Coupled discrete element modeling of fluid injection into dense granular media

Authors


Abstract

[1] The coupled displacement process of fluid injection into a dense granular medium is investigated numerically using a discrete element method (DEM) code PFC2D® coupled with a pore network fluid flow scheme. How a dense granular medium behaves in response to fluid injection is a subject of fundamental and applied research interests to better understand subsurface processes such as fluid or gas migration and formation of intrusive features as well as engineering applications such as hydraulic fracturing and geological storage in unconsolidated formations. The numerical analysis is performed with DEM executing the mechanical calculation and the network model solving the Hagen-Poiseuille equation between the pore spaces enclosed by chains of particles and contacts. Hydromechanical coupling is realized by data exchanging at predetermined time steps. The numerical results show that increase in the injection rate and the invading fluid viscosity and decrease in the modulus and permeability of the medium result in fluid flow behaviors displaying a transition from infiltration-governed to infiltration-limited and the granular medium responses evolving from that of a rigid porous medium to localized failure leading to the development of preferential paths. The transition in the fluid flow and granular medium behaviors is governed by the ratio between the characteristic times associated with fluid injection and hydromechanical coupling. The peak pressures at large injection rates when fluid leakoff is limited compare well with those from the injection experiments in triaxial cells in the literature. The numerical analysis also reveals intriguing tip kinematics field for the growth of a fluid channel, which may shed light on the occurrence of the apical inverted-conical features in sandstone and magma intrusion in unconsolidated formations.

1 Introduction

[2] How a dense granular medium behaves in response to fluid injection is a question of both fundamental and applied research interests to better understand the geological and engineering processes that resulted from fluid overpressurization in unconsolidated sedimentary formations. In sedimentary formations, fluid overpressure may develop from mechanisms such as disequilibrium compaction, tectonic compression, aquathermal expansion or fluid flow, and buoyancy [Osborne and Swarbrick, 1997]. Elevated pore pressure, combined with the buoyancy effect in some cases, provides the driving force that may cause fluid or gas to migrate through the porous media. Manifestations of such diffusive flow include, for example, distributed low-flux hydrate deposits [Tréhu et al., 2006] and seeps of products such as brine, hydrocarbon, and fluid mud on the sea floor [Roberts et al., 1990; Moore and Vrolijk, 1992; Bohrmann et al., 2002]. If triggered by events such as earthquake, submarine landslide, or rapid fluid migration [Truswell, 1972; Jonk, 2010; Hurst et al., 2011], fluid overpressurization may also create localized fluid flow channels associated with formation of geological features such as focused high-flux gas hydrate systems [Tréhu et al., 2006], mud volcanoes [Kopf, 2002; Dimitrov, 2002; Manga and Brodsky, 2006] and sand injectites [e.g., Jolly and Lonergan, 2002; Hurst et al., 2003; Huuse et al., 2010; Hurst et al., 2011]. In general, intrusive features formed from subsurface remobilization and injection, e.g., sand injectites, originate beneath a sealing strata and are observed mostly in mudstones and consolidated sandstones [Hurst et al., 2011]. Nevertheless, observations of sand injectites [Chi et al., 2007, 2012] and also magma intrusion in unconsolidated sandstone formations [Baer, 1991; Summer and Ayalon, 1995; Moreau et al., 2012] have also been reported.

[3] In engineering practices, subsurface fluid injection has been widely employed for applications such as grouting for ground improvement to reduce the liquefaction potential of cohesionless soils, to raise the ground elevation, or to compensate the volume loss due to ground surface settlement [Mitchell and Katti, 1981; Au et al., 2003; Woodward, 2005; Germanovich and Murdoch, 2010]; construction of permeable reactive barriers for soil remediation [Hocking, 1996]; injection of carbon dioxide for geological storage [Bachu, 2000; Hovorka et al., 2004; Lucier et al., 2006] or for enhanced oil or coalbed methane recovery [Orr Jr. and Taber, 1984; Blunt et al., 1993; White et al., 2005]; subsurface disposal of liquid or slurrified solid waste such as drill cuttings [Moschovidis et al., 1998; Schmidt et al., 1999; Keck, 2002; Clark et al., 2005; Guo et al., 2007; Tsang et al., 2008]; and hydraulic fracturing and waterflooding for hydrocarbon recovery [Ayoub et al., 1992; Morales and Marcinew, 1993; Economides and Nolte, 2000; Hustedt et al., 2008; Khodaverdian et al., 2010]. Although the engineering objectives vary in this list of applications, they share a common operation procedure in that clean fluid and/or slurry is injected into the subsurface via a circular wellbore over a certain interval. Fluid may be expected to permeate through the formation only or to create openings in the subsurface. For example, the objective of hydraulic fracturing is to create a localized path to increase the reservoir conductivity and to bypass the near wellbore damage. Meanwhile, in waterflooding, growth of localized features is to be avoided in general in order to maintain the reservoir pressure. Nevertheless, water injection-induced fracturing may have occurred in majority of the performed waterfloods [Hustedt et al., 2005]. The sheer volume of injection in some applications is often enormous. In an operation between 1995 and 1997 in the North Slope of Alaska, eight million barrels of slurried drill cuttings were injected into one single well [Schmidt et al., 1999].

[4] The critical questions that need to be addressed in both the geological systems and the engineering practices are therefore as follows: (1) what are the underlying mechanisms and conditions that govern the growth of the openings or intrusions; and (2) how are the opening geometry and the extent of fluid transport related to the formation and fluid characteristics as well as the fluid overpressure at the source. Better understandings of these fundamental questions are crucial to providing constraints for the geological systems and to improving design and diagnosis for the engineering applications.

[5] Mechanical failure in the form of initiation and propagation of opening mode fluid-driven fractures, i.e., hydraulic fracturing, has been well established as the primary mechanism for growth of intrusive features in cohesive host rocks, and the geometry of the intrusions in terms of the width over length ratio can be described using mathematical models based on Linear Elastic Fracture Mechanics (LEFM) [Pollard, 1973; Rubin, 1995]. LEFM-based models have also been shown to be appropriate for describing hydraulic fracturing in cohesive soils [Murdoch, 1993a] and the first order effect in gas bubble growth in fine grain soft sediments [Nunn and Meulbroek, 2002; Johnson et al., 2002; Barry et al., 2010]. Routine design and diagnosis of hydraulic fracturing for reservoir stimulation and reinjection of drill cuttings also rely on LEFM-based simulators [Mack and Warpinski, 2000; Adachi et al., 2007].

[6] However, questions remain on whether the mechanism of tensile fracturing is applicable in the unconsolidated formations with weak or no cementation and high permeability [Keck, 2002; Smith et al., 2004]. Using the concept of energy release rate, the condition of fracture initiation and propagation, central to the fracture mechanics theory, can be stated as the fracture extends if the energy release rate reaches a critical value [Irwin, 1957]. For a cohesionless granular material, if the capillary effect at the tip of an opening is negligible, the concept of energy associated with creating a new surface area and consequently the condition for an opening to initiate and propagate, cannot be clearly defined.

[7] In this work, we examine the injection process in an initially dry dense granular medium by numerically modeling fluid injection from a circular wellbore using the discrete element method (DEM) [Cundall and Strack, 1979] coupled with a pore network fluid flow scheme [Fatt, 1956a, 1956b, 1956c; Blunt, 2001]. In particular, the effects of the injection rate, invading fluid viscosity, and the granular medium properties, such as the modulus and the permeability on the transitions from permeation flow to the development of localized fluid flow channels, are investigated. The discrete element method (DEM) is a natural tool to model the mechanical response of a granular medium. A particle of an idealized shape, e.g., disk, sphere, or polygon, may be viewed as an individual grain. The macroscale mechanical response emerges from the interaction of particles through their contacts at the microscale (i.e., the particle scale). Meanwhile, the pore network model is capable of resolving fluid flow in between grains. Both permeation flow and the development of localized fluid channels can be modeled as a result of hydromechanical coupling between the pore pressure and the grain displacement fields without having to resort to formulating a growth criterion at the macroscale a priori.

2 Background and Experimental Motivation

[8] Over the past two decades, extensive efforts have been devoted to formulate rigorous solutions for fluid-driven fracture in a linear elastic medium. In addition to the classical square root singularity, e.g., inline image (p - crack tip pressure and f- crack radius or half length) for the toughness-dominated propagation regime [Rice, 1968], depending on the primary energy dissipation mechanisms, a variety of propagation regimes have been identified, e.g., the viscosity-dominated and leakoff-dominated regimes [e.g., Desroches et al., 1994; Savitski and Detournay, 2002; Detournay, 2004; Bunger et al., 2005; Adachi and Detournay, 2008; Kovalyshen and Detournay, 2010; Garagash et al., 2011]. The singularity in the asymptotic solution associated with each regime varies. In the viscosity-dominated regime or the zero toughness regime [Spence and Sharp, 1985; Lister, 1990], the singularity is weaker and follows inline image [Desroches et al., 1994].

[9] Both theoretical and experimental efforts have also been made previously to investigate the injection process in an unconsolidated medium. The unconsolidated nature of the material means that if an opening is created, the tip of the opening cannot sustain large stress contrast, and plastic yielding is expected. Influence of plasticity in hydraulic fracturing has been examined for both cohesive and cohesionless materials [Papanastasiou and Thiercelin, 1993; van Dam et al., 2002; Abou-Sayed et al., 2004; Papanastasiou and Atkinson, 2006; Wu, 2006]. In general, one-dimensional leakoff model [Carter, 1957] is assumed for fluid flow from the fracture faces to the matrix in conventional LEFM-based models. To account for the large permeability typical of the unconsolidated formations, multi-dimensional fluid leakoff models have also been introduced [van den Hoek, 2002; Entov et al., 2007; Mathias and van Reeuwijk, 2009]. Nevertheless, due to the highly nonlinear constitutive behaviors and the strong coupling between mechanical deformation and fluid flow, theoretical analysis of the injection process in the unconsolidated formations remains a great challenge.

[10] Laboratory experiments of fluid injection under triaxial conditions have been conducted in sand and in mixtures of sand and silica flour [Khodaverdian and McElfresh, 2000; Chang, 2004; Bohloli and de Pater, 2006; Bezuijen et al., 2006; Dong and de Pater, 2008; Zhou et al., 2010; Golovin et al., 2010, 2011; Germanovich et al., 2012; Hurt and Germanovich, 2012; Ispas et al., 2012] and in clay [Murdoch, 1993b, 1993c; Bolton et al., 1994; Au et al., 2003; Soga et al., 2004]. While the experiments in clay have mostly produced opening mode tensile fractures, the phenomena in sand are much more complex due to the coupling of fluid infiltration and grain displacements. The morphology of localized features ranges from brittle fracture-like to branched or finger-like. The results are sensitive to most of the geometrical and physical parameters involved in the experiments, among them, the injection rate, invading fluid rheology, leakoff characteristics of the fluid, particle size and distribution, void ratio/packing density, confining stress, and degree of saturation. For experiments with openings resembling brittle fractures and limited leakoff, it is shown [Hurt and Germanovich, 2012] that the pressure decline can be curve-fitted using the solution for the toughness-dominated regime with leakoff [Bunger et al., 2005] using the toughness as a fitting parameter. For the waterflooding experiments [Golovin et al., 2010], a statistical fracture mechanics based model is proposed to predict the probability of initiating a fracture of a given length and orientation [Jasarevic et al., 2010].

[11] Injection experiments are also common analogue models to study magma and sand intrusions [Galland et al., 2007; Mathieu et al., 2008; Rodrigues et al., 2009; Galland et al., 2009; Ross et al., 2011; Mourgues et al., 2012; Abdelmalak et al., 2012] and to investigate buoyant rise of gas bubbles in sediments [Barry et al., 2010; Fauria and Rempel, 2011]. Similar experiments are also performed to model venting dynamics [Mörz et al., 2007; Nermoen et al., 2010; Gay et al., 2012]. In some experiments with fine grain silica flour, the cohesiveness and the relatively low permeability result in the pressure histories showing characteristics of the toughness-dominated behaviors with the square root singularity [e.g., Galland et al., 2007, 2009].

[12] In addition to considering fracturing as the growth mechanism, the injection process in unconsolidated media can also be viewed as a coupled fluid-grain displacement process. Viscous fingering, the instability phenomenon where a more viscous fluid is displaced by a less viscous immiscible fluid in a porous medium or in a Hele-Shaw cell [Saffman and Taylor, 1958], is suggested as a mechanism involved in magma intrusion in the unconsolidated sandstone [Baer, 1991]. Observation of granular fingers, i.e., the conduits created after grains have been displaced by air flow, is first reported in air injection into dry zirconia powder in a Hele-Shaw cell [van Damme et al., 1993]. In recent years, the displacement mechanisms involving both fluid and granular displacements have been investigated experimentally utilizing both radial and linear Hele-Shaw cells filled with spherical glass beads [Johnsen et al., 2006; Cheng et al., 2008; Johnsen et al., 2008a, 2008b; Chevalier et al., 2009; Sandnes et al., 2011; Holtzman et al., 2012]. The displacement mechanisms for two immiscible fluids in rigid porous media can be classified into capillary fingering, viscous fingering, and stable displacement (permeation flow), depending on the capillary number and the mobility contrast [Lenormand, 1989]. When fluid-grain displacement is also involved in the system, transitions from the fluid-fluid displacement regimes for rigid porous media to growth of localized fluid channels occur. In dense granular media, when the applied overpressure or the injection rate increases, a transition from permeation flow to the growth of granular fingers [Johnsen et al., 2006, 2008a, 2008b; Chevalier et al., 2009] and a transition from capillary fingering to capillary fracturing have been observed [Sandnes et al., 2011; Holtzman et al., 2012].

[13] With aqueous glycerin solutions as the invading fluid, the injection experiments in dense dry Ottawa F110 sand (grain fraction around 0.65) [Huang et al., 2012a, 2012b] show that, when the capillary effect is negligible, depending on the interplay between fluid infiltration and grain displacements, the coupled displacement regimes beyond permeation flow can be further divided into an infiltration-dominated regime, a grain displacement-dominated regime, and a viscous fingering-dominated regime. Images representative of these regimes are shown in Figure 1. When the injection velocity v(v=Q/πbDi- the injection rate Q scaled by the cell gap size b and the injection inlet diameter Di) and the invading fluid viscosity ηare relatively small, the infiltration fronts remain nearly circular. Growth of the fluid channels or fingers is either invisible (Figure 1a) or not affecting the infiltration pattern (Figure 1b). As the injection velocity and the invading fluid viscosity increase, the behaviors of fluid flow change from infiltration-governed to infiltration-limited and the granular medium response displays a transition from that of a rigid porous medium to fluid-like behaviors characterized by ramified fluid-grain interface morphology (Figure 1d), reminiscent of pattern formation in viscous fingering in fluid-fluid displacement. Classification of these displacement regimes shares similarities with that of the propagation regimes for hydraulic fracturing in competent rocks [Detournay, 2004; Adachi and Detournay, 2008].

Figure 1.

Four fluid-grain displacement regimes for injection of aqueous glycerin solutions into an initially dry dense granular medium (Ottawa F110 sand) in a Hele-Shaw cell when the capillary effect is negligible: (a) Test A1: the simple radial flow regime (v=8.22 mm/s, η=5 cp), (b) Test A2: the infiltration-dominated regime (v=82.9 mm/s, η=5 cp), (c) Test B3: the grain displacement-dominated regime (v=165.80 mm/s, η=176 cp), and (d) Test C4: the viscous fingering-dominated regime (v=829.52 mm/s, η=942 cp). The black areas are the clean fluid channels; the dark brown areas are the fluid infiltrated areas, and the light brown areas are occupied by dry sand only. Fluid is injected from the center of the bottom plate. [Huang et al., 2012b]

[14] Two dimensionless times are defined to classify the four displacement regimes [Huang et al., 2012b], i.e.,

display math(1)

where is a characteristic length of the injection process; E and k are the small strain Young's modulus and the intrinsic permeability; and η is the apparent viscosity of the granular mixture, assuming the medium can be treated as a nonlinear viscoelastoplastic solid. The two dimensionless times are obtained from the ratios of characteristic times associated with the injection process, hydromechanical coupling and the retardation time of the granular medium. While the dimensionless time τ1determines whether the granular medium response is infiltration-governed or deformation-governed, the time τ2characterizes the viscoelastic response of the granular medium. As τ2 increases, the material behaviors evolve from solid-like to becoming fluid-like. For a dense granular medium, it is expected that τ1<τ2. At large τ2, it is the fluid-like response of the granular medium that results in the viscous fingering type of morphology in Figure 1d.

[15] Take the injection inlet diameter Di as the characteristic process length and the small strain modulus E=66.2 MPa and permeability k=749 mD as the index properties for the dense packing of dry Ottawa F110 sand. The ratio of the area of the fingers (the black area in Figure 1) over the area enclosed by the infiltration front (the brown and the black areas in Figure 1) at late time of injection is used as an index to indirectly measure the injection efficiency in creating open space. Variation of this area ratio λ as a function of the dimensionless time τ1 is shown in Figure 2 from a total of 58 tests. The area ratio λ is close to zero when τ1≲6×10−3 and increases nearly linearly with τ1 when τ1≲0.1. When 0.1≲τ1≲3, the increase in λ becomes nonlinear. When inline image, the area ratio reaches a plateau around λ≃0.45. For this granular medium system, τ1≃6×10−3 may be used to define the transition from simple radial flow to the infiltration-dominated regime; τ1≃0.1 defines the transition from the infiltration-dominated regime to the grain displacement-dominated regime; and τ1≃3 (or inline image, if the apparent viscosity η is known) corresponds to the transition from the grain displacement-dominated regime to the viscous fingering-dominated regime.

Figure 2.

Variation of the area ratio λ as a function of the dimensionless time τ1 from a total of 58 tests; the marker shape corresponds to the Hele-Shaw cell gap size, b1=0.787 mm, b2=1.575 mm, b3=2.362 mm, and b4=3.175 mm; the marker color indicates the fluid type: yellow, corresponding to the 50% c.w. aqueous glycerin solution (η=5 cp), green, 90% (η=176 cp), and red, 100% (η=942 cp); see linear scale plot in the inset [Huang et al., 2012b].

[16] The experimental evidence summarized here serves as the guide for our numerical analysis to investigate the effects of the injection rate, invading fluid viscosity, elastic modulus, and permeability of the medium on the transitions in the granular medium response and fluid flow behaviors in the injection process. In general, since the material properties such as the modulus and permeability are intertwined for a given granular medium, it is unrealistic to isolate the effects of the properties in an experimental analysis. Numerical analysis with its capability of prescribing the properties independently, therefore, has a unique advantage in this regard.

[17] In addition to the time scale argument, emergence of the displacement or failure regimes can also be associated with energy partition in the system. Examples embodying such a concept include, for example, flow in porous media [Homsy, 1987; Lenormand, 1989; Sahimi, 2011], hydraulic fracturing [Savitski and Detournay, 2002; Detournay, 2004; Adachi and Detournay, 2008] and flow of concentrated granular mixtures [Coussot, 2002]. In the injection process with negligible capillary effect, the energy dissipation mechanisms involved in the displacement regimes beyond permeation flow include viscous dissipation for flow through pore spaces, dissipation associated with grain displacements and viscous dissipation for flow through the fluid channels or fingers [Huang et al., 2012b]. The association of the energy dissipation mechanisms with the displacement patterns is also examined in this analysis.

3 Numerical Methodology

[18] The DEM code PFC2D® is employed in this analysis. PFC2D® solves Newton's second law of motion and force-displacement laws at the contact using an explicit finite difference method [Itasca Consulting Group, Inc., 2008]. DEM coupled with the pore network model has been applied to investigate hydromechanically coupled problems, e.g., two hydraulic fractures interaction in a cohesionless medium [Thallak et al., 1991], stress-dependent permeability of sandstones [Bruno, 1994; Li and Holt, 2001], hydraulic fracturing induced microseismicity [Al-Busaidi et al., 2005; Shimizu et al., 2011; Zhao and Young, 2011] and the transition from capillary fingering to capillary fracturing in gas invasion [Jain and Juanes, 2009]. Compared with other DEM-based hydromechanical coupling schemes, e.g., the DEM-lattice Boltzmann method [Cook et al., 2004; Boutt et al., 2007; Han and Cundall, 2012] and the DEM coupled with a fixed coarse grid fluid flow scheme [Tsuji et al., 1993; Shimizu, 2006], the DEM-pore network coupling scheme is capable of resolving fluid flow at the particle scale and has the advantage of simplicity in the formulation.

[19] A prerequisite for numerical analysis with the discrete element method is to establish the scaling relationships between the microscale parameters and the phenomenological description of the materials at the macroscale. Effective mechanical properties of disordered granular media have been studied both analytically and numerically [Duffy and Mindlin, 1957; Chang and Misra, 1990; Bathurst and Rothenburg, 1992; Nemat-Nasser and Hori, 1999; Luding et al., 2001; Qu and Cherkaoui, 2007]. Empirical correlations for permeability of a packed bed, e.g., the Kozeny-Carman correlation, have also been well established [Bear, 1972]. In this section, the contact model, the fluid flow model, and the hydromechanical coupling scheme are introduced. Dependence of the elastic properties on the contact parameters and the permeability on the aperture model prescribed for the flow paths at the particle scale is discussed. The results are then applied in the simulations of the injection process in section 4.

3.1 Contact Model and Effective Mechanical Properties

[20] An elastic and frictional micromechanical contact model is assumed in this study. The contact, which occurs at a point between a pair of particles, can be characterized by the normal and shear stiffnesses, Knc and Ksc, and the friction coefficient μ. Assuming compression positive, the contact model can be expressed as [Itasca Consulting Group, Inc., 2008],

display math(2)

and

display math(3)

where Fn and Fs denote the normal and shear contact forces, respectively; δn is the overlap (δn<0 indicates a gap at the contact) and δs is the slip between the pair of particles. In PFC2D®, the particles are connected in series at the contact, the contact stiffnesses are determined from the particle stiffnesses Kn and Ks. If constant particle stiffnesses Kn and Ks are assigned for all the particles, Knc=Kn/2 and Ksc=Ks/2. For a given particle assembly, if the mean particle radius inline image is much smaller than the characteristic length L of the domain, i.e., inline image, based on dimensional analysis, the effective elastic modulus E and the Poisson's ratio ν of the particle assembly under quasi-static condition can be expressed as [Huang et al., 1999],

display math(4)

where fEand fνare the scaling functions for the elastic modulus and the Poisson's ratio, respectively, and φ is the porosity of the particle assembly corresponding to a reference initial isotropic stress. For a two-dimensional particle assembly, these elastic constants may be interpreted as plane strain properties. Given a particle size distribution, a particle generation scheme, and an initial reference stress state, the porosity φ is more or less fixed. The Poisson's ratio of the particle assembly therefore depends on the stiffness ratio Ks/Kn only. The elastic modulus can be varied independent of the Poisson's ratio by changing the particle stiffnesses while maintaining the stiffness ratio Kn/Ksconstant.

[21] Biaxial compression tests are conducted at various confining stress levels on a rectangular particle assembly of size 50×100 mm. The densely packed specimen is generated using the radii expansion algorithm [Potyondy and Cundall, 2004]. The particles have a uniform radius distribution ranging from r=0.5 to 0.7 mm. The total number of the particles is N=3890. The particle density is ρ=2650 kg/m3 and the contact parameters are as follows: Kn=Ks=83.3 MN/m2 and μ=0.577, corresponding to an intergranular friction angle ϕ0=30°. (Note that the small strain modulus E as a function of Ks/Kn at φ≃17% can be fitted by y=(0.099+1.441x)/(1+2.909x) and the Poisson's ratio ν by y=(0.757−0.248x)/(1+3.745x)). Unless otherwise mentioned, the particle properties specified here are the default values for the simulations in this study. It should be noted that no attempt is made here to match the macroscale properties of the particle assembly to those of the Ottawa F110 sand. Since the stress-strain response of the dense particle assembly is nonlinear, the elastic constants at 50% of the peak stress level are obtained as the index properties. The secant modulus is E 50′≃27.34 MPa and the apparent Poisson's ratio is ν50′≃0.29 at a confining stress σ3=0.1 MPa. Assuming a Mohr-Coulomb failure criterion, i.e., σ1=(1+ sinϕ)/(1− sinϕ)σ3, where σ1 is the axial stress at the peak, the apparent peak friction angle is obtained as ϕ≃29°, slightly smaller than the intergranular friction angle specified at the contact. As the confining stress σ3 increases, the porosity φ at the start of deviatoric loading decreases (note that the biaxial compression test is conducted by first compressing the sample to reach an isotropic stress state and then increasing the deviatoric stress by loading in the axial direction only). Consequently, the modulus E50′ increases while the apparent Poisson's ratio ν50′and the peak friction angle ϕ decreases, see Figures 3 and 4. As the confining stress σ3 increases to 1 MPa, the modulus increases to E50′=39.165 MPa and the Poisson's ratio ν 50′and the peak friction angle ϕ decrease to ν50′=0.175 and ϕ=24.5°, respectively.

Figure 3.

Elastic constants of the particle assembly as functions of the confining stress σ3; the lines are the quadratic fits of the numerically obtained data in dots; E50′/Kny=−0.103x2+0.270x+0.308, ν50′y=0.100x2−0.231x+0.305.

Figure 4.

Peak friction angle of the particle assembly as functions of the confining stress σ3; the line is the quadratic fit of the numerically obtained data in dots; ϕ:y=6.724x2−11.973x+30.038.

3.2 Fluid Flow Model and Hydromechanical Coupling

[22] After a compact particle assembly is generated, the pore network can be established by identifying the domains formed from closed chains of particles as the pore spaces (through connecting center lines of particles in contact) [Itasca Consulting Group, Inc., 2008], see Figure 5a. The number of flow paths is equal to the number of contacts in the domain. Fluid flow between two adjacent pore spaces or domains takes place through the flow path or pore throat at the contact. The Hagen-Poiseuille equation is employed to describe fluid flow through the flow path,

display math(5)

where qp is the flow rate; η is the fluid viscosity; p2p1 is the pressure difference between the two pores; Lp=r1+r2 is the length of the flow path in between two particles of radii r1 and r2; and ais the aperture width.

Figure 5.

Schematics showing (a) pore spaces formed from close chains of particles and (b) the drag force Ffluid as a resultant from the pore pressure.

[23] The solution scheme of the fluid flow calculation is explicit. After a fluid time step Δtf, the volume change ΔVp for each pore can be calculated by summing up the change in the pipe flow volume around this pore. Assuming influx to be positive, the pressure increment Δp at each pore can be updated according to,

display math(6)

where N is the number of flow paths connected to the pore space; Kf is the fluid bulk modulus and Vp is the current pore volume. As expressed in (6), the pore volume change that resulted from mechanical deformation is not considered here in order to save computation time. Such an approximation is reasonable for modeling of fluid injection in an initially dry medium.

[24] Based on the criterion that the pressure increment must be smaller than the original pressure perturbation, (5)(6) yield a local critical time step for the fluid calculation,

display math(7)

A global time step can therefore be obtained from the minimum of the local time steps and a safety factor [Itasca Consulting Group, Inc., 2008].

[25] The hydromechanical coupling is realized by data exchanging at predetermined time steps. For each particle, a resultant drag force Ffluid, obtained from summing up the pore pressure over the particle surface, see Figure 5b, is passed from the fluid calculation to the mechanical calculation. The resultant drag force Ffluid is then applied to each particle in addition to the unbalanced force that resulted from the mechanical contact forces. By solving Newton's second law of motion, a new particle position can be determined. The configuration of the pore structure can be updated accordingly. The hydromechanical coupling is reflected in the change in the aperture a due to mechanical deformation and the addition of the drag forces to the particles. However, since the grain displacement induced pore volume change does not affect the pore pressure, the coupling is one way in the sense of Biot [Biot, 1941].

3.3 Permeability Calibration

[26] The aperture width a may be defined as a function of local particle geometry and deformation [Li and Holt, 2001], i.e.,

display math(8)

where a0 is the zero-force aperture when δn=0; δ0=F0/Knc is the overlap corresponding to the normal contact force F0 when the aperture a=a0/2; and γ is a multiplier. The zero-force aperture a0 is introduced here to remove the paradox in simulating fluid flow in 2-D, so that fluid flow can occur when there is an overlap at the contact. An alternative approach of assuming a cubic packing in the third dimension is used in Jain and Juanes [2009]. The zero-force aperture a0 and the overlap δ0 can be related to the mean particle size, inline image, through

display math(9)

[27] For the fluid flow model described above, permeability of the particle assembly depends on the assembly configuration and the aperture width a of the flow paths. With properly chosen coefficients α, β, and γ, the aperture model in (8) can capture the dependence of permeability on the local porosity as expressed in empirical correlations, e.g., the Kozeny-Carman correlation [Bear, 1972],

display math(10)

where dp is the average particle diameter. It should be noted that previously in studies dealing with relatively low permeability rocks [e.g., Li and Holt, 2001; Al-Busaidi et al., 2005; Shimizu et al., 2011], dependence of the permeability on the assembly porosity is not considered.

[28] To calibrate the permeability against the Kozeny-Carman correlation, a baseline model of length L and width W(L=W=75 mm) is generated with an initial confining stress σ0=0.1 MPa. The particle assembly consists of 4376 particles. Initially, all the pores are fully saturated, but have zero fluid pressure. A fixed constant fluid pressure p0=5 kPa is applied uniformly at the inlet cross section (perpendicular to the length direction), while a zero fluid pressure is maintained at the outlet cross section. The side boundaries, parallel to the length direction, are assumed to be impervious. All the particle positions are fixed in order to eliminate the variation in the pore structure during the simulations.

[29] After the steady state condition is reached, permeability k of the particle assembly can therefore be determined by applying Darcy's law with the mean velocity obtained from averaging the flux over the width of the domain. Since the porosity in (10) is a 3-D porosity, in order to compare the numerical results with the analytical prediction, the 2-D porosity of the particle assembly needs to be converted to a 3-D porosity. Linear interpolation between the porosity at the maximum (hexagonal) and minimum (square or cubic) packing density states for mono-sized packing with disks (2-D) and spheres (3-D) is therefore utilized, i.e.,

display math(11)

The numerical values in (11) are associated with the states of maximum or minimum packing density in 2-D and 3-D, respectively [Borchardt-Ott, 2011].

[30] A simple approach is developed here to determine the calibration constants α, β, and γ. It can be seen from (8) that the coefficients β and γ affect the permeability when the particles overlap or have a gap, respectively. When the particles are just in contact, the most critical parameter is the coefficient α or the zero-force aperture a0. The permeability of the particle assembly can therefore be calibrated by first determining the coefficient α using the baseline particle configuration generated at a relatively low initial confining stress σ0=0.1 MPa, i.e., the condition when both βand γdo not have significant effects. The corresponding porosity of the baseline configuration is φ2−D=0.1646. The coefficient β is then calibrated by matching the permeability when the baseline configuration is further compressed. The coefficient γis lastly determined using particle configurations with substantial percentage of contacts having gaps.

[31] Tests A and B with α=0.45 and 0.55 are first performed with the baseline configuration. Although the contact model defined in (2) and (3) does not sustain a tensile force, when two particles are in close proximity, they are considered in virtual contact with zero contact force. In the baseline configuration, 2975 out of 11,009 total contacts have zero contact forces and 8034 (73%) contacts are in compression. The parameters βand γ are set to β=40,000 and γ=1 (see Table 1). As shown in Figure 6, the results from Tests A and B bracket the prediction from the Kozeny-Carman correlation. We therefore choose α=0.5 and attempt in the second step to determine the coefficient β by matching the permeability as a function of the initial confining stress σ0.

Table 1. Parameters in Tests A and B and Series 1−6 for Permeability Calibration
 Test ATest BSeries 123456
α0.450.550.50.50.50.50.50.5
β40,00040,0000.2720.2400.3200.2720.2720.272
γ1.01.01.21.01.01.41.21.0
Figure 6.

Comparison between the numerical results obtained from Tests A and B and Series 1−3 and the prediction from the Kozeny-Carman correlation; the initial confining stress σ0 refers to the stress level before the fluid flow calculation.

[32] Two series of tests, Series 2 and 3, are performed with β=0.24 and β=0.32, α=0.5, and γ=1. In these series, two new particle configurations are constructed by compressing the baseline sample to initial stress levels σ0=0.5 and 1 MPa. The number of contacts increases with the confining stress. The pore network configuration is therefore refined. At σ0=1 MPa, the porosity is now φ2−D=0.1263 and 9514 out of 11,299 total contacts, i.e., 84%, are in compression. The numerical results from Series 2 and 3 agree well with the prediction from the Kozeny-Carman correlation. The discrepancies increase with the confining stress level. At σ0=1 MPa, the difference between the numerical results and the prediction are 6.9% and 13.7%for Series 2 and 3, respectively. Therefore, [0.24, 0.31] is a reasonable range for coefficient β.

[33] In order to determine the last parameter γ, three additional particle configurations are generated by uniformly stretching the particle assembly initially at the confining stress σ0=0.5 MPa. Large normal and shear bond strengths are assigned to all the contacts so that the total number of contacts remains more or less the same during the stretch. The effect of bond strengths is to maintain the contact between particles when the contact force becomes tensile. Porosity of the particle assembly increases from the initial value of φ2−D=0.146 at σ0=0.5 MPa to φ2−D=0.148, 0.152 and 0.1646. The case with φ2−D=0.1646 has the same porosity as the baseline configuration. However, the contact structures in these two cases with φ2−D=0.1646 are rather different. In the stretched sample, 7500 out of 11,075 total contacts now have gaps, 67.7%compared to 27%in the baseline configuration.

[34] The results from Series 4−6 with α=0.5 and β=0.272, and γ=1, 1.2 and 1.4 are shown in Figure 7. The discrepancy in the numerical results for the three values of γincreases only slightly as the porosity increases.

Figure 7.

Comparison between the numerical results obtained from Series 4−6 and the prediction from the Kozeny-Carman correlation (φ is the 2-D porosity).

[35] The calibration coefficients are therefore chosen to be α=0.5, β=0.272, and γ=1.2. Permeability for the particle configurations at various initial stresses ranging from 0.1 MPa to 1MPa is recalculated with this set of parameters. Comparison between the numerical results and the Kozeny-Carman correlation is shown in Figure 6. The difference is less than 2.6%.

[36] It should be noted that this procedure to determine the calibration constants is indeed arbitrary to a certain degree. It nevertheless reflects the fact that the three coefficients play critical roles at different conditions. Although in principle it is possible to determine the three coefficients through inverse analysis, the effort is likely exhaustive. This three-step approach is a simple and straightforward alternative and requires much less computational effort. In addition, although only limited ranges of porosity are considered in calibrating the permeability, local porosity in the injection simulations is not expected to be greatly different. For the purpose of modeling the injection process in this work, the calibration procedure described above is considered adequate.

4 Fluid Injection Simulations

4.1 Model Setup

[37] Fluid injection from a circular wellbore into an initially dry granular medium is modeled here. The capillary effect is assumed to be negligible. The injection simulations are performed using a hollow circular domain with an outer diameter Do=160 mm and an inner diameter Di=8 mm. The particle configuration at the wellbore vicinity with the pore network added on and the overview of the numerical domain are shown in Figure 8. The assembly consists of 15,605 particles with radii ranging from 0.5 to 0.7 mm (inline image and inline image). The default microscale parameters specified for the particles in the injection tests are: Kn=Ks=83.3 MN/m2, μ=0.577 and ρ=2650kg/m3. Two frictionless circular walls are placed at the inner and outer boundaries. The wall stiffnesses are set to be the same as those of the particles. The assembly is generated with an initial radial confining stress σ0=0.5 MPa. At this level of confinement, the effective material properties are as follows: E50′=34.13 MPa, v50′=0.222, ϕ=25.7°, and k=0.849×10−9m2.

Figure 8.

(a) Overview of the simulation domain; (b) wellbore vicinity with the pore network added on.

[38] Pore pressures are applied as the boundary conditions in the fluid flow calculation. However, a constant injection rate condition is implemented by considering the wellbore as one pore space with the wellbore pressure change Δpw for a given fluid time step determined from

display math(12)

where Vw is the volume of the pore space that represents the wellbore and ΔVw′is the volume increment due to wellbore expansion. If any localized fluid channel develops during the simulation, the opening is merged with the wellbore pore space. However, constant pressure is assumed for the wellbore and any preferential path connected to the wellbore. In other words, no fluid flow is simulated along the preferential paths. A realistic value of Kf=2 GPa is assigned as the fluid bulk modulus for the wellbore. The wellbore volume and the pressure are updated at every five mechanical steps during the mechanical calculation.

[39] Previous DEM coupled analyses in the literature [e.g., Li and Holt, 2001; Al-Busaidi et al., 2005; Shimizu et al., 2011] have mostly dealt with fully saturated conditions and bonded particles, where the solid skeleton is relatively stiff and the pore network structure does not experience significant change in the configuration during the simulations. In this work, the pore network is updated at every mechanical time step to account for the large grain displacements. A pore space can be split into two if a new contact is created in the fluid domain. Two pore spaces can also merge to form a new one if a contact in the fluid domain is deleted. Since we intend to simulate fluid injection into an initially dry medium, the fluid infiltration calculation is implemented in the coupling scheme before the fluid flow calculation of pressure diffusion at each fluid time step. Degree of saturation s as well as the pore pressure are stored at each pore space, s=0 for a dry pore space and s=1 for a fully saturated one. The pore pressure is zero if s<1. Assuming the invading fluid is nearly incompressible, the degree of saturation is only adjusted for the unsaturated pores that have neighboring fully saturated pores at each fluid time step.

[40] As can be seen from (7), the fluid bulk modulus is inversely proportional to the critical fluid time step. The time step becomes rather small if a realistic fluid bulk modulus, Kf=2 GPa for water, is chosen for the fluid calculation. Recognizing that the problem deals with partially saturated fluid flow, we can therefore reduce the fluid bulk modulus to speed up the calculation. A lower value of Kf=750 kPa is chosen as the fluid modulus for the pore spaces excluding the wellbore. Since the calculation in the pore filling step involves only mass conservation or continuity, the fluid bulk modulus may be further reduced and Kf=1 kPa is set for this step.

4.2 Effect of the Injection Rate

[41] Effect of the injection rate on the grain displacement and fluid flow patterns is first examined. A series of eight tests are carried out by increasing the injection flow rate from Q=0.01 to 0.16 m2/s, see Table 2 for the test number and parameters for each test. The corresponding range of the normalized injection velocity is v=Q/πDi=0.398−6.366 m/s. To maintain numerical stability, the fluid time step is decreased as the injection rate increases. Fluid viscosity η=1 Pa s and far field confinement σ0=0.5 MPa are prescribed for the tests in this series.

Table 2. Injection Rate, Fluid Time Step, and Total Simulation Time for Test Series I to Investigate the Effect of the Injection Rate
Test No.Injection RateFluid Time StepSimulation Time
 Q (m2/s)(s)(s)
I10.011.2×10−70.048
I20.021×10−70.04
I30.045×10−80.024
I40.054×10−80.0144
I50.082×10−80.008
I60.12×10−80.0056
I70.121.5×10−80.0036
I80.161×10−80.0012

[42] The displacement patterns at the end of the simulation times for the eight tests are shown in Figure 9. The black circle indicates the initial borehole position. The gray hollow circles represent particles associated with dry pore spaces and the filled circles represent those associated with saturated pore spaces. The filled color indicates the magnitude of the locally averaged pore pressure at the positions of the particles. From red to blue, the pore fluid pressure decreases. The interface between the blue disks and the gray hollow circles can therefore be interpreted as the infiltration front. The grain displacement and fluid infiltration patterns illustrated in Figure 9 are consistent with the experimental results in Figure 1. As the injection rate increases, fluid flow shows a transition from infiltration-governed to infiltration-limited behaviors while the granular medium response evolves from that of a fixed bed to localized failure leading to the growth of fluid channels.

Figure 9.

Displacement patterns from Test Series I at the end of the simulations: (a) Test I1: Q=0.01 m2/s, (b) Test I2: Q=0.02 m2/s (c) Test I3: Q=0.04 m2/s (d) Test I4: Q=0.05 m2/s, (e) Test I5: Q=0.08 m2/s, (f) Test I6: Q=0.1 m2/s, (g) Test I7: Q=0.12 m2/s, (h) Test I8: Q=0.16 m2/s. The black circle indicates the initial borehole position. The gray hollow circles represent particles associated with dry pore spaces and the filled circles represent those associated with saturated pore spaces. The filled color indicates the magnitude of the locally averaged pore pressure at the positions of particles. From red to blue, the pore fluid pressure decreases.

[43] At a relatively low injection rate Q=0.01 m2/s (Test I1), the particle displacements are mostly negligible, and the pressure profile is nearly axisymmetric indicating a radial flow pattern. Consequently, the borehole pressure increases continuously with time, see Figure 10. Axisymmetric fluid flow also results in the particle contact force chains to become radially aligned in the near wellbore area. This case is therefore similar to Test A1 in Figure 1a in the simple radial flow regime.

Figure 10.

Wellbore pressure histories at injection rates Q=0.01, 0.02, 0.04, 0.08, 0.1, and 0.16 m2/s from Test Series I.

[44] As the injection rate increases to Q=0.02 m2/s (Test I2), the infiltration pattern remains circular. Particle displacements at the early time of injection are negligible. However, as fluid injection continues, the wellbore starts to expand slightly with fluid channels of a few particle diameters in length developed near the wellbore, see Figure 9b. Nevertheless, the wellbore pressure increases continuously with almost no local drops. The continuously increasing trend in the wellbore pressure history concurs with the pore pressure distribution in Figure 1b that the fluid flow pattern remains radial. This test case indeed displays behaviors similar to Test A2 in Figure 1b in the infiltration-dominated regime.

[45] As similarly done in the experimental analysis [Huang et al., 2012b], an index λt, which indirectly measures the efficiency of fluid injection in creating open space, may be defined as the ratio between the area of the opening and the infiltrated area at a given time. The area of the opening now includes the area from wellbore expansion and the area of the fluid channels. Meanwhile, the area enclosed by the boundary of fully saturated pores minus the initial wellbore area may be considered as the infiltrated area. The area ratio λt is nearly zero at Q=0.01 m2/s and is less than 4%at Q=0.02 m2/s. The small values of λt in these two cases reflect that the efficiency is rather low when the injection rate is small.

[46] The shape of the infiltration front becomes slightly affected by the growth of the fluid channels when Q=0.04 m2/s (Test I3). The abrupt local drops and the subsequent gradual increases in the pressure curve are the signatures of sudden fluid channel growth events and the arrest periods when fluid flow results in limited grain displacements. Extension of the fluid channels is indeed in spurts. It is interesting to note, however, that in an average sense, the pressure curve as well as the area ratio λt reach plateaus after inline image (inline image may be interpreted as a scaled injection time or the normalized injection volume), see Figures 10 and 11. After the initial increase periods, p/σ0≃2.69 and λt≃10%.

Figure 11.

Variations of the area ratio λt with the scaled time inline image at injection rates Q=0.01, 0.02, 0.04, 0.08, 0.1, and 0.16 m2/s from Test Series I.

[47] As the injection rate increases further, the infiltration fronts follow the channel profiles closely. The depth of infiltration decreases with the injection rate. At Q=0.12 (Test I7) and 0.16 m2/s (Test I8), the depth of infiltration is limited to 2−3 particle diameters. The wellbore pressure histories now exhibit a global trend of an initial increase followed by a decline after reaching a peak around p/σ0∼3. The pressure decline is fairly gradual at Q=0.08 (Test I5) and 0.10 m2/s (Test I6), but relatively steep at Q=0.16 m2/s. The scaled time, corresponding to the peak pressure, decreases as the injection rate increases. In these cases, the area ratio λt increases with time. When Q>0.1 m2/s, the area ratio λt seems to reach a plateau around 50%near the end of the simulations. Since the medium is assumed to be elastic friction (although damping is present in the numerical calculation for stability), these simulations are not expected to be in the viscous fingering-dominated regime.

4.3 Dimensionless Time Scaling

[48] The experimental evidence summarized in section 2 suggests that the injection process in a dense granular medium is governed by the dimensionless time τ1=ηv/Ek. If the wellbore diameter is chosen as the characteristic length, i.e., =Di, τ1=Qη/πEk. Here we intend to verify the scaling of the material properties in τ1 by varying the fluid viscosity, modulus, and permeability of the medium while keeping the dimensionless time τ1 constant. As can be seen from (4), the effective modulus of the medium can be adjusted proportionally at the particle scale through the particle stiffness Kn while keeping the stiffness ratio Ks/Kn and the ratio σ0/Kn constant. The latter ensures that the particle configuration and, consequently, the porosity and permeability of the assembly remains unchanged. Meanwhile, permeability of the particle assembly can be changed through the mean particle radius inline image and the porosity φ according to (10).

[49] Four additional series of tests, Test Series II–IV, are carried out at an injection rate Q=0.08 m2/s. The variables in these tests are summarized in Table 3. Fluid viscosity is varied in Series II while the particle stiffness (or the elastic modulus) and the particle sizes (or the permeability) are varied in Series III and IV. The particle assembly in Series II and III is identical to the baseline configuration in Test Series I. In order to understand whether variation in the assembly configuration has a significant effect, another series (Series IIIb) of tests is conducted with the same microscale parameters as those in Series III, but with a different assembly configuration. In Series IV, the particle radii are uniformly distributed within the ranges of 0.4−0.6 mm (Test IV1) and 0.9−1.1 mm (Test IV2). Since inline image (and consequently the porosity) is not kept constant in these cases, the effective permeability is affected by not only the change in the mean particle radius but also the porosity of the assembly. The numerically measured permeability values in Series IV are k=0.604×10−9m2 for inline image mm and k=2.779×10−9m2 for inline image mm. Change in the particle size and distribution also affects the effective modulus slightly.

Table 3. Test Variables for Series II–IV
 ηKninline imageσ0kE50′
Test No.(Pa s)(MN/m2)(mm)(MPa)(10−9m2)(MPa)
II10.583.30.60.50.84934.13
II2283.30.60.50.84934.13
III1 (III1b)141.650.60.250.84917.08
III2 (III2b)1166.70.610.84968.37
IV1183.30.50.50.60435.40
IV2183.310.52.77932.37

[50] The wellbore pressure histories from Series II are compared with those from Test I3 (Q=0.04 m2/s) and Test I8 (Q=0.16 m2/s) in Figure 12. The pressure histories plotted as a function of the normalized time inline image are nearly identical for the cases with the same Qη. Excellent agreement is also obtained for the comparison among Series III, IIIb and Tests I3 and I8, for the same ratio of Q/E. Although the tests in Series I and IV do not yield identical Q/k(based on the permeability values in Series IV, the injection rates in Series I need to be Q=0.0244 and 0.112 m2/s for the same Q/k), the pressure histories from Series IV are in reasonable agreement with Test I2 (Q=0.02 m2/s) and Test I6 (Q=0.10 m2/s).

Figure 12.

Comparison of the wellbore pressure histories from Series I and II; Test II1: Q=0.08m2/s, η=0.5Pa s, Test II2: Q=0.08m2/s, η=2Pa s, Test I3: Q=0.04m2/s, η=1Pa s, and Test I8: Q=0.16m2/s, η=1Pa s.

[51] The grain displacement and fluid flow patterns from Series II and III also display nearly identical morphology to those cases having the same Qηor Q/E in Series I, see the comparison among Tests II1, III2, and III2b in Figure 13. As expected, with a different assembly configuration, the locations of fluid channels in Series IIIb differ from those in Series I–III. The displacement patterns from Series IV are shown in Figure 14. It is interesting to note that Test IV1 with a smaller mean particle radius in fact produces an opening similar to the bi-wing planar fracture.

Figure 13.

Displacement patterns from Tests (a) II1, (b) III2, and (c) III2b at inline image. The parameters that differ from Test I3 are the following: (Figure 13a) II1: Q=0.08 m2/s, η=0.5 Pa s, (Figure 13b) III2: Q=0.08 m2/s, E50′=68.37 MPa, (Figure 13c) III2b: Q=0.08 m2/s, E50′=68.37 MPa and different particle configurations. The black circle indicates the initial borehole position. The gray hollow circles represent particles associated with dry pore spaces and the filled circles represent those associated with saturated pore spaces. The filled color indicates the magnitude of the locally averaged pore pressure at the positions of particles. From red to blue, the pore fluid pressure decreases.

Figure 14.

Displacement patterns from Series IV: (a) IV1: inline image mm at t=0.0032 s and (b) IV2: inline image mm at t=0.025 s. The black circle indicates the initial borehole position. The gray hollow circles represent particles associated with dry pore spaces and the filled circles represent those associated with saturated pore spaces. The filled color indicates the magnitude of the locally averaged pore pressure at the positions of particles. From red to blue, the pore fluid pressure decreases.

[52] Denote λ as the area ratio at a normalized reference time, i.e., λ=λt(t0). A reference time t0=2.4 ms is chosen for Test I5 (Q0=0.08 m2/s). For all the other cases, the simulation time to calculate the area ratio λ is determined from t=t0(Q0η0Ek/QηE0k0). An interpolated value is used, if time t falls in between two output times. Variation of the area ratio λ as a function of the dimensionless time τ1=Qη/πEkis shown in Figure 15. The agreement among the cases with the same τ1 is indeed remarkable. Furthermore, although the numerical model is rather different from the experimental setup (e.g., the grain size, the inlet configuration, and the material properties), the numerically obtained threshold value τd, which defines the transition from the infiltration-dominated to the grain displacement-dominated regime (τd=0.44 assuming Test I3 as the transition case), compares exceptionally well with the experimental value of τd=0.1 determined from Figure 2. In both the experiments and the numerical simulations, λ reaches around 50% at late time when τ1O(1).

Figure 15.

The area ratio λ at a normalized reference time as a function of the dimensionless time τ1.

4.4 Energy Partition

[53] In this numerical model, the energy components in the particle assembly include the following: (1) the body force work Eb, which in this case comes from the work done by the resultant drag forces exerted onto the particles; (2) the work done by the boundary Ew; (3) frictional work Ef as a result of intergranular slip; (4) kinetic energy of the particles Ek, including the translational and rotational motion; and (5) strain energy Ec, which can be determined from the energy stored at all the contacts. Finally for fluid flow along the flow paths at the contacts, viscous dissipation Ev can be determined from

display math(13)

Since constant pressure is assumed for the fluid channels connected to the wellbore, there is no viscous dissipation from fluid flow along the channels.

[54] Evolution of the energy components from Tests I1, I5, and I8 are presented in Figures 16-18. In all these tests, since the numerical model is subjected to a constant far field confining stress prior to fluid injection, the total strain energy is nonzero initially and remains more or less constant during the simulations. In the case of Q=0.01 m2/s, the input energy is mostly dissipated when fluid flows through the flow paths at the contacts. Since fluid flow results in negligible particle displacements, increments in all other forms of energy are nearly zero. When the injection rate increases to Q=0.08 m2/s, although viscous dissipation Ev is still the largest term, the work done by the drag forces Eb and by the boundary Ew as well as the frictional work Ef gradually increase with time. The work Eb and Ef are the two components of energy dissipation associated with grain displacements. The fact that the frictional work is relatively small indicates that intergranular slip is likely localized (most logically near the tip of the opening). At Q=0.16 m2/s, the work Eb done by the drag forces is comparable with the viscous dissipation Ev, and the sum Eb+Ef is slightly larger than Ev, suggesting that the dissipation associated with grain displacements is becoming the main mechanism of energy dissipation. The intermittent propagation of the fluid channels during fluid injection is also reflected by the step increase in the energy components in Figure 18.

Figure 16.

Histories of the energy components in Test I1 (Q=0.01 m2/s).

Figure 17.

Histories of the energy components in Test I5 (Q=0.08 m2/s).

Figure 18.

Histories of the energy components in Test I8 (Q=0.16 m2/s).

[55] The energy analysis presented here supports our argument that the fluid-displacement regimes in the injection process essentially emerge as a result of the competition among various forms of energy dissipation mechanisms. The fluid-grain displacement regime is infiltration-governed when the input energy is mainly dissipated through flow in the pore spaces. In the grain displacement-dominated regime, the energy dissipated due to grain displacements in the viscous fluid and intergranular slip is comparable to the viscous dissipation from flow in the pore spaces. Although the viscous dissipation from fluid flow along the channels is not counted here, it is expected that the percentage of this portion would increase as the injection rate increases.

5 Discussions

5.1 Growth Mechanisms

[56] In all these test cases (Tests I1–I8), the fluid channels seem to initiate at the same locations. This indicates that growth of the openings from the wellbore is likely governed by the weakest links near the wellbore. An explanation for the initiation of a fluid channel can be found by examining the evolution of the contact force chains. Prior to the fluid loading, the domain is subjected to uniform radial stresses at the inner and outer boundaries. The contact force chains do not show preferential alignment. At the early time of injection, fluid flow in the nearly radial direction results in the major contact force chains to become radially aligned in the near wellbore area. The contact forces in the circumferential direction are less compressive than those in the radial direction. As the pore pressure increases and the wellbore expands, the contact forces in the circumferential direction decrease. If the drag forces that resulted from the pressure gradients exceed the normal force at a contact, the contact between the particles no longer exists, and a fluid channel is created.

[57] When Q≥0.08 m2/s, multiple fingers develop at the end of the injection time due to the presence of an isotropic far field confinement. Growth of these fingers is, however, not simultaneous. As shown in Figures 19a and 20 for Q=0.16 m2/s, there are four seed locations for the development of preferential paths at the early time. The fluid channel to the right of the wellbore is the one actively growing in Figures 19a and 20a. However, in Figures 19b and 20b, the one to the right of the wellbore becomes arrested. Instead, the fluid channel above the wellbore now propagates. The propagation sequence illustrated in Figures 19 and 20 suggests that the growth events alternate between the two fluid channels. This alternating growth scenario is also observed experimentally, but may have been promoted by the existence of the far field confining stress boundary in this case. The far field stress in these simulations is maintained by regulating the average radial stress on the outer wall by expanding or shrinking the confining wall radius using a servo-controlled algorithm. As the fluid channels grow, the stress field no longer maintains the circular symmetry. The side where the fluid channel is actively growing experiences a relatively higher contact stress level. The stress level on the inactive side therefore decreases. As a result, the condition may be met for the fluid channel on the inactive side to grow.

Figure 19.

Displacement patterns for Q=0.16 m2/s (Test I8) at various injection times (a) t=0.2 ms, (b) t=0.4 ms, (c) t=0.8 ms, and (d) t=1.2 ms. The black circle indicates the initial borehole position. The gray hollow circles represent particles associated with dry pore spaces and the filled circles represent those associated with saturated pore spaces. The filled color indicates the magnitude of the locally averaged pore pressure at the positions of particles. From red to blue, the pore fluid pressure decreases.

Figure 20.

Contact force chains for Q=0.16 m2/s at injection time (a) t=0.21052 ms and (b) t=0.21152 ms; the thickness of the force chains represents the contact force magnitude.

[58] From Figures 19c to 19d, widening and branching of the fluid channels can also be observed. In the test cases when Q>0.02 m2/s, the fluid channels all show a certain degree of tortuousness during the propagation or branching near the end of simulation. Indeed as shown in Figure 1, tip blunting and splitting is a ubiquitous feature in the injection experiments. Branching or changing in the propagation direction arises as a result of local contact force anisotropy. As a fluid channel extends, the contact force chains are reorganized to become nearly perpendicular to the fluid channels. Ahead of the fluid channel, if the contact forces in the direction perpendicular to the fluid channel are no longer the least compressive, further extension in the original direction may then become less likely.

[59] The branching mechanism may also be understood from the velocity field as shown in Figure 21. Indeed the particle flow mechanisms near the tips of the two main fluid channels in Figure 21 reveal intriguing details. In the cases when the fluid channels continue to grow in the original direction, the velocity fields, as shown in Figure 21a and in Figure 21b at the tip of the channel to the right of the wellbore, embody the characteristics of a passive failure mode. In other words, the particles move away in the direction of fluid flow. However, in the case when the fluid channel develops a small kink, the velocity field, see Figure 21b at the tip of the fluid channel at the top of the wellbore, actually possesses the characteristics of an active failure mode, where the particles are now moving in the direction against fluid flow. Such granular kinematics shares similarities with the tip splitting instability in viscous fingering between two immiscible fluids [Saffman and Taylor, 1958] and also with normal fault and reverse fault scenarios to describe the formation of an apical inverted-cone structure [Mathieu et al., 2008; Abdelmalak et al., 2012; Gay et al., 2012]. Further analysis is however needed to confirm the link between the tip kinematics and the tortuousness of a fluid channel for general conditions.

Figure 21.

Particle velocity field for Q=0.16 m2/s at injection time (a) t=0.21052 ms and (b) t=0.21152 ms.

5.2 Injection Pressure

[60] The different trends in the pressure curves in Figure 10 suggest that growth of the fluid channels may be stable (Q<0.04 m2/s), in critical equilibrium (Q=0.04 m2/s), or unstable (Q>0.04 m2/s), in an average sense. We now examine the numerically obtained wellbore pressure history from the perspective of continuum mechanics. At early time, the wellbore pressure pw is related to the radius of the infiltration front through the logarithmic expression for radial flow in two dimensions, i.e.,

display math(14)

where rw is the current wellbore radius. For a fixed bed, the wellbore size remains unchanged, rw=rwi(rwi—the initial wellbore radius). The radius of the infiltration front can be related to the injection time through inline image. In both Tests I1 and I2, rwrwi and the wellbore pressure histories indeed follow inline image, although a fluid channel of a few particle diameters in length has developed in Test I2. The continuous rise in the pressure in Test I2 suggests that in applications such as waterflooding, where fluid leakoff is large, fracture initiation from the wellbore does not necessary lead to immediate pressure decline. The wellbore pressure needs to increase in order to expand the wellbore, which implies that the ratio rf/rw also increases as the wellbore expands. If fluid flow results in only wellbore expansion with no growth of any preferential paths, the area ratio inline image is then expected to decrease with time. Such a prediction on the behavior of λt is, however, not observed in this analysis due to limited wellbore expansion.

[61] As the flow rate increases, preferential fluid paths develop. As shown in Figure 10, the initiation or breakdown pressure, however, depends on the injection rate. This may be explained by considering the following two limiting scenarios, namely the low injection rate case when fluid fully penetrates into a flaw of a certain length (inline image for a particle assembly) that exists near the wellbore, and the high rate case when fluid penetration is limited. Given the same wellbore pressure pw, the mechanical resistance is the same. Nevertheless, the drag forces, which are the driving forces to separate the particles, are larger in the low injection rate case where fluid flow fully penetrates into the flaw. The initiation pressure corresponding to the development of localized features therefore increases with the injection rate. Although the pressure histories are obtained here for an initially dry cohesionless material, the rate-dependent behavior shares similarities with that of a saturated brittle material [Detournay and Cheng, 1988].

[62] At large rates, growth of the preferential paths results in steep pressure decline. At Q=0.16 m2/s, the normalized peak pressure pw/σ0≃3. It is interesting to note that this result seems to agree with those from the injection experiments in triaxial cells with saturated sand subjected to large confining stresses, but with limited leakoff [Bohloli and de Pater, 2006; Zhou et al., 2010; Hurt and Germanovich, 2012]. The peak pressure ratio in those experiments is reported to be pw/σ0∼2.5−3 and is not much affected by the fluid rheology, which agrees with the scenario of limited fluid penetration. Limited fluid penetration implies that plasticity solutions for expansion of a circular opening may be applicable for predicting the wellbore pressure prior to the peak. The solution of self-similar cylindrical cavity expansion in an infinite domain [Carter et al., 1986; Huang and Detournay, 2010] is suggested as an upper bound for predicting the peak pressure [Bohloli and de Pater, 2006; Zhou et al., 2010; Hurt and Germanovich, 2012]. If the granular medium can be assumed to be elasto-perfectly plastic following a Mohr-Coulomb yield criterion and the deformation prior to the peak remains axisymmetric, the upper bound pressure from the cavity expansion solution is inline image for the non-associative case with zero dilatancy and inline image for the associative case (assuming an internal friction angle ϕ=25.7°). Since the size of the number model is finite, the upper bound can be reduced by considering the elasto-plastic solution [Jaeger et al., 2007] with full domain yielding, which requires pw/σ0=6.12. Meanwhile, the condition for the onset of plasticity provides a lower bound, namely pw≥1.43σ0. For a mechanical loading pw=3σ0 at the wellbore, the corresponding plastic zone size is rp=3.46rw(note Do/Di=20). That means that, at the peak load, plastic deformation is indeed induced near the wellbore, but is not associated with full domain yielding. The prediction from the elasto-plastic solution may be improved if nonlinearity is better reflected in the constitutive behaviors. Due to limited growth in the length of the fluid channels, analysis of the pressure decline behaviors is not attempted here.

[63] In the displacement regimes studied here, fluid infiltrates ahead of the openings. It is therefore expected that when the injection rate is relatively small, the pressure history prior to the peak does not have a strong dependence on the intergranular friction, since the presence of the fluid pore pressure reduces normal contact forces between particles in the circumferential direction near the wellbore before the onset of fluid channel growth. Indeed, this is confirmed from two additional simulation cases with Q=0.08 m2/s and intergranular friction angles ϕ0=20° and 40° (the default value is ϕ0=30° in previous tests). Although the pressure decline and the opening morphology with ϕ0=20° and 40° differ slightly from the case with ϕ0=30°, the pressure history prior to the peak with ϕ0=20° is very close to the case with ϕ0=30°. The injection pressure is slightly higher when ϕ0=40°. This result as well as those from the energy partition analysis suggest that, in the infiltration-dominated regime (Q≤0.04 m2/s in Series I), the intergranular friction is not the main mechanism of resistance. It is therefore appropriate to use the dimensionless time τ1, the definition of which does not include the friction coefficient, to define the transition from the infiltration-dominated regime to the grain displacement-dominated regime. This is unlike the displacement mechanisms in air displacing a fluid-grain mixture at a low grain fraction with a low injection rate in Sandnes et al. [2011], where the process is governed by the gravity-induced frictional force.

6 Concluding Remarks

[64] The process of fluid injection into an initially dry dense granular medium is investigated systematically using the discreet element method (DEM) code PFC2D® coupled with a pore network fluid flow scheme. The numerical analysis shows that the methodology is capable of reproducing phenomena consistent with those observed in laboratory injection experiments and in engineering applications and geological systems, namely increase in the injection rate causing the fluid flow behaviors to change from infiltration-governed to infiltration-limited and the granular medium response to evolve from that of a rigid porous medium to localized failure leading to the development of preferential paths. The effects of the fluid viscosity and the material properties such as the modulus and permeability are illustrated in the numerical analysis. The association of the displacement regimes with the energy dissipation mechanisms is also shown. The numerically obtained threshold dimensionless time that governs the transition from the infiltration-dominated regime to the grain displacement-dominated regime is in reasonable agreement with that from injection experiments in a Hele-Shaw cell [Huang et al., 2012b]. This threshold value can serve as a guide for engineering design. For example, if maximizing the sweeping efficiency is the objective, e.g., in soil remediation and waterflooding, the operating parameters then need to be chosen to stay in the infiltration-dominated regime. The peak pressures at large rates, when fluid leakoff is limited, compare well with the injection experiments in triaxial cells [Bohloli and de Pater, 2006; Zhou et al., 2010; Hurt and Germanovich, 2012]. The numerical analysis also reveals intriguing tip kinematics field for the growth of a fluid channel, which may shed light on the occurrence of the apical inverted-cone features in sandstone and magma intrusion [Cartwright et al., 2008; Mathieu et al., 2008; Abdelmalak et al., 2012; Gay et al., 2012]. The problem analyzed in this work is most relevant to geological storage in a depleted reservoir due to the assumption of an initially dry medium. The analysis indeed has limitations in assuming isotropic confining stresses and neglecting the capillary effect, which may become important as fluid migrates further away from the wellbore. Nevertheless, the results from the numerical analysis provide valuable insights into the mechanisms and conditions for the growth of preferential paths, which are applicable for the injection processes in general.

Acknowledgments

[65] Acknowledgment is made to the donors of the American Chemical Society Petroleum Research Fund through grant ACS/PRF 49971-DNI9, the Sand Control Client Advisory Board of Schlumberger and the National Science Foundation through grant NSF/CMMI-1055882 for support of this research. F.Z. and H.H. also wish to thank Itasca for the education software loan of PFC2D®; and Joseph Ayoub for the insightful discussions. The authors would also like to thank the two anonymous reviewers for their constructive comments and suggestions which helped improve the manuscript significantly.

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