Predicting the Function of the Dissolution Rate with Depth Using Drilling Data from Shallow Strata at Karst Sites

Abstract

The degree of dissolution in building foundations has been proven to be an exponentially decaying function of depth, which can be characterized by the dissolution rate depth distribution function obtained from survey drilling data. If the dissolution rate depth distribution function can be predicted using shallow drilling data, it would reduce the workload and cost of surveying, and have positive engineering significance. In this study, in the context of the Liuzhou Style Harbor project in Liuzhou City, the dissolution rate depth distribution prediction curve was obtained using drilling data above elevations H₃₀, H₄₀, and H₅₀ (corresponding to dissolution rates of 30%, 40%, and 50%, respectively). The prediction accuracy of the curve was thoroughly analyzed in terms of curve deviation and parameters of the intensive dissolution layer. The conclusion has also been verified by more engineering practices. The results showed that the predicted dissolution rate depth distribution function curve obtained from survey data above elevation H₃₀ was very close to the actual curve, and sometimes matched it. The dissolution rate deviation within the range of the intensive dissolution layer was generally less than 10%, and the deviation in the boundary elevation of the intensive dissolution layer was generally less than 1.0 m. The predicted function was highly reliable, and the prediction deviation met the requirements of engineering practice. Therefore, it is recommended that elevation H₃₀ be used as one of the controlling conditions for the depth of survey drilling in karst building foundation investigations. These research findings can provide a basis for optimizing a karst foundation investigation plan.

1. Introduction

The degree of karst dissolution in the construction foundation in karst areas follows an exponential decay function with increasing depth [1]. By obtaining the depth distribution function of the dissolution degree based on investigation data [2], the following problems can be solved: (i) The average probability and the elevation of rock-embedded piles at different depths are determined [3,4]. This provides a basis for the estimation of the quantity and cost of a rock-embedded pile, and solves the problem of pile foundation and shallow foundation selection failures in detailed surveys or foundation design stages due to the inability to determine the cost of pile foundation. (ii) The current levels of treatment for karst foundations are combined to determine the depth range of the distribution of the intensive dissolution layer (IDL) in the site [5], which represents the complexity and difficulty of the karst foundation, and is a key issue that must be addressed in the survey and design of karst construction foundations. The rock surface undulation and unstable caves in a karst foundation may result in some bores being too deep or too shallow. Shallow bores will affect the accuracy of the function solution, while excessively deep bores will not contribute to the solution of the function. Obviously, if the depth distribution function of the dissolution degree in the site can be reasonably predicted based on shallow drilling data, it will have positive engineering significance by reducing the workload, cost, and duration of karst foundation surveys.

Mankind has an enormous impact on the course of activities that impact the environment—soil in general and rock mass in particular [6,7,8]. There are significant differences in types of karst geological problems, forms of karst hazards, prevention methods, and governance measures in different engineering fields [9,10,11]. This leads to different research content and methods for karst development laws [12,13,14,15]. For large-scale engineering projects such as tunnel engineering [16], geohazards [17,18], reservoirs [19,20], and mining engineering [21], primary techniques applied to study the macroscopic laws governing karst development include geomorphology [22], geological mapping [23], geophysical exploration [24], remote sensing [25], and field surveys [26]. The aforementioned research methods are in line with the characteristics of the industry in the field of study and have made significant contributions to engineering construction in karst regions [27,28,29,30]. However, when these research findings are applied to karst building foundations, they fail to provide the depth variation characteristics of karst foundation dissolution. In this regard, a study [1] based on the karst development mechanism has revealed that the degree of dissolution in karst sites exhibits an exponential attenuation pattern with increasing depth, taking into account factors such as lithology and groundwater dissolution capacity. This breakthrough challenges the previously unquestioned belief among karst professionals that “karst development does not follow any specific function law in the depth direction”. Another study [2] has developed a theoretical and methodological framework for determining the depth distribution function of dissolution in building foundations. The research presented in [31] demonstrates that, without conducting extensive rock drilling, it is possible to predict the variation of dissolution depth in deeper areas of cavernous development sites based on the topographical features of rock surfaces. This has great value in reducing engineering survey work and lowering survey costs. Based on these findings, another intriguing question arises: how can we predict the dissolution depth function of a site based on partial shallow karst survey data? This question has not yet been addressed in the existing literature and constitutes the main objective of this study.

In this paper, the Project Liuzhou Style Harbor, including four high-rise residential buildings and several commercial complexes, is taken as the engineering background. The dissolution rate curves are obtained by respectively fitting the measured dissolution rate above the elevation, at which the dissolution rates are 50%, 40%, and 30%. The predictive effects of these curves are then accessed through analyzing the dissolution rate deviation, elevation deviation, average dissolution rate deviation, and elevation average deviation of each predicted curve within the depth range of the intense dissolution layer (IDL). Finally, 106 engineering site data in karst areas were collected to verify the reliability of the predicted control conditions for predicting the dissolution rate curve.

2. Engineering Overview

The Liuzhou Style Harbor project comprises four high-rise residential buildings and one commercial building. It is situated in the heart of Liuzhou City near the Zhongshan Building and the People′s Cinema.

The site is located on the second-level alluvial terrace on the north bank of the Liujiang River. The ground elevation of the site ranges from 86.34 to 90.38 m, with a general slope angle of less than 10%. There is a covering layer exceeding 15 m at the site, and the debris from the demolition of the old buildings has covered the original ground to a certain thickness. The covering layer thickness and the previous construction activities have covered the original karst geological phenomenon of the site. However, according to the regional geological structure map at a scale of 1:200,000 shown in Figure 1, it is known that the underlying bedrock in the area is dolomitic limestone of the Middle Carboniferous Daibu Formation (C2d), with a trend approximately NE–SW and a dip towards SE at an angle of 12–15°. About 400 m south of the site, there is a secondary inferred fault running approximately east–west, and its tectonic movement occurred before the Mesozoic Cretaceous period. Since the Quaternary period, no obvious signs of new tectonic movement have been found.

The proposed construction site is located on the northern bank of the Liujiang River, situated at the forefront of a U-shaped peninsula. It is approximately 300 m away from the riverbank, and the water level of the Liujiang River remains at an elevation of 77.0–78.0 m throughout the year, indicating a close connection with the groundwater beneath the site. The ground water in the area is divided into two layers: the upper stagnant water and the underground diving water. The upper stagnant water is located at a depth of 2.5–4.0 m and is primarily formed from the infiltration of atmospheric precipitation and domestic drainage into the soil. This layer of water is stored in the fill soil, which has a high water content. In contrast, the underground diving water is found in the pores of intense permeable gravel and pebble beds, as well as in karst cracks of rocks. It has a free water surface and is located at a depth of 10.0–12.0 m with an elevation of 76.40–76.65 m. The water level in this layer is discharged to the base level of Liu-jiang River in the south.

Figure 2 represents two typical geological sections of the site. Based on the drilling data, the stratigraphy of the site from top to bottom is described as follows:

Miscellaneous fill ①: Brown to mottled color, consisting of construction waste such as broken bricks, stones, and a small amount of sticky soil backfill. Some organic matter is present locally. It is distributed on the surface of the site, with a layer thickness ranging from 0.5 to 7.0 m, reaching up to 10.0 m in some areas.

Silt and muddy soil ②: Greyish-black, plastic, and slightly smelly, containing a small amount of organic matter and humus, very wet to saturated. It is distributed in the eastern part of the site, with a burial depth ranging from 1.3 to 9.0 m and a layer thickness from 1.1 to 8.5 m.

Silty clay ③: Brownish-yellow, wet, plastic, and locally hardened or soft-friable. It is widely distributed in most parts of the site, with some lenses embedded in gravels. The burial depth ranges from 0.5 to 18.5 m, with a layer thickness from 0.5 to 14.9 m.

Round gravel and pebbles ④: Moderately dense and locally loose or compact, with particle sizes larger than 2 mm accounting for approximately 50–70% of the content. The primary particle size ranges from 1 to 5 cm, with good particle roundness and moderate sphericity. The composition mainly consists of quartz sandstone, with fine sand as the filling material. It is distributed over a large part of the site, with a burial depth ranging from 5.2 to 30.8 m and a layer thickness from 0.5 to 17.1 m.

Dolomitic limestone ⑤: Whitish-gray and slightly weathered, with some areas showing moderate to strong weathering. The drilling speed is relatively slow due to the hard rock and relatively intact rock formation. The core recovery rate typically ranges from 70–90%. Rock surface is buried under the ground surface at a depth ranging from 14.0 to 37.4 m, with an elevation of exposed rock surface ranging from 49.82 to 74.39 m. The rock surface is characterized by various features, including stalagmites, stalactites, and karst grooves, which are either canine or V-shaped and are often filled with gravel and pebbles. The depth of the grooves ranges from 3.5 to 7.9 m and can reach up to 12.3 to 20.4 m in some areas due to the influence of structural cracks. The rock mass is strongly cut, forming a banded or clustered dissolution and fracture zone in some subareas. Small cavities also develop inside the rock in certain locations. The karst caves vary in shape and height, with cave height ranging from 0.5 to 5.1 m and maximum height ranging from 6.9 to 10.1 m. Some areas contain multiple layers of karst caves in a bead-like shape, which are mostly filled with silty clay mixed with gravel and pebbles.

A total of 150 survey bores were completed at this site, with the total height of the rock being 3006.68 m. Among them, 20 bores revealed karst caves, 13.3% of the total number of bores, and 9 bores revealed karst grooves, 6% of the total number of bores. The cumulative height of the exposed karst caves was 69.10 m, 2.3% of the total thickness of the exposed rock layers.

The site is divided into eight subareas based on the building′s function and shape. High-rise residential buildings A, B, C, and D each constitute one subarea, while the square and commercial podium are divided into four subareas: E1, E2, E3, and E4.

Table 1. Cave development of each subarea.

Subarea	Number of Bores	Number of Bores Revealing Caves	% of Bores Revealing Caves	Total Thickness of Exposed Layers (m)	Total Height of the Cave (m)	% of Cave Height to Total Thickness of Exposed Rock Layers
A	27	3	11.11	512.7	8.5	1.66
B	30	1	3.33	520.5	0.8	0.15
C	27	4	14.81	564.4	10	1.77
D	16	3	18.75	357.4	9.9	2.77
E1	34	9	26.47	759.38	36.4	4.79
E2	21	1	4.76	460.3	1.6	0.35
E3	17	3	17.65	327.7	8.2	2.5
E4	20	3	15.00	326.4	4.4	1.35

shows the caves in each division, and the distribution of bores is presented in Figure 3.

3. Predictive Function Solving

The data required for solving the depth distribution function of the dissolution rate of building foundations are sourced solely from drilling data. These data primarily include the elevation of the exposed rock surface, the bottom elevation of the borehole, the top elevation of the cavity, and the bottom elevation of the cavity. No other experimental data is needed. The diameter of the survey boreholes is typically within the range of 75–110 mm, and the drilling velocity does not affect the collection of the aforementioned data. The boreholes are arranged along the building’s corners, boundary lines, or axis lines, with a spacing of 10–20 m between them. The drilling depth is controlled to ensure the complete exposure of a minimum rock thickness of 5 m. If additional drilling is required near a borehole due to encountering a cavity, the data from these additional boreholes are not used for solving the dissolution rate curve. The data processing method employed in this study is as follows:

3.1. Defining the Statistical Scope and Conducting Elevation Discretization

To determine the statistical range, the length of the statistical interval is set as ΔH, while H_max and H_min are the maximum and minimum elevations of the rock face exposed by all bores in the area. The top statistical elevation (H_b) and the bottom statistical elevation (H_a) are determined using the following formula:

$$\left\{\begin{array}{l}{H}_{\mathrm{a}}=\mathrm{\Delta }H\cdot \mathrm{int}(\frac{H_{\min }}{\mathrm{\Delta }H})\\ H_{\mathrm{b}}=\mathrm{\Delta }H\cdot [\mathrm{int}(\frac{H_{\max }}{\mathrm{\Delta }H})+1]\end{array}\right.$$

(1)

The elevation interval [H_a, H_b] was discretized by length ΔH and arranged in descending order according to the bottom elevation H_i of each interval as follows:

$$H_1>H_2>\dots >H_{\mathrm{i}-1} > H_{\mathrm{i}} > H_{\mathrm{i}+1} >\dots$$

3.2. Calculating the Measured Dissolution Rates

The dissolution rate is defined as the percentage of the dissolution height and the thickness of the stratum. The dissolution height consists of the height of the rock surfaces and the height of the caves [2]. It has been revealed in [1] that the degree of dissolution follows an exponential decay function with depth, and this function can only be characterized by this dissolution rate [2]. The measured dissolution rate r_i of the elevation interval (H_i−1, H_i] is calculated [2] using the following formula.

$$r_i=\frac{{\displaystyle \sum _{k=1}^n[\max (H_{i-1},H_{rk})-\max (H_i,H_{rk})}]+{\displaystyle \sum _{j=1}^m[\min (H_{i-1},{H^{\prime }}_{1\mathrm{j}})-\max (H_i,{H^{\prime }}_{2j})]}}{n\cdot \mathrm{\Delta }H}\times 100\%$$

(2)

where max() and min() are functions used to obtain the maximum and minimum values, respectively, H_rk is the rock surface elevation of the k-th bore, and $H_{1j}^{\prime }$ and$H_{2j}^{\prime }$ are the elevation of the top and bottom of the j-th cave, respectively.

3.3. Solving the Measured Dissolution Rate Curves

The depth distribution curve of the dissolution rate was obtained by fitting the measured dissolution rate [2] above the elevation at which the measured dissolution rate equals 15%, and the fitting formula was given as

$$r=a{e}^{b(H_0-H)}$$

(3)

where H₀ is the starting elevation for depth calculation, which is taken as 75 m in the construction site of Liuzhou Style Harbor. H refers to elevation.

The fitting correlation coefficient R of the dissolution rate depth distribution curve of each subarea is above 0.90, indicating that the dissolution rate of each subarea of the site conforms to the exponential decay law of the depth distribution. The depth distribution curve of dissolution rate in each zone is shown in Figure 4.

3.4. Solving the Predicted Dissolution Rate Curves

By substituting the dissolution rates r of 50%, 40%, and 30% into the dissolution rate depth distribution function obtained in the previous step, the corresponding elevations are calculated as H₅₀, H₄₀, and H₃₀. Data below the elevations of H₅₀, H₄₀, and H₃₀ are filtered out, and only the measured dissolution rates above these elevations are used for curve fitting. The resulting curves are only the predictive curves, denoted as L₅₀, L₄₀, and L₃₀, respectively.

Apart from the prediction curve L₅₀ in Zone E3, which has a fitting correlation coefficient R of 0.893, all other zones have a fitting correlation coefficient R exceeding 0.90 for their respective prediction curves. The prediction function curves L₅₀, L₄₀, and L₃₀ are shown in Figure 4.

4. Deviation Analysis

4.1. Solution for the Prediction Deviation Curve

The intense dissolution layer (IDL) is a defining stratum of the karst foundation. The upper portion of the IDL primarily consists of soil layers, while the lower portion mainly consists of rock layers. Within the depth range of the IDL, it forms a typical soil–rock combination foundation. The IDL represents the depth range that best demonstrates the negative impacts of karst development on engineering construction. The dissolution rate below the lower boundary of the IDL tends to have very small values, with minimal deviation from the predicted values. Including this deviation in the analysis would inevitably lead to a smaller average error and potential distortion of the predicted curve. Moreover, such analysis results fail to address the primary contradiction of the adverse effects of karst development on the foundation. Therefore, the deviation analysis of the predicted curve should focus on the depth range of the IDL. The formula for determining the deviation of each curve within the depth range of the IDL is as follows:

$$\mathrm{\Delta }{r}_i=\left|r_i-r_i^{\prime }\right|$$

(4)

where Δr_i represents the deviation of the predicted dissolution rate at elevation H_i. Meanwhile, r_i represents the dissolution rate at elevation H_i, which can be obtained by plugging in elevation H_i into the fitting curve of the dissolution rate. $r_i^{\prime }$ represents the predicted value of the dissolution rate at elevation H_i, which can be obtained by substituting elevation H_i into the dissolution rate prediction function (L₅₀, L₄₀, and L₃₀).

Let the fitting curve r(H) of the dissolution rate depth distribution function be denoted as

$$r(H)=\mathrm{a}{e}^{\mathrm{b}(H_0-H)}$$

(5)

where a and b are fitting coefficients.

Let the prediction function r′(H) of the depth distribution curve of the dissolution rate be denoted as

$$r\prime (H)=\mathrm{a}\prime e^{\mathrm{b}\prime (H_0-H)}$$

(6)

where a′ and b′ are fitting coefficients.

The prediction deviation of dissolution rate at elevation H can be calculated as Δr:

$$\mathrm{\Delta }r=\left|r(H)-r\prime (H)\right|$$

(7)

By substituting Equations (5) and (6) into Equation (7), the following formula for calculating deviation of prediction curve can be obtained:

$$\mathrm{\Delta }r=\left|\mathrm{a}{e}^{\mathrm{b}(H_0-H)}-\mathrm{a}\prime e^{\mathrm{b}\prime (H_0-H)}\right|$$

(8)

The depth distribution curve of the dissolution rate is mainly used to analyze the distribution characteristics of the IDL, so the deviation of the curve section between the dissolution rate [25%, 75%] in the analysis Formula (8) is of engineering significance.

The depth distribution curve of the dissolution rate is primarily used to analyze the distribution characteristics of the IDL. Therefore, in this case, only the deviation of the curve section between the dissolution rate of 25% and 75% in Equation (7) will be analyzed. To remove the absolute value in Equation (8), we set r(H) = r′(H), which gives us

$$a{e}^{b(H_0-H)}=a\prime e^{b\prime (H_0-H)}$$

(9)

Rearranging the above equation gives us

$$H_p=H_0-\raisebox{1ex}{$\ln \frac{a\prime }{a}$}\!\left/ \!\raisebox{-1ex}{$(b-b\prime )$}\right.$$

(10)

where H_p represents the elevation at the intersection of the fitted curve and the predicted curve of the dissolution rate.

By substituting r = 25 and r = 75 into Equation (7), corresponding elevations of the dissolution rate at 25% and 75% can be obtained and denoted as H₂₅ and H₇₅, respectively. Here, H₂₅ and H₇₅ are the elevation of the bottom boundary and the top boundary of the IDL, respectively. The deviation curves of the corrosion rate depth distribution for the two survey stages within the elevation range [H₂₅, H₇₅] in each zone are shown in Figure 5. The maximum value on the longitudinal coordinate axis in Figure 3 slightly exceeds the top elevation of the IDL, whereas the minimum value on the coordinate axis falls just below the minimum elevation of the IDL. The values d30, d40, and d50 in the graph represent the deviation curves of the curves L₃₀, L₄₀, and L₅₀ within the depth range of the intense dissolution layer.

4.2. The Average Prediction Deviation within the Range of the Intense Dissolution Zone

When H_p is within the range of [H₁₅, H₇₅], the calculation formula for the average deviation of the dissolution rate in the interval of [25%, 75%] can be obtained.

$${(\mathrm{\Delta }r)}_{avg}=\frac{\left|{\displaystyle {\int }_{H_{15}}^{H_p}(a\prime e^{b\prime (H-H_0)}-a{e}^{b(H-H_0)}})dH\right|+\left|{\displaystyle {\int }_{H_p}^{H_{75}}(a\prime e^{b\prime (H-H_0)}-a{e}^{b(H-H_0)}})dH\right|}{H_{75}-H_{15}}$$

(11)

where (Δr)_avg is the average deviation of the predicted dissolution rate.

When H_p is not within the range of [H₁₅, H₇₅], (Δr)_avg can be calculated as

$${(\mathrm{\Delta }r)}_{avg}=\frac{\left|{\displaystyle {\int }_{H_{15}}^{H_{75}}(a\prime e^{b\prime (H-H_0)}-a{e}^{b(H-H_0)}})dH\right|}{H_{75}-H_{15}}$$

(12)

By integrating Equations (11) and (12), we can obtain the formula for (Δr)_avg as follows:

$$\left\{\begin{array}{l}{(\mathrm{\Delta }r)}_{avg}=\frac{\left|{\left[C_1{e}^{b^{\prime }\left(H-H_0\right)}-C_2{e}^{b\left(H-H_0\right)}\right]}_{H_{15}}^{H_p}\right|+\left|{\left[C_1{e}^{b^{\prime }\left(H-H_0\right)}-C_2{e}^{b\left(H-H_0\right)}\right]}_{H_p}^{H_{75}}\right|}{H_{75}-H_{15}} H_p\in \left(H_{15},H_{75}\right]\\ (\mathrm{\Delta }r{)}_{avg}=\frac{\left|{\left[C_1{e}^{b\prime \left(H-H_0\right)}-C_2{e}^{b\left(H-H_0\right)}\right]}_{H_{15}}^{H_{75}}\right|}{H_{75}-H_{15}} H_p\notin \left(H_{15},H_{75}\right]\\ C_1=\frac{a\prime }{b\prime } C_2=\frac{a}{b}\end{array}\right.$$

(13)

4.3. Analysis of the Average Elevation Deviation within the Depth Range of the IDL

The elevation deviation of the curve within IDL refers to the absolute difference in elevation between the dissolution rate curve and the predicted curve at the same dissolution rate. It can be calculated using the following formula:

$$\mathrm{\Delta }{H}_i=\left|H_i-H_i^{\prime }\right|$$

(14)

where ΔH_i represents the corresponding subarea’s bottom elevation when the dissolution rate is r_i, and $H_{\mathrm{i}}^{\prime }$ is the elevation obtained by substituting the dissolution rate r_i into the dissolution rate prediction function (L50, L40, and L30).

4.4. Evaluation of the Prediction Performance

Based on the correlation between the depth prediction curve of the dissolution rate and the fitting curve of dissolution rate, and taking into account the deviation of the boundary parameters of the IDL obtained from the prediction curve, the deviation of the dissolution rate, and the elevation deviation of the predicted curve segment within the IDL, the quality of the curve prediction can be evaluated as follows:

Perfect: The predicted curve almost exactly overlaps with the fitted curve L.

Excellent: The prediction curve is close to the fitting curve L, and the elevation deviation of the boundary of the IDL does not exceed 1.0 m. Additionally, the average deviation of the dissolution rate of the curved section of the IDL does not exceed 10%, and the maximum elevation deviation does not exceed 1.0 m.

Poor: The difference between the predicted curve and the fitted curve L is significant. Alternatively, the elevation deviation of the boundary of the IDL exceeds 3.0 m, the average deviation of dissolution rate within the IDL exceeds 20%, or the maximum elevation deviation corresponding to the same dissolution rate within the IDL exceeds 3.0.

Substandard: Any other cases not meeting the above criteria.

Based on the above specifications, the quality of curve prediction results for each zone of the site is presented in

Table 2. Quality of subarea prediction results.

Subarea	Curve	${{\mathit{H}}_{\mathit{c}\mathit{r}}^{\prime }}_{ }$	Hcr	$\mathsf{\Delta }{\mathit{H}}_{\mathit{c}\mathit{r}}^{\prime }$	ΔHcr	Δr	(Δr)avg	ΔH	Quality
A	L	69.6	67.5
	L50	69.5	65.5	0.1	2.0	8.07–15.07	11.86	0.1–1.2	substandard
	L40	69.5	66.5	0.1	1.0	8.07–9.47	4.68	0.1–0.8	excellent
	L30	69.5	67.3	0.1	0.2	0.4–4.43	0.98	<0.2	perfect
B	L	68.0	62.4
	L50	68.0	64.1	0	1.7	0.76–12.26	9.52	0–1.7	substandard
	L40	68.0	63.5	0	1.1	0.78–7.8	6.08	0.1–1.1	substandard
	L30	68.0	62.6	0	0.2	0.4–3.41	2.63	<0.2	perfect
C	L	72.0	69.3
	L50	71.8	69.5	0.2	0.2	5.4–13.64	11.87	<0.2	excellent
	L40	72.0	69.3	0	0	1.28–2.64	1.75	<0.1	perfect
	L30	72.0	69.3	0	0	1.18–2.52	1.61	<0.1	perfect
D	L	71.7	62.0
	L50	71.4	65.0	0.3	3.0	0.06–10.43	15.14	0–2.8	poor
	L40	71.4	64.5	0.3	2.5	0.51–8.52	5.94	0–2.2	substandard
	L30	71.4	63.0	0.3	1.0	0.31–3.77	2.52	0–0.8	excellent
E1	L	70.7	66.9
	L50	70.7	66.0	0	0.9	1.31–5.97	5.33	0–0.9	excellent
	L40	70.7	66.5	0	0.4	1.95–3.33	2.87	0–0.4	excellent
	L30	70.7	66.9	0	0	0.05–4.19	0.68	<0.1	perfect
E2	L	70.4	66.7
	L50	69.9	66.4	0.5	0.3	0.22–12.01	2.89	0–0.5	substandard
	L40	69.9	66.4	0.5	0.3	0.17–12.02	2.86	0–0.4	excellent
	L30	69.9	66.6	0.5	0.1	0.89–14.46	1.81	0–0.2	excellent
E3	L	71.3	68.3
	L50	71.3	68.3	0	0	2.46–12.85	10.98	0–0.5	excellent
	L40	71.3	68.4	0	0.1	0.52–1.76	1.51	0–0.1	perfect
	L30	71.3	68.5	0	0.2	0.67–2.51	2.16	0–0.2	perfect
E4	L	68.9	64.3
	L50	69.1	63.0	0.2	1.3	0.68–5.98	4.92	0–1.5	substandard
	L40	68.9	64.0	0	0.3	0.19–3.07	1.07	0–0.1	perfect
	L30	68.9	64.2	0	0.1	0.54–3.34	0.89	0–0.1	perfect

.

The $H_{\mathrm{cr}}^{\prime }$ and H_cr in Table 2 represent the top and bottom elevations of the IDL, corresponding to elevations with top dissolution rates of 75% and 25%, respectively. Δ$H_{\mathrm{cr}}^{\prime }$ is the predicted deviation of the top boundary elevation of the IDL, which is the difference between the top boundary elevation of the IDL obtained from the prediction curve (L50, L40, and L40) and the top boundary elevation obtained from the curve L. ΔH_cr is the deviation of the predicted elevation of the bottom boundary of the IDL, which is the difference between the elevation of the bottom boundary of the IDL obtained from the predicted curves (L50, L40, and L40) and the elevation of the bottom boundary obtained from the curve L. Δr, (Δr)_avg, and ΔH are obtained from Equations (7), (13), and (14), respectively.

Table 2 displays $H_{\mathrm{cr}}^{\prime }$ and H_cr values, which indicate the top and bottom elevations of the IDL, respectively. The predicted deviation of the top boundary elevation of the IDL is denoted by $H_{\mathrm{cr}}^{\prime }$, which is the difference between the top boundary elevation obtained from the prediction curve (L50, L40, and L40) and that from the curve L. Similarly, ΔH_cr represents the deviation of the predicted elevation of the bottom boundary of the IDL, which is the difference between the elevation of the bottom boundary obtained from the predicted curves (L50, L40, and L40) and that from the curve L. The values of Δr, (Δr)_avg, and ΔH are obtained from Equations (7), (13), and (14), respectively.

5. Discussion

In Figure 4, it can be seen that the deviation of curve L₅₀ from the measured curve was significantly larger than that of curves L₄₀ and L₃₀, the deviation of curve L₄₀ was slightly larger than that of curve L₃₀, and curve L₃₀ had a very small prediction deviation and was already very close to the measured curve.

Based on Figure 5 and Table 2, the deviations of curves L₅₀, L₄₀, and L₃₀ decrease in sequence within the depth range of the intense etching zone. The average etching rate deviations of curve L₃₀ is not more than 3%, L₄₀ is not more than 6%, and L₅₀ is not more than 12%. The amount of measured etching rate data required for solving curves L₅₀, L₄₀, and L₃₀ increases in sequence; that is, the drilling depth increases. Therefore, increasing the drilling depth appropriately can significantly improve the predictive effect of the prediction curves.

According to Table 2, the deviations in the IDL boundary elevations determined by L₃₀ were small, with a maximum deviation of 0.5 m and an average deviation of 0.11 m in the top boundary elevation, and a maximum deviation of 1.0 m and an average deviation of 0.23 m in the bottom boundary elevation. Such a level of deviation is completely acceptable in engineering practice.

The curve L50 prediction effect in different subareas was evaluated for karst feature analysis in engineering projects. The prediction results in the subarea D, representing 12.5% of the subareas, were poor, while the subareas A, B, E2, and E4, accounting for 50%, were substandard. In contrast, the subareas C, E1, and E3, accounting for 37.5%, exhibited an excellent prediction effect. Given that more than half of the subareas produced poor or substandard results, the use of r = 50% as a predictive control condition is not appropriate.

For the curve L40 prediction effect, substandard prediction results were observed in the subareas B and D, representing 25% of the subareas. In contrast, the subareas A, E1, and E2, accounting for 37.5%, exhibited excellent prediction results. Similarly, the subareas C, E3, and E4, accounting for 37.5%, demonstrated perfect prediction results. The remaining 25% of the subareas produced poor prediction results. Given that 75% of the subareas exhibited perfect or excellent prediction performance, using r = 40% as a predictive control condition can provide reference significance for small engineering projects or projects with low requirements for karst feature analysis.

For the curve L₃₀ prediction effect, the prediction results for the subareas D and E2, accounting for 25% of the predicted sites, were excellent. The remaining 75% of the subareas exhibited perfect prediction results, indicating that the predicted results aligned with the actual situation. Therefore, using r = 30% as the predictive control condition can achieve prediction results that are consistent with the actual situation.

The study further verified that r = 30% is a reasonable predictive control condition by examining 106 engineering projects. Among the 514 subareas in these projects, only three subareas, accounting for 0.58%, produced substandard results. In contrast, 366 subareas, accounting for 71.21%, exhibited excellent prediction results, while 145 sites, accounting for 28.21%, demonstrated perfect prediction results. Overall, the sites with a perfect or excellent prediction effect accounted for 99.42%. Therefore, using r = 30% as a predictive control condition is reasonable and effective for karst feature analysis in engineering projects.

In summary, the prediction accuracy of the L₅₀ curve for the boundary elevation of the IDL is poor and the deviation is large, making it generally unacceptable for engineering practice. The prediction accuracy of the L₄₀ curve is moderate, and the deviation in the boundary elevation of the IDL obtained from it is slightly larger, but this is acceptable in engineering practice. The prediction accuracy of the L₃₀ curve is excellent, and the deviation in the boundary elevation of the IDL obtained from it is small, making it acceptable in most cases.

If the elevation H₃₀ is taken as the control standard for the drilling depth in an engineering survey, the drilling footage of the site can be determined using the following method: ① The drilling depth after entering the rock layer is determined based on the elevation H₃₀ of each subarea. ② If the elevation of the drilling rock layer is lower than H₃₀, the rock surface must be reached before drilling is stopped. ③ For bores shared by two or more subareas on the boundary, the bore depth is determined by the minimum elevation H₃₀ among these subareas. After determining the depth of all drilling bores based on H₃₀, the total footage of the rock layer drilling is calculated as 2128.24 m. The original rock layer drilling footage of the survey was 3006.68 m. By controlling the hole depth with H₃₀, the rock layer footage can be reduced by 878.44 m, saving 29.2% of the rock layer footage, which shows significant economic benefits. Therefore, the elevation H₃₀ can serve as a controlling criterion for drilling depth in karst foundation engineering investigations. In other words, all survey boreholes must have a bottom elevation lower than elevation H₃₀.

6. Conclusions

(1)

The function of the dissolution rate can be reliably predicted from the data above elevation H₃₀, where the deviation of the dissolution rate in the depth range of the IDL is generally less than 10%, and the elevation deviation is less than 1.0 m. The predicted curve closely approximates the real curve, indicating the reliability of the predicted results for engineering practice.

(2)

The function of the dissolution rate predicted from the data above elevation H₄₀ can to some extent reflect the actual depth distribution of the dissolution degree in the site, serving as a reference for assessing the karst development law of medium or small projects.

(3)

If the function of the dissolution rate is predicted from the data above elevation H₅₀, the deviation from the actual engineering situation may be considerable.

(4)

The drilling depth for karst foundation engineering investigation can be controlled based on the criteria that the elevation at the bottom of the borehole should not exceed the elevation H₃₀.