Steady seepage analysis in soil-rock-mixture slope using the numerical manifold method
Introduction
The material parameters of rock and soil are generally significantly different. Hence, the soil-rock mixtures (SRMs) show extremely heterogeneous character in terms of mechanical and hydraulic responses. In China, many SRM slopes [1, [2], [36], [38], [39], [40], [46], [50], [52]] (Fig. 1) have been found in its southwestern area. Slope failure normally results in catastrophe and loss of economy[53], [54], [55], [56], [57]. Based on the statistical data presented in [3], one of the main triggers for SRM slope failure is water. Hence, seepage analysis which can calculate the distribution of water level is essential.
As a typical case, steady seepage problem is very common for SRM slopes. It can further be classified into two subtypes, namely, the confined seepage problem and the unconfined seepage problem. Compared to confined seepage problem, unconfined seepage problem is more complex, since the free surface, which cut the problem domain into the flow domain and the dry domain, is generally not known in advance, and a series of iterations are needed to determine its location.
In the past few decades, various effective numerical methods were developed for steady seepage analysis. Because of the superiorities in treating irregular geometries, the FEM [4, [5], [37]] has been widely used. However, the FE-mesh has to conform the material interfaces and the problem domain boundaries, which makes the preprocessing stage very complicated and time consuming. Furthermore, the performance of a few element types including 4-node quadrilateral element will be influenced by the mesh quality. Note that the generation of high quality FE-meshes is very difficult in the context of the FEM for some geotechnical materials, such as the soil-rock mixtures. To overcome the defects of the FEM, several approaches have been proposed, such as the MMs (meshfree methods) [6], [7], [8], the GFEM (generalized FEM) [9] and the XFEM (extended FEM) [10]. Note that when dealing with hydro-mechanical coupling problems with rock blocks, the XFEM still has difficulties in simulating the movements of discrete rock block system.
Alternatively, the NMM [11], [41], [42], [43] which deals with discontinuous and continuous problem [35], [51]in the same framework, was proposed. One of the most attractive features of the NMM is that regular mathematical meshes are usually employed for the problem yet to be considered without conforming the material interfaces and the problem domain boundaries. Discontinuity can be easily modeled in the NMM, and the jump (or step) function which is used in the XFEM to capture discontinuity, is not needed. Note that when considering intersection of multiple cracks, the use of multiple jump functions makes the problem a headache. Furthermore, contact technique [12] used in NMM has become mature. Due to these advantages, the NMM has been proposed to solve many types of geotechnical problems, such as seepage problems, contact problems, rock fracturing problems, etc. [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [47], [48], [49]. However, application of the NMM for steady seepage analysis in SRM slope is actual limited[44], [45].
In the present paper, the NMM is further extended for steady seepage analysis of SRM slope. In the proposed NMM model, regular triangular meshes are adopted to build the mathematical meshes. The two typical steady seepage problems, namely, the confined seepage problem and unconfined seepage problem, will be solved with the proposed numerical model. Particularly for unconfined seepage problem, an efficient updating strategy for free surface will be incorporated into the proposed numerical model. Based on the proposed numerical model, a series of numerical examples including confined seepage flow through a foundation, unconfined seepage flow through a homogeneous/heterogeneous/SRM dam and a SRM slope will be investigated. The numerical results show that the improved NMM is capable of solving steady seepage problems accurately and effectively.
Section snippets
PDE for steady seepage problems
In this section, the PDE for two typical steady seepage problems, namely, confined seepage problem and unconfined seepage problem, are presented. For convenience, Fig. 2 is adopted to describe the related concepts.
The total water head of an arbitrary point locating within Ωw (the flow domain) is formulated with:where y, p and γw represent the vertical coordinate, pore pressure and unit weight of water, respectively.
According to Darcy's law, seepage flow's velocity in the
A brief introduction to NMM
Two independent cover systems (CSs) are used in the NMM, i.e., MCS (mathematical CS) and PCS (physical CS) [24]. Regular triangular mathematical meshes are employed in this paper to construct a MCS, as shown in Fig. 3.
A MP (mathematical patch) is built through several neighboring triangles which possess a node of common, such as MP1. The union of all the MPs in Fig. 3 is exactly the MCS. A MCS is not essential to conform to the PMs (physical meshes), such as the problem domain boundaries (red
Numerical tests
In this section, a series of numerical examples about confined and unconfined seepage problems are used to test the performance of the proposed numerical model. Note that if not given, the physical units used in the present work are based on the international standard unit system.
Conclusions
In this paper, the NMM which is based on regular triangular mathematical meshes is further developed for steady seepage problems. For unconfined seepage analysis, procedures to determine free surface location are introduced, and incorporated into the NMM code.
Several numerical examples associated to the two typical steady seepage problems (confined and unconfined seepage analyses) are adopted to confirm capability and accuracy of the NMM. In the first example, confined seepage analysis of a
Declaration of Competing Interest
None.
Acknowledgements
This study is supported by the National Natural Science Foundation of China, under the grant no. 11972043.
References (57)
- et al.
Free surface seepage analysis based on the element-free method
Mech. Mechanics Research Communications
(2003) - et al.
A Generalized Finite Element Method for hydro-mechanically coupled analysis of hydraulic fracturing problems using space-time variant enrichment functions
Comput Methods Appl Mech Eng
(2015) - et al.
An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model
Finite Elem Anal Des
(2013) - et al.
Hydraulic fracturing modeling using the enriched numerical manifold method
Appl Math Model
(2018) - et al.
Three-dimensional fracture propagation with numerical manifold method
Eng Anal Bound Elem
(2016) - et al.
An edge-based smoothed numerical manifold method and its application to static, free and forced vibration analyses
Eng Anal Bound Elem
(2018) - et al.
A zero-thickness cohesive element-based numerical manifold method for rock mechanical behavior with micro-Voronoi grains
Eng Anal Bound Elem
(2018) - et al.
Reformulation of dynamic crack propagation using the numerical manifold method
Eng Anal Bound Elem
(2019) - et al.
Modeling unconfined seepage flow using three-dimensional numerical manifold method
J Hydrodyn Ser B (English Ed)
(2010) - et al.
Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method
Comput Methods Appl Mech Eng
(2015)
Frictional crack initiation and propagation analysis using the numerical manifold method
Computers & Geotechnics
Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method
Appl Math Model
Recent advances: theory of variational inequalities with applications to problems of flow through porous media
Int J Eng Sci
On the solution of elliptic free boundary problems via Newton’s method
Comput Methods Appl Mech Eng
A practical method for solving free-surface seepage problems
Comput Geotech
A new contact potential based three-dimensional discontinuous deformation analysis method
International Journal of Rock Mechanics and Mining Sciences
Hydro-mechanical simulation of the semi-saturated porous soil-rock mixtures using the numerical manifold method
Computer Methods in Applied Mechanics and Engineering
A rigorous and unified mass lumping scheme for higher-order elements
Computer Methods in Applied Mechanics and Engineering
Investigation of the sequential excavation of a soil-rock-mixture slope using the numerical manifold method
Engineering Geology
An improved numerical manifold method with multiple layers of mathematical cover systems for the stability analysis of soil-rock-mixture slopes
Engineering Geology
Sequential excavation analysis of soil-rock-mixture slopes using an improved numerical manifold method with multiple layers of mathematical cover systems
Engineering Geology
A high-order numerical manifold method with continuous stress/strain field
Applied Mathematical Modelling
Enriched mixed numerical manifold formulation with continuous nodal gradients for dynamics of fractured poroelasticity
Applied Mathematical Modelling
Modeling the entire progressive failure process of rock slopes using a strength-based criterion
Computers and Geotechnics
Modelling unconfined seepage flow in soil-rock mixtures using the numerical manifold method
Engineering Analysis with Boundary Elements
Stability analysis of soil-rock-mixture slopes using the numerical manifold method
Engineering Analysis with Boundary Elements
Searching for critical slip surfaces of slopes using stress fields by numerical manifold method
Journal of Rock Mechanics and Geotechnical Engineering
Modelling the stability of a soil-rock-mixture slope based on thedigital image technology and strength reduction numerical manifold method
Engineering Analysis withBoundary Elements
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