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The characteristics of rock face dissolution reflect to a certain extent the degree of karst development at the construction site and whether the conditions of karst foundation are good or not, so they can be used as a basis for the design of building foundation. However, huge amounts of reliable rock face data obtained in engineering practice have not been fully utilized, and the studies on rock face dissolution characteristics are still immature, which is an important reason why the dissolution characteristics analysis of karst foundation is often inconsistent with the engineering reality and leads to engineering incidents. With the terminal building at Nanning Wuxu International Airport as an engineering background, this paper defines the rock face dissolution rate as the ratio between the rock face dissolution height and the formation dissolution height (in percentage) and uses it as the index to characterize rock face dissolution degree, and then establishes a method to solve the rock face dissolution rate curve and reveals the distribution pattern of rock face dissolution degree with depth in karst building foundations. According to the results of the study, the rock face dissolution rate of karst building foundations in the karst terrain of Southwest China generally declines exponentially with the depth increasing, and when it is expressed as an exponential function of height, the fitting correlation coefficient can generally be up to more than 0.90. When there are no less than 12 drill holes with spacing of no more than 20m in the site, the rock face dissolution rate curve shows good numerical stability, with the error being generally within 15%, which is sufficient engineering precision.
karst, rock face dissolution degree, rock face dissolution rate, karst foundation
Dissolution is the main factor causing the discontinuous relief of rock faces: on the one hand, the intensity of dissolution differs greatly at different positions of the rock face, and there is much uncertainty in such differences, which is the direct cause of continuous relief; on the other hand, some of the shallowlayer caves may also further aggravate the discontinuous relief of rock faces under the action of dissolution from the outside to the inside – for example, the roof of a shallow karst cave may be thinned and damaged or the side wall of the cave at the free face would eventually form a shape that looks like an eagle beak under the action of dissolution. The discontinuous relief of rock faces reflects the complexity of a karst foundation to some extent [13], so it is an important criterion for evaluating the engineering conditions of karst foundations [4, 5], analyzing the differential settlements of foundations, and the difficulties in pile forming and should be used as one of the bases for the design of building foundations in karst terrain.
There are two types of dissolution in karst foundations  rock face dissolution and cave dissolution. Karst cave has been widely studied in karst engineering geological research [68]. The rate of reaching caves during drilling [911], linear karst rate [12, 13] and karst density [14], etc. are the direct evaluation indicators of cave dissolution in karst terrain. The construction survey of building pile foundations and various geophysical methods are all aimed mainly at the detection of karst caves [1517], and the evaluation of karst cave characteristics [1822] and stability analysis of karst caves [2326] are the main subjects of karst foundation research. The characteristics of rock face dissolution can reflect the karst development in foundations to a certain extent [27], but this indicator cannot properly deal with the discontinuous reliefs of rock faces in complex karst foundations, nor can it characterize the dissolution characteristics of rock faces at construction sites from the overall perspective, so whether it is a reasonable indicator is still worth discussing. Rock face exploration in Karst foundations is a basic task of engineering survey. Most engineering geophysical methods and drilling methods can help obtain survey data like rock face burial depth [2832]. These rock face data are much richer and more reliable than cave data, but they have not been fully and effectively utilized. In summary, there have been a lot of research results on karst caves, but the evaluation theories on and methods for rock face dissolution characteristics still need more research and discussion.
With the terminal building at Nanning Wuxu International Airport as an engineering background, and based on the existing research results, this paper proposes an indicator  rock face dissolution rate – to depict the rock face dissolution degree, develops a method for solving the distribution curve of rock face dissolution rate with depth and at last reveals the distribution pattern of rock face dissolution degree with depth in karst building foundations based on the measured results of rock face dissolution rate at 148 construction sites in the karst terrain of Southwest China.
2.1 Engineering profile
Guangxi Nanning Wuxu International Airport is located south of the former Wuxu Airport in the southwestern suburb of Nanning. It is a regional air link designated in the layout plan of national civil airports. The airport terminal building is located in Yonghong Village, Wuxu Town, Jiangnan District, Nanning City, with a length of about 1080m, a width of 23330m, an area of about 300mu, a ground elevation of about 123.5m and a total floor area of 183,800m^{2}. The terminal building can be divided into the main building and airside concourses, specifically consisting of five functional zones, namely the central hall (zone I), the southern horizontal airside concourse (zone II), the northern horizontal airside concourse (zone III), the southern vertical airside concourse (zone IV) and the northern vertical airside concourse (zone V). In this paper, this building is further divided into 14 subzones according to the plane shape of each functional zone and the number of boreholes surveyed, as shown in Figure 1.
Figure 1. Zoning map of the terminal building
2.2 Regional geological conditions
There is no fault development within 10km in the east and south of the site. There are several faults of unknown nature in the Tertiary and Devonian strata in the hilly area on the northwest side. The nearest one is about 5km away from the site, running in the NE40º direction, with no active fault structure zone passing through. The regional geological structure is relatively stable.
The site is of dissolution peneplain landform, with hills in the east, north and west, and farmland, ponds and woodland in the south. Human activities are frequent, and vegetation coverage is high. There are no obvious karst development morphologies such as karst collapses, funnels and depressions. The terrain is flat with a gentle slope from north to south, whose natural grade is less than 5º.
The overlying soil layer is more than 10m thick, mainly composed of silty clay, clay mixed with breccia and red clay in the Quaternary residual layer (Qel) and limestone in the lower Carboniferous system (C1). The occurrence of the limestone formation is $150^{\circ} \angle 15^{\circ}$.
There is an underground river about 1500m away on the south side. The groundwater flows from west to east, with artificial exploitation, evaporation and underground runoff as its main discharge channels. It is an important source of karst fissure water. The bottom height of a motorpumped well within 5km from the site is about 85120m, and the water yield per well is 1001000t/d. The karst fissure water in the site is abundant, and the rock layer revealed in the survey is mainly located in the vertical vadose zone  shallow horizontal circulation zone.
2.3 Geological conditions of the construction site
The detailed site survey boreholes are evenly arranged along the perimeter lines, corner points and column lines of the building and the central axis of the pedestrian overpass. A total of 362 boreholes were drilled, with spacing being generally between 1525m and about 20m on average. The borehole is generally 58m deep into the intact rock, with a total footage of 4875.4m into the rock formation. 217 caves were exposed in the drilling of 158 boreholes, with a total footage of 371.42m into the caves. The rate of reaching caves during drilling is 088.57%, and the linear karst rate is 018.85%.
The foundation of the building adopts punch filling piles, with a pile diameter of 0.82.0m. Based on the scheme of one pile with one borehole, a total of 1851 boreholes were drilled, with spacing being generally no more than 8m, and about 5m on average. The borehole is all 58m deep into the intact rock, with a total footage of 24706m into the rock. 1556 caves were exposed in the drilling of 940 boreholes, with a total footage of 2666.9m into the caves. The rate of reaching caves during drilling is 29.28% 80.56% and the linear karst rate is 2.9520.99%.
The survey shows that the karst rocks exhibit very different development characteristics  the groundwater conditions are complex, the karst morphologies like funnels, dolines, caves and fissures are well developed, and the rock faces have drastic discontinuous reliefs, indicating it is a strongly developed karst area. The basic information of the detailed survey and construction investigation on each subzone is shown in Table 1.
Table 1. Subzone survey summary table
Sub area 
Detailed investigation 
Investigation in the construction stage 

Boreholes 
Karst caves number 
Drilling depth 
Drilling depth at the intact rock 
Karst curve rate 
Line karst curve rate 
Boreholes 
Karst caves number 
Drilling depth 
Drilling depth at the intact rock 
Karst curve rate 
Line karst curve rate 

pcs 
pcs 
m 
m 
% 
% 
pcs 
pcs 
m 
m 
% 
% 

I1 
32 
10 
35.39 
436.73 
31.25 
8.10 
193 
91 
324.15 
2153.61 
47.15 
15.05 
I2 
46 
21 
59.10 
562.40 
45.65 
10.51 
236 
85 
198.70 
2348.83 
36.02 
8.46 
I3 
46 
6 
10.30 
604.61 
13.04 
1.70 
267 
94 
231.25 
3483.66 
35.21 
6.64 
I4 
32 
11 
31.30 
408.50 
34.38 
7.66 
181 
53 
193.30 
2672.30 
29.28 
7.23 
Ⅱ1 
22 
5 
8.40 
257.00 
22.73 
3.27 
99 
75 
230.19 
1628.91 
75.76 
14.13 
Ⅱ2 
22 
0 
0.00 
356.50 
0.00 
0.00 
72 
58 
53.70 
1093.86 
80.56 
4.91 
Ⅱ3 
16 
4 
6.76 
221.04 
25.00 
3.06 
61 
38 
56.55 
831.18 
62.30 
6.80 
Ⅲ1 
16 
6 
3.50 
191.00 
37.50 
1.83 
61 
35 
30.30 
961.12 
57.38 
3.15 
Ⅲ2 
17 
4 
10.60 
222.80 
23.53 
4.76 
76 
34 
33.10 
1121.18 
44.74 
2.95 
Ⅲ3 
24 
14 
29.50 
292.97 
58.33 
10.07 
99 
59 
138.40 
1476.46 
59.60 
9.37 
Ⅳ1 
19 
6 
17.60 
234.12 
31.58 
7.52 
131 
62 
241.40 
1542.74 
47.33 
15.65 
Ⅳ2 
16 
7 
16.30 
221.10 
43.75 
7.37 
121 
70 
209.00 
1395.81 
57.85 
14.97 
Ⅴ1 
35 
31 
121.47 
644.45 
88.57 
18.85 
138 
107 
414.07 
2506.33 
77.54 
16.52 
Ⅴ2 
19 
33 
21.20 
222.20 
57.58 
9.54 
116 
79 
312.76 
1490.34 
68.10 
20.99 
3.1 Distribution curve of the measured rock face dissolution rate with depth
In this paper, the rock face dissolution rate is defined as the ratio between the rock face dissolution height within a certain depth range and the formation thickness, expressed in percentage as follows:
$r_{s}=\frac{s}{L} \times 100 \%$ (1)
where, r_{s} is the rock face dissolution rate at a certain point within a certain depth range; s is the rock face dissolution height within a certain depth range; and L is the formation thickness within a certain depth range.
The distribution curve of rock face dissolution rate with depth in each subzone of the site can be obtained using the following methods and steps:
(1) Suppose there are n boreholes in the site, and that the rock face elevation of the kth borehole is H_{rk}. According to the positional relationship between the elevation H_{rk} and the interval (H_{i1}, H_{i}] (Figure 2), the rock face dissolution height s_{ik} of the kth borehole within the elevation interval (H_{i1}, H_{i}] is:
$\left\{\begin{array}{l}s_{i k}=0 \quad\left(H_{r k} \geq H_{i1}\right) \\ s_{i k}=H_{i1}H_{r k}\left(H_{i1}>H_{r k} \geq H_{i}\right) \\ s_{i k}=H_{i1}H_{i} \quad\left(H_{r k}<H_{i}\right)\end{array}\right.$ (2)
(a) H_{rk}>H_{i1 }(b) H_{i1}≥H_{rk}≥H_{i}(c)H_{rk}<H_{i}
Figure 2. Positional relationship between rock face and statistical interval
To facilitate the processing of computer programs, formula (1) is written as
$s_{i k}=\max \left(H_{i1}, H_{r k}\right)\max \left(H_{i}, H_{r k}\right)$ (3)
The total rock face dissolution height s_{i }within the height interval (H_{i1}, H_{i}] in the site is obtained as follows.
$S_{i}=\sum_{k=1}^{n} S_{i k}$ (4)
The formation thickness L within the height interval (H_{i1}, H_{i}] is
$\mathrm{L}=n \cdot \Delta H$ (5)
Substitute formulas (4) and (5) into formula (1), and we obtain the rock face dissolution rate r_{si} within the height interval (H_{i1}, H_{i}] in the site:
$r_{s i}=\frac{\sum_{k=1}^{n} s_{i k}}{n \cdot \Delta H} \times 100 \%$ (6)
Substitute formula (3) into formula (6), and we obtain the final calculation formula of rock face dissolution rate r_{si}:
$r_{s i}=\frac{\sum_{k=1}^{n}\left[\max \left(H_{i1}, H_{r k}\right)\max \left(H_{i}, H_{r k}\right)\right]}{(n \cdot \Delta H)}\times 100 \%$ (7)
In this paper, △H is set to be 0.5m to determine the statistical height interval of rock face dissolution rate. According to the above methods and steps, the scatter plot of the measured rock face dissolution rate distribution curve in each subzone of the site can be obtained, as shown in Figure 3 (in the figure, x_real and s_real represent the scatter plots of the measured rock face dissolution rate curve in the detailed survey stage and the construction survey stage, respectively).
3.2 Fitting of the distribution curve of rock face dissolution rate with depth
Cao et al. [33] shows that the dissolution degree of a building foundation in the karst terrain declines exponentially with the depth increasing. The dissolution degree can be characterized by the dissolution rate, which can be approximated as an exponential function in the form of formula (8):
$r=a e^{b\left(HH_{0}\right)}$ (8)
where, r is the dissolution rate at height H, a and b are the fitting coefficients, H is the height and H_{0 }is the height from the rock face.
The dissolution rate in formula (8) is defined as the ratio (in percentage) between the dissolution height within a certain depth range and the formation thickness. The dissolution height consists of two parts  rock face dissolution height and cave dissolution height. Obviously, when the site has no developed cave, the foundation dissolution rate curve is the rock face dissolution rate curve. Therefore, this paper temporarily assumes that the depth distribution function of rock face dissolution rate is consistent with formula (8).
The least square method is used in this paper to fit the measured rock face dissolution rate distribution curve with depth in each subzone of the site according to formula (8). The obvious outlier data at the top and bottom are eliminated during the fitting. The final fitting result is shown in Figure 3 (in the figure x_fit and s_fit are the fitted rock face dissolution rate curve in the detailed survey stage and the construction survey stage, respectively), and the fitting parameters are shown in Table 2 below (unit: m).
Table 2. Range table of fitting data for rock face dissolution rate curves
Subarea 
Detailed survey stage 
Construction survey stage 

Top elevation 
Bottom elevation 
Top elevation 
Bottom elevation 

m 
m 
m 
m 

I1 
110 
101 
110 
104 
I2 
112 
104 
110 
104 
I3 
109 
101 
110 
104 
I4 
110 
105 
110 
105 
Ⅱ1 
109 
96.5 
108 
95 
Ⅱ2 
115 
104.5 
115 
100 
Ⅱ3 
112 
100.5 
112 
104 
Ⅲ1 
110 
104 
110 
94 
Ⅲ2 
109 
105 
109 
103 
Ⅲ3 
109.5 
103 
109.5 
103 
Ⅳ1 
111 
104 
110 
104 
Ⅳ2 
109 
104 
110 
102 
Ⅴ1 
112 
106 
112.5 
101 
Ⅴ2 
110 
103 
110 
104 
3.3 Error curve of rock face dissolution rate
The borehole density during the construction survey stage is generally much greater than that during the detailed survey, and the rock face dissolution rate curve formed based thereon is generally more in line with the actual situation, so it can be deemed that the rock face dissolution rate curve during the construction survey stage reflects the actual rock face dissolution characteristics of the site and can be used as the reference for error analysis of rock face dissolution rate in the detailed survey stage. Accordingly, the error d(r_{si}) of the rock face dissolution rate within the height interval (H_{i1}, H_{i}] in the detailed survey stage is calculated as follows:
$d({{r}_{s}})=\left r{}_{si,x}{{r}_{si,s}} \right$ (9)
where, r_{si,x} and r_{si,s} are the rock face dissolution rates within the height interval (H_{i1},H_{i}] during detailed survey and construction survey, respectively.
According to formula (9), the rock face dissolution rate error curve for each subzone of the site is shown on the left side of the Yaxis in Figure 3. Despite the simple data processing flow, the amount of data to be processed is huge, so a special data processing program needs to be compiled. The curves in Figure 3 are drawn based on the processing results of the selfcompiled data processing program.
Figure 3. Distribution curve of rock face dissolution rate with depth for each subzone of the site
According to Figure 3, when the rock face dissolution rate curve during the construction survey stage of the site is fitted according to formula (8), the fitting correlation coefficient R value is 0.9330.997 and as high as 0.9812 on average, and the fitted curve is in good agreement with the measured curve, indicating that the rock face dissolution rate of the site declines exponentially with the depth increasing (height decreasing). In this study, in addition to the site selected in this paper, the author also collected practical data from 118 karst construction sites in Nanning, Guilin, Liuzhou, Laibin, Hechi, Hezhou and Yulin of Guangxi Province as well as nearly 30 copies of relevant engineering materials from Kunming of Yunnan, Long Gang of Shenzhen, Xiushan County of Chongqing, Guiyang, Zhijin County and Xiuwen County of Guizhou. The strata in the above sites contain various carbonate rocks like limestone, dolomite and marble. The lithological conditions, hydrological conditions and foundation dissolution characteristics of karst development are all quite different, but the function type of the rock face dissolution rate curves is consistent with that for the site studied in this paper, indicating that the rock face dissolution degree of building foundations in carbonate rock areas in Southwest China generally declines exponentially with the depth increasing.
The change pattern of the rock face dissolution rate with depth is consistent with that of the total dissolution rate of the foundation – which is declining exponentially with the depth increasing. In practice, however, the development of caves often shows a weakstrongweak trend with the depth increasing, indicating that rock faces are often the parts that suffer most from dissolution. Rock face dissolution is the main part of foundation dissolution and the main aspect of karst foundation dissolution. The reasons are as follows: the dissolution effect weakens gradually from the outside to the inside, so in a karst foundation, rock faces are the first parts that dissolve, and what is more, the dissolved area of rock faces is also much larger than that of the fissures or caves beneath them. One thing to note is that although rock face dissolution is the main part of the total foundation dissolution, the spatial distribution of karst caves is highly unstable, seriously threatening the safety of building foundations. Therefore, the study on the stability of karst caves in karst foundations is still very important.
It can be seen from Figure 3 that the deviations (errors) d(r_{s}) from the measured rock face dissolution rate are as follows: zone I1 has a deviation of 0.17.22% and an average deviation of 2.86%; zone I2 has a deviation of 0.1718.95% and an average deviation of 9.37%, and the deviation at the height of 109110.5m is over 15%; zone I3 has a deviation of 0.17.16% and an average deviation of 2.59%; zone I4 has a deviation of 0.0911.51% and an average deviation of 2.84%; zone II1 has a deviation of 0.1815.82% and an average deviation of 7.29%; zone II2 has a deviation of 0.0115.82% and an average deviation of 4.14%; zone II2 has a deviation of 0.0115.82% and an average deviation of 4.14%, and the deviation at the height of 110.5m is over 15%; zone II3 has a deviation of 0.3115.65% and an average deviation of 5.42%, and the deviation at the height of 110m is over 15%; zone III1 has a deviation of 0.0317.47% and an average deviation of 3.76%, and the deviation at the height of 108m is over 15%; zone III2 has a deviation of 0.0222.28% and an average deviation of 6.12%, and the deviation at the height of 107.5108.5m is over 15%; zone III3 has a deviation of 0.129.04% and an average deviation of 4.11%; zone VI1 has a deviation of 1.4612.36% and an average deviation of 6.29%, and the deviation at the height of 108m is over 15%; zone VI2 has a deviation of 0.6423.13% and an average deviation of 6.29%, and the deviation at the height of 108109.5m is over 15%; zone V1 has a deviation of 0.2224.92% and an average deviation of 8.21%, and the deviation at the height of 110.5111m is over 15%; zone V2 has a deviation of 0.2217.94% and an average deviation of 6.43%, and the deviation at the height of 107107.5m is over 15%. Therefore, although the density of boreholes in the detailed survey drilling is much smaller than that in the construction survey stage and the dissolution degree of site is high, the deviation from the measured rock face dissolution rate is generally within 15%, with an average of no more than 10%. There are only a few data points with a deviation of more than 15%  usually 14 data points, and they are mainly distributed in the shallow depth range above 107m.
When the rock face dissolution rate curve in the detailed survey stage is fitted according to formula (8), the fitting correlation coefficient R is 0.9370.998 and as high as 0.979 on average, and the fitted curve is also be in good agreement with the measured curve. The fitted rock face dissolution rate curves in the detailed survey and the construction survey stages have basically the same shape. Specifically, the two fitted curves in zone III1 almost overlap; those in zone I1, I3, I4, II1, II2, III3 an V2 are basically the same, with a maximum deviation of no more than 5% at the same height; in zone I2 and VI2, the maximum deviation occurs above the height of 108m, but if this section of curve will basically overlap with that in the construction survey stage if it is shifted down by about 1.0m along the height; the deviation in zone II3 mainly occurs above the height of 104m, and that in zone III2 mainly occurs above 106.5m and that in zone V1 mainly occurs above 108m, and the main deviated sections of the curves in these three subzones will basically coincide with those in the construction survey stage if they are moved down along the height by about 0.5m; and the maximum deviation in zone VI1 occurs below the height of 108m, but its deviation from that in the construction survey stage is still within 10%. Therefore, there is not much deviation between the rock face dissolution rate curves in the detailed survey and the construction survey stages, and the two curves have basically the same shape and changing trend. Considering the deviations of the rock face dissolution rate in the two stages together, it can be seen that the distribution curves of rock face dissolution rate with depth during the two stages in the site are basically the same, indicating that the rock face dissolution rate has good numeral stability, and thus the distribution curve of rock face dissolution rate with depth can be used a reasonable basis for evaluating and analyzing the distribution pattern of rock face dissolution degree with depth.
The rock face dissolution rate curve is a statistical curve. Whether it can reasonably reflect the rock face dissolution characteristics of the site requires a sufficient sample size, which means the density of boreholes within the statistical range must be controlled to ensure there are sufficient boreholes. How to control the density and number of boreholes depends on the characteristics of foundation dissolution. The greater the relief of the foundation rock face, the greater the density of boreholes and the greater the number of boreholes. In general, the density and number of boreholes in construction survey are sufficient to meet the statistical requirement of the rock face dissolution rate curve. Among the 118 copies of materials collected from Southwest China in this paper, 82 contain detailed survey and construction survey data. The statistical analysis and verification results of the 82 copies of materials show that if the average borehole spacing is no more than 20m, and there is no less than 12 boreholes in a site, the error of the rock face dissolution rate is generally within 15%, so this indicator can properly reflect the rock face dissolution degree of the site.
(1) The rock face dissolution rate of karst building foundations in the karst terrain of Southwest China generally declines exponentially with the depth increasing, and when it is expressed as an exponential function of height, the fitting correlation coefficient can generally be up to more than 0.90.
(2) Cave dissolution is the most prominent form of karst that threatens the safety of buildings, but rock faces are the parts that suffer most from dissolution, and rock face dissolution is the main component of site dissolution.
(3) The rock face dissolution rate distribution curve with depth has good numerical stability. The rock face dissolution rate distribution curve with depth obtained during the detailed survey and construction survey stages generally has an error of less than 15%, which meets the precision requirement in engineering.
(4) When there are no less than 12 drill holes with spacing of no more than 20m on a site, the rock face dissolution rate distribution curve with depth can well reflect the actual rock face dissolution of the site.
The authors would like to acknowledge the financial support from the Guangxi Natural Science Foundation (Grant No. 2018GXNSFAA138139, No.2017GXNSFAA198238). The work in this paper was also supported by Guangxi Key Laboratory of New Energy and Building Energy Saving (Grant No. 19J2121, No. 17J215, No. 16J216).
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