Excessive groundwater withdrawal can cause land subsidence and earth fissures. The initiation and propagation of earth fissures are related to tensile failure and crack propagation in soils. Based on fracture mechanics, the crack band model (CBM), one of the smear crack models which is relatively easy to construct and convenient to be integrated into standard finite element codes is used in this paper. The calculated results of CBM are less dependent on the sizes of finite elements. The model was applied to simulate the formation and propagation of earth fissures in the hydrostratigraphic units with a bedrock ridge. The simulated positions and patterns of earth fissures coincide with field observations, suggesting that the modeling approach is adept to simulate the initiation and propagation of earth fissures due to groundwater withdrawal.
Discontinuous problems may arise when performing numerical simulation analysis on earth fissures. It is rather inefficient to treat a discontinuity as an internal boundary when applying a standard finite element method (FEM), especially when formation and propagation of fissures are concerned. This approach usually requires intense work in order to deal with the constantly changing shape and topology of the discontinuity. New methods have been presented aiming to solve such discontinuous problems, such as the extended finite element method (XFEM) (Khoei et al., 2014), the element-free Galerkin method (EFGM) (Belytschko et al., 1995), the discrete element method (DEM) (Camones et al., 2013), the displacement discontinuous method (DDM) (Detournay and Gordeliy, 2011). Most of them behave reasonably well in some particular circumstances but they show limitations in other cases because of their different design nature. On the other hand, it would be quite convenient if a similar approach would be compatible with existing standard finite element methods. The smeared cracking model (SCM) was presented by Rashid (1968), aiming to adapt the standard finite element analysis where instances of discontinuous problems are present. It is relatively easy to build from an existing standard FEM and has been proved successful in describing mode-I fracture behavior. In addition, the method is widely-used in many fracture mechanics field such as concrete fracturing (Li et al., 2007). Early SCM implementations used stress as a failure criterion, which leads to a serious mesh-dependency (Cedolin and Bazant, 1980). One substantial modification has been made since the energy-based criterion from fracture mechanism was considered. Softening process now has been incorporated instead of a pure brittle failure. One typical case is the Crack Band Model (CBM) proposed by Bazant and Oh (1983), which will be applied to simulate earth fissures in this paper.
Constitutive relation in CBM.
Cracking within an element is considered as a three-stage process in the
crack band model: intact stage, micro-fissure stage and cracked stage (Fig. 1).
The critical strains between these three stages are denoted as
When an element is deemed as a cracked or micro-fissure element, this
element is treated as mechanical anisotropy. A reduced modulus perpendicular
to crack face is then used in order to decrease the stress with an
increasing strain. Based on addition of strain which means that the total
strain is the sum of the crack strain and the material strain, the
constitutive relation is modified from
The non-tectonic formation and propagation of earth fissures are often considered to be closely related to groundwater extraction activities. This is clearly the case in Suzhou–Wuxi–Changzhou (SXC) area in Jiangsu Province, China (Zhang et al., 2008; Wu et al., 2003). A simulation is performed based on the crack band model, aiming to find the mechanism of earth fissure formation and propagation under the aquifer system with a bedrock ridge.
Conceptual model (top panel) and the finite element mesh (bottom panel) of the earth fissure example.
According to the local geological conditions and the current developmental
state of earth fissures in the SXC area (Zhang et al., 2008), we can
then propose a conceptual model (Fig. 2) assuming a plane strain problem. The
depth of the whole Quaternary strata is 30 m. Four layers are considered of
alternating clay soil and sandy soil beds. More specifically, the first
layer (on the top surface) and the third one are clay soil; and the other
two beds correspond to the sandy soil. The irregular shape on the lower
right corner of the conceptual model represents the bedrock ridge, which is
cutting off the third and the fourth layers at depth (Fig. 2, top panel). The left
boundary is set as a boundary of fixed water level with no excess pore
pressure and no horizontal displacement. Finally, other boundaries are
assumed impervious. Pumping activities are assumed to lie on both sides of
the bedrock ridge at the fourth layer with a constant flow rate
(100 m
Deformation after consolidation.
Considering its symmetry, the simulation is conducted on the left part, as
shown in the conceptual model in Fig. 2 (top panel). Calculation is carried out
following the Biot's consolidation equations. Since fluid-solid coupling is
concerned, it is notable that the pore pressure should be calculated in an
incremental form instead of total form as given in Eq. (6)
Distribution of minor principal stress during pumping.
The strata deformation after pumping is indicated in Fig. 3. When the pumping started, pore pressure at depth decreased, forming a zone with negative excess pore pressure, which was centered at the pumping point. Effective stress increased accordingly, which has caused a land subsidence bowl on top of the pumping well. It should be noted that the main deformation occurs at the deep clay soil.
Figure 4 shows the minor principal stress distribution as well as earth fissure
formation and propagation at different time during pumping. The red zone
represents the tensile zone and the black lines within represent the earth
fissure simulated by the crack band model. Firstly, a self-weighting
consolidation simulation is performed to determine the initial stress
distribution instead of simply assuming a
This paper discusses the simulation of earth fissuring using the crack band model. The simulation results indicate that the crack band model is useful to model earth fissures generation. An example of earth fissure formation and propagation is simulated based on the crack band model under the usual geological conditions with bedrock ridges in the Su–Xi–Chang area of Jiangsu Province, China.
This paper was financially supported by the Ministry of Land and Resources of China (grant no. 201411096-03), the National Nature Science Foundation of China (grant no. 41572250), and UNESCO and IUGS (no. 641).