Exploration on electrical resistance tomography in characterizing the slurry spatial distribution in cemented granular materials

Abstract

This study investigated the application of electrical resistance tomography (ERT) in characterizing the slurry spatial distribution in cemented granular materials (CGMs). For CGM formed by self-flow grouting, the voids in the accumulation are only partially filled and the bond strength is often limited, which results in difficulty in obtaining in situ samples for quality evaluation. Therefore, it is usually infeasible to evaluate the grouting effect or monitor the slurry spatial distribution by a mechanical method. In this research, the process of grouting cement paste into high alumina ceramic beads (HACB) accumulation is reliably monitored with ERT. It shows that ERT results can be used to calculate the cement paste volume in the HACB accumulation, based on calibrating the saturation exponent n in Archie’s law. The results support the feasibility of ERT as an imaging tool in CGM characterization and may provide guidance for engineering applications in the future.

Graphical abstract

Keywords

electrical resistance tomography / cemented granular material / grouting / spatial distribution / Archie’s law

1 Introduction

Electrically-based methods have been gradually introduced into the field of construction engineering. Compared with other new detection methods based on attenuation of radiation [1–5], electrically-based methods are safe, economical, easy, quick and environmentally friendly, especially for largescale engineering objects. The effectiveness of electrically-based methods is mainly due to the relationships between electrical measurement results and porous properties of the target. For instance, there are quite strong correlations between electrical parameters and water saturation in porous materials [6–11], and electrical parameter change in cement-based materials can match the load–deflection curve quite well in strength tests [12,13]. Also, the electrical parameters of cement-based materials have correspondence with different stages of the hydration reaction [14–16] as well as with alkali-silicate reaction, delayed ettringite formation and the pozzolanic effect [17,18]. Early-age resistivity of cement-based materials can even be used to predict the material properties over a longer period [19,20].

As one type of electrically-based method, electrical resistance tomography (ERT) was originally developed for geophysical exploration [21,22]. In this method, two sets of electrodes, buried in the target body, respectively transmit and receive electrical signals in a pre-set mode, in order to assess the resistivity spatial distribution through subsequent calculations. ERT has been applied in construction engineering such as in the monitoring of seepage in embankment dam [23,24], and in assessment of infiltration processes and crack distribution in cement-based materials [25–29]. However, overall, the application scenarios of ERT in construction engineering are still quite limited.

Cemented granular material (CGM) is a broad category that includes a lot of natural (e.g., sedimentary rock, wheat endosperm) and man-made materials (e.g., grouted soil, asphalt, high explosive) [30]. Briefly, the core feature of CGM is that individual densely-packed granules are bound by cementitious materials into a whole. The voids between granules are partially or totally filled by cementitious materials, which ensures the integrity of the granular system and improves its mechanical properties [30–33]. It’s worth pointing out that self-flow grouting process, which can be treated as one type of unsaturated moisture flow in the earth-rock system, can also be considered as a process of CGM formation. As the world’s largest developing country, China has high demand for comprehensive renovation of existing and potentially dangerous embankment dams, as well as for the construction of small and medium-sized water resources allocation projects. Self-flow grouting can play an important role in strengthening these structures. Through control measures, self-flow grouting can accurately and economically reinforce key areas of earth-rock structures (including embankment dams). The voids between particles of the key areas are always only partially filled after grouting. However, under these circumstances, the heterogeneity of the CGM system is relatively strong, and it is usually impossible to monitor the spatial distribution of the slurry in the particle system (e.g., the geometrical boundary of the slurry, and the grouting compactness in different regions). The CGM system can be far from dense and the bond strength is thereby limited, which results in the difficulty in obtaining in situ samples for quality evaluation of a structure. However, spatial distribution of the slurry not only has an important impact on the overall performance of the earth-rock structure, but also provides the basis for analysis of embarkment dam slope stability against sliding and for assessing the overall seismic resistance of the dam. Therefore, it is important to find a nondestructive monitoring method for CGM systems, in order to obtain the key performance indexes such as grouting compactness, spatial distribution of the slurry, strength of cemented bodies. Such a method can provide the basis for on-site nondestructive monitoring and quality control for future CGM systems such as embankment dams.

In this paper, as in some other recent research [34–37], artificial CGM samples are made from the high alumina ceramics beads (HACB) and cement paste. Based on the development of algorithm and equipment, this paper first monitors the process of grouting cement paste into HACB accumulation with ERT. Next, the saturation exponent n of the HACB accumulation in Archie’s law [6] is calibrated, and ERT results are used to calculate the cement paste volume in the HACB accumulation. This paper supports the feasibility of ERT as a safe, economical, easy, quick and environmentally friendly imaging tool in CGM characterization and offers guidance for future engineering applications.

2 Experiment design

2.1 Materials

In this study, P·I 42.5 cement produced by China United Cement Group Co., Ltd. and Type R216 polycarboxylate-based superplasticizer produced by Beijing Sinoconfix Co., Ltd. are used for cement paste production. The HACB is produced by Zibo Bomai Ceramic Material Co., Ltd. The appearance and particle size distribution of HACB are shown in Fig.1, and some basic performance indicators of HACB are shown in Tab.1. During the experiment, the temperature and the relative humidity are controlled to (20 ± 1) °C and 15% ± 5%, respectively. The proportions as well as fluidity indicators [38] of cement paste used in the experiment are shown in Tab.2.

2.2 Experiment system

The experiment system is composed of computer, signal transmitter, multi-channel electrode switcher, test sample and signal receiver, and the logic diagram of the system is shown in Fig.2.

Two sets of electrodes (12 source electrodes and 36 receiving electrodes) are made of Marine 316 stainless steel with a diameter of 1 mm, as shown in Fig.3. The middle part of the electrode is wrapped with insulating sheath, with only about 1–2 mm exposed at the end. For each source electrode, its one end is connected to the multi-channel electrode switcher by a copper wire, and the other end is compressed to the test sample. For each receiving electrode, its one end is connected to the signal receiver with a copper wire, and the other end is compressed to the test sample. In the experiment, after receiving instruction from the computer, the signal transmitter generates voltage signal, and the multi-channel electrode switcher traverses the voltage signal across every two source electrodes according to a preset scheme (5 s voltage signal for every two source electrodes) in each measurement cycle. The peak value and frequency of the square wave voltage signal are 30 V and 0.5 Hz, respectively. While the signal transmitter and the multi-channel electrode switcher are working, the signal receiver records the potential signal at each receiving electrode. Each measurement cycle typically involves 66 voltage injections (C_{12}^2 = 66) and 36 potential readings for each voltage injection, thereby resulting in a total of 2376 readings. Each measurement cycle takes 330 s to complete.

In this study, artificial CGM samples are made from HACB and cement paste. The mold for test sample preparation in the experiment consists of three parts, including the paste pouring area, sample forming area and paste receiving area, as shown in Fig.4. The sample forming area, as shown in Fig.5(a), is the main part of the mold, and is formed by tightening the upper mold and the lower mold with bolts and nuts, in order to seal the sample forming area to prevent leakage of cement paste during the pouring process. The upper mold is a piece of polymethyl methacrylate sheet with a large number of round holes near its center (shown in Fig.5(b)). For the lower mold, four pieces of polymethyl methacrylate sheets (shown in Fig.5(c)) are respectively adhered perpendicular to the plane of one polymethyl methacrylate sheet (the sheet is exactly the same as the upper mold shown in Fig.5(b)) as side plates, just on the outer boundary of the perforated area of the polymethyl methacrylate sheet. Three source electrode holes and nine receiving electrode holes are drilled on each side plate (shown in Fig.5(c)), and Marine 316 electrodes are threaded through these holes and exposed in the sample forming area for about 1–2 mm during the experiment. The four side plates and the polymethyl methacrylate sheet form an open space, with inner dimensions of 100 mm × 100 mm × 20 mm, for HACB accumulation. The location for the source electrodes and the receiving electrodes, as well as the target body boundary for numerical calculation are shown in Fig.5(d).

The core step of sample preparation is the pouring of fresh cement paste into the mold (also shown in Fig.4). Before pouring, fresh cement paste is stored in a plastic beaker. When pouring the paste from the beaker, a polymethyl methacrylate square tube (with inner dimension of 100 mm × 100 mm) in the paste pouring area acts as an intermediary storage container to ensure the paste flows through the holes on the upper surface of the sample forming area evenly. When the paste flows through the sample forming area filled with HACB, some of the paste adheres to the surface of HACB, and the remaining paste flows into the paste receiving area (a polymethyl methacrylate tray) through the holes on the lower surface of the sample forming area.

2.3 Electrical testing procedure

2.3.1 Monitoring of cement paste grouting

One test group for testing the feasibility of using ERT to monitor cement paste grouting has four main steps.

STEP 1.1. HACB are placed into the lower mold and manually compacted until the upper surface of the HACB accumulation is flush with the upper surface of the lower mold (the porosity is about 43% for all groups), and then the upper mold is fastened to the lower mold with bolts and nuts.

STEP 1.2. 500 mL fresh cement paste is poured into the mold, until no more paste flows out of the holes of the lower mold.

STEP 1.3. One measurement cycle is made with the experiment system and procedure mentioned in Subsection 2.2.

STEP 1.4. The lower surface of the sample forming area is sealed with aluminum foil tape then the bolts and nuts are unscrewed, and the upper mold is removed.

STEP 1.5. A certain volume of cement paste mixed in STEP 1.2 at one selected point of the HACB accumulation is grouted six successive times (about 6.5, 6.5, 6.5, 13.0, 13.0, and 19.5 mL cement paste for each time), and one electrical measurement is made with the experiment system after each grouting.

Between each pair of adjacent times of paste grouting in STEP 1.5, plastic covers are added to the mold to reduce water evaporation during the electrical measurement. It should be noted that the cement paste grouted in STEP 1.5 is the main concern of the experiment, compared to those grouted in STEP 1.2. This is because, in actual engineering grouting projects, natural earth and stone materials have certain conductivity, while the HACB in our experiment can be regarded as insulators under dry conditions. Therefore, the operation in STEP 1.2 gives the HACB accumulation a certain conductivity, so as to better simulate the actual engineering application scenarios of grouting.

2.3.2 Cement paste volume calculation

In this research, for testing the ability of ERT in calculating the cement paste volume in the HACB accumulation, the saturation exponent n in Archie’s law needs to be calibrated first. One test group for n calculation mainly includes four steps:

STEP 2.1. Same as STEP 1.1 in Subsubsection 2.3.1.

STEP 2.2. Same as STEP 1.2 in Subsubsection 2.3.1.

STEP 2.3. Same as STEP 1.3 in Subsubsection 2.3.1.

STEP 2.4. After sealing the lower surface of the sample forming area with aluminum foil tape, unscrewing the bolts and nuts as well as removing the upper mold, cement paste mixed in STEP 2.2 is grouted into the HACB accumulation until all the remaining voids in the accumulation are filled, and then another measurement cycle is made with the experiment system and procedure mentioned in Subsection 2.2.

After the calculation of saturation exponent n, the cement paste volume in the experiment in Subsubsection 2.3.1 can be calculated after some derivation, which are both explained in Subsection 3.2.

2.3.3 Electrical resistance tomography

A nonlinear algorithm is used to calculate the slurry spatial distribution in the HACB accumulation. In forward calculation, the potential at each receiving electrode can be solved by the three-dimensional finite difference numerical simulation when the three-dimensional spatial distribution of resistivity of the model is known. According to Ohm’s law and charge conservation law, on the basis of introducing Dirac delta function, the potential distribution of three-dimensional geoelectric field of point source meets the differential equation below [39]:

(1)\mathrm{\nabla }\cdot [\sigma \left(x,y,\mathit{\text{z}}\right)\mathrm{\nabla }\phi \left(x,y,\mathit{\text{z}}\right)]=-I\delta (x-x_0)\delta (y-y_0)\delta (\mathit{\text{z}}-{\mathit{\text{z}}}_0),

where σ stands for conductivity, φ for potential, I for current intensity, (x₀,y₀,z₀) for the position of point source.

In terms of internal boundary conditions, the internal boundary with resistivity differences in the stable current field satisfies the following relationships:

(2){\phi }_1={\phi }_2,

(3){\sigma }_1\frac{\mathrm{\partial }{\phi }_1}{\mathrm{\partial }{n}_1}=-{\sigma }_2\frac{\mathrm{\partial }{\phi }_2}{\mathrm{\partial }{n}_2},

where {\phi }_{\text{1}}, {\phi }_{\text{2}} stand for the potential of two areas where the conductivities are, respectively, {\mathit{\text{σ}}}_{\text{1}} and {\mathit{\text{σ}}}_{\text{2}}. The quantities n_{\text{1}} and n_{\text{2}} are the normal vector of the boundary positions of the same two areas.

The outer boundary refers to the boundary at infinity of the meshed area and the interface between the measured area and air. The external boundary conditions can generally be divided into three categories, but only the second boundary condition (Neumann boundary condition) is used on the interface between the tested sample and air in this chapter, as shown in Eq. (4), to specify that the normal vector of current density on the interface is zero.

(4)\mathit{\text{σ}}\frac{\mathrm{\partial }\phi }{\mathrm{\partial }n}=0.

The boundary value problems of Eqs. (1)–(4) constitute the basic equations for solving the potential distribution in the three-dimensional geoelectric field of a point source. Then, finite difference method [39,40] is used to solve the potential distribution. When discretizing the volume of the region to be solved, the volume of the element near any mesh node satisfies the equation below:

(5)V_{i,j,k}=\frac{(\mathrm{\Delta }{x}_i+\mathrm{\Delta }{x}_{i-1})(\mathrm{\Delta }{y}_i+\mathrm{\Delta }{y}_{i-1})(\mathrm{\Delta }{\mathit{\text{z}}}_i+\mathrm{\Delta }{\mathit{\text{z}}}_{i-1})}{8},

where \text{Δ}{x}_i, \text{Δ}{y}_i, \text{Δ}{\mathit{\text{z}}}_i are the distances from node (i, j, k) to node (i + 1, j, k), (i, j + 1, k), (i, j, k + 1), respectively. Then, the following further equation can be obtained after the grid element integration of Eq. (1) in any element:

(6){\stackrel{}{\iiint }}_{V_{i,j,k}}\mathrm{\nabla }\cdot \left[\mathit{\text{σ}}\mathrm{\nabla }\phi \right]\text{d}v={\stackrel{}{\iint }}_{S_{i,j,k}}\mathit{\text{σ}}\mathrm{\nabla }\phi \cdot n\text{d}s,

where S_{i,j,k} corresponds to the six faces of the volume unit V_{i,j,k}, and n is the external normal of S_{i,j,k}. For any internal node, Eq. (6) can be further transformed into:

(7)\begin{array}{rl}& C_{\mathrm{t}\mathrm{o}\mathrm{p}}{\phi }_{i,j,k-1}+C_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m}}{\phi }_{i,j,k+1}+C_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}{\phi }_{i-1,j,k}+C_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{\phi }_{i+1,j,k}\\ & +C_{\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{t}}{\phi }_{i,j-1,k}+C_{\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}}{\phi }_{i,j+1,k}+C_{\mathrm{p}}{\phi }_{i,j,k}\\ =& \{\begin{array}{l}I,(x_0,y_0,{\mathit{\text{z}}}_0)\in V_{i,j,k},\\ 0,(x_0,y_0,{\mathit{\text{z}}}_0)\notin V_{i,j,k},\end{array}\end{array}

where C_{\text{top}}, C_{\text{bottom}}, C_{\text{left}}, C_{\text{right}}, C_{\text{front}}, C_{\text{back}}, C_{\text{p}} are the connection coefficients of the upper, lower, left, right, front, and rear grid nodes of grid node (i, j, k) and the connection coefficients of grid node (i, j, k) itself, respectively. After transforming Eq. (7), we can get:

(8){\mathit{K}}^{\prime }\phi =B,

where {\mathit{K}}^{\prime } is the coefficient matrix, \phi is the potential distribution. Then, the potential information can be obtained by solving the large linear equations corresponding to Eq. (8) by the incomplete Cholesky decomposition conjugate gradient method, and the potential difference information can be obtained by further sorting.

The observation data volume d_obs can be obtained by applying voltage signal and making measurement at different positions on the surface of the limited target body. The three-dimensional distribution of resistivity in the target body can be obtained by three-dimensional inversion calculation of this observation data.

A variable m can be defined as the model vector representing the discretized conductivity distribution and F(m) is the forward function. According to the regularized inversion, the objective function of the ith inversion iteration can be defined as [39]:

(9)\begin{array}{rl}\psi \left(m_i\right)=& {\left[d_{\text{obs}}-F\left(m_i\right)\right]}^{\text{T}}{\mathit{V}}^{-1}\left[d_{\text{obs}}-F\left(m_i\right)\right]\\ & +\lambda {\left(m_i-m_0\right)}^{\text{T}}{\mathit{L}}^{\text{T}}\mathit{L}\left(m_i-m_0\right),\end{array}

where V is the variance of the error vector, L is the Laplace matrix, and λ is a regularization parameter and is chosen by cooling scheme [41].

The nonlinear conjugate gradient method is used to inverse the model and get the distribution of three-dimensional resistivity, in order to minimize Eq. (9). The gradient of the objective function can be obtained as follows after calculating the partial derivative of Eq. (9) to the model:

(10)g_i=-2{\mathit{J}}_i^{\mathrm{T}}{\mathit{V}}^{-1}{\mathit{e}}_i+2\lambda {\mathit{L}}^{\mathrm{T}}\mathit{L}\left(m_0-m_i\right),

where J_i and e_i are the Jacobian matrix and the data residual vector in the ith inversion iteration, respectively. Then, the iterative calculation can be carried out according to the search direction p and step size \mathit{\text{α}} determined by Eqs. (11) and (12).

(11){\mathit{p}}_i=-C^{-1}{g}_i+{\beta }_i{\mathit{p}}_{i-1},

(12){\alpha }_i=-\frac{{\mathit{p}}_i^{\mathrm{T}}{g}_i}{2\left[{\left({\mathit{J}}_i{\mathit{p}}_i\right)}^{\mathrm{T}}{\mathit{V}}^{-1}\left({\mathit{J}}_i{\mathit{p}}_i\right)+\lambda {\mathit{p}}_i^{\mathrm{T}}{\mathit{L}}^{\mathrm{T}}\mathit{L}{\mathit{p}}_i\right]},

where C is the precondition factor, {\mathit{\text{β}}}_i is a scalar parameter. The three-dimensional nonlinear conjugate gradient inversion algorithm does not solve the Jacobian matrix directly. Instead, the calculation is simplified by finding the product of the Jacobian matrix (or its transpose) and a vector [39].

3 Experiment result

3.1 Monitoring of cement paste grouting

Two groups of experiment are carried out in this section to ensure credibility, and the selected points of cement paste grouting in STEP 1.5 are on one corner and on the center of the HACB accumulation in the two groups, respectively. In both groups, the mix proportion of the cement paste is the mix proportion No. 3 in Tab.2.

The resistivity distribution inside the sample can be obtained by three-dimensional inversion calculation after the experiment, with the help of a nonlinear conjugate gradient algorithm. Then, the change in the resistivity (\left|\mathrm{\Delta }\mathit{\text{ρ}}\text{/}\mathit{\text{ρ}}\right|, compared to STEP 1.3) at the same position after each time of grouting in STEP 1.5 is obtained. Fig.6 shows the photos of the HACB accumulation and the distribution of \left|\mathrm{\Delta }\mathit{\text{ρ}}\text{/}\mathit{\text{ρ}}\right| on the z = −15 mm profiles in the first experiment group, and Fig.7 shows those of the second group. It should be noted that since STEPs 1.2–1.5 are all completed in 1 h after the production of the fresh cement paste in STEP 1.2, the conductivity of the paste can be regarded as having a fixed value during the experiment [14,15].

Fig.6 and Fig.7 show that the ERT method is rather sensitive to the changes in process of the spatial distribution of the cement paste grouted in STEP 1.5. In the first experiment group, in the early stage of the STEP 1.5, the paste grouted mainly distributes near the selected grouting point, and the distribution area of this paste gradually expands as the paste volume increases, as shown in Fig.6(b)–6(e); \left|\mathrm{\Delta }\mathit{\text{ρ}}\mathit{\text{/}}\mathit{\text{ρ}}\right| has a gradient in the horizontal direction. With further increase of the paste volume, the resistivity decreases in almost all the regions of the sample forming area, while the absolute value of \left|\mathrm{\Delta }\mathit{\text{ρ}}\text{/}\mathit{\text{ρ}}\right| increases roughly uniformly in the horizontal direction. In the second group, similarly, in the early stage of the STEP 1.5, the paste grouted in this step mainly distributes near the selected grouting point, and the distribution area of these paste also gradually expands as the cement paste volume increases, as shown in Fig.7(b)–Fig.7(e). The \left|\mathrm{\Delta }\mathit{\text{ρ}}\text{/}\mathit{\text{ρ}}\right| gradient in the horizontal direction is again obvious in this stage, but not as much as for the first group, which may be because the grouting point is located in the middle of the mold, so the slurry can spread to a larger range in the horizontal direction after the first few grouting operations. With further increase of the paste volume, the spatial range where the resistivity reduces no longer changes.

3.2 Cement paste volume calculation

For porous media, the saturation exponent n is a key parameter that establishes a connection between the pore saturation and the overall conductivity, as given by Archie’s law [6]:

(13)\mathit{\text{σ}}={\mathit{\text{σ}}}_{\text{w}}{\mathrm{\varnothing }}^m{S}^n,

where \mathit{\text{σ}}, {\mathit{\text{σ}}}_{\text{w}} stand for the conductivity of the porous medium and the pore solution, respectively [S/m]; \mathrm{\varnothing } for the total porosity of the porous medium; S for the total pore saturation of the porous medium; m and n for the cementation exponent and the saturation exponent of the porous medium, respectively.

In our experiments, the HACB are the non-conductive solid phase framework in the porous medium, and the cement paste is the electrolyte inside the porous medium. Previous research has shown that, after pouring fresh cement paste into the HACB accumulation bodies, like those used in our experiment, there is a linear relationship between the amount of cement paste which adheres and deposits on the surface of HACB and the yield strength of the cement paste [34,35]:

(14){\mathit{\text{ρ}}}_{\text{m}}=3.95{\mathit{\text{τ}}}_0+3.01,

where {\mathit{\text{ρ}}}_{\text{m}} stands for the ratio of the amount of cement paste which adheres and deposits on the surface of HACB and the total cavity volume of the sample forming area of the mold system. {\mathit{\text{τ}}}_0 represents the yield strength of the cement paste (Pa).

Previous research has also shown that the yield strength {\mathit{\text{τ}}}_0 of cement paste can be obtained by slump flow test [42,43]:

(15){\mathit{\text{τ}}}_0=\frac{225{\rho }_{\mathrm{C}}g{V}_{\text{S}}^2}{4{\text{π}}^2{SF}^5},

where {\rho }_{\mathrm{C}} stands for the density of the cement paste (2088.1 kg/m³ in this experiment), g for the gravitational field strength (9.8 N/kg), V_s for the cavity volume of the slump cone, SF for the slump flow value of the cement paste.

Eleven groups of experiments are carried out in this section, with the cement paste made from the nine mix proportions listed in Tab.2.

The current values measured by the same source electrode combination type in STEPs 2.3 and 2.4 are respectively named I_{1i} and I_{2i}, and it should be pointed out that I_{1i}/I_{2i} can be used to represent the ratio of sample conductivity in STEPs 2.3 and 2.4. Like Subsection 3.1, since STEPs 2.2–2.4 are completed in 1 h after the production of the cement paste in STEP 2.2, the conductivity of the paste can still be regarded as fixed during the whole experiment. Further, literatures show that the main influencing factors of cementation exponent m and the saturation exponent n of the porous medium include particle shape, pore structure, pore connectivity, etc. [7,44,45]. In our experiment, the HACB are packed in a fixed form, so ∅, m, and n can all be regarded as fixed values. The total pore saturation of the HACB accumulation in STEPs 2.3 and 2.4 can be written as S_1 and S_2. Therefore, according to Eq. (13), the conductivity of the HACB accumulation in STEPs 2.3 and 2.4, respectively, named {\mathit{\text{σ}}}_1 and {\mathit{\text{σ}}}_2, satisfy the equations below:

(16){\mathit{\text{σ}}}_1={\mathit{\text{σ}}}_{\text{w}}{\mathrm{\varnothing }}^m{S}_1^n,

(17){\mathit{\text{σ}}}_2={\mathit{\text{σ}}}_{\text{w}}{\mathrm{\varnothing }}^m{S}_2^n.

The voids in the HACB accumulation have been completely filled by the paste in STEP 2.4, therefore, S_2 = 1. so,

(18)\frac{{\mathit{\text{σ}}}_1}{{\mathit{\text{σ}}}_2}=S_1^n.

The average value of I_{1i}/I_{2i} measured by the 66 combinations of source electrodes in each experimental group is taken as the {\mathit{\text{σ}}}_1/{\mathit{\text{σ}}}_2 value in the corresponding experimental group, and S_{\text{1}} satisfies the relationship below:

(19)S_1=\frac{{k\rho }_{\text{m}}}{1-K_0},

where k stands for a correction factor, K_0 for the ratio of V_0 (total volume of accumulated HACB, about 114.6 cm³) to V_{\text{M}} (total volume of the sample forming area, 200 cm³).

Equation (14) is based on the experiment samples with a height of 200 mm [34,35]. In the experiment in Refs. [34,35], there are three components of slurry in the mold, which are the slurry that adheres and deposits in that cases of each of: the HACB accumulation, the lower and the upper surface of the mold, and the side-walls of the mold. As the specimen is sufficiently high in the vertical direction, the quantity of slurry that adheres and deposits to the lower surface and the upper surface of the mold is relatively small compared with that of the slurry that adheres and deposits in the HACB accumulation. Therefore, it can be considered that Eq. (14) mainly takes into account the slurry that adheres and deposits in the HACB accumulation and to the side-walls of the mold. However, as the sample in our experiment is not that high, the quantity of slurry that adheres and deposits to the lower surface and the upper surface of the mold in STEP 2.3 has become a considerable amount, compared with that in Refs. [34,35]. Therefore, the correction factor k is introduced. As mentioned above, the slurry that adheres and deposits to the side-walls of the mold has been taken into account in Eq. (14), and thus this part of the slurry is not considered in the correction. The value of k is taken as 1.142, assuming that the thickness of cement paste that adheres and deposits to the two parts of the mold is the same as the thickness of the cement paste that adheres and deposits to the surface of the HACB. The saturation exponent n can be obtained after the simultaneous solution of Eqs. (14), (15), (18), and (19):

(20)n=\frac{\ln ({\mathit{\text{σ}}}_1/{\mathit{\text{σ}}}_2)}{\ln \left[{\displaystyle \frac{k}{1-K_0}}\left({\displaystyle \frac{3.95\times 225{\mathit{\text{ρ}}}_{\text{C}}g{V}_{\text{S}}^2}{4{\text{π}}^2{SF}^5}}+3.01\right)\right]}.

The values of n are shown in Tab.3. Previous research has shown that the saturation exponent n is obviously different in different porous media, which is roughly in the range of 3.5–5 in hardened cement paste and mortar samples, and in the range of 1.5–3 in porous rocks [7]. It can be seen from Tab.3 that the saturation exponents in different groups are similar. For simplicity, in the subsequent analysis in Subsection 3.2, the saturation exponent of the HACB accumulation body is taken as 1.0.

With the saturation exponent n of the HACB accumulation, the volume of the paste grouted into the HACB accumulation each time in Subsection 3.1 can be calculated, based on Eq. (1) and the electrical conductivity values of each finite element grid obtained from numerical inversion calculation. The total pore saturations of the HACB accumulation in STEP 1.3 and after grouting for one certain time in STEP 1.4 can be written as S_{\text{3}} and S_{\text{4}}, respectively. The conductivity value of a certain finite element grid of the HACB accumulation at the corresponding times can be written as {\mathit{\text{σ}}}_{\text{3}i} and {\mathit{\text{σ}}}_{\text{4}i}. Therefore, considering the saturation exponent n of the HACB accumulation has been obtained (n = 1.0),

(21)S_4=\frac{1}{t}{\sum }_{i=1}^t\frac{{\mathit{\text{σ}}}_{4i}}{{\mathit{\text{σ}}}_{3i}}{S}_3,

where t stands for the total number of finite element grids (t = 12800), and S_3 can be calculated from Eq. (14).

Therefore, the total volume of cement paste grouted in STEP 1.5 after one certain time of grouting is:

(22)V_{\text{c}}^{\prime }=(V_{\text{M}}-V_0)(S_4-S_3).

The V_{\text{c}}^{\prime } obtained by numerical inversion calculation with Eq. (22) and by manual measurement during the experiment for the two experiment groups in Subsection 3.1 are shown in Fig.8. Considering that some {\mathit{\text{σ}}}_{4i}/{\mathit{\text{σ}}}_{3i} values of some grids are obviously higher than the reasonable range, the 1% finite element grids with the largest {\mathit{\text{σ}}}_{4i}/{\mathit{\text{σ}}}_{3i} value are discarded when processing the data. Fig.8 shows that the overall trend of V_{\text{c}}^{\prime } obtained by numerical inversion calculation and by manual measurement is same, but a certain deviation between the results still exists. It is believed that the deviation is mainly due to the limitation of source electrode channels on the accuracy of the inversion results in the vertical direction. There is only one row of source electrodes at one horizontal elevation of the mold, and only one row of receiving electrodes at another horizontal elevation, as shown in Fig.4(c) and Fig.4(d).

4 Conclusions

This study investigated the application of ERT method in characterizing the slurry spatial distribution in CGM. The following conclusions can be drawn from the research.

1) The ERT method can reliably monitor the process of grouting cement paste into HACB accumulation.

2) According to multiple groups of electrical experiments, the saturation exponent n of the HACB accumulation in this research can be taken as 1.0.

3) The ERT method has the ability to establish a quantitative relationship between the electrical monitoring results and the pore saturation in the HACB accumulation.

According to relevant knowledge in the field of geophysics, the algorithm and the equipment developed in this research may have the ability to be applied to the non-destructive monitoring of actual self-flow grouting projects after some necessary improvements are made. That is to say, this research increases the understanding of the spatial distribution of slurry in CGM formed by self-flow grouting, supports the feasibility of ERT as an imaging tool in CGM characterization and may provide guidance for engineering applications in the future.

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