Leakage detection of water reservoirs using a Mise-à-la-Masse approach

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Leakage detection of water reservoirs using a Mise-à-la-Masse approach

C. Ling, A. Revil, F. Abdulsamad, Y. Qi, A Ahmed, P. Shi, Sylvie Nicaise, Laurent Peyras, A. Soueid Ahmed

To cite this version:

C. Ling, A. Revil, F. Abdulsamad, Y. Qi, A Ahmed, et al.. Leakage detection of water reser- voirs using a Mise-à-la-Masse approach. Journal of Hydrology, Elsevier, 2019, 572, pp.51-65.

�10.1016/j.jhydrol.2019.02.046�. �hal-02324247�

Contents lists available at ScienceDirect

Journal of Hydrology

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

Research papers

Leakage detection of water reservoirs using a Mise-à-la-Masse approach

C. Ling

, A. Revil

, F. Abdulsamad

, Y. Qi

, A. Soueid Ahmed

, P. Shi

, S. Nicaise

, L. Peyras

aState Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology), 1#, Dongsanlu, Erxianqiao, Chengdu 610059, Sichuan, PR China

bUniv. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France

cDepartment of Mechanics, Harbin Institute of Technology, Harbin, PR China

dIRSTEA, Aix-en-Provence, France

A R T I C L E I N F O

This manuscript was handled by Peter K.

Kitanidis, Editor-in-Chief, with the assistance of John William Lane, Associate Editor Keywords:

Geophysics Mise-à-la-masse Hydrogeophysics LeakMountain reservoir

A B S T R A C T

Localizing leaks of water and fluids from storage tanks and water reservoirs with geomembranes is an important task for a variety of environmental applications and water resources applications. The minimally intrusive mise- à-la-masse method is used to detect leaks with the current injected inside the reservoir and a return current electrode located remotely. We test a new approach for the inversion of the voltage data using sandbox ex- periments and numerical modeling. A method similar to the self-potential inversion method is proposed to inverse the voltages recorded around the tank or reservoir. A global objective function with a data misfit term and regularization term is minimized to invert the voltages. In the inversion process, a depth-weighting matrix is used to strengthen the depth resolution of the current source, and the minimum support method is used to avoid oversmoothed results in terms of leak detection. The distributions of electrical current density on the walls of reservoir indicate the position of leaks. The results show that the inversion method with source compaction accurately identifies the location of single leaks. For two separated leaks, there is an obvious bias for the deeper hole and the bias increases with its depth. For three holes, the source compaction method generally identifies the location of the three leaks when their depth ranges are similar. When one of the leaks becomes deeper, loca- lization of the deeper one becomes more difficult. The influence of the size of the leak on the inversion results is also investigated. The inversion algorithm overestimates the depth of small leaks while it slightly underestimates the depth of large leaks. For a leak having the form of a crack, the inversion results using the source compaction method agree with the position of the leak and its shape.

1. Introduction

As the world’s most important source of drinking water, the quality of groundwater should be preserved from the spread of various types of contaminants (see World Health Organization, 2006). Underground storage tanks used to store inorganic liquid chemicals, hydrocarbons, and organic compounds can leak because of corrosion and deformation during long-term operations (Jacob and Alexander, 2010). In addition, there is currently an increase of the construction of mountain reservoirs using specific geomembranes (for instance in high density poly- ethylene, HDP) to store water, which is used in the beginning of winter to prepare snow for the ski runs (Poulain et al., 2011).

Large water reservoirs are most often equipped with drainage sys- tems to collect leaks. But many reservoirs do not have this type of drainage equipment and the detection of leaks in the embankment

becomes an important technical problem (Peyras et al., 2008). The leaks associated with a storage tank or a water reservoir or a pond with an insulating and impervious boundary is difficult to detect by visual inspection until the leak poses serious pollution to the surrounding groundwater environments or leakage becomes important. The most common methods of leak detection include volumetric and mass mea- surements, statistical inventory reconciliation, liquid sensing probes, and fiber optic sensing probes (e.g., ADEC, 2000; Colombo et al., 2009).

The volumetric and mass measurement and statistical inventory re- conciliation use an unexplained loss of mass to indicate the presence of a leak, but are not able to detect the positions of leaks (Musthafa et al., 2017). In probe measurements, the liquid or chemical content is esti- mated from the physical properties that are measured by the probes according to the corresponding calibration curves. Probe measurements provide accurate information about a leak but they have to be

https://doi.org/10.1016/j.jhydrol.2019.02.046

Received 22 October 2018; Received in revised form 20 February 2019; Accepted 21 February 2019

⁎

Corresponding author.

E-mail addresses:andre.revil@univ-smb.fr

(A. Revil),

(F. Abdulsamad),

(S. Nicaise),

(L. Peyras).

Journal of Hydrology 572 (2019) 51–65

Available online 27 February 2019

0022-1694/ © 2019 Elsevier B.V. All rights reserved.

T

preembedded in positions close to the leak. The approach is costly when a large number of probes around the storage facility are required.

Cheap, non-invasive, and non-destructive techniques are needed to lo- calize leaks of underground storage tanks and reservoirs. Various active techniques are described in the literature (e.g., Manataki et al., 2014) but their interpretation in terms of leakage is usually ambiguous especially for mountain reservoirs due to the undulating topography.

As a geophysical technique, the mise-à-la-masse method was ori- ginally developed for ore mineral exploration (e.g., Parasnis, 1967;

Ketola, 1972). It was later used in environmental geophysics and hy- drogeology (e.g., Binley et al., 1997; Gan, 2017; Perri et al., 2018) in- cluding for the detection of leaks (Ramirez et al., 1996; Daily et al., 2004). The mise-à-la-masse method is a kind of mix between the re- sistivity and self-potential methods, one being an active technique and the second being a passive technique. The underlying idea is to illu- minate a conductive body at depth by injecting a current that would preferentially flow through this body. The resulting electrical potential distribution distorted by the shape of the conductive body is measured.

In principle, two current electrodes A and B are required and a number of voltage measurements are performed with respect to a reference at the ground surface of the Earth. The voltage measurements themselves are similar, in their principle, to what is done in a self-potential survey (Corwin and Hoover, 1979; Revil and Jardani, 2013). One of the cur- rent electrodes, say A, is used to inject the current in the conductive body while the second electrode, say B, is placed remotely (i.e., far from

the survey area). By using a mobile potential electrode, say M

(we consider a total of n stations covered by this electrode), the potentials relative to the reference potential electrode N (fixed) can be mapped.

The electrical potential map provides some information regarding the shape of the conductive body (e.g., Kirsch, 2006). The mise-à-la-masse method has been used to find out groundwater flow direction (e.g., White, 1994; Pant, 2004; Perri et al., 2018), to delineate conductive tracer plumes (e.g., Osiensky and Donaldson, 1995; De Carlo et al., 2013), and investigate plant roots (e.g., Mary et al., 2018). Binley et al.

(1997) used the mise-à-la-masse method to evaluate the bottom leakage of environmental barriers. In their paper, the genetic algorithm was used to inverse the positions of leaks, but the searching space will highly increase with the number of leaks.

Currently, there is a need to improve the performance of the mise-à- la-masse method for leak detection, especially under the three-dimen- sional (3D) conditions. The goal of the present study is to advance the use of this technique for the detection of leaks in storage tanks and water reservoirs with impervious and insulating geomembranes. In this study, several tests are carried out using a sandbox experiment. For evaluating the position of the leaks on the walls of the reservoir, a method similar to self-potential localized inversion technique discussed

Fig. 1.

Sketch of the sandbox with the cylindrical container used to simulate a leaking reservoir. The scanning electrodes M

are arranged along 16 rays around the current electrode A placed inside the reservoir. The return current electrode B is placed 22.85 cm from the reservoir. The voltage electrode N with distance of 26 cm from the reservoir is used as a reference for mapping the electrical potential distribution. The angle between adjacent rays was set to 0.39 rad (i.e., 22.5°). There were 61 scanning potential electrodes labeled by their number in the figure. This geometry is used for the physical and synthetic (numerical) experiments.

Fig. 2.

Synthetic test geometry and mesh. a. Geometry. The black dots denote the locations of the current electrodes (A and B) and reference potential elec- trode (N). The radial lines around the storage tank denote the scanning po- tential electrodes (M). b. Finite elements mesh of the sandbox with 20,000 elements. The mesh is refined around the reservoir and electrodes to increase the accuracy of solution. Finite elements forward modeling was done to ensure that the refinement of the mesh is optimal. The circular frame shows the area for potential measurement (e.g.

C. Ling, et al. Journal of Hydrology 572 (2019) 51–65

by Haas et al. (2013) is proposed for the first time for the mise-à-la- masse method. These source localization approaches are also similar to what has been proposed in electroencephalography for the localization of brain activity from the voltage distribution measured on the scalp (Grech et al., 2008).

2. Materials and methods 2.1. Experiment setup

In order to evaluate the performance of the mise-à-la-masse method for leak detection, a plexiglass box with the dimensions of 53.7 cm length, 28.5 cm width, and 33.4 cm height was used (see Figs. 1 and 2).

The silica sand was used in the experiment. Its physical properties are reported in Table 1. The dry density and porosity of this sand are 1.4 g/

cm

and 0.37 ± 0.01, respectively. The sand was used to fill the sandbox and its overall thickness was 25 cm. The box was placed in a room with a constant temperature of 20 ± 2 °C.

The tap water used for the experiment has an electrical conductivity of ∼440 μS/cm at 25 °C. In order to minimize the entrapment of gas bubbles in the sandbox, water was added first then the sand grains until the prescribed thickness of sand was reached. The excess water above the sand was pumped out from the tank. Prior to performing the geo- physical measurements, the sand was let settling for 24 h so it could compact by gravity. The electrical resistivity of the silica sand saturated with the tap water was measured and its value is 100 Ω m at room temperature (20 °C).

A plastic reservoir with a height of 8.1 cm and a diameter of 6 cm was used to simulate the storage tank (Figs. 1 and 2). The reservoir was filled with the same tap water used to saturate the sand. The water reservoir reached the top surface of the sand and was located in the center of the sandbox. The height of the reservoir in the sand is 8 cm.

The diameter of the leak was set to 0.4 cm. Experiments were conducted with different leaks in order to study the effect of the number of leaks, their depth, and their size.

Each leak position is described by two indices, an azimuth and a depth in cylindrical coordinates. The azimuth denotes the outward di- rection of leak from the vertical central axis of reservoir with respect to +x axis. The azimuth is set to 0 rad along the +x axis, and increases counter-clockwise in the x-y plane (Fig. 2). The depth is the vertical distance from the leak center to the surface of sandbox. Table 2

summarizes the physical sandbox experiments we performed. The 12 experiments include 4 single-leak tests, 4 dual-leak tests, and 4 triple- leak tests.

2.2. Mise-à-la-masse experiments

A total of 64 stainless steel electrodes with the diameter of 0.3 cm and a length of 0.7 cm were setup on the sand at the top surface of the sand body (Fig. 1a). The positive current electrode A (i.e. current in- jected in the sand) was placed in the water-filled reservoir. The negative (return) current electrode B was placed at the left center of the sand body surface, i.e. in a remote position with respect to the monitoring network of electrodes (Figs. 1 and 2). The reference potential electrode N was located close to the right bottom corner at the sand body surface.

A total of n = 61 scanning electrodes (M

) were used to measure the distributions of potential difference (Fig. 1). These electrodes were ra- dially organized around the reservoir. There are 16 electrodes in a lap and the angle between adjacent electrodes is 0.39 rad (i.e., 22.5°). From inner to outer distance from the reservoir, the distances between the electrode laps and the reservoir boundary were set up to 1 cm, 3 cm, 6 cm, and 10 cm (see Fig. 1a for details). In order to keep the position of the electrodes at the same place, all of electrodes were fixed on a polyvinyl chloride framework. An ABEM Terrameter SAS-1000 re- sistivity meter was used to carry out the mise-à-la-masse measurements.

The supply current I was set at 20 mA. The data of potential differences and corresponding standard deviations were logged automatically. We used a total of 64 electrodes, two of them being used as the current electrodes. Therefore, there was a total of 62 voltage electrodes, one being used as the reference voltage electrode N. Consequently, we had m = 61 voltage stations (Fig. 1).

2.3. Forward modeling

We discuss now the forward numerical modeling step of our ana- lysis. Forward modeling was carried out to simulate the mise-à-la-masse acquisition. Combining Ohm’s law with a continuity equation for the electrical charge in an isotropic medium yields the following elliptic equation (e.g., Rubin and Hubbard, 2005):

= I x x y y z z

·( ) (

) (

) (

) (1)

where is the electrical potential (in V), denotes the electrical con- ductivity (S/m) of the material. I is the volumetric intensity of the in- jected current (in A/m

), denotes the Dirac distribution, and x

, y

, and z

are the spatial coordinates of a given current electrode. In our case, Eq. (1) was numerically solved with two current electrodes A and B with the current of I and –I, respectively. Eq. (1) is solved with the following boundary conditions:

= 0 on

, (2)

Table 2

Summary of the 12 experiments performed in the course of the laboratory investigations. The azimuth denotes the outward direction of leak from the vertical central axis of reservoir with respect to +x axis. The depth is the vertical distance from the leak center to the sand surface.

No. Number of leak (s) Azimuth (rad) Depth (cm) Azimuth (rad) Depth (cm) Azimuth (rad) Depth (cm)

1 1 3.14 1.0 / / / /

2 1 3.14 3.0 / / / /

3 1 3.14 5.0 / / / /

4 1 3.14 7.0 / / / /

5 2 0 1.0 3.14 1.0 / /

6 2 0 1.0 3.14 3.0 / /

7 2 0 1.0 3.14 5.0 / /

8 2 0 1.0 3.14 7.0 / /

9 3 0 1.0 1.57 1.0 3.14 1.0

10 3 0 1.0 1.57 3.0 3.14 1.0

11 3 0 1.0 1.57 5.0 3.14 1.0

12 3 0 1.0 1.57 7.0 3.14 1.0

Table 1

Physical properties of the silica sand used in the sandbox experiment.

Dry density (g/cm³) Porosity Composition (%)

SiO2 Al2O3 TiO2 K2O Others

1.4 0.37 ± 0.01 98 0.9 0.1 0.5 0.5

=

n· 0 on

, (3)

where

and

denote Dirichlet boundary and Neumann boundary, respectively. The quantity n denotes the unit normal vector of the Neumann boundary

. In practice, Dirichlet boundary is referred to as a fixed potential boundary, while Neumann boundary specifies the in- sulation boundary. In our sandbox experiment, the upper surface was in contact with air, and the remaining five faces were in contact with insulated plexiglass. Hence, the six boundaries of the sand body cor- responded to Neumann boundaries.

Using a finite element discretization, Eq. (1) is written in a linear form

= Km (4)

where denotes an n -vector corresponding to the forward electrical potential differences, which could be measured with scanning elec- trodes, m is a m -model vector corresponding to the source current density at each cell on walls of reservoir. n is the number of measured potential stations/data, while m is the number of discretized cells composing the side faces of reservoir. The quantity K denotes the n × m

Contour maps of the electrical potentials obtained experimentally with a single hole at depth of 1 cm (a), 3 cm (b), 5 cm (c), and 7 cm (d). The arrows point to the leak hole, and the azimuth of leak holes is 3.14 rad. The position of the maximum potential reading is located right above the leak. The minimum voltage value is located at the leftmost electrode. The top two figures are shown with a color scale in the −2 to 26 V range while the bottom two figure are shown with a color scale in the −2.6 to 2.8 V range.

C. Ling, et al. Journal of Hydrology 572 (2019) 51–65

kernel matrix. It corresponds to the collection of Green’s functions.

More precisely, each column of K corresponds to the measured elec- trical potential differences by setting a unit source current density at the corresponding cell and zero current density at the remaining cells.

Because the kernel matrix accounts for the boundary conditions, elec- trical conductivity distribution, and electrodes array distribution, the Green’s functions are obtained through numerical modeling using the finite element method with the software Comsol Multiphysics 5.3 (Comsol, 2017; Soueid Ahmed et al., 2018). The numerical calculation was carried m times by successively setting a unit source current den- sity at one cell and zero current density at the remaining cells. and m in Eq. (4) correspond to ( , , ) x y z and I x ( x

) ( y y

) ( z z

) in Eq. (1), respectively. The remaining term at the left hand of Eq. (1) corresponds to K

_{in Eq. (4).}

For the forward modeling; the sandbox was subdivided into ap- proximately twenty thousand tetrahedra (Fig. 2b). In the physical and numerical experiments, the positive current electrode A was placed in the center of the water-filled reservoir. When there is a leak in the side face of the reservoir, all of the electrical current lines will cross this leak because the reservoir is insulating. For the sandbox, the leak acts like a positive secondary current source. Leaks at different positions cause different distributions of potential differences. In order to simulate the potential responses, the side faces of the plastic reservoir were divided into 48 slices along its circumference and 20 slices along its vertical direction. There was therefore a total of 960 cells in the side faces of the reservoir. The size of each cell was 0.4 cm × 0.4 cm. The collection of Green functions with a single leak of 0.4 cm diameter at each cell was obtained thanks to the forward modeling using the finite-element method. The potential difference distributions with two or more leaks is easily obtained using the superposition principle since the problem is linear between the current sources and the voltage distribution.

2.4. Inverse modeling

We propose a method similar to the self-potential inversion method (see Minsley et al., 2007; Jardani et al., 2008; Haas et al., 2013; Soueid

Ahmed et al., 2013). The following objective function with the sum of a data misfit term and a regularization term is subject to minimization (see Tikhonov and Arsenin, 1977).

= +

m W Km W m

P ( )

(

)

(5)

where the first term denotes the data misfit term and the second term denotes the regularization term. The quantity is a regularization parameter under the constraint of 0 < < . The quantity

is the n -vector corresponding to the measured potential difference data. The quantity W

= diag{1/ε

,…, 1/ε

} denotes a square diagonal weighting matrix and the symbol ‘diag’ means that the matrix is a diagonal matrix with elements being the reciprocal of the standard deviations. The other elements are set to zero because the noise on the data is assumed to be uncorrelated. The quantity W

is a weighting matrix in the regular- ization term. It can be associated to the model covariance matrix. In the present case, an identity matrix I (zero-order derivative or minimum norm) for W

was used (Hansen, 1992; Mao and Revil, 2016). The optimized model vector can be obtained with a single step as following (Tikhonov and Arsenin, 1977)

= +

m [ K W W K

(

) ( W W

)] ·[ K W W

(

) ]

d obs

1

(6)

In application, because the electric field strength decays as 1/r

(r being the distance between a source point and an observation point) in a homogeneous half-space, the sensitivity of the potential field decays quickly with the distance to the leak. To provide the deeper cells with the equal probability of obtaining nonzero source current density during the inversion compared to the shallow cells in side faces of re- servoir, a generalized depth weighting function is incorporated into the kernel matrix K . The depth-weighting matrix is designed to the total sensitivity of observations for a unit current source at a particular cell on walls of reservoir. The diagonal elements of the matrix are defined by

= =

=

K i

diag 1, 2, M

j N

ji 1

2 1 4

(7)

The variations in the electrical potential

measured in the sandbox experiment with a single

hole (physical experiment). The blue, purple, red,

and green lines denote the first, second, third, and

forth laps of electrodes around the storage tank,

from inner to outer (Fig. 1a). The dashed lines

denote the azimuths of leaks. (For interpretation of

the references to color in this figure legend, the

reader is referred to the web version of this ar-

ticle.)

and the off-diagonal elements are zero (Mao and Revil, 2016).

For leak detection, a constraint that requires the source model to be localized makes sense. Elsewhere, the source term should be zero. To avoid oversmoothed results in the distributions of secondary current density in the side faces of reservoir, the minimum support (MS) method is used (Zhdanov and Tolstaya, 2004). This iterative step is done to reach a compact source current density. The MS function is defined as

=

+

m MS m

i

M i

1 i k 2 ( 1)

2 2

(8)

where the (k − 1) subscript refers to a prior estimate of current source density distribution m on walls of reservoir. In Eq. (8), is a small threshold number. The MS function allows the greatest of parameters to have amplitude less than while the remaining parameters greater than are not constrained (Zhdanov and Tolstaya, 2004). The value of 0.01

was assigned to in this inversion. With the depth-weighting matrix and MS function, a new diagonal weight matrix is determined as

= + =

m i m

diag

, 1, 2,

i k 2 ( 1)

2 2

(9) The kernel matrix is revised as K = K

, and Eq. (6) is therefore modified to be

= +

m [ K

( W W K

) ( W W

)] ·[

K

( W W

)

] (10) The actual data inversion starts with a model of uniform current density distribution on the walls of reservoir. The initial value of is set to 1, and the value is updated with the ratio of data misfit term to regularization term in Eq. (11). With the new kernel matrix, the ob- jective function is transformed into

= +

m W K m W m

P ( )

(

)

(11)

Fig. 5.

Contour maps of electrical potentials with two holes, the azimuths of which are 0 rad and 3.14 rad (physical experiments). The depth of hole (right) with the azimuth of 0 rad was fixed at 1.0 cm, and the depths of hole (left) with the azimuth of 3.14 rad were set to 1.0 cm (a), 3.0 cm (b), 5.0 cm (c), and 7.0 cm (d).

C. Ling, et al. Journal of Hydrology 572 (2019) 51–65

After each iteration, the current density of sources can be trans- formed back to an unscaled current density according to m =

m . The inversion calculation is repeated until the objective function de- crease to a stable value or the maximum iteration steps are reached. The distributions of unscaled source current density from the last iteration are used to provide the best estimates of the positions of leaks.

3. Experimental results

3.1. Distribution of measured potential differences

The contour maps of the potential difference distribution obtained in the physical experiment with a single leak are shown in Fig. 3. The values of electrical potential difference close to the leak are the highest.

The leak is located right below the maximum voltage measured on the top surface of the sand body. Because the negative current source is located at the leftmost (out of the figure area), the minimum potential value always locates at the leftmost electrode. Fig. 4 presents the var- iations of the electrical potential difference obtained for this case. For a given current, the magnitude of the potential values decreases with the depth of the leak. It becomes therefore more challenging to evaluate the position of deep leaks due to the weak signal obtained at the ground surface (Fig. 3d and 4d).

Fig. 5 shows the contour maps of the measured potential differences with two leaks with azimuths of 0 rad and 3.14 rad. Fig. 6 shows the variations in the electrical potential differences. In order to analyze the impact of hole with the azimuth of 3.14 rad, the depth of hole with the azimuth of 0 rad was fixed at 1.0 cm, and the depths of hole with the azimuth of 3.14 rad were successively set to 1.0 cm, 3.0 cm, 5.0 cm, and 7.0 cm. The maximum values of potential decrease with the number of

leaks. For a single leak and for two leaks located at the same depth, the maximum values are 25.5 V (Fig. 4a) and 12.1 V (Fig. 6a), respectively.

The potential values on the left side of the surface of the sand body are mainly dominated by the electrical anomaly associated with the leak on the left of the reservoir and decrease with depth, while the potential values for the leak on the right side of the reservoir remain unchanged.

When the left leak became deeper, e.g., 5 cm, there is still a visible difference of potentials around it, compared with the distribution of a single leak (Fig. 3a).

Fig. 7 presents the contour maps of electrical potentials obtained experimentally with three leaks. The depth of leak with the azimuth of 1.57 rad was changed from 1.0 cm to 7.0 cm, while the depth of the leaks with azimuths of 0 rad and 3.14 rad was fixed at a depth of 1.0 cm.

Compared to Figs. 3 and 5, the maximum value of the electrical po- tential decreases to 10.1 V. Fig. 8 shows the variations in the electrical potentials. There are three anomalies when the three leaks are at the same depth. When the depth of the leak with the azimuth of 1.57 rad increases, the corresponding maximum values gradually become in- significant because of the overlap of potential signals from another two leaks. The potential signals from two shallow leaks are too strong to cover the signal from the deeper leak.

3.2. Evaluation of leaks

When the leak is deep, the weak signals make the inversion hard to identify its precise position. The inversion processes for a single leak at 7 cm depth are shown in Fig. 9. The distributions of source current density are shown at iterations 1, 3, 6, and 10 in the compaction pro- cesses during the inversion. Iteration 10 corresponds to the last itera- tion, for which the focus target (and therefore convergence) is reached.

Fig. 6.

The variations in the electrical potential

with two holes (physical experiments). The dashed

lines denote the azimuths of holes. The electrical

potential close to the hole with the azimuth of 0 rad

kept almost unchanged. The electrical potentials

close to the hole with the azimuth of 3.14 rad de-

creased with the depth of this hole.

It can be seen that the magnitude of the source current density peaks at the true location of the target. The inversion results show the accurate location of leak, although the electrical potential signal is weak (Fig. 3d).

Fig. 10 presents the distributions of source current density for a single leak hole at different locations. The inversion method with source compaction accurately identifies the location of single leak. The results of two separated leaks are also presented in Fig. 11. Although both of the leaks are identified, there is an obvious bias in terms of locating the deeper leak and the bias increases with the depth of this leak. For a shallow leak, the inversion results are generally accurate due to the strong voltages associated with the leak. The signals from the deeper leak could be in the range of noise so that it is prone to noise, which deteriorate the inversion results.

Fig. 12 shows the distributions of source current density for three

leaks. The source compaction method generally identifies the location for three leaks with the same depth, i.e., 1 cm, although an artifact source appears in the lower right corner. When the middle leak be- comes deeper, it is difficult for the proposed inversion method to identify such a leak. Some erroneous locations for the leakage start showing up. Several obvious artifacts appear in Fig. 12b and c. The deeper leak at the depth of 7 cm totally disappears in the inversion results (see Fig. 12d).

4. Discussion

Besides the parameters set in the experiment, there are still several factors influencing the inversion results such as the distribution of electrical resistivity, and the size of the leak. The distribution of elec- trical resistivity is one of the key factors influencing the results. In the

Contour maps of electrical potentials with three holes, the azimuths of which are 0 rad, 1.57 rad, and 3.14 rad (physical experiments). The depth of holes with azimuths of 0 rad and 3.14 rad was fixed at 1.0 cm, and the depth of hole with the azimuths of 1.57 rad was set to 1.0 cm (a), 3.0 cm (b), 5.0 cm (c), and 7.0 cm (d).

C. Ling, et al. Journal of Hydrology 572 (2019) 51–65

Fig. 8.

The variations in the electrical potential with three holes (physical experiments). The dashed lines denote the azimuths of holes. The electrical potential close to the hole with the azimuths of 1.57 rad decreased with the depth of this hole, while the electrical potentials close to holes with azimuths of 0 rad and 3.14 rad kept almost unchanged. When the depth of the leak with the azimuth of 1.57 rad increases, the corresponding maximum values gradually be- come insignificant because of the overlap of potential signals from another two leaks.

Fig. 9.

Source current compaction processes for a single leak hole at the depth of 7 cm (physical experiment). The black open circles denote the location of leak. Some

parts of these figures are magnified for a better visualization. The spread of source current density reduced to a localized zone and its magnitude increases during the

iterative compaction of the source support area. The isolated peaks of current density indicate the locations of leaks. The inversion results show the accurate location

of leak, although the electrical potential signal is weak (Fig. 3d).

Fig. 10.

Source current distribution for a single leak at different locations (physical experiments). The black open circles denote the location of leak. Some parts of these figures are magnified for a better visualization. The locations of the single leak hole are accurately identified.

Fig. 11.

Source current distribution for two leak holes (physical experiments). The black open circles denote the location of leak. The bias increases with the depth for the deeper leak hole due to the weak signal. For the figure in the lower right corner (d), the source in the upper right corner denotes the shallow hole due to the errors

C. Ling, et al. Journal of Hydrology 572 (2019) 51–65

experiment, a homogeneous resistivity of 100 Ω m was used. In the field experiment, electrical resistivity tomography could be carried out to obtain the distribution of electrical resistivity, which is beneficial for constructing the kernel matrix and producing better source results.

Regarding the issue related to the size of the leak, we supposed that identical size (i.e., diameter 0.4 cm) for the leaks and cells in the ex- periment. In application, we don’t know the actual size of the leak. In order to test the influence of actual size on the evaluation results, we simulated the forward and inversion processes. Random noises of 0–5%

were added to the simulated potential data. The depth of the leak was set to 5.0 cm. The simulated results with a single leak of 0.2 cm, 0.3 cm, 0.6 cm, and 0.8 cm are shown in Fig. 13. All of the sources are located in the interval [3.142–0.131 rad, 3.142 + 0.131 rad] along the cir- cumference. The quantity 0.131 rad denotes the discretization of the azimuthal angle in radian (i.e., 7.5°). The size of the leak has also an obvious influence on the evaluated depth resulting from the inversion.

In general, the algorithm overestimates the depth for the smaller size of hole (i.e., 0.3 cm), while it slightly underestimates the depth for the bigger size of the leak (i.e., 0.6 cm and 0.8 cm). The leak with the diameter of 0.2 mm is wrongly located. The evaluated depth was 6.2 cm and the bias increases to 1.2 cm.

In order to further investigate the influence of the size of the leak on the inversion results, we numerically simulated the potential data and then we contaminated these data with a white noise level of a

maximum of 5% before proceeding to the inversion of these synthetic data. The kernel matrix with cell of 0.4 cm × 0.4 cm was used in the inversion process. Fig. 14 shows the comparison of inversed depth for different size holes. The dotted lines denote the real depths of holes while the scatter points denote the inverted depths. Although the results are influenced by the noise, the inversion method generally over- estimates the depth for the leak hole with diameter less than 0.4 cm.

Compared with the bigger size, the magnitude of depth bias becomes larger for the smaller size of leak.

The potential difference data of inner lap of electrodes (i.e., No.

2–17 in Fig. 1a) are used to illustrate the influence of hole size in

Fig. 15. Compared with potential data with the hole of 0.4 cm diameter,

the peak values of electrical potential (i.e., from 2.36 rad to 3.93 rad)

decrease while other values increase for the leak of 0.2 cm diameter

(i.e., the solid circles in Fig. 15a and d). In the opposite, for a leak of

0.8 cm in diameter, the peak values of the electrical potential (i.e., from

2.36 rad to 3.93 rad) increase while the other values decrease (i.e., the

crosses in Fig. 15a and d). The variable magnitudes for the leak of

0.2 cm diameter are almost twice of that for the leak hole of 0.8 cm in

diameter. For example, comparing to the peak of potential (i.e., 3.28 V

in Fig. 15d) with a leak of 0.4 cm, the potential magnitude decreases by

0.77 V for the leak of 0.2 cm diameter while that increases by 0.44 V for

the leak of 0.8 cm. When the diameter of a leak is kept at 0.4 cm, the

surface potential decreases with the depth of the leak, which has the

Source current distributions for three separated leak holes (physical experiments). The tomograms are determined from the voltage data recorded at the top

surface of the sand body. The black open circles denote the location of leak. The source compaction method generally identifies the location for three leaks with the

same depth (a). When the middle leak becomes deeper, some erroneous locations for the leakage start showing up. It is difficult for the proposed inversion method to

identify the deeper leak.

similar influence with the shrink of the size of the leak (Fig. 15b and e).

On the contrary, compared with the decreased depth of the leak, the increase in the size of the leak has a similar influence on the surface potential (Fig. 15c and f). The fundamental reason of this phenomenon

lies in the fact that the inversion of potential data is ill-posed and does not have a unique solution.

Besides the case of an isolated circular leak, other shape of leak can be investigated. For instance, we evaluate a leak having the shape of a crack, which can be due to for instance the deformation of the geo- membrane over time. The linear superposition of distributed holes is used to model such a crack, and random noises with a level of 0–5% are added to the simulated potential data. Fig. 16 shows the inversion re- sults three holes and five holes used to simulate the crack. The inversion method slightly underestimates the depth of the resulting leak at the tilt direction. In general, the inversion results show however the correct position of such a leak, which indicates that the source compaction method could be used to evaluate the leakages due to elongated leaks in the geomembrane.

5. Conclusions

As an inexpensive, minimally invasive, nondestructive geophysical technique, the mise-à-la-masse method can be used to detect leakages in the underground storage tank or reservoir. A series of sandbox experi- ments were conducted to evaluate a new strategy of leak detection. For evaluating the position of the leaks, a method similar to the self-po- tential inversion method was proposed.

In the forward modeling, the elliptic equation for the electrical potential was solved with the finite element method and the kernel matrix was obtained. In the inversion process, a global objective func- tion with a data misfit term and regularization term was used. A depth- weighting matrix was used to strengthen the depth resolution, and the

The effect of hole size on the evaluation results (synthetic experiments). The depth of all holes was set to 5.0 cm. The results with single leak hole of 0.2 cm (a), 0.3 cm (b), 0.6 cm (c), and 0.8 cm (d) are shown. Random noises of 0–5% were added for the inversion process. In general, the algorithm overestimates the depth for the smaller size of hole, while it slightly underestimates the depth for the bigger size of the leak.

Fig. 14.

Comparison between the true depth of the leaks and the inverted re- sults for five different values of the diameter of the holes (synthetic experi- ments). The dashed lines denote the true depth of the leaks (three different depths are investigated). The three symbols denote the inverted position of the leaks. The crosses correspond to the shallow leaks. The open circles correspond to the intermediate depth for the leaks. Finally, the open triangles correspond to the deepest leaks.

C. Ling, et al. Journal of Hydrology 572 (2019) 51–65

minimum support method was used to avoid the oversmoothed result.

The results show that the inversion method with source compaction accurately identifies the location of single leak hole. For two separated leak holes, there is an obvious bias for the deeper hole and the bias increases with the depth. For three leaks, the source compaction method generally identifies the location when their depths are similar.

When one of the leaks is deeper, it may be more difficult for the in- version algorithm to correctly identify the deeper leak.

The influence of the size of leak on the inversion results is also analyzed. The inversion algorithm overestimates the depth for a small- size leak, while it slightly underestimates the depth of big leaks. For a leak having the shape of a crack (elongated leak), the inversion results

show the correct position and shape of the leak, which indicates that the source compaction method could be efficiently used to evaluate the position of such leaks.

The next step will be to apply this method to the field. In principle the present approach is entirely scalable since the current used in the field can be increased by improving the contact impedance between the electrodes A and B and the soil or the bottom of the reservoir. In ab- sence of leak, it would be impossible to inject any current. In present of leak, the compaction algorithm should be able to localize the position and size of the leak.

Fig. 15.

The influence of hole size (i.e., 0.2 cm, 0.4 cm, and 0.8 cm) on the potential data at depth of 1 cm (a) and 5 cm (d) (numerical experiment). The relationship

between the real potential data of leak hole with diameter of 0.2 cm (b and e) and 0.8 cm (c and f) and simulated data of leak hole with diameter of 0.4 cm. In figures

(b), (c), (e), and (f), the data of x-axis are potentials with 0.4 cm diameter at different depths, while the data of y-axis correspond to the simulated potentials with real

diameter (i.e., 0.2 cm and 0.8 cm).

Acknowledgements

This work was supported by the State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology) (no. 2007DA810083). It is also supported by the project RESBA ALCOTRA funded by the European Community. The postdoc of Abdellahi Soueid Ahmed is funded by EDF through a contract with the CNRS. We thank the two referees for their very useful review of our manuscript.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://

doi.org/10.1016/j.jhydrol.2019.02.046.

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