Sheffer Phd Thesis

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Transcript of Sheffer Phd Thesis

  • Forward modelling and inversion of streaming

    potential for the interpretation of hydraulic

    conditions from self-potential data

    by

    Megan Rae Sheffer

    B.Sc., Queens University, 1995M.A.Sc., University of British Columbia, 2002

    A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OF

    Doctor of Philosophy

    in

    The Faculty of Graduate Studies

    (Geological Engineering)

    The University Of British Columbia

    December, 2007

    c Megan Rae Sheffer 2007

  • Abstract

    The self-potential method responds to the electrokinetic phenomenon of streaming potential

    and has been applied in hydrogeologic and engineering investigations to aid in the evalua-

    tion of subsurface hydraulic conditions. Of specific interest is the application of the method

    to 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 flow problems of complicated geometry, a

    three-dimensional finite volume algorithm is developed to evaluate the self-potential dis-

    tribution resulting from subsurface fluid flow. The algorithm explicitly calculates the dis-

    tribution of streaming current sources and solves for the self-potential given a model of

    hydraulic head and prescribed distributions of the streaming current cross-coupling con-

    ductivity and electrical resistivity. A new laboratory apparatus is developed to measure

    the streaming potential coupling coefficient and resistivity in unconsolidated soil samples.

    Measuring both of these parameters on the same sample under the same conditions enables

    us to properly characterize the streaming current cross-coupling conductivity coefficient. I

    present the results of a laboratory investigation to study the influence of soil and fluid pa-

    rameters on the cross-coupling coefficient, and characterize this property for representative

    well-graded embankment soils. The streaming potential signals associated with preferen-

    tial seepage through the core of a synthetic embankment dam model are studied using the

    forward modelling algorithm and measured electrical properties to assess the sensitivity of

    the self-potential method in detecting internal erosion. Maximum self-potential anomalies

    are shown to be linked to large localized hydraulic gradients that develop in response to

    piping, prior to any detectable increase in seepage flow through the dam. A linear inversion

    algorithm is developed to evaluate the three-dimensional distribution of hydraulic head

    from self-potential data, given a known distribution of the cross-coupling coefficient and

    ii

  • Abstract

    electrical resistivity. The inverse problem is solved by minimizing an objective function,

    which consists of a data misfit that accounts for measurement error and a model objective

    function that incorporates a priori information. The algorithm is suitable for saturated flow

    problems or where the position of the phreatic surface is known.

    iii

  • Table of Contents

    Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

    Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

    List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

    List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

    Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

    Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

    Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

    1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1 Quantitative interpretation of SP data . . . . . . . . . . . . . . . . . . . . 6

    1.2 Thesis objective and outline . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    2 3-D forward modelling of streaming potential . . . . . . . . . . . . . . . . 25

    2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

    2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    2.2.1 Primary flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    2.2.2 Coupled flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

    2.2.3 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

    2.3 Forward modelling methodology . . . . . . . . . . . . . . . . . . . . . . . . 30

    2.3.1 Finite volume solution for self-potential . . . . . . . . . . . . . . . . 31

    iv

  • Table of Contents

    2.3.2 Example: Injection well in a homogeneous halfspace . . . . . . . . . 33

    2.3.2.1 Numerical solution . . . . . . . . . . . . . . . . . . . . . . 34

    2.3.2.2 Analytical solution . . . . . . . . . . . . . . . . . . . . . . 35

    2.4 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    2.4.1 The cross-coupling conductivity coefficient . . . . . . . . . . . . . . 37

    2.4.2 Defining and L property distributions . . . . . . . . . . . . . . . . 37

    2.4.3 Example: Homogeneous lab-scale embankment . . . . . . . . . . . . 39

    2.5 Sources of charge contributing to the self-potential . . . . . . . . . . . . . . 43

    2.5.1 Example: Pumping well in a heterogeneous halfspace . . . . . . . . 45

    2.6 Field example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    3 Apparatus for streaming potential and resistivity testing . . . . . . . . 59

    3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    3.2 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

    3.2.2 Test cell and loading assembly . . . . . . . . . . . . . . . . . . . . . 63

    3.2.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

    3.2.4 Fluid flow system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    3.2.5 Data acquisition and control . . . . . . . . . . . . . . . . . . . . . . 67

    3.2.6 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    3.3 Streaming potential measurements . . . . . . . . . . . . . . . . . . . . . . . 68

    3.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    3.3.2 Calibration and test design . . . . . . . . . . . . . . . . . . . . . . . 71

    3.3.3 Comparison of unidirectional and oscillatory flow test methods . . . 72

    3.4 Resistivity measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

    3.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

    3.4.2 Calibration and test design . . . . . . . . . . . . . . . . . . . . . . . 74

    3.4.3 Comparison of 2- and 4-electrode methods . . . . . . . . . . . . . . 77

    3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    v

  • Table of Contents

    3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    4 Laboratory testing in well-graded soils . . . . . . . . . . . . . . . . . . . . 82

    4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    4.2.1 Streaming potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    4.2.2 Electrical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    4.3 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    4.3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    4.3.2 Streaming potential measurements . . . . . . . . . . . . . . . . . . . 90

    4.3.3 Resistivity measurements . . . . . . . . . . . . . . . . . . . . . . . . 91

    4.3.4 Sample properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

    4.3.5 Sample equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . 94

    4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    4.4.1 Influence of density . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

    4.4.2 Influence of gradation . . . . . . . . . . . . . . . . . . . . . . . . . . 103

    4.4.3 Influence of fluid conductivity . . . . . . . . . . . . . . . . . . . . . 107

    4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

    4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    4.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

    5 Sensitivity of the self-potential method to detect internal erosion . . . 121

    5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

    5.2 The streaming potential phenomenon . . . . . . . . . . . . . . . . . . . . . 123

    5.3 Forward modelling of streaming potential . . . . . . . . . . . . . . . . . . . 125

    5.3.1 Methodology . . . . .