Geomechanical modeling ofstresses adjacent to salt bodies:

INTRODUCTION

AUTHORS

Gang Luo Bureau of Economic Geology,Jackson School of Geosciences, University ofTexas at Austin, Austin, Texas 78713;gangluo66@gmail.com

Gang Luo received his Ph.D. in computationalgeodynamics from the University of Missouriat Columbia in 2009, and then he joined theBureau of Economic Geology. His current re-search interests include wellbore stability analy-sis, poromechanical modeling in stress andpore pressure of sedimentary basins, numericalmodeling in salt tectonics, and finite-elementmodeling in geodynamic processes of crust andlithosphere.

Maria A. Nikolinakou Bureau of Eco-nomic Geology, Jackson School of Geosciences,University of Texas at Austin, Austin, Texas78713; mariakat@mail.utexas.edu

Maria Nikolinakou received her Sc.D. degree intheoretical soil mechanics from MassachusettsInstitute of Technology in 2008. She then workedfor Shell Exploration and Production in reser-voir geomechanics. She joined the Bureau ofEconomic Geology in 2009, where she is nowa research associate. Her current research in-cludes poromechanical modeling of basin sedi-ments, geomechanical pore-pressure prediction,and numerical modeling in salt tectonics.

Peter B. Flemings Bureau of EconomicGeology, Jackson School of Geosciences, Uni-versity of Texas at Austin, Austin, Texas 78713;pflemings@jsg.utexas.edu

Peter B. Flemings is the Jackson Professor ofGeosystems at the University of Texas at Austin.Previously, he taught in the Department of Geo-sciences at Penn State. He received his B.A. de-gree from Dartmouth College and his M.S. andPh.D. in geology from Cornell University. Hisresearch focuses on the study of fluid flow, pres-sure, and stress in sedimentary basins.

Michael R. Hudec Bureau of EconomicGeology, Jackson School of Geosciences,University of Texas at Austin, Austin, Texas78713; michael.hudec@beg.utexas.edu

Mike Hudec received his Ph.D. from the Uni-versity of Wyoming in 1990. After working forExxon Production Research and Baylor Univer-sity, he joined the Bureau of Economic GeologyDrilling into and through salt systems has proven challeng-ing, dangerous, and expensive (e.g., Sweatman et al., 1999;Whitson and McFadyen, 2001; Rohleder et al., 2003; Willson

Copyright 2012. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received September 1, 2010; provisional acceptance January 5, 2011; revised manuscriptreceived March 22, 2011; final acceptance April 11, 2011.DOI:10.1306/04111110144ABSTRACT

We compare four approaches to geomechanical modeling ofstresses adjacent to salt bodies. These approaches are distin-guished by their use of elastic or elastoplastic constitutive lawsfor sediments surrounding the salt, as well as their treatmentof fluid pressures in modeling. We simulate total stress in anelastic medium and then subtract an assumed pore pressureafter calculations are complete; simulate effective stress in anelastic medium and use assumed pore pressure during calcu-lations; simulate total stress in an elastoplastic medium, eitherignoring pore pressure or approximating its effects by decreas-ing the internal friction angle; and simulate effective stress inan elastoplastic medium and use assumed pore pressure duringcalculations. To evaluate these approaches, we compare stressesgenerated by viscoelastic stress relaxation of a salt sphere. In allcases, relaxation causes the salt sphere to shorten vertically andexpand laterally, producing extensional strains above and be-low the sphere and shortening against the sphere flanks. De-viatoric stresses are highest when sediments are assumed to beelastic, whereas plastic yielding in elastoplastic models placesan upper limit on deviatoric stresses that the rocks can support,so stress perturbations are smaller. These comparisons provideinsights into stresses around salt bodies and give geoscientists abasis for evaluating and comparing stress predictions.Part 1Uncoupled modelsGang Luo, Maria A. Nikolinakou, Peter B. Flemings,and Michael R. HudecAAPG Bulletin, v. 96, no. 1 (January 2012), pp. 4364 43

in 2000, where he is codirector of the AppliedGeodynamics Laboratory. His current researchinterests include palinspastic restoration ofsalt structures and early history of the Gulf ofMexico salt basin.

ACKNOWLEDGEMENTS

This work was supported by the Applied Geo-dynamics Laboratory Industrial Associates Pro-gram and UT Geofluids Industrial AssociatesProgram. Members of the Applied GeodynamicsLaboratory consortium include Anadarko, BHPBilliton, BP, CGGVeritas, Chevron, Cobalt, Conoco-Phillips, Ecopetrol, ENI, ExxonMobil, Fugro, GlobalGeophysical, Hess, IMP, INEXS, ION Geophysical,Maersk, Marathon, Mariner, McMoRan, Murphy,Nexen, Noble, Pemex, Petrobras, PGS, Repsol-YPF, Samson, Saudi Aramco, Shell, Statoil, TGS-NOPEC, Total, WesternGeco, and Woodside.Members of the UT Geofluids consortium includeAnadarko, BHP Billiton, BP, Chevron, Conoco-Phillips, Devon, ExxonMobil, Hess, Schlumberger,Shell, Statoil, and Total. We thank AAPG editorStephen E. Laubach and reviewers Tricia Allwardt,Markus Albertz, and Jon Olson for review com-ments and constructive suggestions. We bene-fited from discussions with Glen Bowers andMark Zoback. Lana Dieterich edited the manu-script. Publication was authorized by the Director,Bureau of Economic Geology, Jackson Schoolof Geosciences, The University of Texas at Austin.The AAPG Editor thanks the following reviewersfor their work on this paper: Markus Albertzand Tricia F. Allwardt.

44 Stress Around Salt Bodiesand Fredrich, 2005). One reason is that stresses and fluid pres-sures adjacent to salt may be highly perturbed relative to theirregional values (e.g., Bradley, 1978; Whitson and McFadyen,2001; Rohleder et al., 2003; Dusseault et al., 2004). Predictingthe magnitude of these anomalies is also challenging (e.g.,Rohleder et al., 2003;Willson et al., 2003;Willson and Fredrich,2005).

An obvious solution to this problem is to use geomechan-ical models to simulate stresses and fluid pressures aroundsalt bodies (e.g., Fredrich et al., 2003, 2007; Koupriantchiket al., 2005; Mackay et al., 2008). However, geomechanicalmodeling has yet to produce a clear consensus on what stressfields adjacent to salt should look like, the magnitude of pre-dicted stress effects, or even the proper techniques to use inthe modeling. Two key areas of disagreement in modeling are(1) what constitutive laws are appropriate to describe sedimentssurrounding the salt and (2) how fluid pressures are simulated.These choices, in turn, depend on the scale of the structuresbeing modeled and the time frames being simulated.

At basin scales and geologic time frames, viscous, visco-plastic, frictional-plastic, and elastoplastic constitutive modelsare used to describe sediments during the evolution of salt sys-tems. These models have explored driving forces in salt basins(e.g., Daudr and Cloetingh, 1994; Gil and Jurado, 1998), de-velopments of salt structures and associated faults and folds(e.g., Podladchikov et al., 1993; Poliakov et al., 1993b, 1996;Schultz-Ela, 2003; Dirkzwager and Dooley, 2008), and devel-opment of sedimentary basins (e.g., Gemmer et al., 2004, 2005;Ings et al., 2004). However, when modeling effects of singlesalt structures over short time scales (e.g., the topic of this ar-ticle), most workers have used elastic or elastoplastic consti-tutive laws to model sediments adjacent to salt (Table 1).

In some of these models, fluid pressures are ignored (e.g.,Fredrich et al., 2003, 2007; Sanz andDasari, 2010) or indirectlyincorporated as a reduction in the internal friction angle of sed-iments (e.g., Gemmer et al., 2004, 2005; Ings et al., 2004). Werefer to these models as uncoupled models because they do notconsider fluid flow or interactions between the fluid and solidmatrix. In a companion article (Nikolinakou et al., 2012), wepresent our preliminary attempts at explicitly modeling fluidflow around salt in a fully coupled model.

To the nonspecialist, this range of approaches is confus-ing. How do modeling assumptions affect results? How canresults published by separate authors be compared? Which ap-proaches most closely approximate subsurface geology? In thisarticle, we describe each of the published modeling techniques,illustrate howmodel results differ from one another, and discuss

s De

s

ulatouprd Daure (odeleld.

pressure is not truly included in the models.re prevernistres-streeffecy casthe advantages and limitations of each approach.Furthermore, our discussion of deformation andstress patterns that accompany relaxation of a saltsphere should provide some physical insight intowhy stresses are perturbed around salt and whydifferent stress states occur around different parts ofa salt body.Table 1. Summary of Uncoupled Model Types and Model Case

Model Type Elastic Sediment

Solve governing equationsin total-stress system

Case 1: Ignore pore pressure and sim(e.g., Fredrich et al., 2003, 2007; K2005; Mackay et al., 2008; Sanz anSubtract distribution of pore presshydrostatic pore pressure) from mfield to calculate effective-stress fie

Solve governing equationsin effective-stress system

Case 2: Substitute distribution of pohydrostatic pore pressure) into goand simulate effective stress. Totalcalculated by summing of effectivepore pressure. Resulting total andare the same as those predicted bWe subdivide publisheaccording to (1) whetelastoplastic constituti

th1

approaches and discuss how they have been usedto study deformation and stresses around salt.

Stress applied to thesolid matrix and fluid, item that includes pore fis supported by pore flu

me9e(Terzaghi, 1925; see definitions of variables inTable 2). Effective stress is important because it isused to compute plastic yielding of sediments inelastoplastic models (e.g., Wood, 1990), and thus,control wellbore stability analysis and drilling tra-jectory (e.g., McLean and Addis, 1990; Zoback,2007).

Elasticity does not have yield. However, plas-

ssure (such asng equationss field isss field andtive stressese 1.

Case 4: Substitute distribution of pore pressure(such as hydrostatic pore pressure) intogoverning equations and simulate effectivestress (e.g., Poliakov et al., 1993a, 1996;Dirkzwager and Dooley, 2008).scribed in This Article

Elastoplastic Sediments

e total stressiantchik et al.,sari, 2010).such asd total stress

Case 3A: Ignore pore pressure and simulatetotal stress (e.g., Schultz-Ela et al., 1993;Schultz-Ela and Jackson, 1996; Schultz-Elaand Walsh, 2002; Sanz and Dasari, 2010).

Case 3B: Simulate total stress and considereffects of hydrostatic pore pressure bydecreasing internal friction angle in modeling(e.g., Schultz-Ela, 2003). Note that poreticity has a yield criterion, constraining the magni-

, 2005; Jaeger et al., 2007). DeviatoricpaJsltl.,

Case 1: Elast

oraynre)u, can be expressed as s0ij s ij diju i; j 1; 2; 3 simulate stress relaxation within salt and stresstive stress (Terzaghi, 1total stress,s ij, effectiv25). The relation betweenstress, s

0ij, and pore pressure,

structures. Fet al. (2008drich et al. (2003, 2007) and Mackayused the finite-element method tostress felt by the solidstress supported by thwhole system, including thes called total stress. In a sys-luid, some of the total stressid, reducing the amount ofatrix. The fraction of totalsolid matrix is called effec-

Total Stress

An elastic msolids (no poapproach mfinite-elemeic Models, Calculated in

del assumes that sediments are elastices, and hence, no pore pressure). Thisbe the one most commonly used int modeling of stresses around salt(2) how or whetherfluid pressure (Tabled models into four types,her they assume elastic orve laws for sediments andey simulate the effects of). We define each of these

and Petrinicstress is thetortion (e.g.,and elastoplamodels of saFredrich et art of the stress tensor that causes dis-aeger et al., 2007, p. 40). Both elastictic constitutive laws have been used insystems (e.g., Poliakov et al., 1996;2003; Dirkzwager andDooley, 2008).PUBLISHED MODELING APPROACHES tude of deviatoric stresses (e.g.,Wood, 1990;DunneLuo et al. 45

Table 2. Nomenclature

Variable Description

E Youngs modulusn Poissons ratiog Gravitational accelerationr Bulk densityr Bulk density of sedimentsns arounear viscorent saltd horizonertical docomplex

sed

rsalt Bulk density of salt bodiesrw Bulk density of water

46 Stress Around Salt BodiesUnit Dimensions

Pa ML1T2

m/s2 LT2

kg/m3 ML3

kg/m3 ML3ence me modeltress per ons arouir in the Basin, sostigated b wer-lawnear visc ic salt borence in tatic stre

kg/m3 ML3

kg/m3 ML31 1Gulf of Mexico. Koupriantchik et al. (2005) used dicted by the two salt constitutive laws. Thesedome, a vand a trueme with an overlying tongue,salt-diapir geometry from the

and Maxwell lifound no diffeoelastfinal sdies andsses pre-an idealize tal sheet, an idealized vertical tralia. They inve oth po nonlinear

gated diffe geometries, including a sphere, the Munta diap Officer uth Aus-

law nonlin elastic salt bodies and investi- relaxation and s turbati nd salt of

perturbatio d salt bodies. They used power- the finite-differ thod to salt-stressePlastic Plastic strain magnitude

e3 Minimum principal strain of plastic strain tensor

e2plIntermediate principal strain of plastic strain tensor

e1plMaximum principal strain of plastic strain tensor

sVMplVon Mises stress Pa ML T

s3 Minimum principal total stress Pa ML T

1 2s2 Intermediate principal total stress Pa ML T1 2s1 Maximum principal total stress Pa ML T1 2sh Minimum horizontal principal total stress Pa ML T1 2sH Maximum horizontal principal total stress Pa ML T1 2srz, srq, szq Three shear stress components of total stress tensor Pa ML T1 2sqq Out-of-plane (or hoop) total stress Pa ML T1 2szz = sv Vertical total stress Pa ML T1 2srr Radial total stress Pa ML T1 2sij Stress components of total stress tensor Pa ML T1 2Ds Perturbation of stress Pa ML T1 2s Effective stress Pa ML T1 2s Total stress Pa ML T1 2P Solid pressure Pa ML T1 2u Pore pressure Pa ML T1 2dij Kronecker delta function, when i = j, dij = 1; when i j, dij = 0 1 2K0 Ratio of minimum and maximum principal effective stresses

KT0 Ratio of minimum and maximum principal total stresses

lHR Hubbert-Rubey pore pressure ratio

l Ratio of pore pressure and solid pressure

Ceff Effective cohesion of sediments Pa ML T

C Cohesion of sediments Pa ML T

1 2feff Effective internal friction angle of sediments Degree Degree1 2f Internal friction angle of sediments Degree Degree

h Viscosity of salt bodies Pa s ML T

Effective Stress

Total StressPlasticity has also been used to simulate stress-strainbehavior of sediments around salt. Plasticity intro-duces limits on deviatoric stresses and simulates ir-recoverable or permanent deformation (e.g.,Wood,1990; Roylance, 1996), making elastoplastic mod-els more realistic than elastic models (e.g., Bowers,2007).

In some instances, elastoplastic models ignorethe effects of pore pressure, solving the governingequations in total stress (case 3A). Schultz-Ela et al.(1993) used linear viscoelastic salt and elastoplasticsediments with strain softening in finite-elementmodeling to investigate initiation and developmentof faults near a salt diapir. The same approach wasThis approach, as in case 1, assumes an elastic con-stitutive law for sediments. However, an assumedpore-pressure distribution is included in the gov-erning equations (see equation 2 inAppendix), andthen all computations are performed using effec-tive stress. For solution of elastic problems, cases 1and 2 provide equivalent results. Pore pressure isincluded in the modeling, and the problem issolved in effective stress, but no fluid flow, fluiddissipation, or interaction between fluid and thesolid matrix exists in this case.

Case 3: Elastoplastic Models, Calculated inmodels considered processes related to viscoelasticstress relaxation of salt, but did not consider geo-dynamic processes of salt motion.

All these models are calculated using total stress(see equation 1 in Appendix). As an optional finalstep, an assumed pore-pressure distribution maybe subtracted from the final stresses to produce aresult in effective stress (e.g., Fredrich et al., 2003).However, in this case, pore pressure is assumed,not calculated. Furthermore, the presence of fluidsplays no role in computation. The only purpose ofadding fluid pressure at the end of modeling is topermit comparison with other results calculatedusing effective stress (see cases 2 and 4).

Case 2: Elastic Models, Calculated in Ela (2003) incorporated linear viscoelastic salt andelastoplastic sediments with strain softening to sim-ulate initiation, growth, and evolution of drag foldsaround a salt-diapir system by finite-element mod-eling. This approach has also been applied widely tosimulate the effects of pore pressure on viscoplasticmodels for geologic evolution of a sedimentary ba-sin in salt tectonic systems (e.g., Ings et al., 2004;Gemmer et al., 2005). Note that varying the inter-nal friction angle is only a proxy for pore pressure.Because the model contains neither pore pressurenor pore space, no computations of fluid flow orfluid pressure exist. Also, because the internal frictionangle is everywhere the same and does not changeduring the experiment, it allows for no change ofpore pressure during deformation (see Appendix).

Case 4: Elastoplastic Models, Calculated inEffective Stress

Other elastoplastic models incorporate pore pres-sure by calculating results using effective stress. Theydo so by assuming a pore-pressure field (commonlyhydrostatic) and substituting it into the governingequations (see equation 2 in Appendix), and thenmodels are calculated in effective stress. The modelsare uncoupled because pore pressure is assumed andnot recalculated during the simulation. Dirkzwagerand Dooley (2008) assumed hydrostatic pore pres-sure and used elastoplastic (Drucker-Prager plas-ticity) sediments to simulate stresses and deforma-tion of a salt-based gravity-driven fold and thrustapplied by Schultz-Ela and Jackson (1996) to sim-ulate linkage of fault patterns above and below thesalt in extensional regimes. Similarly, Schultz-Elaand Walsh (2002) applied this approach in explor-ing the development of faults above a salt layer andinteraction between extending overburden andthe underlying salt layer in finite-element models.These models have somewhat limited use becauseof the importance of effective stress in plastic yield.

In other models, the effects of pore pressureare commonly approximated and simulated by as-suming a low internal friction angle (case 3B). Thisangle is commonly called the effective internalfriction angle inmany articles (e.g., Ings et al., 2004;Gemmer et al., 2005). Using this approach, Schultz-Luo et al. 47

We compare these approaches by describing theirdifferences and similarities in model results and

We study salt bodies encased in their surrounding

solve for an equilibrated initial stress field for themodel domain using material constitutive laws,sediments and represent the three-dimensional (3-D)geometry with an axisymmetric model (Figure 1).In all simulations, the radius of the model domainis 20 km (12.4 mi). No normal displacements oc-cur along the outer boundary (r = 20 km [12.4 mi],

Figure 1. The finite-element mesh for the salt sphere andmodel boundary conditions. Dark-gray and light-gray elementsshow the salt structure and sediments, respectively.pointing out model limitations. These results pro-vide a basis for interpreting and comparing pub-lished finite-element models involving salt andoffer insight into stress and deformation patternsaround salt structures.

MODEL SETUP48 Stress Around Salt BodiesFigure 1) and the base (z = 0 km, Figure 1). The topboundary (z = 20 km [12.4 mi], Figure 1) is free.Along the central axis (r = 0 km, Figure 1), axi-symmetric boundary conditions are applied.

Salt is modeled as a linear viscoelastic Maxwellbody (equations 35 in Appendix). Because saltbodies are assumed to be impermeable, they haveno pore pressure. Sediments are assumed to beof either an elastic (cases 1 and 2) or a perfectlyelastoplastic (cases 3 and 4) material using eitherthe Drucker-Prager or the Mohr-Coulomb plasticyield criteria (equations 613 in Appendix). Materialparameters are shown in Table 3. Pore pressure insediments is assumed to be hydrostatic.

We use the commercial finite-element analysissoftware, Abaqus, to simulate processes of visco-elastic salt-stress relaxation in two steps. First, we

Table 3. Material Parameters in Modeling

Material Parameters Salt Sediments

Density r (kg/m3) 2076.5 2220Youngs modulus E (Pa) 3.1 1010 6 109

Poissons ratio n 0.25 0.225Viscosity h (Pa s) 1018 Internal friction angle f () 30Cohesion C (MPa) 0.03Effective internal frictionangle feff ()

15.95 in case 3B

Effective cohesion Ceff (MPa) 0.027 in case 3BWater density rw (kg/m

3) 1000 Gravitational acceleration g (m/s2) 9.81 belt. Poliakov et al. (1993a) assumed hydrostaticpore pressure and applied an elastoplastic (Mohr-Coulomb plasticity) constitutive law to sedimentsto explore the initiation of salt diapirs. Poliakov et al.(1996) assumed hydrostatic pore pressure and usedviscoelastoplastic sediments (Mohr-Coulomb plas-ticity) to simulate brittle deformation within theoverburden above salt diapirs.

In this study, we compare these uncoupled model-ing approaches using two-dimensional (2-D) axi-symmetric finite-element models to explore defor-mation and stresses around a spherical salt bodycaused by viscoelastic stress relaxation of the salt.1

1

2s 01 s 022 + s 01 s 032 + s 02 s 032VM 2 1 2r 1 3 2 3s 1 s s 2 + s s 2 + s s 2

r a measure of deviatoric stresses in three dimensions:less than 0.1MPa (

(Figure 1). First, we compare total-stress modeling(case 1) with effective-stress modeling (case 2) inmodels that have elastic sediments. Next, we in-

model in case 1. The white semi-circle is the interface betweenwhere s1, s2, s3, and s01, s

02, s

03 are principal total

and effective stresses, respectively. During the sec-ond modeling step, viscoelastic salt bodies cannotsustain initial deviatoric stresses, which have de-veloped within the salt during the static initializa-tion process (the first modeling step), so deviatoricstresses or von Mises stress within the salt decayover time. As a response to salt-stress relaxationand force equilibrium, stresses in the surroundingsediments are perturbed.

In both salt bodies and sediments, initial totalhorizontal stresses (sH = srr = sqq) are assumedproportional to the total vertical stress (sV):

KT0 sHsV

2

where KT0 , total-stress ratio, is assumed to be equalto 0.7. In theAppendix,we show that for hydrostatic

conditions with KT equal to 0.7, the equivalenteffective-stress ratio,K0, equals 0.454 (K0 sH=sV ).troduce the effects of plasticity around a sphericalsalt body (case 4). We then further consider theimpact of different plastic yield criteria (case 4)and compare total-stress modeling (case 3) withwe include plasticity in the models (case 4), thegoverning equations must be solved in effectivestress (see equation 2 in Appendix).

MODEL RESULTS

We present four cases (Table 1) to explore defor-mation and stresses around a spherical salt bodyhaving a 1-km (0.6-mi) radius and whose center islocated 5 km (3.1 mi) below the top of the model

salt and sediments. The casenumber is shown in Table 1 andmaterial parameters are shown inTable 3. VE = vertical exaggeration.Figure 2. The final stress fieldis the sum of the initial stress fieldand stress perturbations causedby salt viscoelastic stress relaxa-tion: (A) initial von Mises stressfield, (B) perturbations of vonMises stress, and (C) final staticvon Mises stress field. These re-sults are predicted by the elasticeffective-stress modeling (case 4) using plasticity.Finally, we extend our anWe incorporate hydrostatic pressure in oneof two ways. First, the force equilibrium equation(equation 1 in Appendix) and constitutive laws(see Appendix) are combined, and the system issolved in total stress (cases 1 and 3).Ultimately,whenwe express results, we merely subtract hydrostaticpore pressure from total stress to present effectivestress (case 1). An alternative approach (cases 2and 4), which assumes that pore pressure is equalto hydrostatic pore pressure, includes it in the gov-erning equations (see equation 2 in Appendix) andsolves the system in effective stress. However, when

tion and stresses around an irregular salt sheet in theelastoplastic models (case 4).

Elastic Sediments: Case 1 and Case 2

We begin with an elastic total-stress model (case 1).The final stress state (Figure 2C) is the sum of theimposed initial stress state (Figure 2A) and the stressperturbation that results from viscoelastic stress re-laxation of the salt body (Figure 2B). Before saltrelaxation, von Mises stress increases everywherealysis to explore deforma-

0 0 0Luo et al. 49

by the elastic model in case 1: (A) displacement vectors, (B) vertical strain, (C) horizontal strain, and (D) out-of-plane strain. Red istive).intewith depth because we impose a linearly increasingdifference between vertical and horizontal stresseswith depth (Figure 2A).

contractional strain (positive) and blue is extensional strain (negaincludes both elastic and viscous strain. The white semicircle is theFigure 2. VE = vertical exaggeration.During stress relaxation, salt flows inward from

50 Stress Around Salt Bodiesof the salt causes vertical extension above and belowthe salt and vertical contraction (or vertical short-ening) at the equator (Figure 3B). Contractional

Note that strain in sediments is elastic, but strain within the saltrface between salt and sediments. This case is the same as that instrain is positive in this study. Horizontal contrac-

the poles toward the center and outward from thecenter toward the equator (Figure 3A). Relaxation

tion (or horizontal shortening) at the equator(Figure 3C) and extensional out-of-plane or hoop

Figure 4. Final total stresses for the two vertical profiles through the center of the salt sphere (A) and through the sediments justbeyond the edge of the salt sphere (B), and the horizontal profile through the center of the salt sphere (C) predicted by the elastic modelin case 1. Compressional stress is positive in this study. Gray dashed lines show far-field horizontal and vertical stresses, respectively.Horizontal stress is equal to out-of-plane stress for the vertical profile through the salt center (A). Salt position is shown on axes. This caseis the same as that in Figure 2. Symbols defined in Table 2.Figure 3. The deformation pattern within and around the salt sphere during the processes of salt viscoelastic stress relaxation predicted

strain at the equator (Figure 3D) also exist. Thesecause von Mises stress to increase at the equatorof the salt body (lateral edge) and decrease at thepoles (top and bottom) (Figure 2B). These per-turbations can reach approximately 50% of initialstresses (Figure 2A, B). Finally, after salt stressrelaxation, almost no von Mises stress (

the two vertical profiles in (A) and (B), and

effective stresses for the two vertical profilesin (C) and (D). Gray dashed lines show far-field horizontal and vertical stresses, re-spectively. The assumed hydrostatic pore

pressure profile is shown in (A) and (B).

52 Stress Around Salt BodiesEffective stresses calculated from case 1 andcase 2 are identical. These cases are thesame as those in Figure 7. Symbols aredefined in Table 2.sphere and increase at the equator (Figure 5A, B).Out-of-plane and minimum principal stresses aredecreased around the salt sphere (Figure 5C, D).

Maximum principal stress (s1) is oriented ap-proximately vertically within the vertical cross sec-tion (Figure 6A); however, rotation of maximum

Figure 7. Final vertical stress for the saltsphere predicted by elastic models in cases 1and 2: (A) total stress and (B) effectivestress. Effective stresses are calculated fromeither case 1 or case 2. Effective stressespredicted by the two approaches are equal,and so are total stresses. The white semi-circle is the interface between salt andsediments. The case number is shown inTable 1, and material parameters are shownin Table 3. VE = vertical exaggeration.

Figure 8. Vertical profiles of final stressesthrough the center of the salt sphere andthrough the sediments just beyond the edgeof the salt sphere predicted by the elastic

models in cases 1 and 2: total stresses for

drostatic, the effective-stress field looks extremelyprincipal stress orientation near the salt is significant.

Figure 9. Magnitude of plastic strain (ePlastic) predicted by theelastic-perfectly-plastic model using the inner circle of the Drucker-Prager plastic yield criterion (DP-i) in case 4. White semicircle isthe interface between salt and sediments. Case number is shownin Table 1 and material parameters are shown in Table 3. VE =vertical exaggeration.The orientation of intermediate principal stress (s2)similar to the total-stress plot, although the magni-tudes are approximately half (compare Figure 7A,B). The differences in magnitude between total andeffective stresses are also illustrated in vertical pro-files through the salt body and through the sedi-ments just beyond the edge of the salt (Figure 8).As discussed in the model setup section, the forceequilibrium equation can be solved in total stress(see equation 1 in Appendix, case 1), and thenhydrostatic pore pressure can be subtracted; or theforce equilibrium equation can also be solved in ef-fective stress (see equation 2 in Appendix, case 2).Either way produces identical results (Figures 7, 8).Cases 1 and 2 are thus equivalent.

These results are essentially equivalent to previ-ously published results of relaxation of a salt spherein an elastic medium (e.g., Fredrich et al., 2003;Koupriantchik et al., 2005; Mackay et al., 2008;

Figure 10. Perturbations of vonMises stress (DsVM) predicted by(A) the elastic-perfectly-plasticmodel using the inner circle of theDrucker-Prager yield criterion(DP-i) in case 4, (B) the elasticmodel in case 1, and (C) differ-ence in perturbations betweenthe two models. The whitesemicircle is the interface be-tween salt and sediments. Thecase number is shown in Table 1and material parameters areshown in Table 3. VE = verticalexaggeration.plane around the margins of the salt (Figure 6C).Thus, with depth, the orientation of intermediateprincipal stress rotates from out-of-plane, to hor-izontal, and then back to out-of-plane orientations(Figure 6B). The orientation of minimum principalstress also rotates from horizontal to out-of-planeand back to horizontal orientations with increasingdepth (Figure 6C). These results show that out-of-plane stress near the salt is minimum principalstress and explain that stress changes in Figure 5Cand D are equal just around the salt sphere.

Effective stress is total stress minus pore pres-

sure. Because pore pressure is assumed to be hy-is generally horizontal (or radial) within the verticalplane along the equatorial regions of the salt (bars inFigure 6B); however, it is oriented out of the planeabove and below the salt (dots in Figure 6B). Theorientation of minimum principal stress (s3) is hor-izontal (or radial) within the vertical plane aboveand below the salt, whereas it is oriented out of theLuo et al. 53

panying relaxation of a salt sphere and to provide a nitude of plastic strain is defined by

point of reference against which to compare theelastoplastic results presented next.

Elastoplastic Sediments: Case 3 and Case 4

Effects of PlasticityTo examine the effects of plasticity, we first com-pare results from an elastic-perfectly-plastic model(case 4) with those of the elastic model (case 1 =case 2). For this comparison, we use the inner circleof the Drucker-Prager plastic yield criterion (seeAppendix) with an effective-stress model, assum-ing hydrostatic pore pressure in sediments. In per-54 Stress Around Salt BodiesePlastic 23epl1 2 + epl2 2 + epl3 2

r3

where epl1 , epl2 , and e

pl3 are three principal compo-

nents of the plastic strain tensor.In this example, plastic strain is asymmetric

(Figure 9). In general, areas with high vonMises stressin the elastic model (Figure 10B) also have high vonMises stress in the elastoplastic model (Figure 10A).However, final vonMises stress is significantly lowerin the elastoplastic model (Figure 10A, B) becausesediments yield and plastic laws impose limitations onSanz and Dasari, 2010). We include them to pro-vide physical insight into the deformation accom-

fect plasticity, when stresses in sediments reachthe yield value, plastic strain will occur. The mag-

Figure 11. Final total stresses for the ver-tical profile through center of the salt spherepredicted by the elastic-perfectly-plasticmodelusing the inner circle of the Drucker-Prageryield criterion (DP-i) in case 4 and the elasticmodel in case 1: (A) vertical stress and(B) horizontal stress. Gray dashed linesshow far-field horizontal and vertical stresses,respectively. The assumed hydrostatic porepressure profile is shown. Horizontal stressis equal to out-of-plane stress for this pro-file. These cases are the same as those inFigure 10. Symbols defined in Table 2.Figure 12. Final total stresses for the vertical profile through the sediments just beyond the edge of the salt sphere predicted by the elastic-perfectly-plastic model using the inner circle of the Drucker-Prager yield criterion (DP-i) in case 4 and the elastic model in case 1: (A) verticalstress, (B) horizontal stress, and (C) out-of-plane stress. The assumed hydrostatic pore-pressure profile is shown. The gray dashed lines showfar-field horizontal and vertical stresses, respectively. These cases are the same as those in Figure 10. Symbols are defined in Table 2.

differential stresses. The concentration of von Misesstress at the lateral edge of the salt sphere is thusreduced, and stresses are redistributed (Figure 10C).

In a vertical profile through the salt center, ver-tical stresses predicted by the elastoplastic modeljust above and below the salt are slightly lower thanthose of the elastic model (Figure 11A). In contrast,horizontal stresses in the elastoplastic model arehigher (715MPa [10152176 psi]) than thosepredicted by the elastic model (Figure 11B). In avertical profile through the sediments just beyondthe edge of the salt sphere, the elastoplastic modelpredicts vertical stresses about 10MPa (1450 psi)lower than the elastic model at the salt boundary(Figures 12A, 13A). The elastoplastic model pre-dicts horizontal stresses at the lateral edge slightlyhigher than the elastic model (Figures 12B, 13B);out-of-plane stresses predicted by the elastoplasticmodel at the lateral edge are about 4MPa (580 psi)higher than those predicted by the elastic model(Figures 12C, 13C). All results show the sameeffects of plasticity on stresses around the salt sphere:the elastic model has no limit for shear stresses, butthe elastoplastic model yields at higher deviatoricstresses, limiting stress levels in the model.

Impact of Different Plastic Yield Criteria andModeling ApproachesDifferent plastic constitutive models have a signif-icant impact on resulting plastic strain distributionsand stresses (Figures 14, 15). The inner circle of theDrucker-Prager plastic yield criterion predicts thelargest plastic deformation zone and the largestplastic strain magnitude (Figure 14A). The outercircle of the Drucker-Prager plastic yield criterionpredicts plastic deformation only above and belowthe salt sphere, but noplastic strain at the lateral edgeof the salt sphere (Figure 14B). TheMohr-Coulombplasticity shows a plastic deformation zone at thelateral edge and predicts plastic strain interme-diate between those from the inner and outer cir-cles of the Drucker-Prager plastic yield criterion(Figure 14C). The inner circle of the Drucker-Pragerplastic yield criterion is thus weakest, and the outercircle strongest.

All three elastoplastic constitutive models areeffective-stress models (case 4), and the rock failsin shear at a certain effective stress. A different ap-proach, popular among published studies (e.g.,Schultz-Ela, 2003), is to modify the internal fric-tion angle and cohesion parameters in a total-stressmodel (case 3B) to account for the effect of linearlyincreasing pore pressure with depth. This method-ology is detailed in the Appendix for a hydrostaticpore-pressure profile and the Mohr-Coulomb plas-ticity criterion. A total-stress model (case 3B) withan effective internal friction angle of 15.95 and

Figure 13. Final total stresses for the horizontal profile throughthe center of the salt sphere predicted by the elastic-perfectly-plastic model using the inner circle of the Drucker-Prager yieldcriterion (DP-i) in case 4 and the elastic model in case 1: (A)vertical stress, (B) horizontal stress, and (C) out-of-plane stress. Theassumed hydrostatic pore pressure profile is shown. These casesare the same as those in Figure 10. Symbols are defined in Table 2.Luo et al. 55

in (D). The white semicircle is the interface between salt and sediments. The case number is shown in Table 1 and material parameters areshown in Table 3. VE = vertical exaggeration.effective cohesion of 0.027 MPa (3.9 psi) is equiv-alent to an effective-stress model (case 4) underhydrostatic pore pressure and with an internal fric-tion angle of 30 and cohesion of 0.03MPa (4.4 psi)(see equations 1013 in Appendix). Comparing

results from the twomodeling approaches, we find

56 Stress Around Salt Bodiesthat results from the total-stress system show asmaller plastic strain zone (Figure 14C, D) andmoreconcentration of von Mises stress (Figure 15C, D)than those predicted by the effective-stress system.

The differences between the two modeling ap-

proaches are caused by (1) the difference in frictionFigure 15. Perturbations of von Mises stress predicted by different elastic-perfectly-plastic models: (A) the inner circle of the Drucker-Prager yield criterion (DP-i), (B) the outer circle of the Drucker-Prager yield criterion (DP-o), (C) the Mohr-Coulomb yield criterion (MC)in effective-stress modeling in case 4, and (D) the Mohr-Coulomb yield criterion in total-stress modeling with effective internal frictionangle and effective cohesion in case 3B. The white semicircle is the interface between salt and sediments. These cases are the same asthose in Figure 14. VE = vertical exaggeration.Figure 14. Magnitude of plastic strain predicted by different elastic-perfectly-plastic models: (A) the inner circle of the Drucker-Prageryield criterion (DP-i), (B) the outer circle of the Drucker-Prager yield criterion (DP-o), (C) the Mohr-Coulomb yield criterion (MC) ineffective-stress modeling in case 4, and (D) the Mohr-Coulomb yield criterion in total-stress modeling with effective internal friction angleand effective cohesion in case 3B. Hydrostatic pore pressure is assumed. An internal friction angle of 30 and cohesion of 0.03 MPa(4.4 psi) are used in (A), (B), and (C). An effective internal friction angle of 15.95 and effective cohesion of 0.027 MPa (3.9 psi) are used

lar sCoulensio. Thementangles, which suggests the difference of plastic fail-ure, even for the same stress state, and (2) the as-sumption that the ratio l of pore pressure and solidpressure (P s1s32 ) is equal to Hubbert-Rubeypore-pressure ratio lHR (Hubbert and Rubey, 1959;Rubey and Hubbert, 1959), suggesting that solidpressure equals the weight of density-integral over-burden (see Appendix), which is unsuitable in thiscase. The latter is especially true at the two polesand lateral edge of the salt sphere, where solid pres-sure may deviate from the weight of density-integraloverburden (far-field vertical stress) by a wide mar-gin (Figure 4).

Figure 16. Deformation pattern within and around the irregupredicted by the elastic-perfectly-plastic model with the Mohr-(B) displacement vectors. Red is contractional strain and blue is extstrain, but strain within the salt includes elastic and viscous strainvectors (B). The white curve is the interface between salt and sediare shown in Table 3. VE = vertical exaggeration.In summary, we compare model results fromeling approaches and different plastic constitutivelaws can also predict different results.

A Complex Salt GeometryWe additionally explore a more complex saltgeometryan irregular salt sheetwhichwe solveusing an effective-stress approach assuming hydro-static pore pressure (case 4) in the 2-D axisym-metric model. We use the elastic-perfectly-plasticconstitutive lawwithMohr-Coulomb plasticity forthe sediments.

The outer surface of the irregular salt sheet hasan undulating topography (Figure 16A): where the

alt sheet during the processes of viscoelastic stress relaxationomb plastic yield criterion in case 4: (A) horizontal strain andnal strain. Note that strain in sediments includes elastic and plasticblack box in (A) shows the location of the plot of displacements. The case number is shown in Table 1 and material parameterssalt is thicker, the outer surface is convex; where

cases 3 and 4.We assume hydrostatic pore pressurein the sediments and include plasticity in the sed-iments in modeling. Because both the Drucker-Prager and theMohr-Coulomb plastic yield criteriaare commonly used in numerical models for saltsystems (e.g., Poliakov et al., 1993a, 1996; Schultz-Ela and Walsh, 2002; Dirkzwager and Dooley,2008), we also explore and compare effects of dif-ferent plastic yield criteria: the Drucker-Prager (in-ner andouter circles) and theMohr-Coulombplasticyield criteria in effective-stress models in case 4 (seeequations 69 in Appendix) and theMohr-Coulombplastic yield criterion in a total-stress model in case 3(see equations 1013 in Appendix). In short, elasto-plastic models can predict much different resultsfrom those by elastic models, and different mod-

it is thinner, the outer surface is concave. Hori-zontal extension occurs in the sediments adjacentto where the salt surface is convex; in these loca-tions, the salt is vertically contracted and hori-zontally extended in a manner similar to that ofthe salt sphere (compare Figures 3, 16). In con-trast, a strong horizontal contraction exists in thesediments adjacent to where the salt surface is con-cave (Figure 16A).

Plastic strain occurs at all convex surfaces of theirregular salt sheet (Figure 17A), and von Misesstress decreases (Figure 17B). The von Mises stressalso rises above and below areas where salt is con-cave (warm colors in Figure 17B). The minimumprincipal stress, as a proxy for the fracture gradient,is decreased at all locations where the salt surfaceLuo et al. 57

58 Stress Around Salt Bodiessults explain why large horizontal contraction oc-curs (Figure 16A) and horizontal stresses increasewhere the salt surface is concave (Figure 18B). Ourmodel results are comparable with previously pub-lished results of finite-element modeling around anirregular salt sheet (e.g., Fredrich et al., 2007).

DISCUSSION

Analysis of Model Results

These results show that elastic and elastoplasticmodels predict significantly different stresses aroundsalt. Stress perturbations generated by elastoplasticmodels are less than those generated by elasticmodels (Figure 10). Differences occur when differ-ent plastic yield criteria are used in calculations:the Drucker-Prager inner-circle criterion producesa large plastic strain in our example, whereas theDrucker-Prager outer-circle criterion creates onlysmall amounts of plastic strain. Elastoplastic mod-els have previously been applied to salt basin evo-lution and deformation modeling around salt (e.g.,Schultz-Ela and Walsh, 2002; Schultz-Ela, 2003;Dirkzwager and Dooley, 2008). Plasticity will bet-ter simulate stress perturbations around salt bodies,which are known to involve plastic deformationin nature (e.g., Ratcliff et al., 1992; Davison et al.,2000b; Schultz-Ela, 2003).

The choice of initial stress field can impactmodeling results. We choose to represent the basinin a state of uniaxial strain or under K0 conditions.This is suitable for geologic environments, in whichstresses are generated primarily by depositional pro-cesses. However, many different far-field stressescan be assumed. For example, salt structures canalso exist in environments of horizontal compressionor extension (e.g., Jackson et al., 1994; Stefanescuet al., 2000). An alternative approach would be toassume a critically stressed system, either in ex-tension or compression (e.g., Zoback, 2007).

All of the models show the same general stressand deformation patterns around a relaxing saltsphere. The salt shortens vertically and expandslaterally, producing zones of extension at the topand base of the sphere and a zone of shorteningis convex (cold colors in Figure 17C). However,minimum principal stress is high where the saltsurface is concave (Figure 17C).

In this example, mean stress within the salt con-verges approximately to the vertical stress (Figure 18).The increase in mean stress within the salt causesadditional horizontal compression and contractionin the sediments on the left and right sides of wherethe salt surface is convex (Figures 16, 18). These re-

Figure 17. (A) Magnitude of plastic strain, (B) perturbations ofvon Mises stress, and (C) perturbations of minimum principal stresspredicted by the elastic-perfectly-plastic model with the Mohr-Coulomb yield criterion in case 4 for the irregular salt sheet. Thewhite curve is the interface between salt and sediments. This caseis the same as that in Figure 16. VE = vertical exaggeration.

with this deformation, vertical, radial, and hoopstresses are all reduced at the top and bottom of

Zoback, 2007) and underground excavation andthe sphere. Vertical and radial stresses are both in-creased around the equator of the sphere, accom-panied by a minor reduction in hoop stress. Finally,principal stresses are rotated near the salt, so thatthey are no longer necessarily horizontal or vertical.In the ideal situation of a fully relaxed salt body,deviatoric stresses are zero within the salt, and be-cause of stress continuity at the salt interface, one ofthe principal stresses at the salt interface is alwaysoriented normal to the salt surface and the othertwo principal stresses are within the salt-sedimentinterface (Figure 6).

Comparison with Previous Workaround the equator of the sphere. In accordance

tunneling problems (e.g., Hoek and Brown, 1996).

The pattern of stresses that we have modeled isconsistent with that of stresses encountered aroundnatural salt bodies. For example, Rohleder et al.(2003) showed the observed gradient decrease inminimum principal stress below salt, and othershave reported lost circulation above salt, suggestinglowminimum principal stress (e.g., Bradley, 1978;Whitson and McFadyen, 2001; Dusseault et al.,2004). These observations are consistent with ourpredicted stresses (e.g., Figures 4A, 18A). Ourmodels thus suggest that lateral spreading of salt maybe a cause of low minimum principal stress encoun-tered above and below salt in many wells. Radial andconcentric extensional faults near the salt diapirshave been observed and reported (e.g., Davison et al.,This stress pattern around a salt sphere is analogousto the pattern of stress distribution around a cavity inan elastic material (e.g., Kirsch, 1898; Timoshenkoand Goodier, 1970). When a hole or tunnel is cre-ated, bridging occurs: horizontal and vertical ten-sion occurs above and below the hole, whereashorizontal and vertical compression occurs at thelateral edges of the hole (e.g., Hoek and Brown,1996). In our models, a reduction of vertical stressesabove and below the salt sphere causes stress re-distribution around the salt, and as a result, thesestresses increase along the equator (Figure 5A). Thisapproach has been applied as a way of explainingborehole stresses (e.g., Peska and Zoback, 1995;

Figure 19. Cartoon showing extensional fault patterns aroundsalt sphere predicted by our models.Figure 18. Final total stresses for twovertical profiles through the irregular saltsheet predicted by the elastic-perfectly-plastic model with the Mohr-Coulomb yieldcriterion in case 4: (A) the profile throughsalt convex curves and (B) the profilethrough salt concave curves. The graydashed lines show far-field horizontal andvertical stresses, respectively. The assumedhydrostatic pore pressure profile is shown.This case is the same as that in Figure 16.Symbols defined in Table 2.Luo et al. 59

mum principal stress at salt boundaries. In con-

sity of sediments or salt bodies, and g is gravitational accelera-tion. The relation between total stress sij, effective stress s

0ij, and

pore pressure u can be expressed as s0ij s ij diju (i; j r; q; z)trast, plastic yielding in elastoplasticmodels keepsshear stress lower.

2. Large deviatoric stresses (or vonMises stress) areconcentrated at the lateral edge of a salt sphere,2000a, b; Dusseault et al., 2004). The pattern ofour model stresses near the salt sphere (Figure 6)suggests these extensional faults (Figure 19).

Practical Implications

In the irregular salt sheet that we simulate, mini-mum principal stress drops where salt is thick (atconvex salt surfaces), but not where salt is thin (atconcave salt surfaces) (Figure 17C), suggestingthat a more stable drilling path may lie where thesalt surface is concave. On the basis of our model-ing and other numerical experiments (e.g., Fredrichet al., 2003), we find that mean stress within saltconverges toward far-field vertical stress for thosesalt bodies that arewide and thin (e.g., a salt sheet),whereas mean stress converges approximately tofar-field horizontal stress when salt bodies are nar-row and tall (e.g., a salt dome); salt bodies with amore equal aspect ratio, such as the salt sphere,havemean stress roughly converging to the far-fieldmean stress.

Modeling of stresses around salt will, ultimately,be fully coupled in the sense that pore pressure andfluid flow will be coupled to sediment deforma-tion induced by salt behavior (Nikolinakou et al.,2012), and future models will simulate the geologicevolution of these systems. In systemswith plasticity(unlike elastic models), because the history of de-formation will impact the final state, modelingtheir evolution will become critical. The elasto-plastic simulations that we have presented hereinprovide the needed insight into stresses and de-formation around salt systems.

CONCLUSIONS

1. Elastoplastic models of sediments can simulatestresses around salt more realistically than elas-tic models. Elastic models induce unrealisticallylarge shear stress and unrealistically low mini-60 Stress Around Salt Bodies(Terzaghi, 1925). The last two components in both total-stressvector and effective-stress vector are zero. Here compressionalstress is positive.

Constitutive Laws for Sediments

For elastic sediments, the constitutive equation is Hookes law(e.g., Timoshenko and Goodier, 1970; Zoback, 2007). Whenwe consider solid material (case 1), we express Hookes law interms of total stress. However, when pore pressure is consid-ered (case 2), we express Hookes law with effective stress.

For elastoplastic sediments, we consider Mohr-Coulombplasticity and Drucker-Prager plasticity (e.g., McLean andAddis, 1990; Wood, 1990; Dunne and Petrinic, 2005; Zoback,mean stress is dropped above and below the saltsphere, and minimum principal stress is loweredeverywhere around the salt sphere.

3. A model of an irregular salt sheet shows thatmean stress within the salt converges to the far-field vertical stress and that minimum principalstress is lowest where the salt is thick.

4. By providing insight as to where increased andsuppressed magnitudes of minimum principalstress (fracture gradient) are expected, this studyhas implications for drilling near salt bodies.

APPENDIX

Force Equilibrium Equation

Using the commercial finite-element analysis software, Abaqus,we solve the force equilibrium equation for the axisymmetricproblem in cylindrical coordinate system of (r, q, z). We cansolve the equation in either total stress (equation 1) or ef-fective stress (equation 2):

@srr@r +

@srz@z +

srrsqqr 0

@srz@r +

@szz@z +

srzr + fz 0

srq szq 0

8>: 1

@s 0rr +u@r +

@s0rz

@z +s0rrs

0qq

r 0@s

0rz

@r +@s 0zz + u

@z +s0rzr + fz 0

s0rq s

0zq 0

8>>>>>>>:

2

where fsg srrszzsqqsrzsrqszqT is the total stress vector,fs 0 g s 0rrs

0zzs

0qqs

0rzs

0rqs

0zqT is the effective-stress vector, u is

pore pressure, fz = rg is gravitational body force, r is bulk den-

2007). Similarly, if the material is considered solid (case 3),these laws are written in terms of total stress, whereas if thepresence of pore pressure is included (case 4), the lawsmust beexpressed in terms of effective stress.

Maxwell Linear Viscoelastic Constitutive Law

We simulate salt using a Maxwell linear viscoelastic body,which is similar to that of previous studies (e.g., Schultz-Ela andWalsh, 2002; Schultz-Ela, 2003; Koupriantchik et al., 2005).Incremental strain includes elastic and viscous components:

fdeg fdeeg+ fdevg 3

Incremental elastic strain is given by the elastic Hookeslaw (e.g., Timoshenko and Goodier, 1970), and incrementalviscous strain is given in equation 4 (e.g., Li et al., 2009; Luo andLiu, 2009, 2010):

fdevg Q1fs tgdt; 4

13 16 16 0 0 0

2 3

1 1 1 0 0 06 7tic yield surface, and the outer circle of the Drucker-Pragerplastic criterion inscribes it in the principal stress space andthe p plane (Figure 20). Here, nonassociated plastic flowrules are used in the modeling.

FDP J2

p aI 01 b 6

a1 2 sin f3

p 3+ sin fb1

6C cos f3

p3+ sin f

a2 2 sin f3

p 3 sin fb2

6C cos f3

p 3 sin f

7

FMC sinq +

p3

+

3

p

3cos

q +

p3

sin f

" # J2

p sin f

3I01 C cos f

8

q 13cos1

33

pJ332

9Figure 20. The Mohr-Coulomband the Drucker-Prager plasticyield criteria in the principalstress space (A) and p plane (B).Symbols are defined in Table 2.2J2Q1 1h

6 3 6 16 16 13 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

666664777775; 5

where [Q] is the material matrix in viscosity h, fs tg s trrs tzzs tqqstrzs trqs tzqT is total-stress vector at time t, and dt isthe time increment.

Plastic Yield Criteria

We apply both the Drucker-Prager and the Mohr-Coulombplastic yield criteria: FDP and FMC (equations 69) in themodeling (Abaqus, 2009). We also compare two Drucker-Prager plastic yield criteria: the inner circle of the Drucker-Prager plastic criterion circumscribes the Mohr-Coulomb plas-

where I01 is the first invariant of effective-stress tensor, J2 and

J3 are the second and third invariants of deviatoric stress ten-sor, respectively, f is internal friction angle,C is cohesion, andq is the deviatoric polar angle. The a1 and b1 and a2 and b2 inequation 7 are two groups of parameters for the inner and out-er circles of the Drucker-Prager plastic yield criterion, respec-tively (e.g., McLean and Addis, 1990). Here, compressionalstress is positive.

Simulating Effect of Pore Pressure byReducing Internal Friction Angle

This approach has been used widely in previous studies (e.g.,Schultz-Ela, 2003; Ings et al., 2004; Gemmer et al., 2005).Weuse 2-D Mohr-Coulomb plasticity as an example to describeLuo et al. 61

REFERENCES CITEDthe approach.TheMohr-Coulombplastic yield criterion can begiven by

FMC 12 s1 s3 12s1 + s3 u

sin fC cos f 10

where s1 and s3 are maximum and minimum principal totalstresses, u is pore pressure, f is internal friction angle, andC iscohesion. Pore pressure is commonly assumed to be equal tolP (u = lP), where P is solid pressure (P s1s32 ) and l is theratio of pore pressure and solid pressure (e.g., Gemmer et al.,2004, 2005; Ings et al., 2004). The Mohr-Coulomb plasticyield criterion can be changed to

FMC 12 s1 s3 12s1 + s31 l sin fC cos f

12s1 s3 12 s1 + s3 sin feff Ceff cos feff

11

where feff and Ceff are the effective internal friction angle andeffective cohesion, respectively, and they are given as

sin feff 1 l sin fCeff cos feff C cos f

12

Here, we assume that l is equal to Hubbert-Rubey pore-pressure ratio lHR, and under the conditions of hydrostaticpore pressure, they are given in the equation (Hubbert andRubey, 1959; Rubey and Hubbert, 1959)

l lHR rwrsed13

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3 and rsed = 2220 kg/m3).

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sV rgz 14

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KT0 sHsV

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K0 KT0 r rwr rw

: 18

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