. However, calculated values of ‰ overestimatemeasured values in gbnf (Cu = 2:7) and gb (Cu = 5:7) samples saturated with 500 ›m uid. This is discussed further in Section 4.4.3. Samples saturated with 100 ›m  uiddisplay no signiflcant variation in measured or theoretical values of C or ‰ with gradation.103Chapter 4. Laboratory testing in well-graded soilsFigure 4.8(c) shows that the efiect of changing gradation on L is insigniflcant, and suggestsan average measured value of L = 7 § 1 £ 10¡5A=m2 independent of the saturating  uidresistivity. The exception to this is the gbnf a datum at Cu = 2:7, which exhibits a largervalue of L in keeping with the other gbnf samples shown in Figure 4.6.Figure 4.9 shows the in uence of gradation in samples of glacial till core material (gt a,gt25 and gt0). In these samples, an increase in Cu represents an increase in flnes contentas well as a change in mineralogy. The presence of flne-grained silt and clay material servesto increase the surface conductivity, and leads to a small decrease in the amplitudes of Cand ‰ as shown in Figures 4.9(a) and (b).104Chapter 4. Laboratory testing in well-graded soils2.5 3 3.5 4 4.5 5 5.5 6−6−4−20 C  (mV/kPa)2.5 3 3.5 4 4.5 5 5.5 605001000 ρ  (Ohm m)2.5 3 3.5 4 4.5 5 5.5 600.511.5x 10−4 L  (A/m2 )Coefficient of uniformity, Cu500 Ωm meas. theo. 100 Ω m meas. theo.a)c)b)Figure 4.8: In uence of gradation on the electrical properties in glass bead samples: a)streaming potential coupling coe–cient; b) sample resistivity; c) cross-coupling conductiv-ity. Measured data represent samples gb a, gb b, gbnf a and gbnc a saturated with  uidexhibiting a nominal  uid resistivity of 500 ›m; and samples gbnf d and gbnc b saturatedwith  uid with a nominal resistivity of 100 ›m. Open symbols represent calculated theo-retical behaviour using parameters described in text. Dashed lines represent a linear flt tothe theoretical data points.105Chapter 4. Laboratory testing in well-graded soils0 50 100 150 200 250 300 350 400 450−1−0.50 C  (mV/kPa)0 50 100 150 200 250 300 350 400 4500100200300 ρ  (Ohm m)0 50 100 150 200 250 300 350 400 4500123 x 10−5 L  (A/m2 )Coefficient of uniformity, Cuc)b)a)Figure 4.9: In uence of gradation on the electrical properties of embankment soil sam-ples: a) streaming potential coupling coe–cient; b) sample resistivity; c) cross-couplingconductivity. Measured data represent samples gt a, gt25 and gt0.106Chapter 4. Laboratory testing in well-graded soils4.4.3 In uence of  uid conductivityThe properties of the double layer are in uenced by the concentration and type of ionsin solution, which exhibit difierent adsorption properties and mobilities, as illustrated inMorgan et al. [1989]. Since natural waters typically contain a range of dissolved ions, weuse  uid resistivity as an indicator of the ionic strength of the saturating  uids used in thisstudy.Three glass bead samples of the same gradation were saturated with  uids exhibitingdifierent resistivities for the purposes of evaluating their efiect on the electrical properties.Measured and theoretical values of C, ‰ and L are shown in Figure 4.10 for samples gbnc a,gbnc bandgbnc c. Asexpectedfromtherelationsshownin(4.13)and(4.18), bothmeasuredand predicted data show an increase in the magnitude of C and ‰ with an increase in  uidresistivity. In accordance with (4.20), Figure 4.10(c) shows that measured and predictedvalues of L show very little variation with  uid resistivity, and suggest an average measuredvalue of L = 7:7§0:5£10¡5A=m2 for these three samples, which is within range of thatreported for the data shown in Figure 4.8(c).Measured and predicted values of resistivity compare well over the entire range of  uidresistivity for the gbnc samples shown in Figure 4.10(b). However, a discrepancy betweencalculated and measured sample resistivities in gbnf and gb samples saturated with 500 ›m uid was noted in Figures 4.6(b) and 4.8(b). Figure 4.11 compares the measured sampleresistivity to that calculated using (4.19) for all glass bead samples, and clearly illustratesthis discrepancy. According to (4.19), breakdown of the theory at high  uid resistivitiesmay be caused by too small an estimate of  efi or too large an estimate of F.Efiective conductivity is a function of both  uid and surface conductivity in the porespace. Surface conductivity is calculated using estimates of surface conductance and averagepore radius. The calculation of pore radius using a single representative grain diameter in(4.16) provides a very simple approximation for a well-graded material that exhibits a largerange of particle sizes. However, the predicted surface conductivity was found to be ofconsiderable magnitude in resistive samples. Surface conductivity was found to be largestin samples with the most flnes (gbnc), smallest in the samples with the least flnes (gbnf),and larger than the saturating  uid conductivity in all cases.107Chapter 4. Laboratory testing in well-graded soils0 100 200 300 400 500−4−20 C  (mV/kPa)0 200 400 5000200400600 ρ  (Ohm m)0 100 200 300 400 50000.51 x 10−4 L  (A/m2 )Saturating fluid resistivity (Ohm m)measured theoreticalc)b)a)Figure 4.10: In uence of  uid resistivity on the electrical properties in glass bead samples:a) streaming potential coupling coe–cient; b) sample resistivity; c) cross-coupling conduc-tivity. Measured data represent samples gbnc a, gbnc b and gbnc c. Open symbols representcalculated theoretical behaviour using parameters described in text. Dashed line is a linearflt to the theoretical data points.As discussed in Section 4.3.5, very large changes in  uid conductivity of up to 70%were observed for samples gb a and sg c during the course of sample equilibration. Thissuggests that the measured conductivity of the saturating  uid listed in Table 4.1 may notrepresent the actual pore  uid. Consequently, calculated values of  efi using (4.15) willunderestimate the true conductivity of the pore channel and result in an overestimate ofsample resistivity. Figure 4.11 illustrates the efiect of using measured values of pore  uidconductivity to calculate  efi. Fluid conductivity measurements were made in the uppercell reservoir following testing of samples gb a, gb b, gbnf d, gbnc a and gbnc c. Resistive108Chapter 4. Laboratory testing in well-graded soilssamples gb a and gb b show the largest improvement in predicted resistivity, as indicatedby the arrows in the flgure.0 100 200 300 400 500 60001002003004005006007008009001000Saturating fluid resistivity  ρf  (Ohm m)Sample resistivity  ρ  (Ohm m)measuredtheoreticaltheoretical (ρf  post−test)Figure 4.11: Comparison of measured and predicted sample resistivity versus measuredsaturating  uid resistivity for all samples of glass beads. Theoretical behaviour is calculatedusing Equation (4.19), where formation factor is estimated from F = n¡m and  efi iscalculated using Equation (4.15). Open circles indicate  efi values calculated using thesaturating  uid conductivity; x symbols indicate  efi values calculated using measured pore uid conductivity upon completion of the experiment, where available. Arrows denotechange in the predicted sample resistivity for samples gb a and gb b.Archie’s relation F = n¡m was used to approximate the formation factor in glass beadsamples. However, this relation can not capture the in uence of ionic equilibration onsample resistivity. Figure 4.12 compares Archie’s law to values of formation factor calculatedusing measured resistivities. Measured resistivity data acquired upon  ushing the sample109Chapter 4. Laboratory testing in well-graded soilswith the saturating  uid give formation factors on par with those predicted using Archie’slaw. However, formation factors calculated from the measured sample resistivity followingequilibration are roughly four times smaller at high  uid resistivities.0 100 200 300 400 500 6000123456Saturating fluid resistivity  ρf  (Ohm m)Formation factor, FF = n−mF =  ρ/ρf  (at first contact)F =  ρ/ρf  (at equilibrium)Figure 4.12: Comparison of measured and estimated values of formation factor versus mea-sured saturating  uid resistivity for all samples of glass beads. Theoretical behaviour (+symbols) is calculated using (4.18). Open circles indicate F values calculated from measuredresistivity data acquired upon initial contact between the soil and  uid (not at equilibrium).Open triangles indicate F values calculated from the measured sample resistivity followingequilibration and the saturating  uid resistivity.110Chapter 4. Laboratory testing in well-graded soils4.5 DiscussionThe experimental results conflrm that both C and  are strongly dependent on  uid conduc-tivity, but that L is relatively independent, as suggested by the equations listed in Section4.2. Consequently, it is very important to measure C and  under the same conditions toproperly characterize L. Figure 4.13 illustrates the relative change in C and L with sampleresistivity measured in glass bead samples at difierent times during the sample equilibrationprocess. While C can vary over an order of magnitude, L exhibits a much smaller range ofvariation and appears to be a more constant property of the material.0 500 1000 1500 2000 2500−25−20−15−10−50Sample resistivity  ρ  (Ohm m) C  (mV/kPa)00.511.52x 10−4 L  (A/m2 ) L CFigure 4.13: Comparison of measured streaming potential coupling coe–cient and streamingcurrent cross-coupling conductivity with sample resistivity for samples of glass beads. Datarepresent difierent stages of sample equilibrium.Table 4.3 summarizes the measured electrical properties of embankment soil from thepresent study along with measured data from Friborg [1996], Gray and Mitchell [1967],Pengra et al. [1999] and Morgan et al. [1989] for comparison. These data were chosen for111Chapter 4. Laboratory testing in well-graded soilsthe similarity of materials studied and, in cases where sample resistivity was not measureddirectly, for the suitability of estimating sample resistivity using Archie’s law due to thehigh conductivity of the saturating  uid.Immediately evident in Table 4.3 is that the cross-coupling conductivity varies withinonly one order of magnitude. The cross-coupling conductivity of 1:7£10¡5A=m2 measuredfor glacial till in the present study compares very well with data collected by Friborg [1996],who performed unidirectional  ow streaming potential measurements and separate resis-tivity experiments on till samples. The sand and gravel was found to exhibit an averagecross-coupling conductivity of 2:3£10¡5A=m2. Although no published data were found toenable a direct comparison, this value is within the range of cross-coupling conductivitiesmeasured for rock types listed in the table, which may exhibit similar mineralogy. The sam-ple of glacial till with the flne fraction removed (gt0) was found to exhibit a cross-couplingconductivity and resistivity similar to that measured for the sand and gravel shell material.This has implications for the study of internal erosion and flnes migration in embankmentdams. Although no distinct correlation between L and gradation was observed in the glacialtill samples studied, core material that has lost the bulk of the flne fraction may exhibit Land ‰ values similar to those of shell material at a given site.112Chapter 4. Laboratory testing in well-graded soilsMaterial Source C ‰f ‰ LmV/kPa ›m ›m A/m2well-graded till present study (gt a, gt b) -0.16 68 95 1:7£10¡5(Cu = 417)sandy till Friborg [1996] -0.66 100 840 7:7£10¡6(Cu = 2.3)well-graded till Friborg [1996] -0.45 100 340 1:3£10¡5(Cu = 22.9)well-graded silty Friborg [1996] -0.71 to 100 541 1.3 totill (Cu = 7.3) -0.92 1.7 £10¡5silty clay Gray and Mitchell [1967] y -0.056 to - 333 to 1.5 to-0.61 390 3.2 £10¡5till with no flnes present study (gt0) -0.49 68 228 2:1£10¡5(Cu = 8)sand and gravel present study (sg a, sg b) -0.69 66 296 2:3£10¡5sandstone Pengra et al. [1999] z -0.00549 to 0.5 - 0.89 to-0.00948 1.5 £10¡5limestone Pengra et al. [1999] z -0.00267 to 0.5 - 2.4 to-0.00483 2.9 £10¡6granite Morgan et al. [1989] z -0.198 to 8.5 - 5.8 to-0.217 6.4 £10¡5Table 4.3: Comparison of measured results with published data. Measured results for thecurrent study represent average values derived from sample data as noted. yData derivedfrom electro-osmosis experiments. zSample resistivity was calculated using Archie’s law forthe purposes of calculating L.113Chapter 4. Laboratory testing in well-graded soils4.6 ConclusionA series of streaming potential and resistivity laboratory experiments were conducted tomeasure the streaming potential coupling coe–cient C and resistivity ‰ in order to derivethe streaming current cross-coupling conductivity L for well-graded soil samples. The mainobjectives of the study were to assess the in uence of sample properties on the cross-couplingconductivity and to characterize this property for two representative embankment dam flllmaterials.The oscillatory  ow method was shown to be a valid test method for unconsolidated soilsamples and was found to be the most e–cient technique to acquire streaming potentialdata and achieve sample equilibrium.The results of experiments performed on samples of glass beads suggest that changesin density, gradation and saturating  uid conductivity do not signiflcantly in uence thestreaming current cross-coupling coe–cient. The process of ionic equilibration between thesoil and saturating  uid was found to signiflcantly afiect the measured amplitude of C and‰, and illustrates the importance of measuring these properties under the same sampleconditions to properly characterize L for a given material. The ionic exchange observed insamples saturated with resistive  uids (» 500›m) precluded reliable theoretical estimates ofsample resistivity even when surface conductivity efiects were accounted for. This suggeststhat measurements of pore  uid conductivity are required to estimate sample resistivityusing an Archie’s law-type formulation in resistive environments.Measured values of the streaming current cross-coupling conductivity in glacial till andsand and gravel samples compared favorably with published data for similar materials. Thecompilation of measured results show that L varies within one order of magnitude for typicalgeologic materials, which is much smaller than the typical range of variation of hydraulicconductivity and electrical resistivity. This suggests that in practice it may not be necessaryto measure L with the same rigor as ‰, but that estimates may be su–cient to characterizethe subsurface in forward modelling and inversion investigations of streaming potential.114Chapter 4. Laboratory testing in well-graded soils4.7 ReferencesAdamson, A. W., Physical Chemistry of Surfaces, John Wiley and Sons, New York, 1990.Ahmad, M. U., A laboratory study of streaming potentials, Geophysical Prospecting, 12,49{64, 1964.Archie, G. E., The electrical resistivity log as an aid in determining some reservoir char-acteristics, Transactions of the Society of Petroleum Engineers of the American Instituteof Mining, Metallurgical and Petroleum Engineers (AIME), 146, 54{67, 1942.ASTM, D2434-68: Standard test method for permeability of granular soils (constant head),in Annual Book of ASTM Standards, vol. 4, American Society of Testing and Materials,1991a.ASTM, D4254-91: Standard test methods for minimum index density and unit weightof soils and calculation of relative density, in Annual Book of ASTM Standards, vol. 4,American Society of Testing and Materials, 1991b.ASTM, D2217-85: Standard practice for wet preparation of soil samples for particle-sizeanalysis and determination of soil constants, in Annual Book of ASTM Standards, vol. 4,American Society of Testing and Materials, 1993a.ASTM, D4253-93: Standard test methods for maximum index density and unit weight ofsoils using a vibratory table, in Annual Book of ASTM Standards, vol. 4, American Societyof Testing and Materials, 1993b.BC Hydro, Tech. rep., British Columbia Hydro and Power Authority, Internal report no.PSE187, Burnaby, 2002.Blake, E., and G. Clarke, Subglacial electrical phenomena, Journal of Geophysical Re-search, 104(B4), 7481{7495, 1999.Chandler, R., Transient streaming potential measurements on  uid-saturated porous struc-tures: An experimental veriflcation of Biot’s slow wave in the quasi-static limit, Journalof the Acoustical Society of America, 70(1), 116{121, 1981.115Chapter 4. Laboratory testing in well-graded soilsCorwin, R. F., and D. B. Hoover, The self-potential method in geothermal exploration,Geophysics, 44(2), 226{245, 1979.Corwin, R. F., and H. F. Morrison, Self-potential variations preceding earthquakes incentral California, Geophysical Research Letters, 4(4), 171{174, 1977.de Groot, S. R., Thermodynamics of irreversible processes, Selected Topics in ModernPhysics, vol. 3, North Holland Publishing Company, Amsterdam, 1951.Friborg, J., Experimental and theoretical investigations into the streaming potentialphenomenon with special reference to applications in glaciated terrain, Ph.D. thesis,Lule”a University of Technology, Sweden, 1996.Gray, D. H., and J. K. Mitchell, Fundamental aspects of electro-osmosis in soils, Tech. rep.,Soil Mechanics and Bituminous Materials Research Laboratory, University of California,Berkeley, 1967.Gu, Y., and D. Li, The ‡-potential of glass surface in contact with aqueous solutions,Journal of Colloid and Interface Science, 226, 328{339, 2000.Gu¶eguen, Y., and V. Palciauskas, Introduction to the Physics of Rocks, Princeton Univer-sity Press, Princeton, 1994.Guichet, X., L. Jouniaux, and J. P. Pozzi, Streaming potential of a sand columnin partial saturation conditions, Journal of Geophysical Research, 108(B3), 2141, doi:10.1029/2001JB001517, 2003.Hunter, R. J., Zeta Potential in Colloid Science: Principles and Applications, AcademicPress, Sydney, 1981.Ishido, T., and H. Mizutani, Experimental and theoretical basis of electrokinetic phe-nomena in rock-water systems and its application to geophysics, Journal of GeophysicalResearch, 86(B3), 1763{1775, 1981.Jouniaux, L., andJ.P.Pozzi, Streamingpotentialandpermeabilityofsaturatedsandstonesunder triaxial stress: Consequences for electrotelluric anomalies prior to earthquakes, Jour-nal of Geophysical Research, 100(B6), 10,197{10,209, 1995.116Chapter 4. Laboratory testing in well-graded soilsKeller, G. V., and F. C. Frischknecht, Electrical methods in geophysical prospecting, inInternational series of monographs in electromagnetic waves, vol. 10, edited by A. L.Cullen, V. A. Fock, and J. R. Wait, Pergamon Press, New York, 1966.Kuerbis, R., and Y. P. Vaid, Sand sample preparation: The slurry deposition method,Soils and Foundations, 28(4), 107{118, 1988.Kulessa, B., B. Hubbard, and G. H. Brown, Cross-coupled  ow modeling of coincidentstreaming and electrochemical potentials and application to sub-glacial self-potential data,Journal of Geophysical Research, 108(B8), 2381, doi:10.1029/2001JB001167, 2003.Lorne, B., F. Perrier, and J. P. Avouac, Streaming potential measurements 1. Propertiesof the electrical double layer from crushed rock samples, Journal of Geophysical Research,104(B8), 17,857{17,877, 1999.Mizutani, H., T. Ishido, T. Yokokura, and S. Ohnishi, Electrokinetic phenomena associatedwith earthquakes, Geophysical Research Letters, 3, 365{368, 1976.Morgan, F. D., Fundamentals of streaming potentials in geophysics: Laboratory methods,in Detection of Subsurface Flow Phenomena, Lecture Notes in Earth Sciences, vol. 27,edited by G. P. Merkler, H. Militzer, H. H˜otzl, H. Armbruster, and J. Brauns, pp. 133{144, Springer-Verlag, Berlin, 1989.Morgan, F. D., E. R. Williams, and T. R. Madden, Streaming potential properties of West-erly granite with applications, Journal of Geophysical Research, 94(B9), 12,449{12,461,1989.Ogilvy, A. A., M. A. Ayed, and V. A. Bogoslovsky, Geophysical studies of water leakagesfrom reservoirs, Geophysical Prospecting, 17, 36{62, 1969.Onsager, L., Reciprocal relations in irreversible processes, I, Physical Review, 37, 405{426,1931.Overbeek, J. T. G., Electrokinetic phenomena, in Colloid Science, Irreversible Systems,vol. 1, edited by H. R. Kruyt, Elsevier Publishing Company, Amsterdam, 1952.117Chapter 4. Laboratory testing in well-graded soilsPackard, R. G., Streaming potentialsacross glass capillaries for sinusoidalpressure, Journalof Chemical Physics, 21(2), 303{307, 1953.Pengra, D. B., S. X. Li, and P. Wong, Determination of rock properties by low-frequencyAC electrokinetics, Journal of Geophysical Research, 104(B12), 29,485{29,508, 1999.Reppert, P. M., and F. D. Morgan, Frequency-dependent streaming potentials, Journal ofColloid and Interface Science, 234, 194{203, 2001a.Reppert, P. M., and F. D. Morgan, Streaming potential collection and data processingtechniques, Journal of Colloid and Interface Science, 233, 348{355, 2001b.Reppert, P. M., and F. D. Morgan, Temperature-dependent streaming potentials: 2. Labo-ratory, Journal of Geophysical Research, 108(B11), 2547, doi:10.1029/2002JB001755, 2003.Revil, A., and L. M. Cathles III, Permeability of shaly sands, Water Resources Research,35(3), 651{662, 1999.Revil, A., and A. Cerepi, Streaming potentials in two-phase  ow conditions, GeophysicalResearch Letters, 31, L11605, doi:10.1029/2004GL020140, 2004.Revil, A., and P. W. J. Glover, Nature of surface electrical conductivity in natural sands,sandstones and clays, Geophysical Research Letters, 25(5), 691{694, 1998.Revil, A., P. A. Pezard, and P. W. J. Glover, Streaming potential in porous media: 1.Theory of the zeta potential, Journal of Geophysical Research, 104(B9), 20,021{20,031,1999a.Revil, A., H. Schwaeger, L. M. Cathles III, and P. D. Manhardt, Streaming potential inporous media: 2. Theory and application to geothermal systems, Journal of GeophysicalResearch, 104(B9), 20,033{20,048, 1999b.Revil, A., D. Hermitte, M. Voltz, R. Moussa, J. G. Lacas, G. Bourrie, and F. Trolard,Self-potential signals associated with variations of the hydraulic head during an inflltrationexperiment, Geophysical Research Letters, 29(7), doi:10.1029/2001GL014294, 2002.118Chapter 4. Laboratory testing in well-graded soilsRevil, A., V. Naudet, and J. D. Meunier, The hydroelectric problem of porous rocks:Inversion of the position of the water table from self-potential data, Geophysical JournalInternational, 159, 435{444, 2004.Revil, A., L. Cary, Q. Fan, A. Finizola, and F. Trolard, Self-potential signals associatedwith preferential ground water  ow pathways in a buried paleo-channel, Geophysical Re-search Letters, 32, L07401, 2005.Sears, A. R., and J. N. Groves, The use of oscillating laminar  ow streaming potentialmeasurements to determine the zeta potential of a capillary surface, Journal of Colloidand Interface Science, 65(3), 479{482, 1978.Shefier, M. R., Investigation of geophysical methods for assessing seepage and internalerosion in embankment dams: Laboratory testing of the streaming potential phenomenonin soils, Tech. rep., Canadian Electricity Association Technologies Inc. (CEATI), ReportT992700-0205B/2, Montreal, 2005.Shefier, M. R., and D. W. Oldenburg, Three-dimensional forward modelling of streamingpotential, Geophysical Journal International, 169, 839{848, doi:10.1111/j.1365-246X.2007.03397.x, 2007.Shefier, M. R., P. M. Reppert, and J. A. Howie, A laboratory apparatus for streamingpotential and resistivity measurements on soil samples, Review of Scientiflc Instruments,78, 094502, doi:10.1063/1.2782710, 2007.Suski, B., A. Revil, K. Titov, P. Konosavsky, M. Voltz, C. Dag es, and O. Huttel, Mon-itoring of an inflltration experiment using the self-potential method, Water ResourcesResearch, 42, W08418, doi:10.1029/2005WR004840, 2006.Titov, K., A. Revil, P. Konosavsky, S. Straface, and S. Troisi, Numerical modelling ofself-potential signals associated with a pumping test experiment, Geophysical Journal In-ternational, 162, 641{650, 2005.119Chapter 4. Laboratory testing in well-graded soilsWurmstich, B., F. D. Morgan, G. P. Merkler, and R. L. Lytton, Finite-element modeling ofstreaming potentials due to seepage: Study of a dam, Society of Exploration GeophysicistsTechnical Program Expanded Abstracts, 10, 542{544, 1991.120Chapter 5Evaluating the sensitivity of theself-potential method to detectinternal erosion in the core of anembankment 45.1 IntroductionThe internal erosion of flne-grained material due to seepage forces can compromise thestability of an earthflll dam and cause its ultimate failure. Monitoring of dam perfor-mance has become of critical importance, particularly as structures age and design meth-ods evolve. Conventional monitoring of the hydraulic regime using piezometers and weirsprovides sparse sampling, and these methods may not be su–cient to detect the onset ofinternal erosion. Consequently, a comprehensive investigation tool is needed to complementthese methods.The self-potential (SP) method has been used to successfully delineate anomalous zonesthat correspond with areas of preferential seepage  ow in embankment dams [Ogilvy et al.,1969; Bogoslovsky and Ogilvy, 1970; Black and Corwin, 1984; Butler et al., 1989]. SP dataconsist of voltage difierence measurements, which are typically collected over the surface ofan embankment and in some cases in the impounded reservoir [Ogilvy et al., 1969; Corwin,1990]. Uniform seepage through an embankment gives rise to a background SP response,4A version of this chapter will be submitted for publication. Shefier, M.R. and Oldenburg, D.W. (2007)Evaluating the sensitivity of the self-potential method to detect internal erosion in the core of an embank-ment.121Chapter 5. Sensitivity of the self-potential method to detect internal erosionwhich varies in intensity with reservoir level. Areas of preferential seepage  ow manifestas deviations from this background response. These deviations, or SP anomalies, may beisolated by difierencing repeat SP data sets acquired from single surveys or a continuousmonitoring network of electrodes.The SP method responds to the phenomenon of streaming potential: electrical currentsgenerated by  uid  ow through porous media. This electrokinetic process is the complemen-tary phenomenon to electro-osmosis and is explained using coupled  ow theory [de Groot,1951; Overbeek, 1952]. The streaming potential phenomenon can dominate the SP responsein systems where the hydraulic gradient is the driving force (i.e. no externally-imposedelectrical current sources exist, and temperature and ionic gradients are negligible).The current state of practice is to interpret SP data using qualitative and analyticaltechniques to estimate the location and depth of anomalous sources [Panthulu et al., 2001;Rozycki et al., 2006]. Although these methods enable a spatial interpretation of electricalcurrent sources that explain the geophysical data, they do not give any information aboutthe hydraulic nature of these sources.Numerical modelling techniques based on coupled  ow theory enable a more sophisti-cated approach to data interpretation, where the SP response can be related to hydraulicparameters. We have developed a three-dimensional forward modelling code that predictsthe SP response to laminar  uid  ow in the subsurface [Shefier and Oldenburg, 2007]. Thealgorithm is well-suited to the study of embankment seepage problems, which are three-dimensional in nature due to the irregular topography of the dam and foundation.The present study uses the numerical code to evaluate the SP response to preferentialseepage caused by pipe and construction layer defects within the core of a synthetic modelof a fleld-scale embankment. Section 5.2 provides a description of the streaming potentialphenomenon, and Section 5.3 presents the forward modelling methodology. The geometryof the 2-zone synthetic embankment and 24 defect conflgurations are described in Section5.4.1. Sections 5.4.2 and 5.4.3 give details on the seepage analysis and SP forward modelling.Sections 5.4.4 and 5.4.5 present the residual SP distributions for each defect case, whichare calculated by subtracting the background SP response resulting from seepage throughthe intact embankment. The in uence of the electrical resistivity distribution on the SP122Chapter 5. Sensitivity of the self-potential method to detect internal erosionresponse is discussed in Section 5.4.6. Finally, Section 5.5 discusses practical detectionlimits for the predicted SP anomalies, and Section 5.6 compares the predicted geophysicalresponse to the predicted hydraulic response to evaluate the sensitivity of the method tophysical changes within the embankment.5.2 The streaming potential phenomenonRelative movement between the electrically charged surface of a solid particle and freeions in a saturating solution describe difierent electrokinetic processes, including electro-osmosis and streaming potential. Electro-osmosis is the movement of  uid through a soilmatrix in an applied electric fleld, and has been used in engineering applications to increasethe strength of clays and flne-grained soils [Casagrande, 1983]. Streaming potential is thecomplementary phenomenon in which an imposed hydraulic gradient generates an electricfleld.Figure 5.1 illustrates a conceptual model of streaming potential in the pore space ofa soil. The grain surfaces exhibit a net electrical charge, the magnitude and polarity ofwhich is controlled by surface chemistry. Most geologic materials exhibit a negative surfacecharge when saturated with natural waters displaying typical pH and ion concentrations.An electrical double layer forms at the solid- uid interface, which consists of an adsorbedlayer of tightly bound positive ions and a more loosely bound difiuse outer layer [Adamson,1990]. Under static conditions, the saturated medium is electrically neutral with adsorbedions from the  uid completely balancing the negative surface charge of the solid particles.The onset of  uid  ow pulls positive ions from the difiuse layer in the direction of  ow,which generates a streaming current JS as illustrated in Figure 5.1. The streaming currentmay be described mathematically using a linear  ow law [de Groot, 1951]:JS = ¡Lrh; (5.1)whereL [Am¡2 ] is the streaming current cross-coupling conductivity andh[m] is hydraulichead. A charge imbalance results from the movement of ions in the direction of  ow, whichinduces an opposing conduction current JC that is described using Ohm’s law:JC = ¡ r`; (5.2)123Chapter 5. Sensitivity of the self-potential method to detect internal erosionwhere  [Sm¡1 ] is the electrical conductivity of the saturated porous medium and ` [V] isthe electrical potential. If no signiflcant temperature or ionic gradients exist in the system,the total current J [Am¡2 ] is described using coupled  ow theory as the sum of streamingand conduction currents:J = JC +JS: (5.3)------ ----- -----++++++++++++++++++++++FluidflowJcSOIL GRAINSOIL GRAIN++JsFigure 5.1: Conceptual model of streaming potential, showing streaming current JS andconduction current JC  ow paths in the pore space of a saturated soil.The electrical response to a given hydraulic head distribution is considered steady-state,regardless of the nature of the hydraulic system due to the large difierence in characteristictimes. Consequently, in the absence of any imposed sources of current, the streamingpotential mechanism becomes the sole source for conduction current  ow:r¢ JC = ¡r¢ JS; (5.4)which can be expressed asr¢  r` = ¡r¢ Lrh: (5.5)Although the streaming current is strictly limited to the saturated pore channels, theconduction current permeates the entire medium and is quantifled through measurements of124Chapter 5. Sensitivity of the self-potential method to detect internal erosion` using the self-potential method. The cross-coupling conductivity is deflned as L = ¡C  ,where C is the streaming potential coupling coe–cient. Parameters C and L are typicallycharacterized through laboratory measurements.5.3 Forward modelling of streaming potential5.3.1 MethodologyA seepage analysis is required to determine the distribution of hydraulic head prior tosolving (5.5)fortheself-potentialdistribution. Forwardmodellingofthestreamingpotentialphenomenon is performed using two independent algorithms to solve the hydraulic andelectrical  ow problems. The study region is divided into a discrete rectangular mesh,across which distinct material property units are deflned based on material type. Hydraulicconductivity (K) values are assigned accordingly to each cell in the mesh. A seepage analysisis performed using established software to facilitate the use of mixed boundary conditionsand model calibration. The 3-D flnite difierence code MODFLOW, developed by the USGeological Survey [Harbaugh et al., 2000], is used for conflned or saturated  ow problemsand MODFLOW-Surfact [HydroGeoLogic Inc., 1996] is used for variably saturated  owproblems. Mesh construction, assignment of boundary conditions and the delineation ofconductivity units are facilitated with the use of the Visual MODFLOW graphical userinterface [Waterloo Hydrogeologic Inc., 2004].The self-potential distribution is determined using a 3-D forward modelling algorithmdescribed in detail by Shefier and Oldenburg [2007], which is independent of the softwareused to perform the seepage analysis. The algorithm operates on the same mesh deflned inthe seepage analysis and requires as input a distribution of hydraulic head and a prescribeddistribution of electrical properties, namely streaming current cross-coupling conductivityL, and electrical conductivity  . A discrete form of Equation (5.5) is solved using themethod of flnite volumes, where h, `, L, and  are deflned at cell centres.To solve (5.5), a zero  ux boundary is prescribed for conduction current  ow at theouter edge of the mesh. This requires that the physical extents of the mesh be expandedwith the use of padding cells of prescribed conductivity, in order to remove the in uence125Chapter 5. Sensitivity of the self-potential method to detect internal erosionof the boundary condition on the solution within the study region. The forward modellingproblem is solved using a preconditioned biconjugate gradient stabilized method.5.3.2 Assessing the in uence of unsaturated  owA variably saturated  ow approach is typically required to solve embankment dam seepageproblems using the method of flnite difierences due to the steeply-dipping phreatic surfacethat separates saturated and unsaturated zones. In order to calculate streaming potential,electrical properties must be prescribed for all soil zones in the model. In a variably satu-rated  ow problem, these properties must be assigned as a function of  uid saturation. Theposition of the phreatic surface is determined from the hydraulic head solution and is usedto distinguish between saturated and unsaturated zones.Figure 5.2 illustrates the distribution of electrical properties in a homogeneous embank-ment subject to  ow from a reservoir. Contours of hydraulic head are shown as solid linesbelow the phreatic surface and as dashed lines within the unsaturated zone. Electrical con-ductivity is expressed in terms of resistivity ‰ = 1= [›m], which is the typical parameterused in characterizing geologic systems.L=0=c114fc114L=0=c114airc114L=L=c114 satsatc114L=Lunsath=zL=L =c114 bb c114z=c114 unsatc114Figure 5.2: Illustration of the distribution of electrical properties in a homogeneous em-bankment subject to seepage from a reservoir. Hydraulic head contours are indicated in theflgure, with dashed lines representing unsaturated head values. Subscripts sat and unsatrespectively refer to the saturated and unsaturated zones. Subscript b refers to the founda-tion material or bedrock. The value ‰f represents the electrical resistivity of the reservoirwater.Electrical conduction current  ow permeates the entire porous medium and ‰ is assigned126Chapter 5. Sensitivity of the self-potential method to detect internal erosiona value based on saturation conditions. Streaming current  ow may be handled using asaturated or variably saturated approach. In a saturated approach, we consider that themajority of  uid  ow and consequently streaming current  ow is limited to the saturatedzone and assign Lunsat = 0. In a variably saturated approach, the efiect of streaming current ow in the unsaturated zone is included and we must assign a value for Lunsat that variesas a function of saturation. Limited data exist on the behaviour of the streaming potentialcoupling coe–cient C in partially saturated conditions, with con icting reports of a decreaseGuichet et al. [2003]; Moore et al. [2004]; Revil and Cerepi [2004] or increase [Darnet andMarquis, 2004; Morgan et al., 1989] in the magnitude of C with decreasing  uid saturation.However, it is generally accepted that the streaming potential coupling coe–cient reaches anull value at some minimum critical saturation level where the pore  uid becomes immobile[Revil and Cerepi, 2004; Revil et al., 1999] and that electrical resistivity increases withdecreasing  uid saturation. The net efiect is to force the condition Lunsat < Lsat.A saturated  ow approach to modelling streaming current  ow is warranted in a typicalembankment dam seepage problem due to the nature of the  ow regime. The impoundedreservoir establishes a large hydraulic gradient that governs seepage through the structure.The in uence of the unsaturated  ow component of this seepage is dictated by the sizeof the capillary fringe above the phreatic surface. In free-draining sand and gravel shellor fllter materials, saturation levels decay quickly to residual values and unsaturated  owis restricted to a relatively narrow zone. In the flner-grained core, a signiflcant degree ofsaturation exists above the phreatic surface. However, the contrast in pore pressure betweenthe unsaturated core and downstream shell prevents signiflcant  ow from occurring in thiszone.Anothercomponentofunsaturated owisgravity-driven uid owfromsurfacerecharge.The self-potential response to percolation  ow is superimposed on the response to seepagethrough the embankment. However, the magnitude of the SP response to this vertical  owcomponent is independent of and typically small relative to the response to seepage fromthe reservoir. We neglect the efiect of percolation  ow for the purposes of this study, sincewe are interested in evaluating the change in SP caused by perturbations in the seepagepattern due to internal erosion.127Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.4 Two-zone embankment model5.4.1 Model geometryThe intact embankment model consists of two distinct zones that represent a glacial tillcore and a sand and gravel shell material. For simplicity, the model exhibits a  at baseand vertical abutment contacts. The dam geometry is outlined in Figure 5.3. Defect zonesare included in the model to represent areas of elevated hydraulic conductivity caused bythe loss of flne-grained material from the core. These defects are located at various depthswithin the core and extend parallel to the direction of  ow. Two main defect conflgurationswere studied: a pipe defect represented by a 1 m2 zone, and a construction layer defectrepresented by a 0.5 m-high layer. The width of the layer defect varies between 5 m and20 m parallel to the crest, such that three distinct layer defect types (labelled A, B andC) are deflned. Table 5.1 details the cross-sectional geometry of each of the four defecttypes. Each defect type is evaluated at depths of 15 m, 45 m and 60 m below the crest, asillustrated in Figure 5.3(a). These defects are modelled both as full defects that extend thefull transverse length of the core, and as partial or \half" defects that extend back from thedownstream core-shell boundary to beneath the crest of the dam. The full defect modelsrepresent breakthrough conditions, while the half defect models represent the progressionof internal erosion upstream through the core.Defect type Height Width(m) (m)Pipe 1 1Layer A 0.5 5Layer B 0.5 10Layer C 0.5 20Table 5.1: Defect cross-sectional geometry.128Chapter 5. Sensitivity of the self-potential method to detect internal erosion200m(b) (c)5m0.71 1260m60m45m15m(a)240m100m100m 200mxz yFigure 5.3: Model dam and defect geometry: a) transverse cross-section illustrating thethree defect depths; b) cross-section of dam with a pipe defect; c) cross-section of dam witha layer defect.5.4.2 Seepage analysisA non-uniform 96£90£51 cell mesh was generated within Visual MODFLOW to representthe embankment dam model shown in Figure 5.3. Model x and y dimensions respectivelyrepresent transverse (parallel to the  ow path) and longitudinal (parallel to the crest)dimensions. Distances are stated in metres relative to an origin located at the intersectionof the upstream toe with the right abutment. The mesh was reflned in the vicinity of thethree defect depths, such that cell dimensions range from a maximum of 2:5m£2:5m£2:5mto a minimum of 2:5m£0:5m£0:5m.A variably saturated approach was required to generate a realistic hydraulic head dis-tribution within the saturated zone, and to correctly position the phreatic surface alongthe downstream face of the core. Hydraulic conductivity values were prescribed for all soilzones in the model and are listed in Table 5.2. These values are assumed isotropic and arewithin the range of those typically observed for glacially-derived core and shell material.129Chapter 5. Sensitivity of the self-potential method to detect internal erosionThe relative permeability of all soil zones was deflned using van Genuchten parametersfi = 1 m¡1, fl = 2, and a residual saturation Swr = 0:03. These values are representativeof coarse sand and were chosen to characterize the downstream outer shell material, whichwas found to remain predominantly unsaturated in the analyses.Zone K L ‰(m s¡1) (A m¡2) (›m)Shell 1£10¡5 2:3£10¡5 300(1000)Core 1£10¡9 1:7£10¡5 100(250)Defect 1£10¡5 2:1£10¡5 230(500)Bedrock - - 2500Table 5.2: Physical properties of 2-zone embankment model. Electrical resistivity valuesare assigned based on saturation level: fully-saturated resistivity values are listed withunsaturated resistivity values shown in brackets.Steady-state seepage analyses were performed using the MODFLOW-Surfact numericengine to generate a variably saturated  ow solution. The hydraulic head distribution andtotal seepage  ow were resolved for all embankment defect models subject to an upstreamreservoir level of 55 m.The seepage regime in the intact core model and each of the 24 defect models is charac-terized by a three-dimensional distribution of hydraulic head. Figures 5.4 and 5.5 displayrepresentative transverse cross-sections (y = 100 m) through the centre of the B layerdefects at each of the three defect depths, corresponding to the full and half defects, respec-tively. The presence of a full defect results in a small perturbation of the head distributionthrough the core, which intensifles with the increase in head drop at greater defect depths.The presence of a half defect in the core results in signiflcant variation in the hydraulic headcontours, generating large hydraulic gradients immediately upstream of the defect.Mass-balancecalculationsofthetotalvolumetricseepage owthrougheachembankmentmodel enable the computation of anomalous seepage  ow resulting from the presence of eachdefect type. Table 5.3 lists the increase in seepage  ow rate for all defect models, which130Chapter 5. Sensitivity of the self-potential method to detect internal erosion(a)(b)(c)Figure 5.4: Transverse cross-section of hydraulic head distribution in B layer full defectmodels at defect depths of a) 15 m, b) 45 m, and c) 60 m.was calculated by subtracting the intact core model seepage  ow rate of 0.523 L/minute.Signiflcant incremental seepage was calculated for the full defect models, which provide apervious path through the entire length of the core. The half defect models result in seepagerates only marginally larger than the intact case.Of particular note is the behaviour of the phreatic surface downstream of the defects.Figure 5.4 reveals an elevated water level in the downstream shell, which correlates withdefect depth. This behaviour is a direct result of the increase in seepage  ow, which isshown in Table 5.3 to be quite substantial for breakthrough conditions. Half defect modelsdo not generate the same volume of excess seepage, and consequently the position of thephreatic surface through the core and downstream shell remains essentially unchanged fromthe intact case. This has implications for the predicted SP response, as discussed in Section131Chapter 5. Sensitivity of the self-potential method to detect internal erosion(a)(b)(c)Figure 5.5: Transverse cross-section of hydraulic head distribution in B layer half defectmodels at defect depths of a) 15 m, b) 45 m, and c) 60 m.5.4.4.5.4.3 SP forward modellingThe electrical properties were assigned based on the model grid and material property zonesdeflned in the seepage analysis. Table 5.2 lists the physical property values for each materialtype. Isotropic properties were assumed for each soil unit, with a distinction made betweensaturated and unsaturated zones. Measured values of the cross-coupling conductivity Land resistivity ‰ were obtained from laboratory experiments conducted on saturated em-bankment soil specimens, as described in Chapter 4. A saturated approach was used tomodel streaming current  ow and we assigned a null cross-coupling conductivity above thephreatic surface. Resistivity values that characterize the unsaturated zone within each soil132Chapter 5. Sensitivity of the self-potential method to detect internal erosionDefect type 15m depth 45m depth 60m depthfull defect - pipe 0.317 (61%) 0.477 (91%) 0.511 (98%)full defect - layer A 0.772 (148%) 1.18 (226%) 1.26 (240%)full defect - layer B 1.54 (294%) 2.36 (451%) 2.47 (473%)full defect - layer C 3.06 (586%) 4.71 (901%) 4.84 (927%)half defect - pipe 0.005 (1%) 0.015 (3%) 0.013 (2%)half defect - layer A 0.004 (1%) 0.023 (4%) 0.021 (4%)half defect - layer B 0.007 (1%) 0.031 (6%) 0.026 (5%)half defect - layer C 0.011 (2%) 0.046 (9%) 0.041 (8%)Table 5.3: Predicted volumetric  ow rate increase in L/minute. Anomalous seepage rateswere obtained by subtracting the total seepage  ow rate (0.523 L/min.) through the intactdam. The anomalous increase in seepage is shown as a percentage of the total intact damseepage in brackets.unit were assumed based on published values and empirical estimates. The reservoir wasincluded as a distinct unit in the electrical resistivity model and was assigned a value of 70›m. This value represents the saturating  uid used in the laboratory experiments, whichcorresponds with observed average reservoir water properties at the site from which soilsamples were taken.The original mesh was expanded to apply the zero  ux boundary condition required tosolve for the SP distribution. This boundary was imposed with the use of padding cellssurrounding the original mesh that must be assigned realistic values of electrical resistivity.The embankment was assumed to be  anked by bedrock abutments in the y dimension andunderlain by a bedrock foundation, as illustrated in Figure 5.6. An optimum padded meshsize for this problem was determined to be seven times the size of the original grid, as thisremoves the in uence of the boundary on the solution within the dam. The predicted SPvalues were referenced to a base station coordinate (120,0,60) located at the intersectionof the crest with the right abutment, since self-potential data are potential difierences133Chapter 5. Sensitivity of the self-potential method to detect internal erosionmeasured between two electrodes.2500138076042023013070Ohm.mFigure 5.6: Cut-away view of the electrical resistivity model of the embankment, showingpadding cells over a portion of the discrete mesh.A three-dimensional distribution of self-potential was solved for in each embankmentmodel. Figure 5.7 displays a transverse proflle intersecting the centre of the crest (y = 100m) of the predicted SP response over the surface of the dam and reservoir for the intact coremodel. The distribution of surface measurement points is indicated by arrows on the damschematic in the flgure. Similarly, Figure 5.8 presents a longitudinal proflle along the crest(x = 120 m) for the intact core model. These flgures show the characteristic SP signature touniform seepage through an embankment, which trends from a negative response upstreamto a positive response downstream. This is caused by the accumulation of negative andpositive charge respectively along the upstream and downstream core-shell interface dueto the physical property contrast and the large hydraulic gradient through the core. Forthis embankment model, the SP response trends from a negative minimum of ¡8 mV to amaximum of 38 mV over the surface of the downstream shell.The analysis and interpretation of SP data are aided by some knowledge of the SPresponse to uniform seepage through the structure. In this synthetic case study, the \true"intact core SP response of the embankment is known. Consequently, we can isolate theresidual or anomalous SP response by subtracting the intact core data from the defectmodel data. In a fleld situation, repeat SP survey or long-term monitoring data may becompared to isolate seepage-related anomalies.134Chapter 5. Sensitivity of the self-potential method to detect internal erosion0 50 100 150 200 250−10010203040Distance from upstream toe (m)SP (mV)Figure 5.7: Transverse SP data proflle over surface of intact embankment (y = 100 m).referenceelectrode0 20 40 60 80 100 120 140 160 180 200−505Distance from right abutment (m)SP (mV)Figure 5.8: Longitudinal SP data proflle over surface of crest of intact embankment, lookingupstream (x = 120 m).135Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.4.4 Residual SP response at surfaceThe residual SP data for full and half defect models are presented in proflle form in Figures5.9 and 5.10, respectively. Model results are identifled by defect type at each depth. Trans-verse and longitudinal proflles display residual data along the same surface measurementpoints as shown for the intact case.0 50 100 150 200−2−100 50 100 150 200−0.4−0.200 50 100 150 200−0.4−0.20Distance from right abutment (m)0 50 100 150 200 250−4−20240 50 100 150 200 250−4−2024SP residual (mV)0 50 100 150 200 250−4−2024Distance from upstream toe (m)(a)(c)(e) (f)(b)(d)referenceelectroderefFigure 5.9: Transverse (a, c, e) and longitudinal (b, d, f) proflles of SP residual at thesurface of the full defect models at 15 m depth (a, b), 45 m depth (c, d) and 60 m depth(e, f). Residual data are generated by subtracting the intact model response.The surface SP residual data shown in Figures 5.9(a) and (b) display a small localizednegative perturbation centred downstream of the crest for all full defect types at 15 mdepth, which is barely visible in the 45 m and 60 m depth models. This is due to thesurface proximity of negative sources of electrical charge generated at the defect outlet.Positive charge generated from seepage in ow along the upstream length of the defect doesnot signiflcantly in uence the SP signature at surface due to the high electrical conductivityof the core. This efiect is discussed further below in Sections 5.4.5 and 5.4.6. All full defectmodels display a positive SP residual over the downstream toe, with a maximum residual136Chapter 5. Sensitivity of the self-potential method to detect internal erosion0 50 100 150 200−0.200.20.40.60 50 100 150 200−0.200.2SP residual (mV)0 50 100 150 200−0.4−0.20Distance from right abutment (m)0 50 100 150 200 250−1010 50 100 150 200 250−101SP residual (mV)0 50 100 150 200 250−101Distance from upstream toe (m)(a)(c)(e) (f)(b)(d)referenceelectroderefFigure 5.10: Transverse (a, c, e) and longitudinal (b, d, f) proflles of SP residual at thesurface of the half defect models at 15 m depth (a, b), 45 m depth (c, d) and 60 m depth(e, f). Residual data are generated by subtracting the intact model response.anomaly of over 3 mV observed for the C layer defect at 60 m depth. This anomalousSP response is due to positive sources of charge that develop at the phreatic surface inthe downstream shell. The SP residual displays a positive correlation with defect size anddepth, since these parameters control the increase in seepage  ow and consequently the risein the phreatic surface downstream of the defect.The half defect models generate residual SP anomalies of smaller amplitude than thefull defect models, with a maximum response of < 1 mV at surface, as shown in Figure5.10(e). This is attributed in part to the negligible change in the position of the phreaticsurface downstream of the defects due to the small increase in seepage  ow rate. The largehydraulic gradients surrounding the upstream end of the half defects generate signiflcantpositive electrical charge within the core, as discussed in Section 5.4.5. The in uence of thischarge is visible to a slight degree in the 15 m depth proflles due to the proximity of thedefects to surface, but is not visible in proflles corresponding to the deeper defect models.137Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.4.5 Residual SP response at depthThe small residual anomalies visible at the surface of the embankment suggest that  owthrough the defects generates a negligible response. However, the residual SP response atdepth tells a difierent story. Figures 5.11 and 5.12 illustrate images of the residual SPdistribution in vertical cross-section through the centre of the full and half pipe defects.Evident in the flgures is the focused anomalous response surrounding each defect within thecore. Flow into the defect generates a positive SP response; out ow from the defect resultsin a negative SP response. In the full pipe defects shown in Figure 5.11, this translatesto positive SP anomalies ranging from 2.5 to 13 mV surrounding the upstream length ofthe defect, and negative SP anomalies on the order of several mV centred at the defectoutlet. For all full layer defect models, maximum anomaly amplitudes range from 13 mVto ¡7 mV. In general, the residual anomalies surrounding the defects do not propagate tothe surface of the embankment due to the low resistivity of the core material. However, asnoted in Section 5.4.4, the positive residual anomaly caused by the rise in phreatic surfacein the downstream shell does propagate to the surface due to the high resistivity of theunsaturated shell material. These positive responses are evident in the cross-sections shownin Figure 5.11 and the surface proflles shown in Figures 5.9(a), (c) and (e).The large hydraulic gradients that exist immediately upstream of the half defects gen-erate positive SP anomalies of signiflcant magnitude, as shown in Figure 5.12. Positive SPanomalies ranging from 12 to 65 mV surround the upstream end of the half pipe defects,which are representative for all half layer defects. The magnitude of the hydraulic gradientand resultant SP response are positively correlated with defect depth. These high amplitudeSP signals do not propagate to the surface of the embankment due to the high electricalconductivity of the core, as evidenced by the surface proflles shown in Figure 5.10. Thein uence of the electrical properties of the core is further examined in Section 5.4.6.138Chapter 5. Sensitivity of the self-potential method to detect internal erosion7.0-4.313-2.32.5-4.0mVmVmV(a)(b)(c)Figure 5.11: SP residual in transverse cross-section intersecting the centre of the full pipedefect: a) 15 m depth; b) 45 m depth; c) 60 m depth.139Chapter 5. Sensitivity of the self-potential method to detect internal erosion46-0.765-0.712-0.7mVmVmV(a)(b)(c)Figure 5.12: SP residual in transverse cross-section intersecting the centre of the half pipedefect: a) 15 m depth; b) 45 m depth; c) 60 m depth.140Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.4.6 In uence of the electrical resistivity distributionThe self-potential fleld is generated by the distribution of streaming current sources, whichare concentrated at physical property boundaries within the embankment. The maximumSP residual anomalies are caused by streaming current sources located at the physicalboundaries of the defect. The sign and amplitude of these sources are dictated by thedirection and magnitude of the gradients in head and physical properties at the boundary.The SP signal decays with inverse distance away from the source. However, the distributionof electrical resistivity in the embankment dictates the amplitude of the electrical potentialfleld. The higher the resistivity, the larger the self-potential signal. The core is the leastresistive soil zone within an embankment, since it typically contains electrically conductivesilts and clays. Consequently, for a given streaming current source amplitude, the SPresidual signal strength is governed to a large extent by the resistivity of this zone.A twofold increase in the resistivity of the core material is examined to illustrate theresultant efiect on the SP response. This change in resistivity is well within the range forglacial till material. It can be seen to represent a minor change in mineralogy, such asa decreased clay content, or a difierence in temperature. The electrical properties usedin this study were obtained from laboratory measurements made at 22–C, as presented inChapter 4. However, below a depth of 10 m, the earth temperature remains constant withinone or two degrees of the mean air temperature if we neglect the in uence of geothermalefiects [Stevens et al., 1975]. An assumed mean air temperature of 10–C translates to anapproximate 30% increase in the resistivity of saturated core material, if we use empiricalmodels [Archie, 1942; Arps, 1953] to estimate the in uence of a decrease in  uid temperatureon the bulk resistivity.Figures 5.13 and 5.14 illustrate the efiect of increasing the resistivity of the core mate-rial from 100 ›m to 200 ›m. As seen in the flgures, a twofold increase in core resistivityefiectively doubles the residual SP response observed at surface and at depth within theembankment. This has implications for the detection of SP anomalies given typical mea-surement sensitivities, as discussed in Section 5.5.141Chapter 5. Sensitivity of the self-potential method to detect internal erosion−5 0 5 10 150102030405060SP residual (mV)Depth (m)0 50 100 150 200 250−1012Distance from upstream toe (m)SP residual (mV)0 50 100 150 200−0.4−0.3−0.2−0.10Distance from right abutment (m)SP residual (mV)200 Ohm.m100 Ohm.m(a)(c)(b)Figure 5.13: Efiect of increased resistivity of core on the SP residual resulting from a fullpipe defect at 45 m depth: a) surface transverse proflle, b) surface longitudinal proflle, c)vertical proflle beneath centre of crest intersecting defect.0 50 100 150 200−0.4−0.3−0.2−0.10Distance from right abutment (m)SP residual (mV)0 50 100 150 200 250−0.500.511.5Distance from upstream toe (m)SP residual (mV)0 20 40 60 80 1000102030405060SP residual (mV)Depth (m)200 Ohm.m100 Ohm.m(a)(c)(b)Figure 5.14: Efiect of increased resistivity of core on the SP residual resulting from a halfpipe defect at 45 m depth: a) surface transverse proflle, b) surface longitudinal proflle, c)vertical proflle beneath centre of crest intersecting defect.142Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.5 Practical SP anomaly detection limitsThe amplitude of an SP anomaly that can be reasonably detected from fleld measurements isgoverned by the ratio of signal strength to noise level at a given site. Sources of noise includetelluric currents, spurious currents caused by other operations at site, poor electrical contactbetween the electrodes and the soil, electrode drift, and the in uence of non-streamingpotential source mechanisms on the measured self-potential. These noise sources can beminimized with proper fleld data acquisition procedures [Corwin, 2005].SP data may be acquired in a single survey using a portable pair of electrodes, orthrough continuous monitoring of a permanently-installed array of electrodes. A singlesurvey typically involves making surface measurements of SP using a flxed base electrodeand a roving electrode that is used to occupy survey stations across the site. Single surveysmay be performed once or repeated to isolate a residual SP response. Continuous monitoringinvolves measuring the SP fleld across multiple electrodes that are installed near surfaceor at depth within the embankment. Data sets are acquired in a relatively short period oftime and may be compared over difierent time scales to isolate the residual response.Given that appropriate equipment and acquisition methods are employed, the chosensurvey method will in uence the measurement sensitivity. The measured self-potential fleldofiers a snapshot of current hydraulic conditions within the embankment. Fluctuationsin reservoir level and seasonal variations in water properties generate temporal changesin resistivity and the self-potential fleld [Johansson and Dahlin, 1996; Johansson et al.,2005]. These temporal changes contribute to the total SP response to seepage throughthe dam. Where single surveys characterize current seepage conditions at the time ofmeasurement, continuous monitoring can more accurately track the temporal response uponwhich anomalous variations are superimposed. In this way, continuous monitoring canresolve smaller residual changes that develop over time from the background response.The limit of detection varies between sites depending on the signal to noise ratio. How-ever, as a rough guideline, SP anomalies on the order of 10 mV are generally detectablefrom single survey data made at the surface of an embankment, while smaller anomaliesmay be detectable from ofishore or borehole measurements. Although steel casing can in- uence downhole SP measurements [Darnet et al., 2003], these efiects are not addressed143Chapter 5. Sensitivity of the self-potential method to detect internal erosionhere. Detection limits for continuous monitoring data are more di–cult to estimate, sincethe development and application of these systems is in its early stages. However, one mightexpect that residual anomalies of several milliVolts would be detectable with careful dataacquisition and processing procedures. Given these detection limits, the predicted SP resid-ual responses at surface presented in Section 5.4.4 would not typically be detectable usingcurrent survey techniques. As discussed in Section 5.4.6, an embankment with a more re-sistive core would give rise to larger SP anomalies that could be more easily detected fromsurface measurements. However, the stronger signal levels at depth presented in Section5.4.5 would surely be detectable by measurements made within the vicinity of the defects.Further examination of the spatial range of detection for anomalies at depth is included inSection 5.6.5.6 Comparison of SP and hydraulic response to defectsAs discussed in Section 5.4.5, the SP anomaly at depth generated by a half defect is of muchlarger amplitude than that generated by a full defect through the core. This suggests thatthe SP response actually peaks prior to any signiflcant increase in seepage  ow that occursupon breakthrough of the defect into the upstream shell.To examine this further we studied the development of a pipe defect at a depth of 45 mwithin the core. Figure 5.15 illustrates the change in hydraulic head distribution in trans-verse cross-section through the centre of the dam, as the pipe progresses upstream throughthe core. Forward modelling was performed for incremental defect lengths of 25%, 50% and75% of the core and for breakthrough conditions. Figure 5.16 shows the corresponding SPresidual in vertical proflles below the crest at each defect length. These proflles may be seento represent data acquired from a vertical string of electrodes installed permanently withinthe core or via borehole measurements. Figure 5.16(a) represents a vertical proflle thatintersects the centre of the defect, while Figures 5.16(b), (c) and (d) respectively representvertical proflles ofiset along the crest by distances of 5 m, 10 m and 15 m from the centreof the defect, as illustrated by the schematic shown in the flgure.The SP residual responses clearly show that the maximum SP anomaly occurs prior to144Chapter 5. Sensitivity of the self-potential method to detect internal erosionbreakthrough of the pipe into the upstream shell. Maximum anomalies range from 46 mV atthe defect to over 5 mV a distance of 15 m from the defect, which exceed the expected fleldsensitivity for measurements made at depth. The maximum SP residual observed at eachseparation distance from the defect is compared to the total increase in seepage throughthe dam for each defect length in Figure 5.17. Assuming that a 15% increase in seepagecould be detected using an average, manual weir system and as little as a 7% increase couldbe detected using a more advanced system, we would not detect the presence of the pipeprior to breakthrough with standard seepage  ow measurements. As seen in Table 5.3,the increase in seepage  ow resulting from any of the partial defects at any depth wouldtypically be undetectable by standard weir measurements. However, the SP response within10 to 15 m of a partial defect is above a reasonable detection threshold of 5 mV. It is alsoimportant to note that weir measurements do not give a clear indication of the locationwhere preferential seepage is occurring. These results demonstrate that the progressionof internal erosion through the core should be detectable by SP measurements made atdepth, prior to the onset of a signiflcant increase in seepage  ow that is detectable at thedownstream toe.The formulation of the coupled  ow equation (5.5) and the results shown in Figures5.15 to 5.17 demonstrate that it is hydraulic gradient, not seepage  ow rate, that dictatesthe amplitude of the streaming current source and resulting streaming potential anomaly.This link suggests that SP measurements may complement pore pressure measurements toalert to changes in seepage patterns within the dam. Figures 5.18 and 5.19 display theresidual SP and hydraulic head responses with depth through the core, in the presence ofboth pipe and B layer full and half defects. Figure 5.18 represents vertical proflles thatintersect the centre of the defect, while Figure 5.19 represents vertical proflles ofiset alongthe crest by a distance of 10 m from the centre of the defect. Clearly evident in the flguresis the correlation between the SP and head responses, where the SP residual proflle closelymirrors the residual hydraulic head proflle.The response to the pipe and B layer defects are very similar when the measurementproflle intersects the centre of the defect, as shown in Figure5.18. At a distance of 10 m awayfrom the centre of the defect, the response to the pipe defects is signiflcantly attenuated145Chapter 5. Sensitivity of the self-potential method to detect internal erosionwhereas the B layer continues to exert a response due to the proximity of the outer edge ofthe defect to the vertical proflle.Assuming that a 1 m of head variation from background is signiflcant enough to signalan alert, and choosing 5 mV as an alert threshold for a change in SP, we can assess andcompare the two measurement methods in terms of detection ability. For a vertical proflleintersecting the centre of the defect, both SP and head measurements at depth would signalan alarm for all full and half defects, with the exception of the full defects at 15 m depth,shown in Figure 5.18(a) and (b). For measurements made 10 m away from the centre ofthe defect, only the half defects at 45 m and 60 m depth would be detectable by both SPand head readings, as shown in Figures 5.19(i) to (l). Given our chosen threshold levels, afull B layer defect at 45 m depth and full pipe and B layer defects at 60 m depth wouldgenerate an alert in head but not in SP.An important consideration in the acquisition and analysis of SP and hydraulic headdata is the transient nature of these data sets. Both SP and head respond to  uctuationsin reservoir level, tailwater level, as well as regional groundwater patterns. The lag timebetween the source and the impact of these variations at difierent locations within the damwill also in uence the behaviour of the SP and pore  uid pressure responses. Consequently,a performance history is required to assess normal background behaviour for a given siteand the sensitivity of difierent instrumentation before determining thresholds that alertto anomalous behaviour. However, since SP and pore pressure responses should behaveconcurrently, SP measurements can be used in conjunction with pressure measurements tosubstantiate observed behaviour or be used on their own to alert to changes in seepagepatterns.146Chapter 5. Sensitivity of the self-potential method to detect internal erosion(a)(b)(c)(d)Figure5.15: Transversecross-section ofhydraulicheaddistributionforthepipedefectmodelat 45 m depth, as pipe progresses upstream through the core: a) 25% of core length; b) 50%of core length; c) 75% of core length; d) 100% of core length (breakthrough conditions).147Chapter 5. Sensitivity of the self-potential method to detect internal erosion−5 0 5 100102030405060SP residual (mV)Depth (m)−5 0 5 100102030405060SP residual (mV)Depth (m)−5 0 5 10 15 200102030405060SP residual (mV)Depth (m)−50 10 20 30 40 500102030405060SP residual (mV)Depth (m)referenceelectrode(a) (b) (c) (d)ref25%50%75%breakthroughss = 0 s = 15 ms = 5 m s = 10 mFigure 5.16: Proflles of SP residual with depth beneath the centre of the crest marking theprogression of the pipe defect through the core. Vertical proflles are located at incrementalofisets (s) from the centre of the defect in a direction parallel to the crest: a) intersectingthe defect (s=0); b) 5 m separation; c) 10 m separation; d) 15 m separation.148Chapter 5. Sensitivity of the self-potential method to detect internal erosion01020304050600 25 50 75 100Pipe propagation distance through core (%)Maximum SP residual (mV)0255075100Volumetric flow rate increase (%)intersecting defect (s = 0)s = 5 ms = 10 ms = 15 mFlow rateFigure 5.17: Comparison of the maximum SP residual and increase in seepage during theprogression of the pipe defect at 45 m depth through the core. SP residual data representthe maximum anomaly measured at depth beneath the centre of the crest, at difierent ofisetdistances (s) from the centre of the defect in a direction parallel to the crest.149Chapter 5. Sensitivity of the self-potential method to detect internal erosion−5 0 5 10 150102030405060SP residual (mV)Depth (m)−10 −5 0 50102030405060Residual head (m)Depth (m)0 20 40 60 800102030405060SP residual (mV)Depth (m)−40 −20 00102030405060Residual head (m)Depth (m)0 20 40 600102030405060SP residual (mV)Depth (m)−30−20−10 0 100102030405060Residual head (m)Depth (m)−5 0 5 10 150102030405060SP residual (mV)Depth (m)−10 −5 0 50102030405060Residual head (m)Depth (m)−5 0 5 100102030405060SP residual (mV)Depth (m)−4 −2 00102030405060Residual head (m)Depth (m)−2 −1 00102030405060SP residual (mV)Depth (m)0 0.5 10102030405060Residual head (m)Depth (m)ref(a)(i)(h)(g)(f)(e)(d)(c)(b)(l)(k)(j)refpipeB layerFigure 5.18: Proflles of SP residual and residual hydraulic head with depth beneath thecrest for pipe and B layer defect models. Vertical proflles intersect the centre of the defects.Residual data are shown for full defect models at 15 m depth (a, b), 45 m depth (c, d) and60 m depth (e, f); for half defect models at 15 m depth (g, h), 45 m depth (i, j) and 60 mdepth (k, l).150Chapter 5. Sensitivity of the self-potential method to detect internal erosion−2 0 20102030405060SP residual (mV)Depth (m)−1 0 10102030405060Residual head (m)Depth (m)−5 0 5 10 150102030405060SP residual (mV)Depth (m)−10 −5 0 50102030405060Residual head (m)Depth (m)0 10 200102030405060SP residual (mV)Depth (m)−15−10−5 0 50102030405060Residual head (m)Depth (m)−5 0 5 100102030405060SP residual (mV)Depth (m)−4 −2 00102030405060Residual head (m)Depth (m)0 2 40102030405060SP residual (mV)Depth (m)−2 −1 0 10102030405060Residual head (m)Depth (m)−2 −1 00102030405060SP residual (mV)Depth (m)0 0.50102030405060Residual head (m)Depth (m)(a)(i)(h)(g)(f)(e)(d)(c)(b)(l)(k)(j)pipeB layerref refFigure 5.19: Proflles of SP residual and residual hydraulic head with depth beneath thecrest for pipe and B layer defect models. Vertical proflles are ofiset by 10 m along the crestfrom the centre of the defects. Residual data are shown for full defect models at 15 m depth(a, b), 45 m depth (c, d) and 60 m depth (e, f); for half defect models at 15 m depth (g,h), 45 m depth (i, j) and 60 m depth (k, l).151Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.7 Discussion and ConclusionThe streaming potential signal generated by seepage through an embankment was evaluatedfor a series of synthetic models using a three-dimensional forward modelling algorithm. Thepurpose of the study was to assess the sensitivity of the self-potential method in detectingpreferential  ow paths caused by internal erosion.The current study makes use of an embankment geometry and physical property valuesassumed to represent a general case. Defect properties were chosen to re ect a realisticapproximation of preferential seepage  ow paths that could develop in the early stagesof internal erosion. However, the self-potential distribution is highly dependent on theembankment geometry and physical property distribution, such that the amplitude andcharacter of a measured SP anomaly may vary signiflcantly from the results presented here.Pipe and construction layer defects located at three depths within a zoned embankmentwere studied for difierent propagation distances through the core. Seepage through an in-tact core model of the embankment generated a peak-to-peak SP response of 45 mV. Thepresence of defects within the core resulted in anomalous variations from this backgroundresponse. Full defect models representing breakthrough conditions generated residual anom-alies of up to 3 mV at the surface of the embankment, and a dipolar anomaly at depthexhibiting maximum amplitudes of ¡7 mV to 13 mV. Half defect models representing theupstream progression of piping through the core generated residual anomalies of less than1 mV at surface and maximum anomalies ranging from 12 mV to 65 mV at depth.The numerical modelling results are useful in demonstrating how the SP method mayrespond to the presence of defect zones within an embankment, and provide guidance forsurvey design to maximize detection. The studied defects would not be detectable fromsurface measurements made using portable electrodes in a single or repeat SP survey con-flguration. Measurements are required at depth in order to detect su–cient signal. The halfdefect results showed that signiflcant SP anomalies are generated around the upstream endof the defects due to the large hydraulic gradients that develop prior to breakthrough. Acomparison of the predicted SP response and  uid  ow rate from the upstream migrationof a pipe defect in the core clearly showed that the maximum SP anomaly occurs prior toany detectable increase in seepage. The localized SP response mirrors the pressure response152Chapter 5. Sensitivity of the self-potential method to detect internal erosionat depth, and shows the capacity of the method to detect changes in hydraulic gradient.Consequently, SP measurements made using buried electrodes could complement existingpiezometric measurements, and may provide a cost beneflt.The results have implications for the application of the self-potential method to damsafety investigations. Single or repeat surveying has been successful in delineating preferen-tial seepage paths [e.g., Butler et al., 1989] and is a useful investigative tool that may be usedto characterize overall conditions and identify larger-scale anomalies. However, continuousmonitoring afiords an increased sensitivity to the development of seepage-related anomaliescaused by internal erosion, since temporal changes can be tracked more efiectively. Ideally,electrodes should be installed beneath the surface to capitalize on larger signal levels. Inexisting structures, the installation of electrodes at depth can risk damaging the core bydrilling, so pre-existing boreholes may prove useful. Electrodes and cabling may be installedduring upgrading of existing structures or the construction of new embankments.Self-potential surveying is a tool to be used in conjunction with standard monitor-ing methods and other geophysical techniques to help characterize seepage conditions inembankments. Future developments in instrumentation may ofier improved measurementsensitivity, but optimizing electrode placement along with data acquisition and processingstrategies will improve the efiectiveness of the method.153Chapter 5. Sensitivity of the self-potential method to detect internal erosion5.8 ReferencesAdamson, A. W., Physical Chemistry of Surfaces, John Wiley and Sons, New York, 1990.Archie, G. E., The electrical resistivity log as an aid in determining some reservoir char-acteristics, Transactions of the Society of Petroleum Engineers of the American Instituteof Mining, Metallurgical and Petroleum Engineers (AIME), 146, 54{67, 1942.Arps, J. J., The efiect of temperature on the density and electrical resistivity of sodiumchloride solutions, Transactions of the Society of Petroleum Engineers of the AmericanInstitute of Mining, Metallurgical and Petroleum Engineers (AIME), 198, 327{330, 1953.Black, W. E., and R. F. Corwin, Application of self-potential measurements to the delin-eation of groundwater seepage in earth-flll embankments, Society of Exploration Geophysi-cists Technical Program Expanded Abstracts, 3, 162{164, doi:10.1190/1.1894185, 1984.Bogoslovsky, V. A., and A. A. Ogilvy, Application of geophysical methods for studyingthe technical status of earth dams, Geophysical Prospecting, 18, 758{773, 1970.Butler, D. K., J. L. Llopis, and C. M. Deaver, Comprehensive geophysical investigation ofan existing dam foundation, The Leading Edge, 8(8), 10{18, 1989.Casagrande, L., Stabilization of soils by means of electro-osmosis: State of the art, Journalof the Boston Society of Civil Engineers, 69(2), 255{302, 1983.Corwin, R. F., The self-potential method for environmental and engineering applications,in Geotechnical and Environmental Geophysics, vol. 1, edited by S. H. Ward, pp. 127{145,Society of Exploration Geophysicists, Tulsa, 1990.Corwin, R. F., Investigation of geophysical methods for assessing seepage and internalerosion in embankment dams: Self-potential fleld data acquisition manual, Tech. rep.,Canadian Electricity Association Technologies Inc. (CEATI), Report T992700-0205B/1,Montreal, 2005.Darnet, M., and G. Marquis, Modelling streaming potential (SP) signals induced by watermovement in the vadose zone, Journal of Hydrology, 285, 114{124, 2004.154Chapter 5. Sensitivity of the self-potential method to detect internal erosionDarnet, M., G. Marquis, and P. Sailhac, Estimating aquifer hydraulic properties from theinversion of surface Streaming Potential (SP) anomalies, Geophysical Research Letters,30(13), 1679, doi:10.1029/2003GL017631, 2003.de Groot, S. R., Thermodynamics of irreversible processes, Selected Topics in ModernPhysics, vol. 3, North Holland Publishing Company, Amsterdam, 1951.Guichet, X., L. Jouniaux, and J. P. Pozzi, Streaming potential of a sand columnin partial saturation conditions, Journal of Geophysical Research, 108(B3), 2141, doi:10.1029/2001JB001517, 2003.Harbaugh, A. W., E. R. Banta, M. C. Hill, and M. G. McDonald, Mod ow-2000, the U.S.GeologicalSurveyModularGround-WaterModel-UserGuidetoModularizationConceptsand the Ground-Water Flow Process, Tech. rep., United States Geological Survey, Open-File Report 00-92, Washington, 2000.HydroGeoLogicInc., MODFLOW-SURFACT Version 2.2 Documentation, HydroGeoLogicInc., Herndon, 1996.Johansson, S., and T. Dahlin, Seepage monitoring in an earth embankment dam by re-peated resistivity measurements, European Journal of Environmental and EngineeringGeophysics, 1, 229{247, 1996.Johansson, S., J. Friborg, T. Dahlin, and P.Sj˜odahl, Long term resistivityand self potentialmonitoringofembankmentdams: ExperiencesfromH˜allbyandS˜advadams, Sweden, Tech.rep., Elforsk/Canadian Electricity Association Dam Safety Interest Group, Report 05:15,Stockholm, 2005.Moore, J. R., S. D. Glaser, H. F. Morrison, and G. M. Hoversten, The streaming potentialof liquid carbon dioxide in Berea sandstone, Geophysical Research Letters, 31, L17610,doi:10.1029/2004GL020774, 2004.Morgan, F. D., E. R. Williams, and T. R. Madden, Streaming potential properties of West-erly granite with applications, Journal of Geophysical Research, 94(B9), 12,449{12,461,1989.155Chapter 5. Sensitivity of the self-potential method to detect internal erosionOgilvy, A. A., M. A. Ayed, and V. A. Bogoslovsky, Geophysical studies of water leakagesfrom reservoirs, Geophysical Prospecting, 17, 36{62, 1969.Overbeek, J. T. G., Electrokinetic phenomena, in Colloid Science, Irreversible Systems,vol. 1, edited by H. R. Kruyt, Elsevier Publishing Company, Amsterdam, 1952.Panthulu, T. V., C. Krishnaiah, and J. M. Shirke, Detection of seepage paths in earth damsusing self-potential and electrical resistivity methods, Engineering Geology, 59, 281{295,2001.Revil, A., and A. Cerepi, Streaming potentials in two-phase  ow conditions, GeophysicalResearch Letters, 31, L11605, doi:10.1029/2004GL020140, 2004.Revil, A., H. Schwaeger, L. M. Cathles III, and P. D. Manhardt, Streaming potential inporous media: 2. Theory and application to geothermal systems, Journal of GeophysicalResearch, 104(B9), 20,033{20,048, 1999.Rozycki, A., J. M. Ruiz Fonticiella, and A. Cuadra, Detection and evaluation of horizontalfractures in earth dams using the self-potential method, Engineering Geology, 82, 145{153,doi:10.1016/j.enggeo.2005.09.013, 2006.Shefier, M. R., and D. W. Oldenburg, Three-dimensional forward modelling of streamingpotential, Geophysical Journal International, 169, 839{848, doi:10.1111/j.1365-246X.2007.03397.x, 2007.Stevens, H. H., J. F. Ficke, and G. F. Smoot, Water temperature: In uencial factors, fleldmeasurements, and data presentation, in Techniques of Water-Resources InvestigationsBook 1, United States Geological Survey, Washington, 1975.Waterloo Hydrogeologic Inc., Visual MODFLOW version 4.0 User’s Manual, WaterlooHydrogeologic Inc., 2004.156Chapter 6Three-dimensional inversion ofself-potential data to recoverhydraulic head 56.1 IntroductionSelf-potential (SP) data resulting from mineral exploration, geothermal and hydrogeologicalinvestigations have been interpreted using a number of quantitative techniques. Geometricsource methods range from curve-matching [de Witte, 1948; Y˜ung˜ul, 1950] and parametriccurves [Paul, 1965; Bhattacharya and Roy, 1981; Atchuta Rao and Ram Babu, 1983] toleast-squares inversion [Fitterman and Corwin, 1982; Abdelrahman and Sharafeldin, 1997].Source imaging [Di Maio and Patella, 1994; H˜ammann et al., 1997; Patella, 1997] andsource inversion methods [Shi, 1998; Minsley et al., 2007] have also been developed torecover parameters of an electrical current source distribution that explain the measuredpotentials. Although these methods vary in complexity, they provide a means of interpretingSP data that is essentially independent of the underlying physical source mechanism. Thiscan be beneflcial in problems where the source mechanism is unknown or poorly understood.The SP method may be used to study streaming potentials in order to assess subsurface ow conditions. Fluid  ow through porous media generates electrical current  ow, andthe resultant electrical potentials can be used to evaluate characteristics of the hydraulicregime. To extract information about the hydraulic system, a coupled  ow model is requiredto explain the interaction between the electrical and  uid  ow systems.5A version of this chapter will be submitted for publication. Shefier, M.R. and Oldenburg, D.W. (2007)Three-dimensional inversion of self-potential data to recover hydraulic head.157Chapter 6. 3-D inversion of self-potential dataA number of studies have focused speciflcally on the inversion of self-potential datato recover parameters related to the hydraulic system, when electrokinetic coupling is thedriving mechanism. Revil et al. [2003] developed a parametric inversion routine using aweighted least-squares approach to flnd the depth to the water table. Their method isbased on a pseudo-potential formulation of the coupled  ow problem [Nourbehecht, 1963;Fitterman, 1978] and uses an approach flrst presented by Fournier [1989] where electroki-netic sources are represented by a sheet of vertical dipoles at the phreatic surface. Birch[1998] and Revil et al. [2003] used this same approach to image the position of the watertable. Sailhac and Marquis [2001] used a wavelet-based technique to invert for electrokineticsource parameters in 2-D assuming a homogeneous conductivity distribution and identicalboundary conditions for electrical and hydraulic  ow flelds.These techniques assume that any heterogeneities in the electrical conductivity distrib-ution have a negligible efiect, particularly between saturated and unsaturated zones. Theassumption of a homogeneous conductivity distribution in a truly heterogeneous systemcan have ramiflcations on the recovered model [Minsley et al., 2007; Shefier and Oldenburg,2007]. Revil et al. [2004] included the efiect of an electrical conductivity contrast betweensaturated and unsaturated zones into their parametric inversion routine to recover the depthto the water table. Darnet et al. [2003] used a weighted least-squares approach to invertfor the hydraulic conductivity, cross-coupling conductivity and thickness of a homogeneousaquifer subject to  ow from a pumping well. Their approach accounts for the electricalconductivity contrast between the aquifer and well casing, and shows that this can have asigniflcant efiect on the resulting potential fleld distribution at surface.In this paper we present an inverse method to recover a three-dimensional hydraulichead distribution from measured self-potential data. Our approach enables us to extractpertinent information about the underlying hydraulic regime from the geophysical data, andpermits us to include the efiect of heterogeneous conductivity distributions. The method isfor particular application to systems where streaming potential is the dominant mechanismgiving rise to the SP response. The recovered head distribution may be interpreted directlyto identify zones of preferential  ow caused by subsurface heterogeneity, or used as input toan inversion routine to recover other hydraulic parameters, such as hydraulic conductivity.158Chapter 6. 3-D inversion of self-potential dataAs a flrst step, we have developed an e–cient 3-D forward modelling algorithm that en-ables us to determine the SP distribution resulting from a saturated or variably saturated ow model and a known distribution of electrical properties, namely the electrical conduc-tivity and cross-coupling conductivity [Shefier and Oldenburg, 2007]. The study region isdivided into a discrete rectangular mesh, in which hydraulic head and electrical proper-ties are deflned for each cell. Discrete equations are formulated using the method of flnitevolumes and the forward problem, which is a linear relationship between the electrical po-tentials and hydraulic head, is solved using a preconditioned biconjugate gradient stabilizedmethod.In the inverse problem we are supplied with measured electrical potentials and our goal isto estimate the causative 3-D model of hydraulic head. We assume that electrical propertiesare not a function of hydraulic head, i.e. the medium is fully-saturated or the positionof the phreatic surface is known a priori. The earth model is divided into rectangularcells, each of which is assigned constant values of the electrical properties. In practice,these must be obtained through additional analysis, but the cross-coupling coe–cient mayusually be estimated since it varies over a small range, and the electrical conductivity maybe characterized using DC resistivity survey techniques. Since the number of available SPdata is usually less than the number of grid cells we are faced with a typical underdeterminedinverse problem. We solve the inverse problem by minimizing an objective function thatconsists of a data misflt and a model objective function. The data misflt is weightedaccording to the measurement error. A priori information is incorporated into the solutionvia the model objective function. We use a Tichonov-type regularization technique tobalance fltting the data with fltting a chosen reference head model.Section 6.2 gives an overview of the streaming potential phenomenon and coupled  owtheory. Section6.3describestheformulationoftheinverseproblem, includingadiscussionofboundary conditions, physical properties, and an active cell approach used to isolate relevantregions in the problem domain. Finally, Section 6.4 presents the inversion of measured SPdata from a laboratory tank model of seepage under a cut-ofi wall, and illustrates how thesolution is improved by incorporating prior information about the problem.159Chapter 6. 3-D inversion of self-potential data6.2 Coupled  ow model of streaming potentialStreaming potential is a function of the properties of the solid- uid interface in a saturatedporous medium. An electrical double layer forms at the surface of the solid grains, whichconsists of a tightly adsorbed inner layer and a weakly-bound difiuse outer layer of ions.These act together to balance the surface charge of the solid. Fluid  ow through the porespace drags ions from the difiuse layer in the direction of  ow and generates a streamingcurrent. This movement of ions causes a charge imbalance within the pore space, whichdrives an opposing conduction current that permeates the porous medium.The total charge  ux, or current densityJ [Am¡2 ], can be expressed in terms of the sumof primary and secondary potential gradients in a system [Onsager, 1931]. In the absenceof signiflcant thermal or ionic concentration gradients, electrical current  ow is describedby:J = JC +JS = ¡ r`¡Lrh; (6.1)where JC and JS are respectively the conduction current and the streaming current densi-ties [Sill, 1983]. The  ow of conduction current is governed by the distribution of electricalconductivity  [Sm¡1 ] and electrical potential ` [V] in the subsurface. The streaming cur-rent is governed by the distribution of cross-coupling conductivity L [Am¡2 ] and hydraulichead h [m].The electrical response to a given hydraulic head distribution is considered a steady-state process, regardless of whether the hydraulic system exhibits steady-state or transientbehaviour. In the absence of any imposed current sources on the volume, ` is referred toas the self-potential and is described by:r¢  r` = ¡r¢ Lrh: (6.2)6.3 Inversion Methodology6.3.1 Discrete formulationWe wish to analyze observed self-potential data to recover a subsurface distribution ofhydraulic head in systems where streaming potential is the driving mechanism for the self-160Chapter 6. 3-D inversion of self-potential datapotential fleld. We represent the study region using a discrete mesh of rectilinear grid cells,in which values of `, h,  and L are deflned at cell centres. Each self-potential datumcorresponds to an observed potential difierence at a given coordinate with respect to areference position. The self-potential data at surface or at points within the subsurface arerelated to the subsurface distribution of hydraulic head through the relation:d = Gh; (6.3)where h is a vector of hydraulic head values corresponding to each cell, and d is a vector ofself-potential data. The matrix G is the discrete representation of (6.2) and is deflned as:G = ¡QfDSGg¡1fDLGg; (6.4)where S and L are matrices that respectively describe the distribution of electrical con-ductivity and cross-coupling conductivity using a flnite volume approximation. Matrices Dand G are respectively the divergence and gradient operators, and Q is a linear operatorthat relates each datum to the self-potential ` according to:d = Q`: (6.5)Self-potential data yield information on the hydraulic gradient. To recover a unique solutionfor h, we choose to specify zero mean potential in formulating the discrete equations. Theforward problem deflned in (6.3) is linear provided that L and  do not vary as a functionof h. The implications of this assumption are further discussed in Section 6.3.3.We are faced with solving an underdetermined inverse problem in which an inflnitenumber of solutions exist, since the number of data is typically much smaller than thenumber of discrete model cells that deflne the study region. Consequently, we deflne amodel objective function in which we can include prior information about the hydraulic headdistribution. We seek to minimize the model objective function ˆm to recover the simplesthydraulic head model that leads to an acceptable flt with the observed self-potential data.The model objective function is deflned as:ˆm = fisZVws(h¡href)2dV + fixZVwx @h@x¶2dV +fiyZVwy @h@y¶2dV + fizZVwz @h@z¶2dV ; (6.6)161Chapter 6. 3-D inversion of self-potential datawhere the flrst term works to minimize the difierence between the recovered solution h anda reference model of hydraulic head href, and the remaining three terms seek to minimizethe recovered hydraulic head gradient in each spatial dimension. The functions ws, wx,wy and wz are spatially-dependent weighting functions that regulate the minimization ona cell-by-cell basis. The scalar coe–cients fis, fix, fiy and fiz govern the relative weightingof each term in the objective function. These values are determined through a dimensionalanalysis. The discrete form of (6.6) is deflned on the mesh described above and is expressedas:ˆm = kWs(h¡href)k22 +kWx(h)k22 +kWy(h)k22 +kWz(h)k22 ; (6.7)or more simply asˆm = kWs(h¡href)k22 +kW(h)k22 ; (6.8)where Ws and W are matrices that contain discrete approximations of the functions oper-ating on the model.The chosen formulation of the model objective function enables us to incorporate dis-crete hydraulic head observations and information about the nature of the subsurface  owregime to help constrain the solution. Observed head values obtained through point mea-surements or associated with boundary conditions for the problem can be incorporated intothe reference model. In areas where we have confldence in our reference model, we canpenalize discrepancies between the recovered model and the reference model by choosingappropriately large ws values. Prior knowledge of the orientation of hydraulic gradientscan be included by preferentially weighting the appropriate spatial term. For example, inlow-topography groundwater  ow problems where the preferential  ow direction lies in aplane approximately parallel to the ground surface, we can encourage a head model with aminimum vertical gradient below the phreatic surface by choosing a relatively large valuefor fiz.We must estimate the noise present in the observed self-potential data in order to achievean acceptable flt between the observed data and the predicted data from our model. Thisnoise originates from measurement errors, which include instrument sensitivity and po-sitional errors, as well as from spurious SP signals caused by telluric currents or othersubsurface source mechanisms. Our discrete approximation of the earth can also lead to162Chapter 6. 3-D inversion of self-potential dataerrors between the observed and predicted data. Since the noise is typically unknown, weassume it exhibits a Gaussian distribution and choose to flt the data within a noise envelopecharacterized by a standard deviation † deflned for each datum dobs:ˆd =NXj=1ˆGh¡dobsj†j!2; (6.9)where N is the number of data points and ˆd is the data misflt. The discrete form of thedata misflt is expressed as:ˆd = kWd(Gh¡dobs)k22 ; (6.10)where diagonal elements of the matrix operator Wd are equal to 1=†j. Since ˆd is a chi-squared variable, we seek a target data misflt ˆd = ˆ⁄d, where ˆ⁄d » N.The inverse problem is solved by flnding a hydraulic head model that minimizes ourmodel objective function and satisfles our data misflt criterion. The inverse problem is posedas an unconstrained optimization problem, where we seek to minimize a global objectivefunction:ˆ = ˆd +flˆm ; (6.11)where fl is a regularization or \trade-ofi" parameter. The discrete form of the global objec-tive function ˆ is expressed as:ˆ = kWd(Gh¡dobs)k22 +flfkWs(h¡href)k22 +kW(h)k22g : (6.12)Calculating the gradient of (6.12) with respect to head and setting rhˆ = 0 results in:'GTWdTWdG +fl£WsTWs +WTW⁄“h =GTWdTWddobs +flWsTWshref : (6.13)Equation 6.13 is of the form Ax = b and is solved using a preconditioned method of con-jugate gradients. A cooling method is used to determine the optimum value fl⁄ that resultsin ˆd = ˆ⁄d.6.3.2 Boundary conditionsThe problem domain is represented by a discrete 3-D mesh of grid cells. The outer edgeof the mesh is specifled as a zero  ux boundary for conduction current  ow (r`¢ ^n = 0).163Chapter 6. 3-D inversion of self-potential dataConsequently, the mesh must be large enough to impose the zero  ux boundary conditionfar from areas in which signiflcant hydraulic head gradients are expected to occur. Problemsof irregular topography are handled by specifying null values of electrical conductivity inair cells. This efiectively propagates the no- ow boundary at the limit of the rectilinearmesh down to the earth’s surface.Unlike conduction current  ow, which propagates through any electrically conductivemedium, convection current  ow is restricted to the limits of the porous medium in which uid  ows. Null values of the cross-coupling conductivity are specifled in grid cells that donot correspond to the porous medium, such as air, external bodies of water, or infrastructuresuch as well casings.6.3.3 Physical property modelsTo solve (6.13), the distribution of cross-coupling conductivity and electrical conductivitymust be prescribed in the problem domain. The conductivity distributions are representedmathematically by piecewise constant functions, which can vary signiflcantly from one cellto the next.The electrical conductivity of the subsurface may be characterized using a geophysicalfleld survey technique such as DC resistivity. This approach is preferable, particularly if wehave limited prior information about subsurface structural features. Empirical estimates[e.g. Archie, 1942] or laboratory-measured values may be assigned to a given material unitif we have some lithologic or stratigraphic information.The cross-coupling conductivity is a function of the electrical conductivity of the satu-rated medium and a streaming potential coupling coe–cient C through the relation L =¡C , where C is typically characterized through laboratory measurements [e.g. Ishido andMizutani, 1981; Morgan et al., 1989; Jouniaux and Pozzi, 1995]. The streaming poten-tial coupling coe–cient may also be estimated using difierent forms of the Helmholtz-Smoluchowski equation [e.g Overbeek, 1952], which note the dependence of this parameteron the saturating  uid conductivity and in some cases surface conductivity. Consequently,care must be taken in estimating L using independent measurements of C and  , to en-sure that these values correspond to similar saturating  uid properties for a given material164Chapter 6. 3-D inversion of self-potential data[Shefier et al., 2007].As stated in Section 6.3.1, the inverse problem is linear assuming that L and  donot vary as a function of h. This condition is satisfled in conflned or fully-saturated  owproblems. In unconflned  ow problems, a phreatic surface delineates the boundary betweensaturated and unsaturated zones, as shown in Figure 6.1. This surface deflnes where pore uid pressure is atmospheric and hydraulic head is equal to the elevation z above somedatum. Since L and  vary with the degree of saturation Sw, we must know the positionof the phreatic surface a priori to prescribe values for L(Sw) and  (Sw).waterearthairh=zFigure 6.1: Schematic of the spatial  ow domain.Electrical conduction current  ow permeates the entire medium and  is assigned a valuerepresentative of average unsaturated or partially saturated conditions above the phreaticsurface. Streaming current  ow may be treated using a saturated or variably saturatedapproach. In a saturated approach, we consider streaming current  ow to be limited tothe saturated zone and L is assigned a null value above the phreatic surface. In a variablysaturated approach, the in uence of streaming current  ow in the unsaturated zone isincluded. The behaviour of the streaming potential coupling coe–cient has been studiedfor partially saturated conditions [Guichet et al., 2003; Revil and Cerepi, 2004; Moore et al.,2004] and tends to decrease in magnitude with saturation level. However, the behaviour ofstreaming current  ow in the unsaturated zone is not fully understood. In this paper weassume that the phreatic surface acts as a zero  ux boundary for streaming current  ow.165Chapter 6. 3-D inversion of self-potential data6.3.4 Active cell approachIt is not reasonable to recover a hydraulic head model in regions of the mesh where it isnot physically realistic to do so. Examples of this include air cells in problems of irregulargeometry, and cells corresponding to impervious non-porous structures such as well-casings.Similarly, it is not reasonable to recover a hydraulic head model above the phreatic surfacein problems where we are only concerned with  ow in the saturated zone. Consequently, weseparate the domain into active and inactive regions for the purposes of solving the inverseproblem. We solve for the active component of the hydraulic head vector and specify a nullhead value in inactive regions.Considering only active model components, (6.13) becomes:'GATWdTWdGA +fl£WsATWsA +WATWA⁄“hA =GATWdTWddobs +flWsATWsAhrefA ; (6.14)where the subscript A denotes the active vector or matrix. Solving this equation efiectivelyreduces the size of the problem, and ensures that head is only recovered in cells where it isphysically reasonable to do so.The active component hA of the head vector is isolated using the expression:hA = RATh ; (6.15)where RA is a matrix operator that specifles active cells in the mesh. The complete headvector is reconstructed for the purposes of evaluating the data misflt using:h = RAhA +RIhI ; (6.16)where hI is the inactive head vector component and RI identifles inactive cells in the mesh.Similarly, the active columns of matrix G are isolated using the expression:GA = GRA ; (6.17)and the complete matrix is deflned by:G = GARAT +GIRIT: (6.18)166Chapter 6. 3-D inversion of self-potential dataA zero  ux boundary imposed by null conductivity values inG is upheld even when inactivecells are removed from the matrix. This is a result of the flnite volume approximation, whichimplements harmonic averaging to calculate the conductivity at cell faces. The boundarycondition is therefore applied at the interface between active and inactive regions, and isnot afiected by the removal of inactive cells.Special consideration must be given to the construction of WA, since this matrix con-tains a discrete approximation of the flrst derivative. We do not wish to minimize the modelgradient across boundaries between active and inactive cells. Therefore, we must removethe in uence of inactive cells on adjacent cells in addition to removing inactive columns ofthe matrix.6.4 Flow under a cut-ofi wallThe example of seepage under a cut-ofi wall is used to demonstrate a successful applicationof the inversion algorithm using data acquired under controlled conditions. We use thisexample to illustrate how the solution is improved by incorporating prior information aboutthe problem into the model objective function, and by including data acquired at depthbelow the ground surface.Following a discussion of the laboratory measurements and details of the discrete model,we present the results of inverse modelling as four progressive case studies. Each case incor-porates additional information about the problem. The recovered head models obtained byinverting SP data collected at the earth surface and at depth are evaluated for each case.In addition, we compare the recovered head models to observed hydraulic head values toevaluate the quality of the result.6.4.1 Laboratory measurementsA laboratory-scale model of a cut-ofi wall was constructed to measure self-potential andhydraulic head in a saturated homogeneous system subject to steady-state seepage. Figure6.2 presents a schematic of the acrylic tank showing tank dimensions, the position of thecut-ofi wall, and manometer port locations. The acrylic tank was fllled with uniform quartz167Chapter 6. 3-D inversion of self-potential datasand compacted in layers up to a height of 38 cm. Steady-state seepage was initiated bymaintaining constant surface water levels of 70.8 cm and 40 cm respectively upstream anddownstream of the cut-ofi wall.733817.41.225.061.313.716.536.8 45.064.329.01070.84016.5 22.1 22.09.927.644.55 6 789 1011xyz×SP referenceFigure 6.2: Schematic of the laboratory cut-ofi wall model, indicating tank dimensions andthe position of manometer measurement ports, labelled as 5 to 11 (all dimensions in cm).Self-potential data were acquired using a high-impedance voltmeter and a pair of non-polarizing Ag-AgCl electrodes. One electrode served as a flxed reference and was positionednear the upstream edge of the tank at the surface of the upstream water column, as indicatedin Figure 6.2. The second electrode was used to occupy survey stations along the surface ofeach water column, and within seven  uid-fllled manometer tubes located along the frontface of the tank. Data were collected with a measurement sensitivity of 0.01 mV.Hydraulic head measurements were obtained by manually recording water levels in theseven manometers. These readings were compared with the recovered head model to helpevaluate the quality of inversion result, as discussed in the sections that follow.Water was circulated through the system until a state of ionic equilibrium was reached168Chapter 6. 3-D inversion of self-potential databetween the soil and  uid, to ensure that streaming potential was the primary coupled  owmechanism in the system. A state of chemical equilibrium was conflrmed by  uid resistivityand self-potential measurements that were stable with time.The streaming potential coupling coe–cient and resistivity were measured directly inthe tank to derive a cross-coupling conductivity of L = 7 £ 10¡5A=m2 for the saturatedquartz sand using the relation L = ¡C=‰. A streaming potential coupling coe–cient of C =9 mV/m (0.9 mV/kPa) was obtained by calculating the linear flt between the measured self-potential and hydraulic head values under steady-state  ow conditions. The resistivity ofthe saturated sand was characterized as ‰ = 130 ›m through DC resistivity measurementsusing a Wenner array conflguration. A  uid resistivity of 40 ›m was measured using aportable conductivity meter. These quantities were used to build the discrete physicalproperty models described in the next section.6.4.2 Discrete model developmentSolution of the inverse problem requires that we approximate the study region using adiscrete model and prescribe electrical property values to each cell in the mesh. The tankwas modelled using a 37 £ 5 £ 35 cell mesh with a uniform cell dimension of 2 cm. Thelimits of the mesh were chosen to correspond with the tank dimensions, since the sides ofthe acrylic tank impose a no- ow boundary for both conduction and convection current ow.The upper surface of the sand acts as a no- ow boundary for convection current  ow,since the porous medium is completely saturated. Consequently, all air and surface watercells were assigned a null cross-coupling conductivity, and the sand cells were assignedthe measured value noted above. Cells used to represent the acrylic cut-ofi wall wereassigned a null cross-coupling value. The air-water interface serves as a no- ow boundaryfor conduction current  ow and air cells were therefore assigned a null electrical conductivity.Water and sand cells were assigned the measured values noted above. Cells representingthe acrylic cut-ofi wall were assigned a null conductivity.In all inversion models, the relative spatial weighting term fiy was set 3 orders of mag-nitude higher than fix and fiz in the model objective function to re ect the planar nature169Chapter 6. 3-D inversion of self-potential dataof seepage  ow under the cut-ofi wall. The data misflt was calculated using a standarddeviation of † = 0:05mV to represent observed noise levels for this experiment. A solutionto the inverse problem was achieved by minimizing the model objective function subject tothe data misflt being equal to the number of data points used in the inversion. Table 6.1lists the optimum regularization parameter and corresponding data misflt and model normfor each inverse solution. In cases 1, 2 and 3 where no prior head information is supplied(i.e. fis = 0), the solutions are referenced to a known head at a single point on the surface ofthe upstream water column to enable a comparison between recovered and observed heads.In case 4, prior head information is included by setting ws to be 4 orders of magnitudehigher in cells where the hydraulic head is known.Case Data set flo fl⁄ ˆd ˆm kh¡hobsk1 surface 200,000 21,868 8 0.01107 0.00113subsurface 200,000 22,968 7 0.00702 0.00072all 200,000 28,145 15 0.01149 0.001502 surface 200,000 29,225 8 0.00801 0.00194subsurface 200,000 25,863 7 0.00589 0.00012all 200,000 35,330 15 0.00843 0.000303 surface 200,000 85,317 8 0.00322 0.00199subsurface 200,000 60,437 7 0.00290 0.00050all 200,000 122,830 15 0.00329 0.000424 surface 200,000 n/a 0.88 0.00396 0.00029subsurface 1,500,000 207,110 7 0.00399 0.00027all 1,500,000 494,630 15 0.00397 0.00024Table 6.1: Parameters that characterize the inversion results for each case of modelling ow under a cut-ofi wall, listed according to input data set: initial (flo) and optimum (fl⁄)regularization parameters, data misflt (ˆd) and model norm (ˆm). The difierence betweenobserved and recovered head values at the seven manometer ports is quantifled using an l-1norm measure. Note: ˆd < ˆ⁄d for all values of fl in case 4 using only surface data.170Chapter 6. 3-D inversion of self-potential data6.4.3 Case 1: Homogeneous subsurface approximationAs a flrst pass at interpreting the SP data, we consider that we have no prior informationabout the head distribution and no knowledge of the vertical extent or integrity of the cut-ofiwall below the ground surface. In this case we assume a homogeneous subsurface. Figure 6.3illustrates the distribution of the various input parameters specifled in the inverse problem.Active and inactive regions of the model are identifled in Figure 6.3(a). Cells representingair and the acrylic cut-ofi wall above the surface of the sand were set as inactive for thepurposesoftheinversion, sinceitisunrealistictosolveforheadinthesecells. Theprescribedcross-coupling conductivity and electrical resistivity distributions re ect the homogeneoussubsurface assumption and are shown in Figures 6.3(b) and (c).inactiveactive(a)L = 7 ×10-5A m-2(b)ρ= 130 Ω mρ= 40 Ω m(c)Figure 6.3: Cut-ofi wall model input - homogeneous subsurface approximation: a) activemodel cells; b) cross-coupling conductivity distribution; c) electrical resistivity distribution.Three separate inversions were performed using data collected along the surface of thewater columns, at points below surface corresponding to manometer locations, and thecombination of surface and subsurface data sets. Figures 6.4, 6.5 and 6.6 present theobserved data locations and the results of each inversion.Figure 6.4 presents the inversion results for the self-potential data set collected only atsurface. Figure 6.4(a) denotes the eight SP data locations along the surface of upstreamand downstream water columns. Figure 6.4(b) shows a vertical cross-section through the171Chapter 6. 3-D inversion of self-potential datarecovered hydraulic head model, which is smoothly-varying and ranges from 72 cm in theupstream water column to 40 cm downstream. A hydraulic head equipotential coincideswith the known location of the cut-ofi wall below surface. The flt between observed andpredicted SP data is reasonable, as shown in Figure 6.4(c). However, a comparison betweenobserved and recovered head values at each of the manometer ports in Figure 6.4(d) re-veals that a relatively poor flt exists at manometer ports 7 and 9, positioned immediatelyupstream and downstream of the known location of the wall.The inversion results for self-potential data collected at points below surface are pre-sented in Figure 6.5. Figure 6.5(a) denotes the seven SP data locations, which correspondto the position of manometer ports along the side of the tank. The recovered model resultshown in Figure 6.5(b) displays signiflcantly difierent behaviour than that obtained usingthe surface data, with more structure in the vicinity of observed data locations. Althougha good SP data flt exists, Figure 6.5(d) shows that the recovered model underpredicts theobserved head in manometers 5, 6 and 7 downstream of the known location of the wall.Figure 6.6 presents the inversion results for the complete SP data set consisting ofboth surface and subsurface data points. Figure 6.6(b) displays the recovered head modelobtained by including all 15 data points as indicated in Figure 6.6(a). The recovered headmodel is smoothly varying and comparable to that from the inversion of surface data.However, small head perturbations in the vicinity of manometer ports 7 and 9 may indicatea tendency for the head contours to curve inwards toward the known location of the wall.Figure 6.6(c) shows a good SP data flt, but Figure 6.6(d) shows that the recovered modelunderpredicts the observed head downstream of the known location of the wall.Inconsistencies between the recovered models suggest that the homogeneous subsurfaceapproximation is inadequate. The inversion of surface data and all data points result insimilar hydraulic head models, where a head equipotential coincides with the known locationof the cut-ofi wall (Figures 6.4(b) and 6.6(b)). These models are plausible only if the barrieris pervious with a hydraulic conductivity essentially equivalent to that of the soil.172Chapter 6. 3-D inversion of self-potential data151413124321(a)m0.720.670.6130.560.5070.4530.4(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.4: Cut-ofi wall model results - homogeneous subsurface approximation - surfacedata: a) observation locations; b) vertical cross-section of recovered head model; c) com-parison of observed and predicted SP data at each measurement location; d) comparison ofobserved and recovered model head values in each manometer.173Chapter 6. 3-D inversion of self-potential data111098765(a)m0.720.670.6130.560.5070.4530.4(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.5: Cut-ofi wall model results - homogeneous subsurface approximation - subsurfacedata: a) observation locations; b) vertical cross-section of recovered head model; c) com-parison of observed and predicted SP data at each measurement location; d) comparison ofobserved and recovered model head values in each manometer.174Chapter 6. 3-D inversion of self-potential data151413121110987654321(a)m0.720.670.6130.560.5070.4530.4(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.6: Cut-ofi wall model results - homogeneous subsurface approximation - all data:a) observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.175Chapter 6. 3-D inversion of self-potential data6.4.4 Case 2: Heterogeneous subsurface approximationWe will now incorporate a priori information about the existence and depth of the cut-ofi wall. Without any prior information about the geometry of this structure, we assumea vertical barrier exhibiting the same electrical properties as the segment of the cut-ofiwall that exists above ground. In this instance, the material is acrylic and so we assumenull values of cross-coupling conductivity and electrical conductivity in subsurface cellsthat represent the barrier, as illustrated in Figures 6.7(b) and (c). The chosen electricalproperties of the barrier make this structure impervious to electrical current  ow. However,by setting these cells active for the purposes of the inversion, as indicated in Figure 6.7(a),we are assuming that the wall is hydraulically pervious to some degree.inactiveactive(a)L = 7 ×10-5A m-2(b)ρ= 130 Ω mρ= 40 Ω m(c)Figure 6.7: Cut-ofi wall model input - heterogeneous subsurface approximation: a) activemodel cells; b) cross-coupling conductivity distribution; c) electrical resistivity distribution.Figures 6.8, 6.9 and 6.10 present the inversion results respectively for surface data, sub-surface data, and the complete data set. The recovered models exhibit similar characteris-tics to those presented in Figures 6.4 to 6.6 for the homogeneous subsurface approximation.However, as in Case 1, the recovered model resulting from the inversion of subsurface datapoints is not consistent with those resulting from the inclusion of surface data. The lack ofany signiflcant head gradient across the known location of the wall suggests that the walldoes not impede  uid  ow, which we know not to be true.176Chapter 6. 3-D inversion of self-potential data151413124321(a)0.730.670.6230.570.5170.4630.41m(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.8: Cut-ofi wall model results - heterogeneous subsurface approximation - surfacedata: a) observation locations; b) vertical cross-section of recovered head model; c) com-parison of observed and predicted SP data at each measurement location; d) comparison ofobserved and recovered model head values in each manometer.177Chapter 6. 3-D inversion of self-potential data111098765(a)m0.730.670.6230.570.5170.4630.41(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.9: Cut-ofi wall model results - heterogeneous subsurface approximation - sub-surface data: a) observation locations; b) vertical cross-section of recovered head model; c)comparison of observed and predicted SP data at each measurement location; d) comparisonof observed and recovered model head values in each manometer.178Chapter 6. 3-D inversion of self-potential data151413121110987654321(a)m0.730.670.6230.570.5170.4630.41(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.10: Cut-ofi wall model results - heterogeneous subsurface approximation - all data:a) observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.179Chapter 6. 3-D inversion of self-potential data6.4.5 Case 3: Inactive wall approximationIf the cut-ofi wall acts as a ideal no- ow barrier below surface, then it is not physically real-istic to solve for head in these grid cells. Consequently, we now set the wall cells as inactivefor the purposes of the inversion, as shown in Figure 6.11(a). We choose to incorporateadditional information about the geometry of the wall by specifying a wall thickness con-sistent with that observed above surface. In a fleld investigation, prior information aboutthe geometry and material properties of the cut-ofi wall could be garnered from designdrawings.inactiveactive(a)L = 7 ×10-5A m-2(b)ρ= 130 Ω mρ= 40 Ω m(c)Figure 6.11: Cut-ofi wall model input - inactive wall approximation: a) active model cells;b) cross-coupling conductivity distribution; c) electrical resistivity distribution.Figure 6.12 presents the inversion results obtained using surface data. The presenceof the impervious cut-ofi wall imposes a no- ow boundary that is clearly represented inthe recovered head distribution shown in Figure 6.12(b). Observed and recovered headscorrelate well, but Figure 6.12(d) suggests that the recovered head model may exhibit asmaller dynamic range. The inversion of subsurface data and all data points shown inFigures 6.13 and 6.14 results in head models that are consistent with Figure 6.12(b). Thisgives confldence that the inactive wall approximation is a valid representation of the cut-ofiwall. However, the dynamic range of these models is over 2 cm smaller than the knownhead distribution. This suggests that including known head data may improve the solution.180Chapter 6. 3-D inversion of self-potential data151413124321(a)m0.710.630.6170.570.5230.470.43(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.12: Cut-ofi wall model results - inactive wall approximation - surface data: a)observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.181Chapter 6. 3-D inversion of self-potential data111098765(a)m0.710.630.6170.570.5230.470.43(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.13: Cut-ofi wall model results - inactive wall approximation - subsurface data: a)observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.182Chapter 6. 3-D inversion of self-potential data151413121110987654321(a)m0.710.630.6170.570.5230.470.43(b)2 4 6 8 10 12 14−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.14: Cut-ofi wall model results - inactive wall approximation - all data: a) observa-tion locations; b) vertical cross-section of recovered head model; c) comparison of observedand predicted SP data at each measurement location; d) comparison of observed and re-covered model head values in each manometer.183Chapter 6. 3-D inversion of self-potential data6.4.6 Case 4: Incorporating known head valuesThe subsurface representation of the cut-ofi wall using inactive cells discussed in Section6.4.5 signiflcantly improves the inversion result. We now seek to further improve the solutionby incorporating observed values of hydraulic head as a constraint.The observed water levels in the upstream and downstream water columns were includedin the reference model href, as shown in Figure 6.15(a). In a fleld situation, these observa-tions could be obtained from water depth measurements made from surface. In water cells,the relative weighting parameter ws was set 4 orders of magnitude higher than in adjoiningcells to encourage a close flt between the recovered and reference head values in the watercolumns.Figure 6.16 presents the inversion results using surface data. The recovered modelexhibits a very similar character to that shown in Figure 6.12(b), except that the dynamicrange of head is consistent with that observed. This is evidenced by the close flt betweenobserved and recovered head values, as shown in Figure 6.16(d). The combination of correctmodel geometry, stipulated reference head values at surface, and a lack of SP data at depthserve to limit the solution such that the SP surface data are essentially not required torecover a model that satisfles the constraints. A very small data misflt is achieved regardlessof the chosen regularization parameter, as indicated in Table6.1. This was not the case whenthe inversion was performed using subsurface data. Figures 6.17 and 6.18 respectively showthe inversion results from subsurface and all data points. The recovered head models arenearly identical and similar to that obtained using surface data alone. The close flt betweenrecovered and observed head values at manometer locations make 6.18(b) the preferredsolution to represent the hydraulic head distribution around the cut-ofi wall.184Chapter 6. 3-D inversion of self-potential datah = 0h = 0.4 mh = 0.54 mh = 0.708 m(a)inactiveactive(b)L = 7 ×10-5A m-2(c)ρ= 130 Ω mρ= 40 Ω m(d)Figure 6.15: Cut-ofi wall model input - reference head in water column: a) reference hy-draulic head model; b) active model cells; c) cross-coupling conductivity distribution; d)electrical resistivity distribution.185Chapter 6. 3-D inversion of self-potential data151413124321(a)0.7080.6560.6050.540.5030.4510.4m(b)1 2 3 4 5 6 7 8 9 101112131415−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.16: Cut-ofi wall model results - reference head in water column - surface data: a)observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.186Chapter 6. 3-D inversion of self-potential data111098765(a)m0.7080.6560.6050.540.5030.4510.4(b)1 2 3 4 5 6 7 8 9 101112131415−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.17: Cut-ofi wall model results - reference head in water column - subsurface data:a) observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.187Chapter 6. 3-D inversion of self-potential data151413121110987654321(a)m0.7080.6560.6050.540.5030.4510.4(b)1 2 3 4 5 6 7 8 9 101112131415−0.500.511.522.53DatumSelf−potential (mV)predicted dataobserved data(c)5 6 7 8 9 10 110.40.450.50.550.60.650.70.75Manometer portHydraulic head (m)recovered modelobserved(d)Figure 6.18: Cut-ofi wall model results - reference head in water column - all data: a)observation locations; b) vertical cross-section of recovered head model; c) comparison ofobserved and predicted SP data at each measurement location; d) comparison of observedand recovered model head values in each manometer.188Chapter 6. 3-D inversion of self-potential data6.5 ConclusionWe have developed an inversion method that recovers a 3-D model of hydraulic head frommeasured self-potential data in systems where streaming potential is the dominant signal.Our approach permits an interpretation of subsurface  ow patterns directly from the geo-physical data. The recovered hydraulic head model may be studied on its own to identifysubsurface  ow patterns, or used as input to a subsequent inversion to recover other hy-draulic parameters, such as hydraulic conductivity.The inverse problem was formulated as an unconstrained optimization in which weminimize a model objective function subject to fltting the measured data. This approachenables us to cope with the inherent non-uniqueness of the underdetermined problem. Theability to account for heterogeneous cross-coupling conductivity and electrical conductivitydistributions was included as these can impact the quality of the solution. The modelobjective function was designed to allow the inclusion of prior information about the natureof the hydraulic head distribution and observed head values by means of a reference modeland 3-D weighting functions. This additional information can help to constrain the solutionand improve the quality of the recovered model.The inversion algorithm was successfully applied to study  ow under a cut-ofi wall in alaboratory tank model, where the recovered head distribution was found to be consistentwith observed head values. This example served to illustrate how prior information isincorporated into the inverse problem and how it can improve the solution.189Chapter 6. 3-D inversion of self-potential data6.6 ReferencesAbdelrahman, E. M., and S. M. Sharafeldin, A least-squares approach to depth determina-tion from self-potential anomalies caused by horizontal cylinders and spheres, Geophysics,62(1), 44{48, 1997.Archie, G. E., The electrical resistivity log as an aid in determining some reservoir char-acteristics, Transactions of the Society of Petroleum Engineers of the American Instituteof Mining, Metallurgical and Petroleum Engineers (AIME), 146, 54{67, 1942.Atchuta Rao, D., and H. V. Ram Babu, Quantitative interpretation of self-potential anom-alies due to two-dimensional sheet-like bodies, Geophysics, 48(12), 1659{1664, 1983.Bhattacharya, B. B., and N. Roy, A note on the use of a nomogram for self-potentialanomalies, Geophysical Prospecting, 29, 102{107, 1981.Birch, F. S., Imaging the water table by flltering self-potential proflles, Ground Water,36(5), 779{782, 1998.Darnet, M., G. Marquis, and P. Sailhac, Estimating aquifer hydraulic properties from theinversion of surface Streaming Potential (SP) anomalies, Geophysical Research Letters,30(13), 1679, doi:10.1029/2003GL017631, 2003.de Witte, L., A new method of interpretation of self-potential fleld data, Geophysics, 13(4),600{608, 1948.Di Maio, R., and D. Patella, Self-potential anomaly in volcanic areas: The Mt. Etna casehistory, Acta Vulcanologica, 4, 119{124, 1994.Fitterman, D. V., Electrokinetic and magnetic anomalies associated with dilatant regionsin a layered earth, Journal of Geophysical Research, 83(B12), 5923{5928, 1978.Fitterman, D. V., and R. F. Corwin, Inversion of self-potential data from the Cerro Prietogeothermal fleld, Mexico, Geophysics, 47(6), 938{945, 1982.190Chapter 6. 3-D inversion of self-potential dataFournier, C., Spontaneous potentials and resistivity surveys applied to hydrogeology in avolcanic area: Case history of the Cha^‡ne des Puys (Puy-de-D^ome, France), GeophysicalProspecting, 37, 647{668, 1989.Guichet, X., L. Jouniaux, and J. P. Pozzi, Streaming potential of a sand columnin partial saturation conditions, Journal of Geophysical Research, 108(B3), 2141, doi:10.1029/2001JB001517, 2003.H˜ammann, M., H. R. Maurer, A. G. Green, and H. Horstmeyer, Self-potential imagereconstruction: Capabilities and limitations, Journal of Environmental and EngineeringGeophysics, 2(1), 21{35, 1997.Ishido, T., and H. Mizutani, Experimental and theoretical basis of electrokinetic phe-nomena in rock-water systems and its application to geophysics, Journal of GeophysicalResearch, 86(B3), 1763{1775, 1981.Jouniaux, L., andJ.P.Pozzi, Streamingpotentialandpermeabilityofsaturatedsandstonesunder triaxial stress: Consequences for electrotelluric anomalies prior to earthquakes, Jour-nal of Geophysical Research, 100(B6), 10,197{10,209, 1995.Minsley, B. J., J. Sogade, and F. D. Morgan, Three-dimensional source inversion of self-potential data, Journal of Geophysical Research, 112, B02202, doi:10.1029/2006JB004262,2007.Moore, J. R., S. D. Glaser, H. F. Morrison, and G. M. Hoversten, The streaming potentialof liquid carbon dioxide in Berea sandstone, Geophysical Research Letters, 31, L17610,doi:10.1029/2004GL020774, 2004.Morgan, F. D., E. R. Williams, and T. R. Madden, Streaming potential properties of West-erly granite with applications, Journal of Geophysical Research, 94(B9), 12,449{12,461,1989.Nourbehecht, B., Irreversible thermodynamic efiects in inhomogeneous media and theirapplications in certain geoelectric problems, Ph.D. thesis, Massachusetts Institute of Tech-nology, Cambridge, 1963.191Chapter 6. 3-D inversion of self-potential dataOnsager, L., Reciprocal relations in irreversible processes, I, Physical Review, 37, 405{426,1931.Overbeek, J. T. G., Electrokinetic phenomena, in Colloid Science, Irreversible Systems,vol. 1, edited by H. R. Kruyt, Elsevier Publishing Company, Amsterdam, 1952.Patella, D., Introduction to ground surface self-potential tomography, GeophysicalProspecting, 45(4), 653{681, 1997.Paul, M. K., Direct interpretation of self-potential anomalies caused by inclined sheets ofinflnite horizontal extensions, Geophysics, 30(3), 418{423, 1965.Revil, A., and A. Cerepi, Streaming potentials in two-phase  ow conditions, GeophysicalResearch Letters, 31, L11605, doi:10.1029/2004GL020140, 2004.Revil, A., V. Naudet, J. Nouzaret, and M. Pessel, Principles of electrography appliedto self-potential electrokinetic sources and hydrogeological applications, Water ResourcesResearch, 39(5), 1114, doi:10.1029/2001WR000916, 2003.Revil, A., V. Naudet, and J. D. Meunier, The hydroelectric problem of porous rocks:Inversion of the position of the water table from self-potential data, Geophysical JournalInternational, 159, 435{444, 2004.Sailhac, P., and G. Marquis, Analytic potentials for the forward and inverse modeling ofSP anomalies caused by subsurface  ow, Geophysical Research Letters, 28(9), 1851{1854,2001.Shefier, M. R., and D. W. Oldenburg, Three-dimensional forward modelling of streamingpotential, Geophysical Journal International, 169, 839{848, doi:10.1111/j.1365-246X.2007.03397.x, 2007.Shefier, M. R., J. A. Howie, and P. M. Reppert, Laboratory measurements of the streamingpotential coupling coe–cient and resistivity in well-graded soils, 2007.Shi, W., Advanced modeling and inversion techniques for three-dimensional geoelectricalsurveys, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, 1998.192Chapter 6. 3-D inversion of self-potential dataSill, W. R., Self-potential modeling from primary  ows, Geophysics, 48(1), 76{86, 1983.Y˜ung˜ul, S., Interpretation of spontaneous polarization anomalies caused by spheroidalorebodies, Geophysics, 15(2), 237{246, 1950.193Chapter 7ConclusionThe main objective of the research presented in this thesis was to investigate what use-ful information about hydraulic parameters and embankment integrity could be garneredfrom self-potential data in systems where streaming potential is the driving mechanism.This objective was fulfllled through the development and application of forward and inversemodelling algorithms, and a laboratory investigation of the electrical properties that governthe streaming potential phenomenon. Figure 7.1 highlights the key contributions of the the-sis research and illustrates how they combine to advance the current state of self-potentialdata interpretation. The motivation for this work stems from the application to embank-ment dam seepage monitoring and detection, which demands a quantitative interpretationof hydraulic parameters from the geophysical data.A three-dimensional forward modelling algorithm was developed using a coupled  owmodel of streaming potential to study variably saturated  ow problems of complicatedgeometry. This approach is necessary for the study of practical embankment seepage prob-lems. The forward model enables us to study the self-potential response to a given seepageregime, and serves as a basis for the inverse problem.The practical interpretation of measured SP data requires an inverse methodology. Aninversion algorithm was developed to recover a three-dimensional distribution of hydraulichead from self-potential data. This was the flrst documentation of such an approach, andit enables us to extract pertinent hydraulic information from the geophysical data. Therecovered head model may be used on its own to interpret  ow patterns, or used as in-put to another inversion routine to recover other hydraulic parameters such as hydraulicconductivity or  ow rate.Realistic estimates of the electrical properties are required for an efiective SP datainterpretation using the numerical modelling techniques. A lack of available data on the194Chapter 7. Conclusionelectrical properties of soils prompted a laboratory study to measure these parameters. Anew laboratory apparatus was designed and developed to perform measurements of thestreaming potential coupling coe–cient C and electrical resistivity ‰ on the same specimenin order to characterize the streaming current cross-coupling coe–cient L for well-gradedsoils. This apparatus was used in a laboratory investigation to study the in uence of soil and uid properties on L and to characterize this property for typical embankment soils. Theresults of this investigation showed that L does not vary considerably for typical geologicmaterials. This suggests that we may not have to characterize L with the same rigor as‰ at a given site when interpretating SP data using the forward and inverse modellingtechniques.Realistic estimates of electrical coupled flow properties3-D forward model of streaming potentialInterpretation of hydraulic conditions from SP data3-D inversion for hydraulic headFigure 7.1: Key contributions to the quantitative interpretation of hydraulic conditionsfrom SP data.The following sections discuss the practical contributions of the research to SP surveydesign and data interpretation, and comment on areas of future work, with speciflc focuson the study of embankment dams.7.1 Practical contributions7.1.1 Survey designSelf-potential surveys can be conducted as single or repeat surveys, where a fleld crew isdeployed to re-occupy survey stations, or in an automated monitoring framework using apermanently-installed array of electrodes. Practical fleld survey design for single or repeatsurveys is often guided by soil conditions and access on the upstream and downstream faces195Chapter 7. Conclusionof the dam. Most often, the dam crest ofiers the easiest access for SP data collection.However, as shown by the examples in Chapters 2 and 5, SP data at the crest often showthe least amount of variation in response to changing hydraulic conditions. The results ofChapter 5 showed that SP measurements at depth are most favourable for detecting smallvariations in seepage patterns caused by internal erosion. However, the speciflc hydraulicboundary conditions, structure and physical properties will dictate the seepage regime andresulting SP distribution at a given site.Efiective placement of electrodes in the design of a fleld survey or monitoring networkwill maximize the ability of the method to detect changes that are of particular interest.A mathematical description of primary and secondary sources of charge was derived inChapter 2 to facilitate a conceptual understanding of the polarity and accumulation patternof charge caused by  uid  ow and physical property boundaries in the subsurface. Thisequation can be used to give a flrst pass estimate of the polarity and distribution of chargefor a given system in a practical fleld setting. The forward modelling algorithm enablesa thorough examination of the self-potential distribution resulting from a comprehensiveseepage analysis in 3-D. This code can be used to evaluate the suitability of self-potentialmeasurements for a given site, and to design optimum electrode placement and surveyconflgurations.7.1.2 SP data interpretationThe forward and inverse modelling routines developed in Chapters 2 and 6 can be appliedto interpret hydraulic conditions from SP data, given some knowledge of the structure andelectrical properties of the subsurface.The cross-coupling coe–cient is a fundamental parameter required in numerical mod-elling, which had not been studied previously for well-graded soils. The results of thelaboratory investigation described in Chapter 4 concluded that the streaming current cross-coupling coe–cient does not vary over a wide range based on the results of testing andpublished data in the literature. However, a limited number of soil types were studiedand the reported values and trends may not apply to every site. The lab apparatus andtesting methods described in Chapter 3 and documented in detail in a report by Shefier196Chapter 7. Conclusion[2005] provide both guidance and a means with which to measure the streaming currentcross-coupling coe–cient. If laboratory measurements are not possible, theoretical relationsmay be used to estimate values of L. Special considerations for the practical application oftheoretical estimates, particularly in resistive environments, are discussed in Chapter 4.The interpretation of SP data acquired from a conventional survey provides a snapshotof hydraulic conditions, and is most useful for characterizing global or background seepageconditions at a given site. A comparison of data acquired from repeat surveys can help toidentify seepage-related anomalies. However, the interpretation of SP anomalies from dataacquired in a conventional survey at an embankment site can be very challenging. Thesurface expression of SP anomalies caused by preferential seepage within the dam may bevery small, as suggested by the results of Chapter 5. Noise in the data can further hinderdetection. Furthermore, it is di–cult to characterize and assign physical property distrib-utions that re ect actual conditions. Fluctuations in the reservoir level lead to transienthydraulic conditions within the core of the embankment. Seasonal variations in temperatureand reservoir properties can lead to variations in the electrical properties. The boundaryconditions at the time of a conventional survey may not re ect current conditions insidethe dam.These di–culties suggest that a monitoring approach to SP data collection and interpre-tation is required. Chapter 5 illustrated that the localized hydraulic head and SP responsesto the development of a pervious defect peak prior to any detectable increase in seepage ow from the dam body. This suggests that SP electrodes installed in conjunction withor in place of piezometers could support or provide evidence of a problem prior to failureof the core. This is one example of how SP measurements could be integrated into a damseepage monitoring system. Long-term monitoring afiords an increased sensitivity to thedevelopment of localized gradients caused by internal erosion, since temporal changes canbe tracked more efiectively. Deviations from an established background response can thenbe evaluated using numerical methods to determine their hydraulic signiflcance. Acquiringdata in a monitoring framework also afiords the opportunity to study the temporal SP re-sponse to transient hydraulic behaviour and seasonal variations, and to determine whetherthese factors assist or hinder the detection of anomalous behaviour.197Chapter 7. ConclusionThe simplest means of interpreting SP data acquired from a monitoring network wouldbe to track the SP signals with time, much like pressure measurements are typically mon-itored. A range of variation typical of normal operating conditions could be establishedfrom observations over time and predictive modelling. A steady deviation in the SP signallevel outside of this normal range would then trigger a warning that conditions inside thedam may be changing.A more comprehensive picture of changing hydraulic conditions could be obtained byintegrating inverse modelling into the interpretation methodology. Recovered head modelsfrom successive inversions could be used to identify areas where localized gradients maybe developing. These models could be used to evaluate the location and severity of theanomalous region in conjunction with other investigative tools.7.2 Future workThe ultimate goal of this research is to investigate how SP measurements can be incorpo-rated into an embankment dam seepage monitoring and detection methodology to extractpertinent information about hydraulic conditions. A flrst step is to interpret an SP dataset to infer the hydraulic conditions within an embankment. Of particular interest is theapplication of the inversion algorithm to a fleld-scale problem.A sensitivity analysis must be conducted using the inversion algorithm to evaluate thein uence of the prescribed electrical property distributions on the recovered hydraulic headmodel. The sensitivity of the solution to the electrical conductivity and the cross-couplingconductivity, in particular, will determine how much efiort should be put towards charac-terizing these properties.The main challenge limiting the development of SP interpretation strategies in embank-ment dam problems is a lack of comprehensive fleld data sets. Conventional surveys havebeen performed at a number of sites, but to date SP monitoring has only been performedon a handful of sites using a single line array of electrodes along the crest. This suggestsa strong need for carefully designed, site-speciflc monitoring array installations to establishbaseline readings and enable further research into the efiectiveness of SP for dam seepage198Chapter 7. Conclusionapplications.The capability of the SP method to detect small streaming potential signals is governedin a large part by the measurement sensitivity. An evaluation of SP fleld data acquisitionand processing methods must be performed to determine if improvements can be made toreduce noise levels in the data, particularly for monitoring applications.The contribution of the unsaturated  ow component to the measured SP signal maybe signiflcant in certain problems. Consequently, it is of interest to examine how best toincorporate this  ow contribution in the forward modelling algorithm by assigning values ofL as a function of saturation. In the inverse problem, allowing the electrical properties tovary with saturation renders the problem non-linear. The beneflt of solving the non-linearproblem would have to be evaluated following further application of the linear algorithm.In some cases, streaming potential may not be the sole source mechanism, as is assumedin the present research, and other sources will have to be accounted for. External sourcesof current, due to cathodic protection devices or other infrastructure, will have a signiflcantefiect on measured SP data and should be stopped or minimized prior to data collection.Other natural source mechanisms are a function of mineralogy,  uid properties and siteconditions. Consequently, the relative importance of each mechanism must be weighed ona site-by-site basis. Since the measured SP signal results from the superposition of sourcesin the subsurface, other source mechanisms could be studied by removing the streamingpotential component from the data, assuming the primary  ows are decoupled. The forwardmodelling algorithm can be used to predict the streaming potential component or otherindividual coupled  ow mechanisms to aid in the interpretation.Temperature and resistivity measurements have been implemented to monitor and de-tect anomalous seepage in embankments [Johansson and Dahlin, 1996; Sj˜odahl, 2006; Jo-hansson, 1997]. Estimates of anomalous seepage  ow rates may be interpreted from tem-perature data, temporal variations in the electrical resistivity of the core may be interpretedfrom the inversion of resistivity data, and the current research showed that hydraulic headmay be interpreted from the inversion of self-potential data. It is of great interest to in-vestigate the joint interpretation of these data to give a more comprehensive picture ofsubsurface conditions.199Chapter 7. Conclusion7.3 ReferencesJohansson, S., Seepage monitoring in embankment dams, Ph.D. thesis, Royal Institute ofTechnology, Sweden, 1997.Johansson, S., and T. Dahlin, Seepage monitoring in an earth embankment dam by re-peated resistivity measurements, European Journal of Environmental and EngineeringGeophysics, 1, 229{247, 1996.Shefier, M. R., Investigation of geophysical methods for assessing seepage and internalerosion in embankment dams: Laboratory testing of the streaming potential phenomenonin soils, Tech. rep., Canadian Electricity Association Technologies Inc. (CEATI), ReportT992700-0205B/2, Montreal, 2005.Sj˜odahl, P., Resistivity investigation and monitoring for detection of internal erosion andanomalous seepage in embankment dams, Ph.D. thesis, Lund University, Sweden, 2006.200Appendix ADiscrete approximation using theflnite-volume methodPoisson’s equation for the self-potential distribution with no imposed external sources ofcurrent  ow is deflned as:r¢  r` = ¡r¢ Lrh : (A.1)To avoid having to evaluate the derivative of conductivity at discontinuous boundaries, werestate (A.1) as two flrst-order equations with unknown variables J and `:r¢ J = f (A.2)and ¡1J ¡r` = 0 ; (A.3)where J is the conduction current density. The source term f describes sources of streamingcurrent:f = ¡r¢ Lrh : (A.4)The weak forms of (A.2) and (A.3) are:ZV(r¢ J) dV =ZVf dV; (A.5)and ZV( ¡1J ¡r`) dV = 0: (A.6)The discrete forms of (A.5) and (A.6) are derived using a cell-centred flnite-volume methodon a staggered grid. The self-potential ` is deflned at cell centres, with normal componentsof current density J deflned at cell faces, as shown in Figure A.1.201Appendix A. Discrete approximation using the flnite-volume methodJxi+1=2;j;kJyi;j+1=2;kJzi;j;k+1=2Ái;j;k¾i;j;k¢yj¢zk¢xiFigure A.1: Discrete grid cell showing cell dimensions, location of components of currentdensity J at cell faces, and location of potential ` and conductivity  at the cell centre.The 3-D volume is divided into rectangular cells with nx rows, ny columns and nzlayers respectively in the x, y and z directions. Each cell is deflned by the coordinate atits centre (xi;yj;zk), where the position of cell boundaries in each dimension are denotedas fxi¡1=2 xi+1=2g;fyj¡1=2 yj+1=2g and fzk¡1=2 zk+1=2g. The length of the cell in eachdimension is denoted as ¢xi;¢yj and ¢zk, respectively. This arrangement corresponds tonx⁄ny⁄nz cells with a total of (nx+1)(ny+1)(nz+1) cell faces. Consequently, quantitiesmay be identifled by their location in terms of grid coordinates (e.g. `i;j;k ; Jyi;j¡1=2;k).To solve (A.5) and (A.6) on the staggered grid, values of  and L must be determinedat cell faces. Integrating (A.6) over a cell volume shows that it is the harmonic average ofconductivity that is needed. This is also consistent with our physical understanding thatthe overall conductivity of two conductors in series is quantifled by their harmonic average.A.1 Discrete approximation of equation (A.5)Invoking the divergence theorem, (A.5) becomesISJ ¢ ^n dS =ZVf dV; (A.7)202Appendix A. Discrete approximation using the flnite-volume methodwhich may be expressed in discrete form for the cell at (xi;yj;zk) as(Jxi+1=2;j;k ¡Jxi¡1=2;j;k) ¢yj ¢zk +(Jyi;j+1=2;k ¡Jyi;j¡1=2;k) ¢xi ¢zk +(Jzi;j;k+1=2 ¡Jzi;j;k¡1=2) ¢xi ¢yj = „fi;j;k ¢xi ¢yj ¢zk (A.8)or(Jxi+1=2;j;k ¡Jxi¡1=2;j;k)¢xi +(Jyi;j+1=2;k ¡Jyi;j¡1=2;k)¢yj +(Jzi;j;k+1=2 ¡Jzi;j;k¡1=2)¢zk =„fi;j;k ; (A.9)where „fi;j;k is the volume-averaged value of f represented at the cell centre. This discreteequation may be written in matrix form as‡¡ 1¢xi 1¢xi·0@Jxi¡1=2;j;kJxi+1=2;j;k1A+‡¡ 1¢yj1¢yj·0@Jyi;j¡1=2;kJyi;j+1=2;k1A+‡¡ 1¢zk 1¢zk·0@Jzi;j;k¡1=2Jzi;j;k+1=21A = „fi;j;k : (A.10)At the outermost edge of the mesh, we assume Neumann boundary conditions for J:J ¢^nj@› = 0:This condition eliminates J on the outer boundaries, reducing the number of unknownvalues to (nx¡1)(ny ¡1)(nz ¡1). Consequently, the numbering scheme for componentsof J may be rewritten in terms of cell number. Cells are numbered consecutively start-ing at cell 1 and increasing in x then y then z directions. Jx1 corresponds to the cellface at (x1+1=2 ;y1 ;z1), Jxnx¡1 is at (xnx¡1=2 ;y1 ;z1). Jxnx is in the second row of cells at(x1+1=2 ;y2 ;z2), and Jx(nx¡1)nynz is the flnal element of Jx at (xnx¡1=2 ;yny ;znz). A similarpattern describes Jy and Jz.Based on this numbering scheme, we can expand (A.10) to include all cells from 1 tonx⁄ny ⁄nz, and write the matrix equation asDxJx +DyJy +DzJz = f; (A.11)where the matrices Dx, Dy and Dz are flnite difierence operators, and vectors Jx, Jy andJz contain components of current density. Considering the flrst term of (A.11), non-zero203Appendix A. Discrete approximation using the flnite-volume methodelements of Dx are of the form §1=¢x1, and the matrix has dimensions fnx⁄ny ⁄nzg£f(nx¡1)⁄ny ⁄nzg. Matrices Dy, Dz show similar characteristics in y and z dimensions,respectively. Vector Jx takes the formJx =0BBBBBBBBBBBBB@Jx1Jx2......Jx(nx¡1)nynz1CCCCCCCCCCCCCA;and similarly for Jy and Jz. The source vector is described byf =0BBBBBBBBBBBBB@„f1„f2......„fnxnynz1CCCCCCCCCCCCCA:Equation (A.11) may be more simply written asDJ = f; (A.12)where the matrix D is the discrete form of the divergence operator r¢, consisting of[Dx Dy Dz]. Its dimension is fnx ⁄ ny ⁄ nzg £ f(nx ¡ 1) ⁄ ny ⁄ nz + nx ⁄ (ny ¡ 1) ⁄nz + nx⁄ny ⁄(nz ¡1)g. The vector J is the concatenation of vectors Jx, Jy and Jz.A.2 Discrete approximation of equation (A.6)Equation (A.6) may be split into three integral equations:RV ( ¡1Jx ¡ @`@x) dV = 0; (A.13)RV ( ¡1Jy ¡ @`@y) dV = 0; (A.14)and RV ( ¡1Jz ¡ @`@z) dV = 0: (A.15)204Appendix A. Discrete approximation using the flnite-volume methodSince components of J are located at the boundaries between adjacent cells, the volumeintegration is carried out over a new volume. This volume is deflned between cell centresin a given dimension, as shown in Figure A.2 for the x-dimension. The expanded integralform of (A.13) deflned on this new volume isZ xi+1xiZ yj+1=2yj¡1=2Z zk+1=2zk¡1=2 ¡1 Jx dxdydz =Z xi+1xiZ yj+1=2yj¡1=2Z zk+1=2zk¡1=2@`@x dxdydz ; (A.16)which may be written in discrete form as(„ i+1=2;j;k)¡1Jxi+1=2;j;k =‡¡ 1¢„xi+1=21¢„xi+1=2·0@ `i;j;k`i+1;j;k1A: (A.17)The length of the new volume in the x direction is deflned by¢„xi+1=2 = xi+1 ¡xi:The quantity „ i+1=2;j;k is the harmonic average of conductivity located at the interfacebetween cells centred at (xi;yj;zk) and (xi+1;yj;zk). It is deflned as„ i+1=2;j;k = 2  i;j;k  i+1;j;k i;j;k + i+1;j;k: (A.18)¢yj¢zk¢¹xi+1=2¢xi ¢xi+1Figure A.2: Dimensions of the new cell volume over which (A.6) is solved in the x direction.Shaded area indicates the new volume that straddles the interface between cell centres.Assembling (A.17) for all cells and adopting the revised numbering scheme that accountsfor the grid boundary conditions, we can write the matrix equation asS¡1x Jx = Gx`: (A.19)205Appendix A. Discrete approximation using the flnite-volume methodThe inverse conductivity matrix takes the formS¡1x =0BBBBBB@1=„ x1 ¢¢¢ 0... 1=„ x2......0 ¢¢¢ 1=„ x(nx¡1)nynz1CCCCCCA;where „ x denotes the harmonic average value of  in the x direction. Matrix Gx is a flnitedifierence operator of dimension f(nx ¡ 1) ⁄ ny ⁄ nzg £ fnx ⁄ ny ⁄ nzg, whose non-zeroelements are of the form §1=¢„xi+1=2.Similarly, the discrete forms of (A.14) and (A.15) may be written asS¡1y Jy = Gy`; (A.20)and S¡1z Jz = Gz`: (A.21)Equations (A.19), (A.20) and (A.21) are joined to form the matrix system0BBB@S¡1xS¡1yS¡1z1CCCA0BBB@JxJyJz1CCCA =0BBB@GxGyGz1CCCA`; (A.22)which may be simplifled and expressed asS¡1J = G`: (A.23)The matrix G is the discrete form of the gradient operator r, with a dimension off(nx¡1)⁄ny⁄nz + nx⁄(ny¡1)⁄nz + nx⁄ny⁄(nz¡1)g£fnx⁄ny⁄nzg. The matrixS¡1 corresponds to the inverse of the harmonic average of conductivity, with a dimensionof f(nx¡1)⁄ny ⁄nz + nx⁄(ny ¡1)⁄nz + nx⁄ny ⁄(nz ¡1)g2.Since matrix S¡1 is invertible, we can multiply both sides of (A.23) by S to obtain anexplicit algebraic equation for J:J = SG`: (A.24)To solve for `, (A.24) is substituted into (A.12) to give the discrete form of Poisson’sequation:DSG` = f; (A.25)206Appendix A. Discrete approximation using the flnite-volume methodwheref = ¡DLGh: (A.26)The matrix L in (A.26) represents the harmonic average of cross-coupling conductivity.207

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