4.1. Experimental Results
In the compaction process of a soil-rock mixture, fine particles are bonded to form an aggregate structure, and the pore distribution is usually bimodal. The SWCC curve is usually a double-step shape consisting of two “S” shapes combined up and down [
30].
Compaction degree, also known as compaction coefficient, is the ratio of the actual dry density of the fill to the maximum dry density, and the compaction coefficient of each specimen is calculated according to Equation (3).
in the formula:
—compaction coefficient.
—target dry density of the specimen (g/cm3).
—the maximum dry density of the prepared specimen (g/cm3).
Since there is a large error between the actual compaction degree of the specimens produced by the compaction method and the target compaction degree, the soil-water characteristic surface model of coupled moisture content-matrix suction-compaction degree is used to fit the test data, and the specific expression of the mathematical model is:
in the formula:
, ,—fitting parameters.
—mass moisture content (%).
—Compaction degree.
—Substrate suction (kPa).
With the semi-logarithmic horizontal coordinate indicating the matrix suction and the vertical coordinate indicating the mass moisture content, the soil-water characteristic surface plot of the soil-rock mixture is shown in
Figure 4, and the surface fit was 0.873. The fitted values of
P1,
P2,
P3,
P4,
P5 were 2.8450, 1.0178, 1.1654, 0.0004, and 0.9827, respectively. There was good agreement between the test data and the model, and the model fit was successful.
As shown in
Figure 5, the soil-water characteristic curves were obtained from the surface in
Figure 4 at compaction degrees of 0.8, 0.85, 0.90, 0.95, and 1.0, respectively, by controlling the compaction degree. From the figure, it can be seen that each curve has an obvious double-step shape, describing the drainage processes of large and small pores in the boundary effect section, transition section, and unsaturated residual section, respectively. Similarly to Wang’s [
6] and Su’s [
24] studies, the two steps on the curve reflected the soil-water characteristics of large and small pores in the soil from top to bottom, respectively. The shapes of the SWCC curves for different compaction degrees were almost the same. Under the condition of equal suction, the mass moisture content of the soil-rock mixture was larger as the compaction degree increased; under the condition of equal mass moisture content, the suction of the soil-rock mixture was larger as the compaction degree increased.
With the semi-logarithmic vertical coordinate indicating the matrix suction and the horizontal coordinate indicating the electrical resistivity,
Figure 6 shows the relationship between the electrical resistivity
of the soil-rock mixture specimen and the matrix suction
. From the figure, it can be seen that in the 0 ≤
< 100 kPa range, the width of the data point distribution was 50, which meant that a certain determination
corresponding to the
had a large range of values, and the functional correspondence was not precise enough.
4.2. Electrical Resistivity Comprehensive Parameter
The matrix suction of unsaturated soils is the result of the combined action of soil structure, particle composition, particle arrangement, and moisture content, etc. Therefore, the structural parameters of soil electrical resistivity were combined to form a electrical resistivity comprehensive parameter that could be used to represent the electrical resistivity characteristics of the soil and reflect the intrinsic variation law of matrix suction [
31]. Zha [
32] proposed the electrical resistivity comprehensive parameter in the study of electrical resistivity parameters of unsaturated soils, and its expression is as follows.
in the formula
—electrical resistivity comprehensive parameter.
—electrical resistivity structure factor. , represents the electrical resistivity of soil-rock mass, and represents the electrical resistivity of pore water.
—shape factor; there is a functional relationship between the shape factor and the structure factor of the soil-rock mass. .
—porosity.
—The anisotropy parameter, which is taken as 1 in this paper.
The electrical resistivity comprehensive parameter
of the calculated values and the measured values of electrical resistivity are shown in
Figure 7; a linear fit of the two was performed using Origin software, and the fitting formula was
where
,
are fitting parameters, and the curve values in the figure are 0.4 and 0.03, respectively.
From
Figure 7, it can be seen that the fitting degree between electrical resistivity comprehensive parameter
and electrical resistivity
was 0.97, and the linear relationship was significant. The electrical resistivity comprehensive parameter could be used to represent the electrical resistivity characteristics of the soil-rock mixture.
4.3. - Curve
- (1)
Fitting of the - curve
Combining the electrical resistivity comprehensive parameter
with the substrate suction,
nonlinear fit was performed. The fitting toolbox of Origin was used to find the best basic functional relationship between
and
, and the composite optimization of the base functional relationship equation was conducted. The optimized mathematical model is as follows.
where
,
and
are fitting parameters that are related to the physical properties of the soil.
The model was used to fit
and
, and the fitting results of the fit are shown in
Table 2. Adjusted R-Square, the corrected coefficient of determination, was used to determine the degree of fit of a multiple linear regression equation. The closer the values of R
2 and adjusted R-Square are to 1, the better the fit of the regression line to the observed values. Root-mean-square error (RMSE) indicates the standard error, which is the square of the ratio of the deviation between the predicted value and the true value to the number of observations
. RMSE can be used to measure the deviation of the observed value from the true value, which well reflects the precision of the measurement. Each decision index indicates the success of curve fitting.
With
as the semi-logarithmic vertical coordinate and
as the horizontal coordinate, the fitted curve is plotted as shown in
Figure 8. It can be seen from the diagram that there is a certain exponential function relationship between the substrate suction
and the electrical resistivity comprehensive parameter
. The value of
increases with the increase in
, and the overall increase rate of
is also increasing. When
,
-
curve is very steep and
increases rapidly from 0 kPa to 1000 kPa; when
, the
-
curve slows down, but the increment of
is 9000 kPa, and the increment of
is not weakened; when
, the slope of the
-
curve continues to increase, and the increment of
reaches 90,000 kPa.
- (2)
Physical meaning of the - curve
According to the trend of the
-
curve, the curve can be divided into three segments: with the increase in
, the order is
,
and
;
at the intersection of
and
is 1000 kPa, and ψ at the intersection of
and
is 10,000 kPa, as shown in
Figure 9a. By fitting the measured mass moisture content and matrix suction, combined with the previous analysis of the SWCC curve, it can be found that when 1000 ≤
< 10,000 kPa, the SWCC curve is exactly the oblique part of the small pore, that is, the stage when the drainage of the large pores in the soil has been completed and the small pores are about to start draining in large quantities. The SWCC curve is divided into the three segments
,
and
by using the points corresponding to
at 1000 kPa and 10,000 kPa, respectively, as shown in
Figure 9b.
As can be seen in
Figure 9,
reflects the entire large pore drainage phase of the
-
relationship curve; at this stage, the water discharged from the large pores is about half of the total, and the moisture content
of the soil at this time is higher, so the
reflecting the electrical resistivity characteristics takes a low value, approximately equal to 2.
reflects the large-scale drainage stage of the small pores of the
-
relationship curve; in this stage, the water discharged from small pores is about the same as that from large pores, but the increment of
is about four times that of
, which shows that
increases with the decrease in
, and the rate of increase increases accordingly.
reflects the unsaturated residual phase of the SWCC curve; in this phase, the weak decrease in the moisture content
of the soil increases the
and
rapidly, and the rate of increase exceeds that of the
phase. Overall, the matrix suction and electrical resistivity keep increasing as the water in the soil decreases, and the lower the moisture content of the soil, the higher the corresponding growth rate of matrix suction and electrical resistivity.
In general, it is seen from
Figure 9a,b that the pore structure and water content are the main factors affecting electrical resistivity
and matrix suction
. Additionally, the change in pore structure and the increase in water content during the water absorption process are the main reasons for this phenomenon. However, further research is needed on the change in pore structure of soil-rock mixtures during water absorption; this problem will be further studied from the perspective of meso-structure in future studies.
4.4. Parameter Sensitivity Analysis of - Model
As shown in
Figure 10, the model parameters are
= 0.0001,
= 2.
= 0.8,
= −400, and the control variable method is used to plot
-
curves under the influence of each parameter.
From
Figure 10a, it can be seen that when the substrate suction
is less than 1000 kPa, the parameter
takes different values of the curves to overlap, and the change in
has no effect on the
segment curves. In the
segment, the slope of
-
is constant, and with the change in
, the left and right shifts. It can be seen that the change in parameter
affects the horizontal width of the
segment of the curve; when
increases, the horizontal width of the
segment decreases.
From
Figure 10b, it can be seen that when the substrate suction
is less than 1000 kPa, the curves of different parameter
values coincide and the change in
has no effect on the
segment curves. When
is greater than 1000 kPa, the parameter
changes affect the curves of the
and
segments, and with the increase in
, the horizontal width of the
segment decreases and the slope of the
segment increases.
From
Figure 10c, it can be seen that with the increase in
, the
-
curve in the
the segment is shifted upward, and the shifted curves keep overlapping toward the starting and ending directions. Therefore, the change in parameter
has an impact on
,
and
but it mainly affects the height of
, and the degree of influence gradually decreases toward the two ends of the curve.
Through this experimental study, the mathematical model of the matrix suction- electrical resistivity comprehensive parameter (-) of soil-rock mixture was developed, and the results can be applied to hidden danger detection and quality evaluations in geotechnical engineering. However, from the fitting of the -model to the experimental data, the points with larger matrix suction were concentrated near the model curve, while the points with smaller matrix suction were distributed and more scattered, and the application of the model was not perfect. This problem can be studied further in the future.