Slope deformation prediction has important significance for slope prevention and control. Based on historical time series, the trend of displacement variation can be predicted in advance, and according to the development trend, risk warnings and treatment measures are proposed. The use of the mathematical model to predict slope deformation has been proved to be feasible by many studies; therefore, the choice of the predictive model and the practicability of the model are crucial issues in the prediction of slope deformation, and the mathematical prediction model used should be less complicated considering the practicality of the model. In view of slope deformation prediction, a fractional-order calculus gray model based on the coupling of gray theory and the fractional derivative method is proposed, which takes a deep foundation pit slope in Chongqing, Southwest China, as the study object. The fractional-order gray model is compared with the traditional gray models; therefore, the results show that the accuracy of slope deformation prediction based on the gray coupling model of cumulative displacement and fractional calculus is significantly higher than that of the conventional gray model, and its error is in the acceptable range compared with the actual monitoring data, which can meet the needs of engineering application. Compared with the traditional gray theory method, the gray coupling model of fractional-order calculus only increases the fractional derivative order, which is verified to be feasible, and can be used as a reference method for slope deformation prediction. It has a certain theoretical basis and a good application prospect in slope deformation prediction.
Slope disaster prediction and forecasting is the frontier problem and research focus of current environmental geology research and is also one of the critical aspects for people to understand the natural environment [
At present, the slope deformation prediction is based on slope displacement monitoring data, reservoir water level change, rainfall monitoring, etc., and therefore, using certain mathematical methods or means to obtain the slope deformation prediction model is one of the main technical difficulties in the slope engineering. The mechanical knowledge still cannot solve the problem of slope deformation prediction properly. Therefore, mathematical methods or approaches are adopted to predict the deformation of the slope, in order to obtain the deformation trend. Foreign scholars started related research earlier. For example, Jibson [
Using mathematical models and intelligent evaluation methods to accurately predict slope deformation and at the same time integrating the mechanical theory interpretation with the deformation prediction has important theoretical significance and practical engineering value. The fractional-order calculus method is an extension of the classical calculus method with memory and genetic functions, especially in the complex system description; it boasts the advantages of simple and fast modeling, easy to implementing, and accurate positioning of description and has become one of the most important tools for mathematical modeling of complex mechanical and physical processes. There are relatively few studies in this area, and among the engineering application foundations, there are a few reports on fractional creep and fractional plasticity. Wang et al. [
Assume that the slope deformation monitoring data are
Derivate parameters
The time response of the descendant into the fractional gray model (see Appendix Lemma 5 for details) is
According to the above mathematical lemma and derivation, the detailed steps of the fractional-order calculus gray model of slope deformation are as follows: First of all, process the original displacement data, and subtract the initial value from the existing time series data so that the initial value is 0. This is to directly apply the solution formula of Lemma 5. In this step, it should be noted that the postprocessing time series data cannot be negative because usually the cumulative displacement for slope deformation prediction is adopted, and thus, it can be satisfied. Determine the order of the fractional order, and apply the parameters of the lemma gray model. Substitute the parameters into the solution of the albinism differential equation of the gray model, adding the initial values to all the simulated values, thus obtaining the predicted value of the slope deformation.
In the above derivation, the derivative order
In order to facilitate visual reading and make it easy to use, the Matlab toolbox is called to compile the corresponding program, and the implementation process is shown in Figure
Fractional-order calculus gray model implementation process.
The deep foundation pit slope project of an international garden in Chongqing is located in the Northern New District of Chongqing. It covers an area of about 1,200 mu (1 mu ≈ 666.66 m2), of which the pure water surface area of the lake is about 160 mu. The terrain falls on the lakeside like a gentle slope, wherein Region B covers 300 mu, which is located in the central lakeside hinterland of the project.
This region belongs to the subtropical monsoon climate area, its annual average rainfall is 1100 mm, and the rainfall is mostly concentrated from May to September, taking up 70% of the annual rainfall; the long-time average annual temperature is 18.6°C, the extreme minimum temperature is 4.5°C, and the extreme maximum temperature is 42.5°C; the long-time average annual relative humidity is 80%, and the annual distribution is the largest in December, namely, 87%, and the smallest in August, namely, 74%; and the frost period generally lasts 10∼20 days, and fog days are up to 20∼35 days, and the number of sunshine hours is as many as 1384.2∼1542.8 hours.
The area is a type of denuded hilly landform, with no faults passing through it. The occurrence of rock formation is SW310°∠9° in the exposed area of the bedrock near the area, and there are two sets of fissures in the rock mass. The structural fissures are not developed, with no faults passing through it, and the development degree of fissures is simple. The slope angle of the terrain is 2∼28°; the average slope angle is about 22°; the topography and geomorphology are relatively complex; the dip angle of the rock layer is gentle, with no faults passing through it; the fissures are not developed; the rock mass takes on the shape of the medium thick to thick layer; the lithology and soil layer combination are binary combination, and the exposed bedrocks are the mud rocks and sand stones of Middle Jurassic Upper Shaximiao Formation (J2s); the quaternary soil layer includes el-dlQ (Q4el + dl), Qml (Q4ml), and Qapl (Q4al + pl); the soil layer is 0.50 to 4.7 m thick (2.7 m on average), and the dip angle of the geosynthetic interface is 3∼14 degrees; and the local geotechnical interface is forward and nonemptive, and dip size is ≥20°, accounting for 28.8% of the planning area. The relationship between the rock mass penetrating the structural plane and the oblique (edge) is relatively complicated, and the influence of surface water and groundwater on the rock mass is small, the basic intensity of earthquakes of the VI degree. Human activities that affect the geological environment are not strong. There are no bad geological phenomena such as dangerous rock collapse, landslide, debris flow, and ground collapse, and thus, the geological environment is complex to some degree.
According to the engineering geological conditions, three kinds of protection methods are used to support and protect the slope of the deep foundation pit, such as protection treatment of the lattice anchor rod or ordinary slide-resistant pile or slide-resistant pile with prestressed anchor cable by combining with the C25 shotcrete panel. In the process of construction, monitor the horizontal and vertical displacements of the slope top, monitor the anchor rod and prestressed anchor cable press, and make a macroscopic inspection of the Earth’s surface.
The safety grade of the foundation pit slope in Region B of an international garden in Chongqing is Grade I, the whole excavation panorama of the deep foundation pit slope is shown in Figure
Deep foundation pit slope, monitoring layout, and engineering geological section: (a) whole panorama of deep foundation pit slope; (b) monitoring floor plan of foundation pit slope B district treatment project; (c) Building No. Six engineering geological section.
Actual measurement data of JC-1 and JC-2 monitoring points of a deep foundation pit slope.
Select the first 50 time series data of the JC-1 monitoring point to build the model, and predict the displacement changes of the following 26 time series; and select the first 30 time series data of the JC-2 monitoring point to build the model, and predict the displacement changes of the following 15 time series. Among them, the JC-1 monitoring point is located at the top of the deep foundation pit, and the deep foundation pit of the JC-2 monitoring point is located at the bottom position. The sudden change of the monitoring point JC-1 at 10∼20 days is mainly due to the local temporary loading effect at the top of the slope.
According to the measured data of JC-1 and JC-2 monitoring points, using the fractional calculus gray prediction model and gray prediction model, the
(a) JC-1 and (b) JC-2 monitoring point deformation predictions.
(a) JC-1 and (b) JC-2 monitoring point deformation prediction relative errors.
Analysis of the influence of the Different from the traditional gray model, the fractional-order gray model introduces the derivative order
Parameter
Slope deformation is closely related to soil properties and engineering geological environment. Rainwater erosion and slope load may directly accelerate slope deformation. Therefore, the analysis of the development trend of slope deformation needs to consider the internal and external factors. It is often combined with the existing monitoring data to use mathematical models to carry out nonlinear calculations and predict the trend of change, which can provide technical support for slope management and promote the disaster prevention and reduction. With the fractional-order calculus method, based on the deformation monitoring data time series, from the theoretical derivation point of view, the fractional-order calculus gray model is proposed after analysis and derivation, and the applicable conditions and ranges are nonnegative sequences, and the initial value is 0. The steps of using the fractional-order calculus gray prediction model are given. Using the Matlab language to compile the corresponding program, these corresponding functions can be realized one by one. Based on the established fractional-order calculus gray model method, the monitoring data of a deep foundation pit slope in Chongqing, Southwest China, was selected as an example to verify the model. The calculation results show that the fractional-order calculus method has better prediction effect than that of the gray model, and the latter has a maximum relative error of 37.96%, while the former has a maximum relative error of 17.63%. These results show that the fractional-order gray model has a certain application prospect in the prediction of mountain slope deformation. Compared with the measured data, the relative errors are within the acceptable range, which can provide reference for similar projects.
Assume that the nonnegative sequence is
In Lemma 1, when
The fractional calculus gray model is
In the formula,
The Least Squares estimate of Lemma 3 satisfies
The solution of the fractional partial differential equation
The equation is Laplace positive transformation:
So,
Laplace transformation by
Comparing Formula (
The inverse of Equation (
Proof completed.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
Li Li and Yue Qiang are the first authors.
The study is performed within the Changjiang River Scientific Research Institute of Changjiang Water Resources Commission Open Research Program (program no. ckwv2016393/KY),the Open Foundation of Jiangxi Engineering Research Center of Water Engineering Safety and Resources Efficient Utilization (Grant no. OF201605), Three Gorges Reservoir Ecological Environment Protection and Disaster Prevention (2011 collaborative innovation center of Chongqing), Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant nos. CXTDX201601034 and KJQN201801224), Science and Technology Talent Special Fund Project of Wan-Zhou District, Chongqing (Grant no. 2016027), Three Gorges Reservoir Area Project Structure Disaster Prevention and Reduction and Safety Research Innovation Team of Chongqing Three Gorges University, Outstanding Scientific and Technological Achievements into Cultivation Project (Grant no. 17zh01), and Field Grade Major Breeding Program of Chongqing Three Gorges University (Grant no. 15ZP04)