Simulation of Geomechanical Behavior during SAGD Process using COMSOL Multiphysics

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1 Siulation of Geoechanical Behavior during SAGD Process using COMSOL Multiphysics X. Gong 1, R.Wan *2 Departent of Civil Engineering, University of Calgary, Calgary, Alberta, Canada. *Corresponding author: 2500 University Dr. NW, Calgary, Alberta, Canada. Eail: Abstract: Stea-Assistant-Gravity-Drainage (SAGD) has becoe one of the ost coonly used theral recovery processes to extract heavy oil fro oil sand reservoirs, especially in Northern Alberta, Canada. However, due to geological history, interbedded shales (IBS) in the reservoir act as barriers due to their low pereability and as such hinder the progression of the stea chaber. Cap-rock integrity also needs to be ensured to prevent stea breakthrough. Therefore, a coprehensive investigation of coupling between theral ultiphase fluid flow and geoechanics is required in odeling SAGD processes to properly address the above issues. In this paper, the THM behavior of the heavy oil reservoir was studied through a proper constitutive odeling of the porous edia. Both hoogeneous and non-hoogeneous reservoir conditions were investigated based on stress path and stress/strain field analyses. Keywords: SAGD, Cap-Rock, Interbedded Shale (IBS), Thero-ElastoPlastic Constitutive Model 1. Introduction Stea-Assisted-Gravity-Drainage (SAGD) is one of the ost effective theral recovery ethods used to recover heavy oil fro oil sand reservoirs of Northern Alberta, Canada. This process involves the injection of stea at high teperature ( 225 C ) into the oil sand foration through an upper injection horizontal well, with a lower horizontal well acting as the producer. Basically, the echanis is such that, as stea condenses in contact with the surrounding foration it transfers heat to the latter thereby significantly reducing the viscosity of heavy oil in the pore space. As a result, the heated heavy oil with higher obility will flow under gravity into the lower production well. Nuerous theoretical and nuerical studies based on so-called dead oil odel, black oil odel, copositional odel, aong others, have been conducted following well-established ultiphase fluid flow theory. For instance, Shutler (1969, 1970) studied a three-phase dead oil odel and applied to the steaflooding strategy. Jensen (1991) developed a steaflooding siulator using a siple threephase and two-coponent dead oil odel to study fractured reservoir. Hansauit (1990) also developed a three-diensional copositional siulator to investigate stea injection process. However, all the above-entioned studies focus on well-established reservoir siulation without considering iportant geoechanical effects. On the other hand, it is well-known that stea injection during SAGD process significantly alters both pore fluid pressure and teperature fields, which in turn affects the effective or skeleton stress in the reservoir. As such, geoechanical effects and ultiphase fluid flow are tightly intertwined and this coupling needs to be systeatically considered in any nuerical siulation of SAGD, especially if aterial failure is one of the iportant issues to be investigated. In this paper, the ain objective is to investigate the geoechanical behavior of oil sand and interbedded shale (IBS) during SAGD process using an advanced constitutive odel. Specifically, plastic deforations, dilation and potential unstable failure conditions of both oil sand and shale during the SAGD process are evaluated through the respective stress strain relationships and an effective stress path analysis. In addition, localization and aterial instability phenoenon (Wan, et al. 2011, Wan, et al. 2012) are also studied. A generalized density-stress-fabric dependent elasto-plastic (socalled WG) odel initially developed by Wan and Guo (1998) is eployed in this paper to show the relevance of geoeechanics and how it is effectively used to evaluate aterial failure issues in SAGD process with IBS.

2 2. Matheatical Forulation In this section, the physical process of coupling non-isotheral ultiphase fluid flow with geoechanical deforation and stress in the reservoir is atheatically forulated following basic balance laws. For ultiphase fluid flow, ass and energy conservation equations are advocated, whereas for geoechanics, Biot s consolidation (Biot, 1941) type of quasi-static equilibriu forulation with elasto-plastic constitutive relation is used. As for the elasto-plastic constitutive relation, a densitystress-fabric dependent elasto-plastic (WG) odel initially developed by Wan and Guo (1998) and generalized thereafter is briefly suarized. 2.1 Assuption It is iportant to recall soe assuptions that are ade for the derivation of the current coupled ultiphase flow and geoechanics odel, i.e. Three phase (water, oil and gas) and two coponents (water and oil) dead oil odel is assued; Capillary pressure is ignored; Local therodynaic equilibriu is assued; Darcy's law and sall strains are assued. 2.2 Multiphase fluid equations Water ass balance: S S t w w g g S v S v q q w w w g g g w g Oil ass balance: S S q t v o o o o o o Darcy's law: kr v d K p gz w, o, g Energy balance: 1 su s S U t w, o, g 1 sh svs pt S H v w, o, g Saturation constraint: S S S Geoechanics Equilibriu equation: w o g x ij j b 0 Principle of effective stress: ' p ij ij avg ij i Stress strain relation: ' ep th 0 ij D ij ij ij 2.4 Elasto-Plastic Constitutive Law Here, the generalized density-stress-fabric dependent WG odel is used to evaluate the granular geoaterial behavior. The yielding function is derived fro ulti-surface plasticity and Critical State theory, with flow rule derived fro Rowe's stress dilatancy theory (Rowe, 1962). The basic forulation is suarized below. Yielding Function: F q Mp where M is the slope of yielding surface in p q plane which can be written as: 6sin M 3 sin with as the obilized friction angle. Hardening/Softening Law: p e sin sincs 0 p ecs where p is the deviatoric plastic strain; e is the current void ratio; e cs is the critical void ratio; cs is the critical state friction angle; 0 and n are aterial properties. n

3 Plastic Potential: G q sin p where is the obilized dilatancy angle expressed by: sin sinf sin 1 sin sin 3. Nuerical Studies The ain focus here is to study geoechanical behaviors of oil sand and interbedded shale (IBS), and their influence on the SAGD process under a fully coupled fluid flow and reservoir deforation finite eleent analysis. Three ain case studies have been siulated and systeatically copared as follows: (1) Hoogeneous reservoir using Mohr- Coulob odel as constitutive law for oil sand reservoir; (2) Hoogeneous reservoir using WG odel as constitutive law for oil sand reservoir; and (3) Randoly distributed inclined IBS using WG odel for both oil sand and IBS. Various aspects of the coputations are discussed, naely: effective stress paths at strategic points in the reservoir, and plastic shear strain levels and potential localization, including unstable aterial zones. Both teporal and spatial aps of the various above-entioned features enhance the understanding of the geoechanical behavior of the reservoir under a thero-hydroechanical setting, especially in the presence of IBS. Figure 1 shows the general configuration of the finite eleent siulation as well as the initial and boundary condition. Strategic points A to F are chosen within the reservoir to follow the evolution of various controlling field variables. Four interbedded shale (IBS) intrusions, 20 long and 1 thick inclined at 10 to the horizontal, are placed hypothetically within the reservoir. The finite eleent esh coprised of 3850 and 64,140 quadratic eleents for the hoogeneous reservoir and the IBS ebedded case respectively. For the COMSOL odeling, the user defined PDE s are the ost suitable odules for this highly coupled and nonlinear proble. The nuerical siulation of the SAGD process span over a period of 2.5 years approxiately. f Figure 1: SAGD siulation configuration and operation conditions (figure not to scale) 3.1 Stress Path and Stress Strain Figures (2a) and (2b) show the effective stress paths followed by strategic points A-F in the reservoir as coputed using MC and WG odels, respectively. For MC odel, the effective stress paths initially turn to the left with decreasing effective ean stress and increases in deviatoric stress as a result of an upsurge in pore fluid pressures for points A-D located above the well. At this stage, the effect of pressure is doinant due to faster pressure propagation relative to that of teperature. However, after about 200 days, the effect of teperature significantly coes into play as deviatoric stresses increase ainly because of the theral expansion and the confineent of the reservoir. The strength of aterial is ultiately reached at the plastic liit whereby the effective stress path essentially tangents the Mohr-Coulob failure envelope; see Figure (2a). Large plastic strains are thus developed due to aterial failure as illustrated in the stress strain plots in Figure (2c) where a slight increase in deviatoric stress causes large increases in plastic shear strain. As teperatures further rise within the reservoir, ean effective stresses can only increase, aking the stress path deviate fro the failure envelope. For points E and F which are located away fro the central axis of the well, unloading of deviatoric stresses occurs due to the fact that theral expansion akes the horizontal stress increase at these points, thus causing the deviatoric stress to decrease fro its initial value. Turning to the WG odel results, see Figure (2b), siilar initial behavior is observed for points A to D initially. However, siilarities stop here since both effective stress paths and stress strain behaviors are very different fro the MC case after the initial part. This is because of distinctive features of the WG odel that incorporates progressive obilization of dilatancy through strain hardening and softening,

4 aong others. As the effective stress path reaches a certain level consistent with the peak strength of the aterial, it turns back to the left again with a reduction in deviatoric stress. The Critical State Line shown in Figure (2b) represents an ultiate state at which the critical state friction angle is 25, an intrinsic aterial strength paraeter for oil sand. Fro the stress strain plot (Figure (2d)), it can be seen that aterial points A to C are subjected to an initial strain hardening phase followed by strain softening after a peak has been reached. This is a very typical behavior of geoaterials and oil sand whereby dense granular aterials dilate and exhibit peak strength followed by strain softening around which point, unstable aterial behavior occurs through strain localization (see Wan, et al. 2012). The MC odel does not contain the basic ingredients to capture such phenoena. (c) (d) (a) (b) Figure 2: Effective stress path, stress strain and voluetric strain coparison for MC and WG odels: (a) Stress path for MC odel; (b) Stress path for WG odel; (c) Stress strain for MC odel; (d) Stress strain for WG odel 3.2 Plastic Strain and Strain Localization Figures (3a) and (3b) give the set of plots of teperature and plastic strain at soe particular ties. It can be found that before the stea chaber touches the top of the reservoir, the axiu plastic shear strain front is slightly ahead of teperature. Banded shear zones with large plastic shear strains are fored along a V shaped configuration. Thus, it can be concluded that potential localization failure and aterial instability zones are highly possible in these V shaped shear banded zones which eanate fro the shoulders of the stea chaber at the reservoir top.

5 4. Conclusions (a) Teperature at 167 days (b) Plastic shear strain fro 120 to 167 days Figure 3: Teperature, increental plastic shear strain and second order work plots at various ties for WG odel 3.3 Interbedded shale case Figures (4a) and (4b) show, respectively, the teperature field plots at 1080 days and the increental plastic shear strain field at a particular tie increent for non-hoogeneous reservoir case. Although the IBS has extreely low pereability and porosity, the propagation of stea and teperature is not copletely blocked by the IBS, but advances with a slight degree of penetration into the IBS. However, the propagation of both fluid flow and teperature is postponed, thereby slowing the progress of heating in the foration in contrast to the hoogeneous reservoir case. Herein, the difference in pereability between oil sand and IBS is about 6 orders of agnitude. Such a large difference akes both the stea and teperature to evolve through the gap between IBS in the reservoir and ultiately for the "zig-zag" flow pattern. (a) Teperature at 1080 days (b) Plastic shear strain fro 996 to 1080 days Figure 4: Teperature, increental plastic shear strain and second order work plots at various ties for WG odel The paper provides a coprehensive analysis of aterial failure in a reservoir under SAGD process fro a fully coupled geoechanics and ultiphase fluid flow setting. Aong other things, the intent is to highlight the display of rich geoechanical features afforded by the use of a coprehensive constitutive odel. The essential findings of the paper can be suarized as follows. The coprehensive and robust elastoplastic constitutive WG odel is capable to capture the coplex stress strain response of oils and and shale. Linear/nonlinear elasticity or elastic-perfectly plastic Mohr-Coulob odels severely hapers their use in analyzing geoechanics in SAGD process. Strain localization failure odes and potentially unstable aterial behavior were successfully siulated using the WG odel. Results of the IBS reservoir siulation show a eandering flow pattern around the IBS due to the extreely low pereability and porosity of the shale. The blocking of stea flow by the IBS postpones the whole SAGD operation when copared with the hoogeneous reservoir case. 5. Acknowledgeents This work is jointly funded by the Coputer Modelling Group (CMG) and the Natural Science and Engineering Council of Canada (NSERC) through a CRD grant. The authors also thank CMG Ltd. for generous support in the for of free access to an-hours and STARS software. 6. References 1. Biot, M.A., "General theory of three-diensional consolidation". Journal of Applied Physics, Vol. 12, pp (1941) 2. COMSOL Multiphysics. COMSOL Multiphysics User Guide (Version 4.3a), COMSOL, AB. (2012) 3. Hansauit, V., "The developent and application of a three-diensional copositional siulator for stea injection processes". Ph. D Thesis, The Pennsylvania State University. (1990)

6 4. Jensen, T.B., "Stea flooding of fractured reservoirs: Experiental and nuerical studies". Ph. D Thesis, University of Wyoing. (1991) 5. Mohaadi, M., Gong, X., and Wan, R.G., "Laboratory and Constitutive Modeling of Colorado Shale at High Pressure and Teperature". ARMA th US Rock Mechanics/Geoechanics Syposiu. San Francisco, CA, USA, June. (2013) 6. Rowe, P.W., "The stress-dilatancy relation for static equilibriu of an assebly of particles in contact". Proc. R. Soc. London. 269, pp (1962) 7. Shutler, N.D., "Nuerical, Three-Phase Siulation of the Linear Steaflood Process". Society of Petroleu Engineers J., pp (1969) 8. Shutler, N.D., "Nuerical Three-Phase Model for the Two-Diensional Steaflood Process". Society of Petroleu Engineers J., pp (1970) 9. Wan, R.G., and Guo, P.J., "A Siple Constitutive Model for Granular Soils: Modified Stress-Dilatancy Approach". Coputer and Geotechnics, Vol. 22, No. 2, pp (1998) 10. Wan, R.G., Pinheiro, M., and Guo, P.J., "Elastoplastic odelling of diffuse instability response of geoaterials". Int. J. Nuer. Anal. Mech. Geoech, 35, pp (2011) 11. Wan, R.G., Pinheiro, M., Daouadji, A., Jrad, M., and Darve, F., "Diffuse instability with transition to localization in loose granular aterials". Int. J. Nuer. Anal. Mech. Geoech, Published on line, DOI: /nag (2012)