Modeling of stress-dependent static and dynamic moduli of weak sandstones

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jb009019, 2012 Modeling of stress-dependent static and dynamic moduli of weak sandstones Liming Li 1 and Erling Fjær 1,2 Received 11 November 2011; revised 28 March 2012; accepted 6 April 2012; published 18 May [1] Static and dynamic moduli of a sandstone may be different and stress-sensitive. Knowledge about this is important for the interpretation of seismic data and sonic logs. We use a numerical model which is based on the discrete element method (DEM) to study the static and the dynamic mechanical behavior of weak sandstones. We developed a constitutive contact law and implemented it into the DEM model in order to properly simulate the deformation of weak sandstones. The contact law includes the effect of nonlinear elasticity and plasticity at grain contacts. The DEM model can also capture the effect of the closing of pre-existing cracks, the formation of new cracks and slip at uncemented grain contacts due to stress alteration. Using such a model, we simulated sandstone specimens under laboratory experimental conditions. We calculated the static and the dynamic moduli of the DEM model at different stress states. The modeling results qualitatively agree with some published laboratory observations. By applying different stress paths and stress histories, we studied how the moduli were influenced by the stress state, stress path, and stress history. In particular, the simulations show the growing difference between the static and the dynamic moduli as failure is approached. The simulations thus support the assumption that theoretical insight in the grain-scale deformation mechanisms helps to understand and explain the difference between static and dynamic moduli, as well as for the failure process. Citation: Li, L., and E. Fjær (2012), Modeling of stress-dependent static and dynamic moduli of weak sandstones, J. Geophys. Res., 117,, doi: /2011jb Introduction [2] Static and dynamic moduli are terms that are used to distinguish between elastic moduli derived from stress-strain relations in a rock mechanical test (static moduli) and those derived from wave-velocity measurements (dynamic moduli). The static moduli and the dynamic moduli may be very close for some materials such as steel, if the materials deform within a linearly elastic regime. However, they are often different for rocks. [3] Note that in this paper we are only considering dry rocks. Hence we are not considering the apparent difference between static and dynamic moduli that may appear in saturated rocks due to the fact that wave propagation is an undrained process while static deformation can be drained. [4] For many rocks, a perfectly elastic regime is difficult to identify or does not exist at all in a simple compressive loading test. This implies that the commonly used static moduli for rocks are often not purely elastic moduli, unless particular unloading or cyclic loading tests are performed 1 SINTEF Petroleum Research, Trondheim, Norway. 2 Norwegian University of Science and Technology, Trondheim, Norway. Corresponding author: L. Li, Department of Formation Physics, SINTEF Petroleum Research, S. P. Andersens vei 15 B, Trondheim NO-7465, Norway. (Liming.li@sintef.no) Copyright 2012 by the American Geophysical Union /12/2011JB [e.g., Heap and Faulkner, 2008]. For some types of sandstone, even if the unloading test is performed, it does not ensure a perfectly elastic regime to be measured in the stressstrain curve as well [Fjær, 2009]. Moreover, as we will show in this paper, both the static moduli and the dynamic moduli for sandstone can be stress-sensitive. In practical rock characterization, there are different methods to measure and calculate the static moduli of rocks. For example, Brady and Brown [2006] listed three most common methods to calculate the tangent Young modulus, the average Young modulus and the secant Young modulus, respectively, from a stressstrain curve of an unconfined compressive test. It is important to be aware of what we are using and what we should use for different applications. [5] In this paper, for the static modulus we mean the tangent modulus which is the slope of a stress-strain curve at a point. The tangent moduli relate the increments of the components of the stress on the rock and the corresponding increments of strain components, started from any initial conditions. They are important parameters for geomechanical modeling when we want to compute the stress alteration and the deformation of the underground rock due to, for example, reservoir depleting or well drilling. [6] The static moduli of weak sandstones in hydrocarbon reservoirs are difficult to obtain because coring in weak sandstone reservoirs can be difficult. The coring process may also cause damage to the cores. Hence the core data might be unreliable if they are not properly interpreted [Nes et al., 1of16

2 Figure 1. Dynamic Young s modulus (E dynamic ), static Young s modulus (E static ) and axial stress (s z ) versus axial strain, from a laboratory triaxial compression test (using 12 MPa confining stress) on Castlegate sandstone. This rock is an outcrop with 28.8% porosity. It consists of 70% quartz, 30% feldspar and rock fragments, and no clay. The unconfined strength of the rock is about 16.5 MPa. 2002]. Rock mechanical core testing is also resourcedemanding as well as destructive and therefore it is a limited activity even if the core material is available. In practice, most data of reservoir rock moduli are derived from in situ measured elastic wave velocities. They are dynamic moduli. [7] Static moduli and dynamic moduli of weak sandstones could be quite different and their relationships are quite complex. Apart from the difference and the complexity induced by the measurement methods of the static moduli as mentioned previously, the stress-dependency of the moduli may also be different. When static moduli are plotted against dynamic moduli the data from sandstones reveal extensive scatter [Wang, 2001]. One of the reasons might be that the measurements have been carried out under different stress or pressure conditions. It has already been found that both static moduli and dynamic moduli may vary significantly with stress state, stress path and stress history [Walsh, 1965; Ita et al., 1993; Dvorkin et al., 1996]. [8] Based on their observations, previous researchers assumed that the stress-dependence of the weak sandstone moduli is due to the closure of microcracks in the rock, the new microcracking events, and contact area change when the grains are compressed against each other [e.g., Digby, 1981; Walton, 1987; Ita et al., 1993; Sayers, 2002; Fjær, 2006, 2009]. These factors may have complex effects on both the static and the dynamic moduli. The observed stressdependency is a combined result of those factors. [9] Figure 1 shows data from a test on Castlegate sandstone which is an outcrop and is often used as an analogue of reservoir sandstones. The specimen was under triaxial compressive test with 12 MPa confining stress. The measured dynamic modulus E dynamic and the static modulus E static are plotted versus axial strain in Figure 1. This started from the condition of 12 MPa hydrostatic stress. The axial stress is also plotted versus axial strain in Figure 1 for reference. [10] It is observed that the dynamic modulus increases with the axial stress when the axial stress is low. The modulus increase with stress is assumed the effects of the closure of microcracks which have already existed in the rock before loading, and the increases of the contact area between the grains. New microcracking events induced by the stress changes may occur from a very low stress level and cause the moduli to decrease. However, this effect is not significant when the stress is low relative to the rock strength. When the axial stress increases further the microcracking events increase and may increase dramatically to some stage as indicated by acoustic emission events monitoring [e.g., Tembe et al., 2008]. When the specimen approaches global failure, the effect of the microcracking events may become dominant. This causes the dynamic modulus to eventually drop a little. [11] There is a discrepancy between the static (tangent) modulus and the dynamic modulus from the beginning when the triaxial test starts from 12 MPa hydrostatic stress condition. This can be ascribed to plastic deformation which is affecting the static modulus but not the dynamic modulus. The plastic deformation increases significantly when the microcracking events increase dramatically. Therefore, the discrepancy between the dynamic modulus and the static modulus also increases significantly. At the peak of the stress-strain curve, the static (tangent) modulus drops to 0, while the dynamic modulus still remains high. [12] It seems that we could logically explain our observations based on the assumed mechanisms for the stressdependency of the moduli and the discrepancy between them. However, it will still be interesting to further investigate these issues in order to eventually quantify the link between the static moduli and the dynamic moduli at different stress conditions. Numerical experiments using a micromechanics-based model were therefore performed. [13] The discrete element method (DEM) [Cundall and Strack, 1979] has been extensively used to study the mechanical behavior of uncemented and cemented granular materials such as sands and sandstones [Bruno, 1994; Li and Holt, 2002; Makse et al., 2004; Holt et al., 2005; García and Medina, 2006; Boutt et al., 2007; Ng, 2009]. A DEM model usually uses an assembly of spherical elements (or circular elements if the model is 2-dimensional (2D), as illustrated in Figure 2). Note that we used only 3-dimensional (3D) models in this work. However, some of the illustrations are shown in 2D just for simplicity. A discrete element does not necessarily always represent a single grain of the granular material, although sometimes it is used to represent a single grain. Generally, in a 3D DEM model the physical medium is discretized using an assembly of spherical elements. In other words, a discrete element represents a piece of the physical medium which may contain many grains. By using the spherical elements, it is simple in computation to determine whether two elements are in contact or not. Therefore, the elements in a DEM model are allowed detaching from each other, and being pressed against each other during a simulation. This is the main reason that the DEM is convenient in simulating dynamic problems such as those involving complex fractures and large deformation. [14] The interactions between the elements in a DEM model can be defined in the so-called contact law in order to simulate the mechanical behaviors of different physical media. Different schemes can be implemented to mimic bonds between those elements [Potyondy and Cundall, 2004]. Such a bonded model can be used to simulate rock and rock-like materials. The rock is discretized using bonded 2of16

3 Figure 2. (a, b) Illustration of the crack closure and opening in a DEM model which simulates a piece of sandstone. A crack shown in Figure 2a may be closed under stress as shown in Figure 2b, while a new crack may be opened as shown in Figure 2b. elements in 3D space. The bond breakage mimics microcracking and fracturing in the rock. Two elements which are originally in contact are allowed to detach from each other during the simulated process if the elements are unbonded, or the bond between them is broken. They can also slide over each other. New contacts between element pairs can be detected and established during the simulation. [15] Li and Fjær [2008] used such a 3D DEM model and simulated the stress-dependent static and dynamic moduli of sandstones. In this paper we will present in detail how the DEM model has been formulated. The discussions on the modeling results will be extended, with comparison to results of laboratory experiments with sandstone cores and comparison to an empirically based analytical model [Fjær, 1999, 2009]. This model describes a relation between static and dynamic moduli for weak rocks, based on the assumption that the total strain increment induced by a stress increment is the sum of an elastic term and a non-elastic term. The elastic term is proportional to the stress increment, and the proportionality constant is assumed to be the elastic stiffness which can be derived from acoustic velocities for a dry material. Consequently, the non-elastic term is assumed to vanish for small amplitude, oscillating deformations. Considering a uniaxial stress increment, the non-elastic term consists of two parts: [16] The first part is proportional to the stress increment. The proportionality constant is found to decrease with increasing stress during initial loading. It is assumed that the physical origin of this term is a damage process typically located to the grain contacts, since the stress concentrations in these areas may exceed the elastic limit even at low external stresses. At higher stress levels, the contact area will increase and the stress concentration is reduced. [17] The second part is proportional to the total strain increment in the direction of loading. The smooth curvature of the stress-strain curve near the peak stress point for a ductile material is closely related to this dependency. The proportionality constant appears to be proportional to the total shear strain, and inversely proportional to the square root of the mean stress. For a unidirectional loading the term will grow monotonously because of the dependency on total shear strain. Eventually, this term will cause the static stiffness of the rock to vanish (and even become negative), and hence it is the origin of shear failure of the rock sample. It is assumed that the physical origin of this term is local shear failure, for instance shear failure of a grain contact or frictioncontrolled sliding along a closed crack. Such events of local failure implies that the remaining parts of the intact rock have to take larger loads and thus deform accordingly. It can also be argued [Fjær, 1999] that the stress and strain dependency of this term is consistent with the assumptions that local shear failure is controlled by local shear stress according to the Griffith [1921] criterion, and that shear strain is the most relevant global parameter for representing local shear stress. [18] In a standard triaxial test, consisting of a hydrostatic loading phase followed by a uniaxial loading phase, the first part of the non-elastic term may play a significant role in the hydrostatic phase while the second part is absent in this phase. During the uniaxial loading phase, both parts will be active, but the second part will gradually take over and totally dominate the rock behavior as failure is approaching. Both parts of the non-elastic term are found to depend on stress history, in qualitative agreement with the proposed physical origins: upon unloading or reloading both parts drops to zero and gradually picks up again, and only regain their previous level when the previous maximum stress level is exceeded [Fjær, 2009; Fjær et al., 2011]. [19] In this paper, we will demonstrate how the DEM model, with the above mentioned micro-mechanisms implemented, can reproduce the macro-properties of the sandstone. 2. DEM and a Constitutive Contact Law for Weak Sandstones [20] The DEM model presented in this paper is based on a commercial code, Particle Flow Code in 3 Dimensions (PFC3D) from Itasca Consulting Group, Inc. [Itasca, 2008]. The fundamentals of the DEM implemented in that code have been described in the software manuals [Itasca, 2008] and in a paper by Potyondy and Cundall [2004]. In their paper, the model was used to simulate granite. 3of16

4 Figure 3. A contact in the unbonded state: (a) The total overlap u n is composed of an elastic part u n,e and a plastic part u n,p. (b) The total overlap u n is smaller than the residual plastic deformation u n,p. The contact normal force is 0, even though the elements still overlap. (c) Two elements are separated. There is no contact between the two elements any more. [21] The DEM uses an explicit time stepping scheme to solve the motion equation. The rock is discretized using bonded spherical elements. Each element has mass. It translates and rotates according to Newton s law of motion. The shape and the radius of the element will not change automatically during the simulation. However, the elements can overlap if they are compressed against each other (e.g., Figure 3). The constitutive contact law determines interactions between the element pairs due to the relative displacements. Beam-like bonds (Figure 4) are applied to neighboring element pairs so that the modeled material can resist tension and rotation. In the DEM model, new contacts can be established during the simulation and bonds are allowed to break (Figure 5). This can be associated with closure of preexisting cracks and opening of new cracks in a real rock. [22] An important development in our work is that we formulated a particular constitutive contact law based on a phenomenological analysis, aiming at making the DEM model fit the laboratory experimental data (mainly of triaxial tests) of natural sandstone specimens. According to the contact law, the contact between two elements will be in one out of three possible states: 2.1. Unbonded State [23] Figure 3 illustrates a contact in the unbonded state. This state may be applied when the elements are used to represent, for example, cohesionless sand, or when there is a Figure 4. A contact in the bonded state: (a) The total overlap u n is composed of an elastic part u n,e and a plastic part u n,p. (b) The total overlap u n is smaller than the residual plastic deformation u n,p. This contact is in tension even though the two elements overlap. (c) Two elements are separated. The bond still exists. There is a tensile force between two elements. 4of16

5 Figure 5. A contact in the broken state: (a) The total overlap u n is composed of an elastic part u n,e and a plastic part u n,p. This situation is different from the unbonded contact shown in Figure 3a. One difference is that the parameters are different. Another difference is that this contact can provide resistance to bending moment but with a reduction factor. (b) The total overlap u n is smaller than the residual plastic deformation u n,p. The contact forces and moment vanishes, even though the two elements still overlap. (c) Two elements are separated. There is no contact between the two elements any more. crack between two elements which represent sandstone. Either the crack has existed at the onset of the simulation or the crack is formed and opened during the simulation. The contact in the unbonded state then captures the effect when a crack is closed during the simulation later on. An unbonded contact can be in one of two situations. In the first situation shown in Figure 3a, the overlap between the two elements u n is composed of two parts: the elastic part u n,e and the plastic part u n,p. In the situation shown in Figure 3b, the total overlap u n between the two elements is actually smaller than the residual plastic deformation u n,p. In Figure 3c, two elements become completely separated. There is no contact any more. [24] For an unbonded contact, we assume the contact normal force: ( F n ¼ k cðu n u n;p Þ 2 if u n > u n;p ðnormal compressionþ 0 if u n u n;p ðno elastic compressionþ ; where k c is a constant. [25] Equation (1) is different from the Hertzian contact law where the contact normal force between a pair of identical spheres is given by [e.g., Johnson, 1985] F n ¼ 2G pffiffiffiffi s 2r u 1:5 n 3ð1 n s Þ ; ð2þ where G s and n s are the shear modulus and Poisson s ratio of the material, respectively, and r is the radius of the spheres. [26] u n has an exponent 1.5 in equation (2), while we use an exponent 2 in equation (1). Effective medium models based on the Hertz-Mindlin contact law predict that the relation of wave velocity V of unconsolidated sands and the applied stress s follows a power law, V s 1/6 [Walton, 1987; Mavko et al., 1998]. However, experiments have ð1þ shown that this relation should be V s 1/4, approximately [Domenico, 1977; Zimmer et al., 2007]. This indicates that an exponent 2 as used in equation (1) could be adequate. [27] By comparing with the Hertz-Mindlin contact law, we assume prsmall 2 k c ¼ a G s k ðr 1 þ r 2 Þð1 n s Þ ; where a k is a constant for all contacts in the same DEM model, r 1 and r 2 are the radii of the two elements associated with this contact, and r small is the smaller one of r 1 and r 2. [28] Plasticity implies that the rock builds up less stress for a given deformation than it should have done according to pure elasticity. Upon subsequent unloading, the rock will behave elastically with a stress reduction related to the reduction in deformation. In our model, plasticity is accounted for through the plastic overlap parameter u n,p. [29] We update the plastic overlap in equation (1) in an accumulated way. It is updated in every time step Dt. The current plastic overlap u n,p t equals to the plastic overlap in the previous time step u n,p t Dt plus the increment of the plastic overlap during this time step Du n,p : u n,p t = u n,p t Dt + Du n,p, while 8 < p a p b Du n;p ¼ p c F n þ 1 þ p b Du n if Du n > 0 ; ð3þ : 0 if Du n 0 where p a, p b and p c are constants which control the plastic behavior of the model, Du n is the increment of the total overlap between two grains in a time step. [30] According to equation (3), the plastic overlap u n,p increases only when the total overlap u n increases (Du n >0 when two elements are pressed against each other). The increment of the plastic overlap Du n,p is a fraction of the 5of16

6 increment of the total overlap Du n. Note that time-dependent deformation is not considered in this contact law. [31] The calculation of the shear stiffness at the contact of a pair of elastic spheres has been first studied by Mindlin and Deresiewicz [1953] and has been proven to be a complex theoretical problem. It depends on the loading path and loading history. Here we simplify the calculation and assume that the shear stiffness is a fraction of the normal stiffness. Therefore, we calculate the increment of the shear force in vector form as DF s ¼ 8 < k cðu n u n;p Þ Du s if u n > u n;p ðnormal compressionþ b ub : : 0 if u n u n;p ðno elastic compressionþ ð4þ where k b = E b pr 2 small ðr 1 þr 2 Þ, and E b is Young s modulus of the cementing material. [36] For the bonded contact, we calculate the increment of the plastic deformation as Du n;p ¼ p bdu n if DF n Du n > 0 : 0 otherwise The condition DF n Du n > 0 means that plastic deformation occurs only if the magnitude of the contact normal force increases (either in tension or in compression). [37] If F s Du s < 0, the relative shear displacement tends to increase the magnitude of the shear force. The increment of the shear force is ð7þ 8 ð1 h p Þ k cðu n u n;p Þþk b >< Du s if u n > u n;p ðnormal compressionþ DF s ¼ b b ð1 h p Þ k ; b >: Du s if u n u n;p ðnormal tensionþ b b ð8aþ Du s is the increment vector of shear displacement, while b ub is a constant (assumed to be the same for all unbonded contacts in the model). [32] We allow slip between a pair of unbonded elements. The slip condition is F s ¼ f ub F n ; where f ub is the friction coefficient for all unbonded contacts Bonded State [33] In order to properly capture the mechanical behavior of weak sandstone, the formulation for a contact in the bonded state in this paper is also different from some previously published models [e.g., Potyondy and Cundall, 2004; Wang et al., 2006; Abe et al., 2006]. Both the nonlinear elasticity and plasticity have been considered in the contact law in this work. In order to properly capture the strength behavior of weak sandstones, the bond may fail in tensile, shear and compressive modes. The simulated sandstone may fail under hydrostatic compressive stress. [34] Figure 4 illustrates a contact in the bonded state. If two bonded elements are compressed against each other (Figure 4a), the contact stiffness is the combined result of the grain-grain contact stiffness and the stiffness provided by the cement at the grain contact. If two bonded elements are pulled apart, either when the total overlap is larger than 0 but smaller than the residual plastic overlap (Figure 4b) or when there is a gap between two elements (Figure 4c), the tensile stiffness equals to the stiffness of the bond material only. [35] We calculate the normal force at a bonded contact as ð5þ where h p and b b are constants for all bonded contacts in the model. [38] If F s Du s 0, the magnitude of the shear force decreases as a result of the relative shear displacement. We assume that the plastic shear deformation keeps unchanged in this case and the change of the shear force is then given by 8 >< DF s ¼ >: k cðu n u n;p Þþk b b b Du s if u n > u n;p k b b b Du s if u n u n;p : We calculate the increment of the bending moment as DM ¼ p 4 r4 small k bdq; ð8bþ where Dq is the increment of the rotation angle. We neglect the twisting moment. [39] A bond may fail in three different modes: [40] 1. A bond fails in the tensile mode, if the average normal stress in the bond contact s n = F n s pr 2 t, where s t is small the tensile strength of the bond material. (Note that the tensile stress is negative in this paper, while the tensile strength is positive.) [41] 2. A bond fails in the compressive mode, if F n s n = s c, where s c is the compressive strength of pr 2 small the bond material. [42] 3. A bond fails in the shear mode, if the average shear stress in the bonded contact t = C + f b s n, where C is F s pr 2 small ð9þ F n ¼ k cðu n u n;p Þ 2 þ k b ðu n u n;p Þ if u n > u n;p ðnormal compressionþ k b ðu n u n;p Þ if u n u n;p ðnormal tensionþ ; ð6þ 6of16

7 [46] We assume that the bending moment may exist at a broken but closed contact and will be given by DM ¼ ( p 4 h wrsmall 4 k cbrokendq if u n u n;p ðnormal compressionþ ; 0 if u n < u n;p ðno elastic compressionþ ð12þ where h w is a reduction factor less than 1. [47] Slip may happen at a contact in the broken state. The slip condition is F s ¼ f broken F n ; ð13þ Figure 6. Illustration of the force-displacement relationship in the normal direction. Note that the contact normal force is 0 if the contact is in the unbonded or the broken state, and u n < u n,p. the cohesion of the bond material, and f b is the internal friction coefficient of the bond material Broken State [43] If a bonded contact fails in any one of the above described three modes, the contact turns into the broken state. Figure 5 illustrates a contact in the broken state. The behavior of a broken contact is similar to but different from an unbonded contact: The formulas of the contact force calculations are similar. However, the parameters in the formulas may take different values. Moreover, in the situation shown in Figure 5a, the total overlap is larger than the plastic overlap. The broken contact behaves as a closed crack. Some rotation resistance still exists at the contact in this situation. Once a broken contact comes to the situation shown in Figure 5c, the element pair becomes completely separated, the contact will be reset as an unbonded contact if the element pair comes into contact again later on. [44] Therefore, for a broken contact, the contact normal force is written as F n ¼ kcbroken ðu n u n;p Þ 2 if u n > u n;p ðnormal compressionþ 0 if u n u n;p ðno elastic compressionþ ; pr 2 small G s ð10þ where k cbroken = a kbroken ðr 1 þr 2 Þð1 n s Þ, a kbroken is a constant for all contacts where a bond existed but is broken. We calculate the increment of u n,p according to equation (3). [45] The increment of the shear contact force is given by where f broken is the friction coefficient for a contact in the broken state. [48] Figure 6 is an illustration of the relationship of the normal force and the normal displacement according to the contact law presented here. 3. Modeling of Rock Specimen [49] In order to generate the model, we first followed the procedure described by Potyondy and Cundall [2004]. A random assembly of unbonded spherical elements was generated in a box under relatively low hydrostatic compressive stress, 0.1 MPa. Linear contact law has been used at this stage. With a servo-mechanism applied on the model boundary to maintain a constant stress, some additional calculation cycles have been run to ensure the elements to be close to equilibrium and without much locked in stress in the model. Then the linear contact law was replaced by the contact law presented in this paper. For a contact if two elements are in touch or overlap, they are assumed in the bonded state. The initial overlap is assumed as plastic overlap. Therefore, there is no force at any contact in the model at this moment. [50] Figure 7 shows the half part of the model. The elements are confined by six plates which simulate a polyaxial cell. The front half part of the model has not been shown so that the numerical transducers inside the model can be seen. The model has been used to simulate static deformation and elastic wave propagation in a sandstone specimen. Elastic wave propagation is studied by using two groups of model elements as numerical transducers. One of the transducers is creating a disturbance which is later picked up by the other transducer, and the wave velocity is calculated from the time lag [Li and Holt, 2002]. In order to avoid the boundary effects which may induce noise in the wave signal, the transducers have been placed inside the numerical specimen, instead of on the ends of the specimen as in the laboratory experiments. [51] The input parameters used in the model are listed in Table 1. All the models presented in this paper are square 8 < DF s ¼ k cbrokenðu n u n;p Þ Du s if u n > u n;p ðnormal compressionþ b broken ; : 0 if u n u n;p ðno elastic compressionþ ð11þ where b broken is a constant for all broken bonds. 7of16

8 Figure 7. A DEM model (only half of the model is shown) confined by six plates. The model has been used to simulate a sandstone specimen in a polyaxial cell. The front part of the model is not shown, so that the transducers inside the model can be visible. The bonds are not visible either. The direction later referred to as axial is the z-direction shown. cuboids in shape. The dimensions are mm. The simulated results with units such as stress or strain do not change if the geometric parameters of the model are scaled up or down simultaneously, as long as the mechanical input parameters are kept unchanged. The longest dimension (in the z-direction as shown in Figure 7) is twice of the other two dimensions (the x- and the y-direction as shown in Figure 7). The aspect ratio is the same as the specimens of the sandstone cores as we usually use in the laboratory. However, the specimens in the laboratory are usually cylindrical in shape. In this paper we will call the longest direction of a modeled specimen the axial direction. Therefore, it will be comparable with the axial direction of a cylindrical core. The other two directions of a modeled specimen will be called the lateral directions which will be comparable with the radial direction of a cylindrical core. [52] The model used 4509 elements, with the radii uniformly distributed between mm. The model size is rather small in terms of element number. Generally, the DEM model can be grid sensitive since the model is not homogeneous and it is used to simulate phenomena involving inhomogeneous deformation such as fracturing and strain localization. Koyama and Jing [2007] studied the representative elementary volume (REV) in order to simulate a square rock specimen using a 2D model. According to their study, the proper model size would be more than elements in the 2D square model. If such a 2D study can be used to estimate the REV for the 3D cuboid model, the proper model size will be millions of elements. It will be very computation demanding using such a model size for the study presented in this paper. Often the 3D DEM model used to simulate a rock specimen uses tens of thousands of elements [e.g., Schöpfer et al., 2009]. The model presented in this paper will be used to qualitatively study the micromechanism relevant to the stress-dependency of the moduli, mainly before the global failure of the rock specimen. We do not make any quantitative prediction or conclusion based on the modeling results. [53] The apparent porosity of the model was about 39%. It is calculated from the total volume of the spherical elements and the model dimensions, without counting the volume of the bonds which mimic the cement. The apparent porosity is a parameter of the mesh, without a direct link to the porosity of the real rock to be simulated. However, the apparent porosity of the model was used in choosing the element density. In order to simulate a Castlegate sandstone specimen with density 2000 kg/m 3, the element density was set to 3310 kg/m 3. Therefore, the element density is different from the grain density of the sandstone. In fact, the element mass is a numerically lumped mass at the center of the element. The assumption was that the mass of the sand grains and the cement between the grains were lumped on the discrete points of the element centers. This is a common way in numerical discretization, just like, for example, the finite element method [e.g., Zienkiewicz and Taylor, 2005]. [54] A limitation with the model is that so far there is no direct way to derive the input parameters from laboratory measurements on natural rocks. In order to simulate a particular rock, a calibration study has to be done. Some simulations (for example, of triaxial compression tests) have to be performed. The simulation results will be compared with Table 1. Parameters Used in the DEM Model Parameter Value Element radii (mm) Element density (kg/m 3 ) 3310 a a k a kbroken G s (GPa) 41 n s 0.1 E b (GPa) 4 b ub 5 b b 2 b broken 8 p a 0.9 p b 0.3 p c 0.15 h p 0.1 h w 0.1 f ub 0.3 f broken 0.1 s t (MPa) s c (MPa) C (MPa) f b a The density of the elements was calculated by matching the density of the model with the density of Castlegate sandstone, 2000 kg/m 3. 8of16

9 Figure 8. Comparison of the simulation results and the laboratory measurements on Castlegate sandstone from the triaxial compressive tests: peak axial stress versus confining stress. laboratory data. Then the parameters will be adjusted, until acceptable agreement between the simulation results and the laboratory data is achieved. A model using a perfect set of parameters should be able to quantitatively reproduce the mechanical response of the natural sandstone on different aspects and on different conditions, provided that the micromechanisms which control the mechanical behavior of the sandstone have been properly mimicked in the model. This means, for example, under the triaxial compressive situation the model should be able to predict the peak axial stresses of the sandstone under different confining stresses, the stressstrain curves of the sandstones under different confining stresses, and the elastic wave velocities of the sandstone under different stress conditions. The more the model matches the available data from the natural rock, the more reliable the predictions can be made from the model. [55] However, a very good quantitative match of a model with a sandstone specimen may take quite some time for computation and analysis. In this work, we intended to perform only a qualitative mechanism study without spending enough time to quantitatively match all parameters of a particular type of sandstone. Some of the input parameters for the model were therefore chosen based on modeling experience. Only three parameters: s t, s c and C have been adjusted in a process of trial and error, by comparing the peak axial stress versus confining stress of Castlegate sandstone in triaxial compressive tests. Therefore, we do not claim any simulation results in this work quantitatively represent any particular type of natural sandstone. [56] Using the DEM model with the input parameters listed in Table 1, we performed simulations of triaxial tests with 2 MPa, 5 MPa and 15 MPa confining stresses. Some parameters calculated from the simulation results were compared with those calculated from the laboratory experiments with Castlegate sandstone specimens as shown in Figures 8 and 9 (The laboratory experimental data with Castlegate sandstone were presented by Li et al. [2008]). Figure 10 shows the stress-strain curves of the simulated triaxial tests. [57] From the comparison of the modeling results and laboratory measurements shown in Figures 8 and 9, we can see that the axial peak stresses predicted with the model match the lab data quantitatively well. The E 50 is the slope of the axial stress versus axial strain curve at 50% of the peak differential stress (the axial stress minus the confining stress), while the n 50 is the slope of the negative lateral/radial strain versus axial strain at 50% of the peak differential stress. The values of those parameters do not agree quantitatively very well. However, a trend which shows that the E 50 increases with increasing confining stress while n 50 decreases with increasing confining stress is consistent from both the modeling results and the lab data. It indicates that the model makes qualitative predictions on these respects. 4. Simulation of Wave Propagation [58] The discrete element method which has been used here is a dynamic method. The forces acted on the elements in the model are seldom in perfect equilibrium. The element velocities and the interactions between the element pairs are updated in every time step. An artificial damping mechanism is used in order to reduce oscillations in the DEM model [Potyondy and Cundall, 2004]. When the DEM model is used to simulate, for example, a triaxial compressive test, the loading platen velocities should be sufficiently low. The element velocities are kept low enough so that the effect of the kinetic energy can be ignored. The response of the model can be interpreted as the static behavior. The model thus works in the quasi-static mode. As described by Potyondy and Cundall [2004], for a particular model, the platen velocities should be chosen to ensure the quasi-static test conditions by demonstrating that reducing the platen velocities does not alter the measured macro-properties. [59] Using a small and realistic damping coefficient, however, the model can work in the dynamic mode where an elastic disturbance is propagating between the numerical transducers. Thus wave propagation can be explicitly simulated [Li and Holt, 2002]. 9of16

10 Figure 9. Comparison of the simulation results and the laboratory measurements on Castlegate sandstone from the triaxial compressive tests: The tangent Young modulus and the Poisson ratio at 50% of the peak differential axial stress versus confining stress. [60] Wave propagation can be simulated at any stress level. The model works in the quasi-static mode where a stress change is imposed. Still in the quasi-static mode, we cycle the model a few more time steps with fixed displacements or constant stresses on the boundary. This is to ensure that the elements in the whole model come close to equilibrium. Thus the velocities of the elements become low relative to the excitation magnitude to be applied. After that, we reduce the damping coefficient to make the model work in the dynamic mode so that wave propagation can be simulated. [61] In the model, as shown in Figure 7, we selected a group of elements as a transmitter and another group as a receiver. We can then create the excitation by applying a single cycle sine velocity pulse to the transmitter elements. The disturbance propagates through the model, either as a P wave or as an S-wave, depending on the direction of the excitation velocity applied to the transmitter elements and the propagating direction of the observed wave (i.e., the axis between the transmitter and the receiver). We monitor the average velocity of the receiver elements in order to detect the arrival of the disturbance. The wave travel time is identified as the interval between the first peak of the excitation applied to the transmitter and the first peak of the monitored average velocity of the receiver elements. The phase velocity of the wave is calculated from the wave travel time and the distance of the travel path, in exactly the same way as in laboratory tests. In the simulations presented in this paper, the applied excitation frequency was 600 khz for the P wave simulation and 400 khz for the S-wave simulation. 5. Stress-Dependent Static and Dynamic Moduli of the Model [62] We performed simulations of hydrostatic compression (same stress in all directions), triaxial compression (increasing stress in one direction, constant stress in the other two) and uniaxial compaction (increasing strain in one direction, no strain in the other two) using the same assembly of discrete elements as shown in Figure 7 and the set of parameters listed in Table Hydrostatic Compression [63] The model was compressed hydrostatically to 15 MPa. In Figure 11a, the mean stress, the static bulk modulus K static and the dynamic bulk modulus K dynamic are plotted against the volumetric strain. The static bulk modulus K static was calculated from the increment of the mean stress Ds m and the increment of the volumetric strain Dɛ v : K static ¼ Ds m Dɛ v : ð14þ Figure 10. Axial stress versus axial/lateral strain from the simulation results of the triaxial compressive tests with confining stress (s h ) 2, 5 and 15 MPa. 10 of 16

11 Figure 11. Simulation results of a hydrostatic compression test: (a) mean stress (s m ), dynamic bulk modulus (K dynamic ) and static bulk modulus (K static ) versus volumetric strain; (b) coordination number (N coord ) and axial plastic deformation (ɛ c,pz ) versus volumetric strain. The dynamic bulk modulus K dynamic was calculated from the wave velocities V p and V s : K dynamic ¼ rv 2 p 4 3 rv 2 s ; ð15þ where r = 2000 kg/m 3 is the average density of the model. [64] During the simulations, we also studied the evolution of the coordination number and the development of the plastic deformation at the contacts. We define the coordination number of an element as the number of the neighbors which are in actual contact (hence load bearing) with this element. The plastic deformation in the model could be a combined result of the plastic part of the overlap, u n,p at the contacts, the plastic part of the tangential displacement at the bonded contacts, slip at the unbonded or broken contacts, and bond breakages. [65] Based on u n,p, an average of the axial component of the normal plastic deformation at the contacts has been calculated as P u n;p n r ɛ c;pz ¼ 1 þ r 2 j Dz j Dl ; ð16þ n where Dl is the distance between the centers of two elements which are associated with this contact, Dz is the component of Dl in the z-direction, and n is the number of the actual contacts in the whole model. We use ɛ c,pz as an indicator of the plastic deformation at the contacts in the whole model. [66] In Figure 11b, the average coordination number and ɛ c,pz are plotted against volumetric strain. A nonlinear stress - strain relationship is observed in the simulation of the hydrostatic compression test (Figure 11a). Both the dynamic modulus and the static modulus increase with increasing volumetric strain, due to the nonlinear contact model. Figure 11b shows that the coordination number also slightly increases with increasing volumetric strain. This contributed to the increasing moduli. [67] The difference between the dynamic and the static moduli exists already at the beginning of the loading path, and it increases with increasing volumetric strain. This difference appears to be induced by the plastic deformation at the contacts, u n,p. [68] The static modulus increases faster than the difference between the static and dynamic moduli however, so that the ratio between the static and the dynamic modulus is also increasing with increasing volumetric strain Triaxial Compression [69] Simulations of triaxial compressive tests were performed with constant confining stresses 2 MPa, 5 MPa and 15 MPa. Using the simulation results of triaxial compressive tests, we calculated the static elastic parameters E static and n static from the axial stress increment Ds z, the axial strain increment Dɛ z, and the lateral strain increment Dɛ h : E static ¼ Ds z Dɛ z ; n static ¼ Dɛ h Dɛ z : ð17þ ð18þ The dynamic elastic parameters E dynamic and n dynamic were calculated from the wave velocities in the axial, i.e., the loading direction as E dynamic ¼ rv 2 s ð3v 2 p 4V 2 s Þ ðv 2 p V 2 s Þ ; ð19þ n dynamic ¼ V 2 p 2V 2 s 2ðV 2 p V 2 s Þ : ð20þ Equations (17) (20) are valid for isotropic media. The DEM model under anisotropic stress state is not isotropic as shown by, for example, Li and Holt [2002] when wave propagation have been simulated in both the axial and the radial directions. Therefore the parameters calculated from equations (17) (20) are only approximate estimations. However, it may still be relevant to use them for a qualitative comparison. [70] Figure 12a shows a comparison of the static elastic parameters and the dynamic elastic parameters, based on the simulation results of the triaxial compression test with 2 MPa confining stress. [71] It is observed that both E dynamic and E static increase initially with increasing axial strain (Figure 12a). There is a difference between the static and dynamic moduli from the beginning of the simulation. The difference increases slightly 11 of 16

12 lateral expansion is suppressed in this case due to high confinement and hence n static does not increase so much as in the low confinement case. The onset of microcracking is seen to occur at somewhat higher axial strain in the high confinement case than in the low confinement case (0.7% versus 0.5%, respectively) Uniaxial Compaction [75] Simulations of uniaxial compaction tests were performed by fixing the lateral boundary (so that there is no lateral strain), and compressing the sample in the axial direction. We performed two simulations with different initial hydrostatic stresses: 5 MPa and 10 MPa, respectively. [76] The results from the simulation of uniaxial compaction test with 5 MPa initial hydrostatic stress are shown in Figures 14a 14c. The static Poisson ratio and the uniaxial compaction modulus are calculated as n static ¼ Ds h Ds h þ Ds v ; H static ¼ Ds z Dɛ z ; ð21þ ð22þ Figure 12. Simulation results of the triaxial test with 2 MPa confining stress: (a) dynamic Young s modulus (E dynamic ), static Young s modulus (E static ) and axial stress (s z ) versus axial strain; (b) dynamic Poisson s ratio (n dynamic ) and static Poisson s ratio (n static ) versus axial strain; (c) coordination number (N coord ), crack number (N crack ) and plastic deformation (ɛ c,pz ) versus axial strain. with increasing stress during the first part of the loading path. A dramatic change starts from about 0.5% axial strain, then both E dynamic and E static drop, and the difference between E dynamic and E static increases significantly, mainly due to the sharp drop of the static modulus. [72] The dynamic and the static Poisson s ratios, n dynamic and n static respectively, are almost equal at the beginning of the simulation (Figure 12b). They both increase with increasing axial strain, but n static increases a bit faster than n dynamic. Later, the difference between n dynamic and n static increases dramatically. Again the onset of the rapid change occurs at about 0.5% axial strain. [73] A clear onset of microcracking (mimicked by the bond failure in the model) occurs at about 0.5% axial strain as indicated by the increase of the number of cracks (bond breakages) N crack (Figure 12c). Thus, the dramatic alteration of the elastic parameters appears to be linked to the development of microcracks. [74] Figures 13a 13c show the results from the simulation of a triaxial test with 15 MPa confining stress. The results are qualitatively similar to the results obtained in the simulation of the triaxial test with 2 MPa confining stress. However, Figure 13. Simulation results of the triaxial test with 15 MPa confining pressure: (a) dynamic Young s modulus (E dynamic ), static Young s modulus (E static ) and axial stress (s z ) versus axial strain; (b) dynamic Poisson s ratio (n dynamic ) and static Poisson s ratio (n static ) versus axial strain; (c) coordination number (N coord ), crack number (N crack ) and plastic deformation (ɛ c,pz ) versus axial strain. 12 of 16

13 [77] Figure 17 shows the stress path of the two uniaxial compaction test simulations. The horizontal stress is plotted against the axial stress in Figure 17. It is seen that the curves from the two simulations are almost parallel at lower stresses. However, at higher stresses, an obvious bend-over can be seen for the curve from the simulation that started at 10 MPa hydrostatic stress condition. [78] In Figure 18, the wave velocities in the axial direction are plotted against the axial stress. For the same axial stress, higher initial hydrostatic stress results in higher S-wave velocity, while such an effect is less significant for the P wave velocity. The reason is probably that the S-wave velocity depends on the shear stiffness, which appears to be sensitive to the stresses not only in the propagating direction but also in the polarizing direction. [79] In the model, this stress sensitivity is only caused by the nonlinearity of the force-displacement curve of the grain contacts (and possibly also a small increase in the coordination number) and is apparently small. For a rock that Figure 14. Simulation results of the uniaxial compaction test started from 5 MPa initial hydrostatic stress: (a) dynamic uniaxial compaction modulus (K dynamic ), static uniaxial compaction modulus (K static ) and axial stress (s z ) versus axial strain; (b) dynamic Poisson s ratio (n dynamic ) and static Poisson s ratio (n static ) versus axial strain; (c) coordination number (N coord ), crack number (N crack ) and plastic deformation (ɛ c,pz ) versus axial strain. while the dynamic uniaxial compaction modulus is calculated as H dynamic ¼ rv 2 p : ð23þ It was observed (as shown in Figures 14a and 14b) that the static Poisson ratio is smaller than the dynamic Poisson ratio. After the onset of microcracking (Figure 14c), the curve of the static Poisson ratio becomes bumpy, and drops slightly. This phenomenon can be seen even more clearly in Figures 15a 15c which show the simulation results of the uniaxial compaction test started from 10 MPa hydrostatic stress. This may be due to the local failure in a compactive mode, which reduces the lateral response of the model. Figure 16 shows the local volumetric strain calculated in a slice of the model, in the uniaxial compaction test started from 10 MPa hydrostatic stress condition. Figure 15. Simulation results of the uniaxial compaction test started from 10 MPa initial hydrostatic stress: (a) dynamic uniaxial compaction modulus (K dynamic ), static uniaxial compaction modulus (K static ) and axial stress (s z ) versus axial strain; (b) dynamic Poisson s ratio (n dynamic ) and static Poisson s ratio (n static ) versus axial strain; (c) coordination number (N coord ), crack number (N crack ) and plastic deformation (ɛ c,pz ) versus axial strain. 13 of 16

14 Figure 16. A plot of the volumetric strain in a slice (through the ZOX plane) in the model. Strain localized can be seen in this plot. contains some initial cracks that may close due to increasing confinement stress, the effect is likely to be larger. 6. Discussion [80] The modeled behavior of the static and dynamic bulk moduli (Figure 11) shows that there is a difference between the static and the dynamic moduli already at the start of the loading path. The ratio of the static to the dynamic modulus is however decreasing with increasing stress (although the difference is increasing). This is qualitatively in agreement with the observed behavior for weak sandstones [Fjær, 1999]. According to the empirically based analytical model of Fjær [1999, 2009], the parameter 1/P defined by Figure 18. Wave velocities versus axial stress derived from the uniaxial compaction simulations. should increase linearly with stress during hydrostatic loading. Figure 19 shows that this is also the case for the numerical simulation. Hence the simulated behavior reproduces closely the empirically based analytical model. This suggests that the physical model which forms the basis for the numerical simulations is relevant. [81] No microcracks were observed during the simulation of the hydrostatic test. Therefore, the difference between the static and the dynamic moduli appears to be caused by the plastic deformation at the grain contacts for this loading path. This plastic deformation is seen to occur already at the start of the loading path. Hence the rock is never truly elastic during uniform static loading, not even at the lowest stress levels. This shows that there may be large stress concentrations at the grain contacts, even if the global stress level is low. [82] Also for the static and the dynamic Young moduli the results of the numerical modeling (Figures 12 and 13) are qualitatively in agreement with laboratory observations P ¼ K dynamic K static 3K static K dynamic ð24þ Figure 17. Simulation results of the uniaxial compaction tests with 5 and 10 MPa initial hydrostatic stress: horizontal stress versus axial stress. Figure 19. 1/P (defined by equation (24)) versus mean stress for the simulated hydrostatic test shown in Figure 11. The dashed line is a linear trend line, in accordance with the analytical model of Fjær [2009]. 14 of 16