International Conference on Geomechanics, Geo-energy and Geo-resources
IC3G 2016
A DISCRETE ELEMENT MODEL TO CONSIDER THE EFFECT OF PORE PRESSURE FOR INTACT ROCK G. Ronga*, G. Liua, Y. Tiea, J.H. Chena, and W. Dub a
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan, China Changjiang Project Supervision & Consultant Company, Ltd. (Hubei),Wuhan, China *
[email protected] (corresponding author’s E-mail)
b
Abstract: A discrete element model to consider the effect of pore pressure for intact saturated rock is presented. In this model, the pore pressure is calculated by the volumetric strain of pore and we apply the pore pressure on rock grains in each time step. The compression of pore and change of permeability are both considered in this model. A program using this model is developed based on three-dimensional discrete element method (PFC3D). Triaxial coupling tests for intact saturated rock under different water pressures are performed using this program. The result shows that the deviatoric stress-axial strain and lateral strain curves are consistent with that from experiments. The strength degradation due to pore water pressure also can be obtained in the numerical test. Keywords: Rock, Pore water pressure, Discrete element method, Triaxial test, Hydro-mechanical coupling
Introduction Rock masses exist widely in the crustal environment. As rock is influenced by lengthy diagenesis, complex tectonic stress, man-made engineering disturbance and several factors, there are many non-uniform pores, micro-cracks, joints and other defects in rock interior. The existence of these pores and joints determines the macroscopic anisotropy of the rock.The study found that the failure process of brittle rock is accompanied by internal pore compaction, crack initiation, propagation and interaction. And the compaction of pore, microcracks damage and extension would affect rock’s strength characteristics and heterogeneity (Bieniawski, 1967; Wawersik and Brace, 1971; Tapponnier and Brace, 1976). In the complex geological environment, the interaction between deep rock and groundwater is very intense, so the deformation and failure process of rock are closely related to the effect of pore water pressure. Based on theoretical analysis and experimental data, Bruno and Nakagawa studied the effect of pore water pressure on crack initiation and propagation. They found that crack propagation is affected by pore water pressure of crack tip and gradient distribution of pore pressure at the same time (Bruno and Nakagawa, 1991).By Fast Lagrange method (three dimensional and continuum) to analyse the localized influence of pore pressure on the deformation, results show that the shear zone tends to be less apparent and rock samples tend to be tensile failure with the increase of pore water pressure. By applying Force-Seepage-Damage (FSD) coupled analysis software to stimulate the whole process of crack initiation and propagation under the effect of pore water pressure, results show that pore water pressure plays an important role in rock deformation, failure process and failure mode (Tang et al., 2003). Hydro-mechanical (HM) coupling has always been a hot and difficult problem in rock mechanics. At present, research of this problem mainly focuses on two aspects: one is the study of the HM coupling of intact rock. For intact rock, there is no through crack and joint in the interior, the HM coupling effect mainly includes the change of permeability and pore water pressure caused by pore compression. In the elastic deformation stage, rock sample’s permeability decreases slightly with the increase of stress. After entering the elastic-plastic stage, with the propagation and penetration of new cracks, the permeability increases. And it reaches the maximum value near the peak stress point. In the residual stage, original cracks begin to close, and the permeability begins to decrease (Peng et al., 2003). By applying automatic random cell model of HM coupling to stimulate rock samples’ failure process, Zhou analysed the micro-mechanism of rock’s mechanical properties and variation of permeability coefficient caused by pore water pressure (Zhou and Feng, 2006). On the other hand, HM coupling behavior of fractured rock mass is extensively studied. For fractured rock mass, crack propagation and its permeability are mainly dominated by joints and fractures. Zhou simplified structural plane as interface layer, and studied its characteristics of closure and dilation. The interface layer deformation model is built, which consider elastic, elastic-plastic and post-peak mechanical properties (Zhou et al., 2008).
Methodology Meso-pore structural characterization of intact model The pore water is normally stored inside pores of intact rock. So if a model intends to reflect the distribution of pore water pressure, first it needs to accurately characterize the distribution of internal porosity. In general, the pore distribution in the natural rock is very heterogeneous. Figure 1 is a slice image of sandstone observed by microscope. It can be seen that quartz grains which are composed of sandstone are well sorted, and most of the pores are intergranular pores (Vergés E et al., 2011). Mercury injection test is a common method for
International Conference on Geomechanics, Geo-energy and Geo-resources
IC3G 2016
measuring the porosity of rock, but this method is easy to damage the pore structure. In contrast, CT scanning is a commonly used method to detect the porosity without damage. Through the binarization processing of CT image, calculating the ratio of the black pixel area and the cross-sectional area of the specimen can obtain the pore distribution of each scanning section.
Figure 1. Petrographic microscope images of sandstone samples with plane polarized (A) and crossed polars (B)( Vergés E et al., 2011) In Figure 2, a part of spherical region in specimen is enlarged and defined as a measurement sphere. Okada et al. established the concept of measurement sphere when studying sand’s undrained shear process (Okada and Ochiai, 2007). Based on this, this paper introduces the concept of porosity and considers the pore compression process of shearing process. It redefines the scope of measurement sphere, optimizes the algorithm, and improves the operation efficiency. As shown in Figure 2, the measurement sphere is constituted by rock particles (gray), cementing material (white) and pores. Suppose the cementing material is wrapped around the rock particles with a certain thickness. Because compression modulus of rock particles and cementing material are larger than that of fluid, in the deformation process these solid particles could be broken. However, it is assumed that the total volume of solid particles is constant, for different position of measurement sphere the porosity n is defined as,
Vv [1] V Vv is the void volume of measurement sphere, V is the volume of measurement sphere. For saturated rock, water is filled with pores of the rock. n is equivalent to the volume percentage of the fluid inside the ball. For the convenience of narration, in the derivation process it is assumed that the fluid in the pores is water. n
Soil particle
Measurement Sphere
Figure 2. Schematic plot of measurement sphere The process of pore water flow in rock Before the penetrating fractures form in the rock mass, the seepage in rock is mainly in micro-voids and microcracks. Considering the measurement sphere as a basic unit of fluid calculation, the whole sample is discretised into a series of measurement spheres and the radius of each measurement sphere is determined by the particle size. Due to the difference of pore water pressure, there will be flow exchange between different measurement spheres. The exchange process satisfies Darcy's law, so in time of t , the flow between two measurement spheres A and B is,
International Conference on Geomechanics, Geo-energy and Geo-resources
PB PA t gLAB
Q kSn
IC3G 2016 [2]
Eq. [2] is called flow equation. S is the contact area of sphere A and B. n is the average porosity of A and B. If rock particles and cementing materials are uniformly distributed within the measurement sphere, its flow area can be estimated by multiplying the contact area with the average porosity. 𝐿𝐴𝐵 is the distance between the center of A and B. k is permeation coefficient of the measurement sphere (unit: m/s). For any of the two measurement spheres A, B in the model, If its radius are 𝑅𝐴 and 𝑅𝐵 , the centre distance of A and B is d AB . When d AB ≥𝑅𝐴 + 𝑅𝐵 , A and B do not contact and there is no flow exchange (contact area is 0). When the contact areaS of A and B can be calculated according to its relative positions, it is assumed that radius of 𝑅𝐴 is greater than the radius of 𝑅𝐵 , the centre distance between A and B is expressed by d AB , Then S can be calculated by the following formula, the radius of contact area (yellow region)is calculated by Heron formula, and then S can be calculated. Figure 3 gives a brief description when RA d AB RA RB . 0 if d AB RA RB 4 R t S (Rt RA )(Rt RB )(Rt d AB ) if RA d AB RA RB 2 d AB RB 2 if d AB RA
Rt 0.5( RA RB d AB )
[3]
Figure 3. Contact area of measurement sphere A and B For each measurement sphere, the inflow and outflow of pore water will cause the change of pore water pressure. In time of t , since the flow resulting in the change of pore water pressure, can be calculated by the total inflow ,
Ew Q nV In Eq. [4], Ew is the compression modulus of pore water. p
[4]
The coupling process of pore water and rock particles In the process of rock’s deformation and failure, it needs to go through the process of pores compaction and micro-cracks closure. For water-saturated rock, the pore pressure caused by stress and deformation will cause the increase of pore water pressure to a certain extent. Pore water pressure also acts on the rock particles, and then hinders the further compression of the pore structure. Based on a unit of measurement sphere, here is the derivation of effect of pore water pressure increment and pore water pressure on rock mass caused by pore deformation. Assuming a measurement sphere A contains N rock particles, cementing materials are wrapped around rock particles with a certain thickness. For any rock particle p (p=1,2,... N) inside A, the velocity is 𝑉𝑝 , the centre position is 𝑋𝑝 . The velocity 𝑉̅ and centre position 𝑋̅ are defined as average values of N of rock particles’ velocities and centre positions. N
v
v p 1
N
p
[5]
International Conference on Geomechanics, Geo-energy and Geo-resources
IC3G 2016
N
x
x
p
p 1
[6]
N
Eq. [6] can calculate the center position of measurement sphere. A local coordinate system is established with this center as the coordinate origin, and the relative velocity position of each particle can be obtained. For the rock particle p within the sphere, The relative velocity 𝑉𝑃′ and relative position 𝑋𝑃′ can be calculated by the velocity of the particle 𝑉𝑝 , the position of the particle 𝑋𝑝 and the velocity 𝑉̅ and the center position 𝑋̅ of the measurement sphere. 𝑉𝑃′ = 𝑉𝑃 − 𝑉̅ [7] 𝑋𝑃′ = 𝑋𝑃 − 𝑋̅ [8] On the other hand, for the position of particle p within the sphere, according to the principles of elastic mechanics, the relative velocity can also be estimated by measuring the velocity gradient of measurement sphere and relative position of the point, referred to 𝑉𝑃𝑟 as estimated relative velocity. 𝑉𝑃𝑟 can be expressed as, v rp = Gx p
[9]
The Strain rate tensor of measurement sphere can be expressed as,
1 [10] (G + G T ) 2 Comparing Eq. [7] and Eq. [9], in a measurement sphere, it requires the calculation error minimization of relative velocity and estimates relative velocity. Here is the algorithm of error μ. ε=
N
μ ( v rp v p )2
[11]
p1
Porosity variation will result in Variation of permeability of rock mass, the change of permeability and porosity is characterized by the following exponential relation.𝐾0 is the initial permeability, 𝑛0 is the initial porosity. David pointed out that the index of Boise sandstone α is about 4.6 (David Cet al., 1994). The test sample in this paper is quartz sandstone. For simplicity sake, we take α=5. [12] K K0 (nnew / n0 ) Measurement sphere only consists of fluid and particles as we regard it as saturation. The buoyancy effect is negligible compared with the effect of pore pressure. For the rock particles of full immersion in the measurement sphere, the pore water pressure from this measurement sphere is balanced (Figure 4). Only particles located at the edge of the measurement sphere, it will be subject to imbalanced water pressure by measurement sphere. In order to simplify the calculation, it is assumed that the width of pressure imbalanced area is m times of the radius of sphere (0
