Comput MechDOI 10.1007/s00466-015-1123-8

ORIGINAL PAPER

A fully coupled porous flow and geomechanics model for fluiddriven cracks: a peridynamics approach

Hisanao Ouchi Amit Katiyar Jason York John T. Foster Mukul M. Sharma

Received: 25 September 2014 / Accepted: 15 January 2015 Springer-Verlag Berlin Heidelberg 2015

Abstract A state-based non-local peridynamic formula-tion is presented for simulating fluid driven fractures inan arbitrary heterogeneous poroelastic medium. A recentlydeveloped peridynamic formulation of porous flow has beencoupled with the existing peridynamic formulation of solidand fracturemechanics resulting in a peridynamicmodel thatfor the first time simulates poroelasticity and fluid-drivenfracture propagation. This coupling is achieved by modelingthe role of pore pressure on the deformation of porous mediaand vice versa through porosity variationwithmediumdefor-mation, pore pressure and total mean stress. The poroelasticmodel is verified by simulating the one-dimensional consol-idation of fluid saturated rock. An additional porous flowequation with material permeability dependent on fracturewidth is solved to simulate fluid flow in the fractured region.Finally, single fluid-driven fracture propagation with a two-dimensional plane strain assumption is simulated andverifiedagainst the corresponding classical analytical solution.

Keywords Peridynamic theory Hydraulic fracturing Coupled fluid flow and geomechanics Crack propagation Non-local formulation

H. Ouchi A. Katiyar J. York J. T. Foster M. M. Sharma (B)Department of Petroleum and Geosystems Engineering, TheUniversity of Texas at Austin, 200 E. Dean Keeton St.,Stop C0300, Austin, TX 78712-1585, USAe-mail: msharma@mail.utexas.edu

A. Katiyare-mail: akatiyar@austin.utexas.edu

J. T. Fostere-mail: john.foster@utsa.edu

1 Introduction

The propagation of fluid driven cracks is important in avariety of applications such as hydraulic fracturing (HF),which refers to fluid pressure induced deformation, dam-age and fracture propagation in a porous material. Thisprocess has become particularly important for the stimu-lation of unconventional hydrocarbon reservoirs. In hetero-geneous, anisotropic and highly fractured poroelastic geo-logic settings, three-dimensional (3-D) fractures may initiatein a non-preferred direction, become non-planar and multi-stranded, and compete with neighboring growing fractures[16]. The prediction of such a complex fracture geometry(i.e. length, width and height) and created complex networkis crucial for the design of hydraulic fracturing treatments[7] of unconventional reservoirs. Several HF models start-ing from 2-D analytical- Perkins-Kern-Nordgren (PKN) andKristonovich-Geertsma-Daneshy (KGD)- and pseudo-three-dimensional (P3-D) models have been developed [8]. TheP3-D HF models [912], being comparatively fast, sufficethe need of engineering design and on-job evaluation. Planarand non-planar 3-D models substantially improve simula-tion predictions over P3-D models [1321] however may notincorporate all of the above mentioned complexities impor-tant to HF.

Fully coupled 3-D rock mechanics and porous flow mod-els can in principle predict complex fracture geometry. How-ever, there are limitations to previously developed 3-D HFmodels [22]. 3-D HF models based on classical continuummechanics [7,2325] and linear elastic fracture mechanics(LEFM), besides being computationally expensive [22], areunable to capture material heterogeneity, the non-linear con-stitutive response of pressurized fluid-saturated-poroelasticrocks as well as the non-linear process zone physics arounda fracture tip. There is evidence that previous models do

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not reproduce observations from the field; for instance, oftenthe treating pressures required to open and extend fracturesexceed the pressures predicted by these models [26,27]. Thiscould be a result of the creation of multiple, non-planar frac-tures or LEFMs neglect of the actual material state in theso called process zone surrounding the fracture tip. Thisprocess zone could include inelastic deformations due tomicro-cracking and/or pore collapse and highly anisotropicstress states. Bazant [28,29] has shown that quasi-brittlematerials exhibit a fairly deterministic length-scale causedby stress redistributions and energy release associated withthe growth of a large process zone size or the propaga-tion of a long stable fracture. This suggests the need for amodel that incorporates the process zone physics with anappropriate constitutive model and process zone size withan inherent length scale or non-locality. State of the artcomputational methods attempt to extend the BEM or FEMmethods i.e. extended finite element models (XFEM) or theuse of cohesive elements. However, previously developedpartial differential equations (PDEs) based models do notincorporate a length-scale and result in computational meth-ods that are incapable of resolving the finite damage fieldsor process zone size. Another common method uses theDiscrete Element Method (DEM), which severely restrictsthe fracture propagation to only element boundaries andcauses extreme mesh dependence. A 3-D HF model basedon DEM provides physical insight at microscopic scales[3032] but does not present much improvement over theclassical continuum mechanics based models for field-scaleproblems.

We here develop a comprehensive 3-DHFmodel based ona recently developed nonlocal theory of continuum mechan-ics known as peridynamics [33,34] which was proposed toovercome some of the intrinsic limitations of the correspond-ing local theory in dealing with discontinuous displacementfields. The essence of the peridynamic model is that inte-gration, rather than differentiation, is used to compute theforce at a material point. Since the spatial derivatives are notused, the equations remain equally valid at points or surfacesof discontinuity [33]. The peridynamic theory has been suc-cessfully applied to diverse engineering problems [3537]involving autonomous initiation, propagation, branching andcoalescence of fractures in heterogeneous media. However,peridynamics has not been applied to simulate fluid pressuredriven deformation and damage of porousmedia. Turner [38]utilized the theory of interacting continua [3941] and pre-sented a formulation for incorporating the effects of porepressure in the state-based peridynamic formulation of solidmechanics. However, to avoid the complexities of a fullycoupled geomechanics and fluid flow model, he assumedthe fluid pressure to be either known or opted to solve forit numerically or analytically through classical means. Inour fully implicitly coupled, 3-D peridynamic formulation of

fluid driven deformation and damage,we followhis approach[38] to include the effects of pore pressure into the peridy-namic equation of motion however we also solve for the fluidpressure in the medium as well as in the fracture space usingour recently developed peridynamic porous flow formulation[42]. The additional steps of the coupling for the appropriatefluid-solid interaction in an arbitrary heterogeneous mediumare presented in the mathematical modeling section.

In the subsequent section, we present a review of the gov-erning equations of the state-based peridynamic formulationof solid mechanics. We then present our model to simulateporoelasticity and material failure in peridynamics. A briefoverview of the peridynamic porous media flow formula-tion is presented followed by our approach to couple it withthe peridynamic formulation of solid mechanics and fluidflow inside the fracture space. We also report a novel wayto implement the non-local traction boundary condition andverify it by comparing the deformation of a 2-D elastic bodyin plain-strain obtained using the peridynamic theory andthe classical theory. In the last section, we verify the peri-dynamic poroelasticity model through comparison with theclassical 1-D consolidation problem.Wedemonstrate that thenonlocal model can recover well-known classical solution ofporoelasticity. Finallywe simulate the propagation of a singlefracture in a poroelastic medium. The peridynamic solutionof fracture geometry and injection pressure in 2-D plain-strain is verified by comparing the numerical results with the2-D analytical KGD model [43,44]. The stress field arounda fracture of known length and fluid pressure from peridy-namic formulation is verified with the well-known analyticalsolution by Sneddon [20].

2 Mathematical model

In this paper, we use both lower-case and upper-case lettersfor scalars, e.g. , , t , A, V , lower case boldface letters forvectors, e.g., x, u, , upper-case bold face letters for second-order scalar valued tensors, e.g., K and blackboard letters forsecond-order tensors otherwise, e.g., K.

In the state-based peridynamic formulation, mathematicalobjects called peridynamic states have been introduced forconvenience [45]. Peridynamic states depend upon positionand time, shown in square brackets, and operate on a vec-tor, shown in angled brackets, connecting any two materialpoints. Depending on whether the value of this operation is ascalar or vector, the state is called a scalar-state or a vector-state respectively. To differentiate, peridynamic scalar statesare denoted with non-bold face letters with an underline andperidynamic vector states are denoted with bold face letterswith an underline. The mathematical definition of these peri-dynamic states is provided wherever they have been used inthis work.

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Fig. 1 Schematic diagram of aperidynamic material point xand its interactions (bonds) in itsnon-local neighborhood(horizon) Hx of length

2.1 Review of the state-based peridynamic formulation forsolid mechanics

Peridynamic theory is a non-local continuum formulationof the classical solid mechanics theory. The classical con-servation of linear momentum at position x in the referenceconfiguration and time t takes the following form

u[x, t] = [x, t] + b[x, t], (1)

where, is the mass density, u is the vector-valued displace-ment field, is the first PiolaKirchoff stress tensor and b isthe body force density. Peridynamic theory assumes a mater-ial to be composedofmaterial points, each interactingwith allits neighboring points, within a nonlocal region known as thehorizon (Fig. 1). Each interaction pair of a material pointwith its neighboring material point is referred as a bond(Fig. 1). The peridynamic equation of motion at a point thusinvolves an integral functional summing up the force den-sities in all the connected bonds over the horizon in placeof the divergence of the stress field in the classical theory(1). The current model utilizes the generalized state-basedperidynamic formulation that overcomes several limitationsof the bond-based model, able to model only an isotropic,linear, micro-elastic material resulting in an effective Pois-son ratio of 1/4 in 3-D [45,46]. The state-based peridynamicapproach allows constitutive material response at a point todepend collectively on the deformation of all the bonds con-nected to that point in addition to providing the ability toincorporate any classical constitutive model into peridynam-ics through the method of correspondence [45]. In the rest ofthis document we drop the explicit dependence on time, t tomake the notation concise. The generalized state-based peri-dynamic equation of motion for a material point x is givenas

u[x] =Hx

(T [x] T [x] )dVx + b[x] (2)

x

x'

x x'

x x'

(a) (c)

(b)

Fig. 2 a Schematic diagram of reference and deformed positions ofa material point x and its neighbor x and resulting peridynamics ref-erence and deformed position vector states, b schematic diagram ofnon-ordinary state-based force vector states T and c schematic diagramof ordinary state-based force vector states T

whereHx is the neighborhood of x with characteristic length (Fig. 1), T is the peridynamic force vector state, x is theposition vector of the neighboring material points of x, anddVx is the differential volume of x inside the horizon ofx. The relative position of material points x and x in thereference configuration is given by the reference positionvector state X denoted as bond vector

X = x x = , (3)and the relative position of the same material points in thedeformed configuration is given by the deformed positionvector state Y

Y = y (x) y (x) = + , (4)where the relative displacement is given as

= u (x) u (x) . (5)The above mentioned states are geometrically shown inFig. 2. The bond length in the reference and the deformed

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configurations are given by the following scalar states respec-tively

x = and y = + . (6)

The constitutive material model determines how the peridy-namic force state T depends on the peridynamic deformationstate Y . In peridynamics, a material is called ordinary (Fig.2) if

T [x] = t [x] [

+ +

], (7)

where the last term in the bracket is the unit vector that pointsfrom the deformedposition ofx toward the deformedpositionof x. t , the peridynamic scalar force-state, represents themagnitude of the force density in a bond. For demonstration,we choose the constitutive model of a linear peridynamicsolid for which the scalar force-state in 3-D is defined as

t [x] = 3K m

x + 15Gm

ed ,

ed = e x3

, (8)

where is the influence function, K and G are the bulk andshear modulus of the solid. Inside the nonlocal neighborhoodof a point, influence function provides an additional mecha-nism to modulate the nonlocal contributions in computationof volume-dependent quantities [47]. The elongation scalarstate e, weighted volume m, and dilatation are defined by

e = y x = + , (9)m = x x =

Hx

x x dVx , (10)

= 3mx e = 3

m

Hx

x edVx . (11)

For a small deformation, is a measure of the volumetricstrain. As highlighted by Silling [45], for an isotropic defor-mation of the form Y = (1+ )X for all x, with constant|| 1, = 3 which is same as the trace of the linearizedstrain tensor in the classical theory. Since this work reportsthe numerical results for 2-D plane strain problems, the fol-lowing are the corresponding definitions in 2-D:

t [x] = 2(K G3

)

mx + 8G

m ed, (12)

ed = e x3

, (13)

m = x x =Hx

x x d Ax , (14)

= 2mx e = 2

m

Hx

x ed Ax , (15)

where d Ax is the differential area of x inside the horizon ofx. Note that the horizon in 2-D is in form of a disk.

2.2 State-based peridynamic formulation of poroelasticity

To be able to simulate the poroelastic response of a porousmedium, an effective force-state was introduced to the peri-dynamic formulation [38] by modifying the isotropic part ofthe force scalar state as follows,

t [x] =

2

(K G3

)

m x +2pp

m px + 8Gm ed for 2D, 3K m x +

3ppm px + 15Gm ed for 3D.

(16)

Here is the Biot coefficient, pp is the pore fluid pressureand and p are influence functions. The above couplingfor a 3-D problemwas derived using the theory of interactingcontinua [38].

2.3 Material failure modeling in peridynamics

For the generalized state-based peridynamic formulationused here, Foster et al. [48] proposed an energy-based fail-ure criterion for a bond to break. Based on this model, whenthe energy density stored in a bond due to relative displace-ment of the associated material points exceeds the prede-termined critical energy density value, the bond breaks irre-versibly keeping only the contribution due to fluid pressureafterwards. The force scalar state of a broken bond is

td [x] =

2ppm px for 2D,

3ppm px for 3D.

(17)

The total energy density stored in a bond, w , is obtainedby integration of the dot product of the force density vectoracting on the bond

(T [x] T [x] ) and the relative

displacement vector of the two material points (x and x)forming the bond

w = (t f inal)0

(T [x] T [x] ) .d, (18)

where (t f inal

)is the final scalar value of the relative dis-

placement of the two material points. The peridynamic crit-ical energy density stored in a bond is obtained by summingthe energy required to create unit fracture area in peridynam-ics and equating it to the energy release rate, E . FollowingFig. 3, in peridynamics, to open a fracture surface of unitarea, the energy released to break all the bonds connectingeach point A along 0 z to each point B in the sphericalcap in 3-D and the cylindrical cap in 2-D across the fracturesurface is summed up by the integral in Eq. (18) using thecoordinate system centered at A

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Fig. 3 Evaluation of strain energy release rate in peridynamics. Foreach point A along the dashed line, 0 z , the work required tobreak the bonds connecting A to each point B in the spherical cap issummed using a spherical coordinate system centered at A

E =

0

20

z

cos1( z )0 wc

2 sin d d d dz for 3 D 0

z

sin1( z )sin1

(z

) wc d d dz for 2 D.

(19)

The above integral, when evaluated and solved for the criticalenergy density stored in a bond wc, we obtain

wc ={ 4E

4for 3 D

9E43

for 2 D (20)

This bond failure criterion is numerically implementedthrough the influence function in the constitutive materialmodel

={0 w > wc otherwise . (21)

In a peridynamic formulation of solidmechanics, materialfailure is modeled through a scalar field referred as damagedefined as the fraction of broken bonds at a material point inits horizon

d (x) =Hx dVx

Hx dVx . (22)

Damage, d at any point and time varies from 0 to 1, with 1representing all the bonds attached to a point broken. Oncebonds start to break and stop sustaining any tensile load, asoftening material response results leading to crack nucle-ation. However, only above a critical damage value, when anumber of bonds fail and coalesce onto a surface, fracturepropagates.

2.4 State-based peridynamic formulation of porous flow

We recently developed a state-based formulation of flow ofsingle Newtonian fluid of constant and small compressibilityin porous medium with following governing equations [42]

t([x][x]) =

Hx

(Q [x] Q [x] ) dVx

+R [x] I [x] , (23)Q [x] =

2

0

(K [x] 14 tr(K[x])I

)

4([x]

[x]) , (24)where is the fluid density, is the medium porosity, R andI are the mass source and sink respectively per unit volumeof the material point per unit time, Q is the peridynamicmass flow scalar state that maps a bond onto mass influxdensity in that bond, K is the material permeability tensor,0 is the density of the fluid at reference pressure p0, isthe fluid viscosity, is the flow potential and is a scalingfactor dependent on the horizon size and the dimension ofthe problem given as,

=

32

[Hx

4i24 d Ax

]1= 8

2for 2D,

32

[Hx

4i24 dVx

]1= 45

43for 3D.

(25)

Coupling of the porous flow formulation with the solidmechanics formulation in peridynamics is achieved by mod-ifying the porosity and permeability of a material point asa function of deformation, represented by dilatation. Firstfor porosity coupling, a novel porosity formulation devel-oped by Tran et al. [49] has been implemented in the currentperidynamic formulation. This formulation allows simulat-ing porosity as a function of volumetric strain that in turn is afunction of pore pressure, mean total stress and temperature.With an isothermal assumption, the reservoir porosity to beimplemented numerically takes the following form:

n+1 [x] = n [x] [1 Crpp]+Cb

(1 nv

) [pp +m

],

n [x] = Vnp

Vb0, (26)

where n represents a time step, Vb0 is the initial bulk volumeof a material point, V np is the pore volume of a material pointat nth time step,Cr andCb are the isothermal compressibilityof solid grains and bulk rock respectively, m is the totalmean stress, nv is the volumetric strain at nth time step and represents the differential change in a physical quantitywith time. Using Biots notion of effective stress in Eq. (27),we modify the above formulation of porosity by rewritingthe total mean stress in terms of the pore pressure and theeffective stress, resulting in Eq. (28)

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m = Klocal + pp, (27)n+1 [x] = n [x] [1 Crpp]

+ (1 local)[Crpp +local

], (28)

where peridynamic local dilatation local based on immediateneighborhoodHxlocal has been used to replace the volumetricstrain

local ={ 2

m

Hxlocal

xe d Ax for 2D,3m

Hxlocal

xe dVx for 3D. (29)

The variation in pore space through dilatation would affectthe medium permeability as well. However, in the currentwork, we have not modeled medium permeability as a func-tion of deformation unless the deformation is large enoughresulting in a fracture space.

2.5 State-based peridynamic formulation of fracture flow

To simulate fracture flow, we assume a critical damage para-meter dcr and define a critical stretch of a bond, scr

scr = + w=wc

. (30)

We assume that a fracture space is created in any two adjacentmaterial points when they separate beyond the critical stretchscr and d > dcr for both the points. Such material points arereferred as dual points, not only acting as porous mater-ial but also representing the fracture space for fluid to flow(Fig. 4) in between them.

In the current formulation, like the 2-D fluid flow insidea fracture opening modeled through lubrication theory, wehave chosen the nonlocal porous flow formulation with per-meability determined based on fracturewidth. The governingintegral equation based peridynamic formulation to solve forfracture flow is,

t

( f [x] f [x]

) =Hx

(Q [x] Q [x] ) dVx

+R [x]+ I [x] , (31)Q [x] =

2

f

f

22( f

[x] f [x]) ,

(32)

where f is the fluid density at the fracture fluid pressure, f is the scalar permeability of the dual point for fluid flowinside fracture, R is the mass generation per unit volume perunit time, f is the flow potential in the fracture space ofthe dual point with fracture fluid pressure p f and is thescaling factor given in Eq. (25). f is the porosity of thedual material point representing the added fluid volume tothe fracture space defined as

f ={0 for d dcrlocal local (scr ) for d > dcr , (33)

Fig. 4 Schematic diagram for two adjacent dual points with existingbulk volume of the porous medium and the added fracture fluid volumein between

where local is the local dilatation based on the instantaneous

stretch(

s = +)of the bonds attached to the dual

point while local (scr ) is the local dilatationwheremaximumstretch of the bonds attached to the dual point is restricted tothe critical stretch beyond which a bond breaks. local (scr )represents maximum volumetric change to a porous materialpoint before it becomes a dual point.

The scalar permeability of the dual point, f is determinedthrough fracture width w that is approximated based on f

f = w2

12, w = Vb0 f

Ap, (34)

where Ap is the surface area of the one of the faces of mate-rial point assumed cubic. The convective transport of fluidbetween fracture space and porous space of the dual node ismodeled though a source term I [x] in the governing Eq. (31)

I [x] = k AV

d

dl= Apk

( f [x] [x]

)V

(l p2

) (35)

where l p is the characteristic length of a material point.For a dual point, the definition of scalar force-state is

modified such that the peridynamic areal force density, anequivalent of traction force from classical theory, on the dualpoint normal to assumed fracture surface is equal to fracturefluid pressure in the dual point. For the bonds contributing

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to the areal force density at the dual point, the force densityis obtained based on only fracture fluid pressure at that dualpoint and the scalar force-state is given by:

td [x] ={ 2p f

m px for 2D,3p fm px for 3D.

(36)

2.6 Imposing boundary conditions

In a 3-D peridynamic formulation, boundary information isprovided in the form of volumetric constraints instead ofsurface constraints as in an equivalent local formulation [50].These volume constraints are imposed in a nonlocal regionalong the domain boundary constraining the solution in anonzero volume.

2.6.1 Imposing nonlocal displacement boundary condition

The peridynamic analogue of the classical displacementboundary condition is the displacement loading condition[33]. A fictitious region of thickness is assumed immedi-ately outside the computational domain in which the knownboundary displacement information is specified. The integralterm in the governing equation of motion for the materialpoints in the thickness inside the boundary is evaluated forthe specified displacements in the fictitious region outsidethe boundary.

2.6.2 Imposing nonlocal traction boundary condition

The traction boundary condition, in peridynamics, is imposedthrough the force density term in the governing equation ofmotion. Such a condition is referred as force loading condi-tion [33]. As mentioned above, the force density represent-ing the traction boundary force can be numerically imposedeither in just the boundary nodes or in several layers of nodesinside a nonzero volume of thickness along the boundary.Ha and Bobaru [51] have investigated the numerical imple-mentation of the traction boundary condition in peridynamicsand demonstrated the applicability of force density to just asingle layer of boundary nodes. They also discuss the appli-cation of areal force density, the peridynamic analogue ofthe classical traction vector, to determine the required forcedensity at the boundary nodes. However for simplicity, theyconvert the constant traction boundary vector into the forcedensity by just using nodal area andvolumeand found that theperidynamic solution converges to the corresponding classi-cal one in the limit horizon size going to zero.

In the current work with porous media, we present a dif-ferent method to impose the traction boundary condition. Forthis, we use Sillings [33,52] definition of peridynamic arealforce density at a point x in the direction of a unit vector n.Assume the plane defined by n divides the body B into two

Fig. 5 Evaluation of peridynamics equivalent of traction force, theareal force density (x,n). For each point A along the dashed line,0 z , the force density in the bonds connecting A to each pointB in the spherical cap is summed using a spherical coordinate systemcentered at A

sub regions B+ and B as shown in Fig. 5 and given by

B+ = {x B : (x x) .n 0} andB = {x B : (x x) .n 0} , (37)and let L be given by

L = {x B : x = x sn, 0 s } . (38)The peridynamic areal force density at pointx in the directionof unit vector n in B is defined by

(x,n) =L

B+

(T[x] x x T [x] x x) dVx dl

(39)

where dl is the differential path length overL.We impose thenonlocal traction boundary condition in the nonzero volume of thickness inside the boundary. A fictitious regionB ofthickness is assumed to exist outside the domain boundary.To impose the traction boundary condition T at any pointx in , we assume mechanical equilibrium in the mediumwhile the force density in the bonds connecting the materialpoints x in with points in B to be characterized by themagnitude of traction as following

t[x] x x = t [x] x x

=

2Tm px x for 2D,3Tm px x for 3D.

(40)

For an ordinary peridynamic material, it can be shown thatfor the scalar force-state in Eq. (40), the peridynamic arealforce density (x,n) at the domain boundary comes out tobe the same as the required traction force T . For any point x

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in , the integral term in the governing equation of motioncan be divided in two partsHx

(T[x] x x T [x] x x) dVx

=Hx\B

(T[x] x x T [x] x x) dVx

+Hx\B

(T[x] x x T [x] x x) dVx (41)

where the first integral on the right hand side is known interms of traction boundary force and is used as a force densityb[x]to impose the nonlocal traction boundary condition.

2.6.3 Imposing nonlocal boundary conditions for porousflow

The nonlocal boundary conditions for the porous flow prob-lem has been discussed in a previous study [42] where anonlocal region of thickness outside the domain boundaryis used to impose the boundary conditions. In the previousstudy [42], the domain boundary was used as a mirror planefor the pressure field to achieve the no-flowboundaries. How-ever, based on the results in Fig. 10(c) of this study [42], theno flow condition can also be achieved by imposing the zeropermeability in the nonlocal region outside the domain. Inthe current work, we used the latter for imposing the no-flowboundary condition.

3 Results and discussion

3.1 Traction boundary condition

We proposed above a novel way to impose traction boundaryconditions. Here, we simply verify this approach by com-paring the deformation of a 2-D elastic body in plain-strainobtained using the peridynamic theory and the classical the-ory. A 2-D square domain of length L = 100m in a Cartesiancoordinate system, with Youngs modulus E = 30 Gpa andPoissons ratio = 0.25 under confining boundary stressTx = 12 Mpa and Ty = 18 Mpa in the x and y directionsrespectively equilibrates to the following plane stain solu-tion:

x = Tx (1 ) TyE/ (1+ ) = 1.875 10

4,

y = Ty (1 ) TxE/ (1+ ) = 4.375 10

4, (42)

where x and y are the strains in the in the x and ydirections respectively. The peridynamic strain for uni-form discretization of x = y = 1m and horizonsize = 3m, results in x = 1.879 104 and y =

Fig. 6 Computational domain for simulating 1D consolidation prob-lem

4.371 104 which deviates less than 0.3 % from the clas-sical solution. This supports the validity of the proposedmethod of imposing the non-local traction boundary con-dition.

3.2 Poroelasticity

Next we validate the coupling of the peridynamic formu-lations of porous flow and solid mechanics by solving theclassical 1D consolidation problem and comparing the peri-dynamic solution with the corresponding analytical solution[53]. Consider a fluid-filled poroelastic layer extending fromthe surface z = 0 down to a depth z = h and resting onthe surface z = h (Fig. 6). At time t < 0, all the bound-aries are no-flow boundaries, boundaries in x and y direc-tions are constrained so as to be unable to deform in thelateral direction and the pore fluid pressure is assumed to bep0. A normal traction of magnitude Tz is then applied at theupper surface, resulting in the deformation of the poroelas-tic layer, and an increased pore pressure, pref is inducedin the layer as a result of the Skempton effect. At t = 0,the top surface at z = 0 is made open to the atmosphere(p

boundary= 0.1MPa). Gradually, the pore fluid drains out

of the upper surface, and the pore pressure relaxes untilit drops down to atmospheric pressure. As this occurs, thelayer continues to deform vertically downward. Due to themedium being constrained in the lateral direction, the onlynon-zero displacement is the vertical displacement w(z, t).The analytical solution for the normalized pore fluid pressureis

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(pre f p(z, t)

)(

pre f pboundar y) = 1

n=0

(1)nerfc

2n + zh(

8t/teq) 12

+erfc2 (n + 1) zh(

8t/teq) 12

(43)

where erfc (x) is the co-error function, equilibration time

teq = 4Sh2m , and S is the storage coefficient definedas

S =[

C f 1Km

] +

[1 2 (1 2)

3 (1 ) ]

Kb. (44)

The analytical solution for the normalized displacement atposition z and time t is

w (z, t) = 1(Kb + 4G3

) [Tz (z h)

+hpre f

n=1,3,..

8

n22cos

(n z2h

)e

(n22 t2teq

) .(45)

To solve this problem using our peridynamic poroelasticformulation, we consider a 2-D domain of height h = 162 mand breadth b = 108 m. The simulation parameters are fluidviscosity = 103 kg m1s1, fluid compressibility C f =5 1010 pa, medium permeability m = 6 1015 m2 =6 mD, medium porosity = 0.02, bulk modulus of rockKb = 20 Gpa, shear modulus of rock G = 12 Gpa, bulkmodulus of the rock matrix material Km = 400 Gpa, rockdensity = 3 103 kg m3, and magnitude of normal trac-tion Tz = 10 Mpa. We use uniform discretization of gridspacing x = z = L/n for the two-dimensional domainwhere computational nodes are placed at the center of thesquare cells of length x . Each node is assigned an areaequal to the square cell area

(x2

). A mid-point numeri-

cal integration is used to convert the governing integral Eqs.(2) and (23) for a node xi into coupled algebraic equationsinvolving neighbor nodes x j inside the horizon of the nodexi . In 2-D, the resulting algebraic equation is shown in Eqs.(46) and (47)

u [xi ] =k

j=1

{T [xi ] x j xi

T [x j ] xi x j } Ax j + b [xi ] , (46) [xi ] [xi ]+ [xi ] [xi ] =

kj=1

{Q [xi ] x j xi

Q [x j ] xi x j } Ax j + R [xi ]+ I [xi ] . (47)Here Ax j is the area of the neighbor node x j covered insidethe horizon of node xi . Note that while approximating theintegral with a summation, the contribution from the node

area Axi is omitted for being non-existent. For neighbor nodewhose square area is completely inside the horizon of thenode xi , Ax j = x2. For nodes whose area is intersected bythe horizon boundary, we have used the algorithm of Bobaruand Duangpanya [54] to calculate Ax j . Similarly the porousflow equation is obtained in its algebraic form.

The coupled peridynamic formulation is solved implic-itly and the pressure and vertical displacement was obtainedalong z direction in themiddle of the domainwhere boundaryeffects can be fairly neglected. We define the relative differ-ence in fluid pressure along z direction as the following tocompare peridynamic solution with the analytical solution:

Rel differencep =p

analytical(z) p

peridynamics(z)2

panalytical

(z)2 .

(48)

The peridynamic formulations are non-local and includea length-scale determined by the horizon size . For afixed horizon size, as the domain discretization is refined,m = /x increases to approach infinity and the exact non-local solution is obtained. For a peridynamic solution witha fixed horizon, m should be large enough to minimize theerror in numerical approximation but also small enough forcomputational ease. This requires an mconvergence test toobtain a suitable m value. Further the exact non-local peri-dynamic solution is different from that obtained from theclassical local model. The peridynamic solution of problemswithout singularities approaches the one from the classicallocal model as the horizon 0, while keeping m fixed orincreasing with rate slower than the rate at which decreases[54,55].

We perform an m-convergence test for fixed values ofL/18, L/27 and L/36 and investigate the variation in relativedifference between numerical peridynamic solution and theexact analytical solutionwith increasingm in Fig. 7. For each value, m is increased from 2 to 5 and the relative differ-ence substantially decreases as m increases, highlighting theerror in numerical approximation due to the smaller numberof nodes in the horizon for smaller m. For m 4, the rela-tive difference increases only slightly for all values. Thissuggests that an exact nonlocal solution is reached and maynot be substantially different than the exact local solution asfor m = 5, reducing did not reduce the relative differencein fluid pressure. Therefore, based on the data in Fig. 7, wechoose m = 5 for the presented peridynamic solution.

In Fig. 8, the variation in the normalized pore pressurewith normalized position in the z-direction is compared withthe exact analytical solution at different non-dimensionaltimes t = tteq = tm4Sh2 . The peridynamic solution is ingood agreement with the exact analytical solution, however,comparatively larger deviation at larger time. In Fig. 9, wecompare normalized displacement in the column with non-

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Fig. 7 m-convergence curves for the relative difference in pore fluidpressure along the column at x = 0 between the exact local solutionand the peridynamic numerical solution with horizon sizes and numberof nodes along a radial direction inside the horizon, (m = /x)

Fig. 8 Variation in normalized pore fluid pressure (pre f p(z,t))(pre f pboundar y) with

vertical position in column normalized with initial column height atdifferent non-dimensional time t = tteq = tm4Sh2

dimensional time. The peridynamic solution is again in goodagreement with the corresponding analytical solution.

3.3 Single fracture propagation

For the model validation, we simulate single fracture propa-gation in a homogeneous 2-D poroelastic domain in a plane-stain settingdue to injectionof aNewtonian andcompressiblefluid through a point source (Fig. 10). The initial pore fluidpressure is assumed to be p0. With respect to porous flow, allthe boundaries are no-flowboundaries.At time t < 0, the cor-ners of the 2-Ddomain are fixed to limit the rotational degreesof freedom and normal compressive stress of magnitude Tx

Fig. 9 Variation in displacement normalizedwith initial columnheightat with vertical positions in the columnwith non-dimensional time t =

tm4Sh2

andTy are applied at the xandyboundaries respectively.An equilibrium condition is achieved resulting in negativevalues of the dilatation due to the domain being compressedfrom its reference configuration. In the present problem, thelocal material points surrounding the mass injection location(Fig. 10) are modeled as the dual points and are now onwardsreferred as injection dual points. Note that the dual pointsalso represent the fracture space which suggests that an ini-tial fracture length equal to the grid spacing exists from thebeginning. According to the boundary stresses, the fracturepropagation is expected along x-axis. At time t = 0, mass isinjected in the injection dual points through the source termR [x] in the fracture flow governing equation. The fracturepressure at the injection dual points increases causing theinjection dual points to displace away from each other. Sincethe stress in y-direction is lower, the points displace more iny-direction and a fracture surface along x-axis begins to open.The dual points displaced along y-direction pull the neigh-boring points along x-axis away from each other resultingin bonds across x-axis to fail and damage increasing. Onceany two adjacent material points separate beyond the criticalstretch scr and d > dcr for both the points, these materialpoints are transformed into dual points, additional fractureflow equation is solved for these points and fracture pressureis predicted. The newly formed dual points across the x-axiswith higher fracture pressure displace in the y-direction awayfrom the x-axis resulting in damage evolution and fracturepropagation.

The peridynamic solution is compared with the approx-imate analytical solution obtained from the KGD model[44,56]. TheKGDmodel assumes themedium to be homoge-neous, isotropic and linear elastic, injection fluid to be New-

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Fig. 10 2D Computational domain with boundary stress, injectionlocation and dual material points to simulate fluid driven fracture prop-agation

tonian and fluid flow in the fracture to be laminar and neglectsgravity. The approximate solutions for fracture length l f ,maximum fracture opening wmax and net pressure pinjectionat the injection point were derived as

l f = 20.539

(E q3

) 16

t23

, (49)

wmax = 2.36(

q3

E

) 16

t13 and, (50)

pinjection

= 1.09(

E 2) 1

3t

13 , (51)

where q is the mass injection rate per unit fracture height, E

is the plane-strain elasticity modulus(

E = E12

), t is time

and all the variables and numbers are in MKS units. Sincethe KGD model does not incorporate fluid leak-off from thefracture,we considered amediumwith very lowpermeabilityof the order of nano-Darcy to simulate the same. Since thepressure gradient for fluid to flow in the fracture (throughlubrication theory) decreases by at least the third power of thedistance from fracture tip, theKGDmodel assumes a constantaverage pressure throughout the fracture length except nearthe tip where fluid pressure drops to zero. To capture suchan assumption with our coupled formulation where fracturepermeability is obtained through fracture width, the fracturelength to be simulatedmust be large.However that adds to thecomputational cost. Therefore, we modeled this assumptionby considering infinite conductivity for fluid flow inside thefracture with permeability f being function of local damage

f = fmax(

d dcr1 dcr

)ndamage(52)

Table 1 Simulation parameters for fracture propagation in 2-D

Dimension of the 2-D domain in X direction Lx = 40 mDimension of the 2-D domain in Y direction L y = 32 mBoundary stress in x direction Tx = 12 MpaBoundary stress in y direction Ty = 8 MpaYoungs modulus of the domain = 60 Gpa

Shear modulus of the domain = 24 Gpa(Poissons ratio = 0.25)

Reference pore pressure p0 = 3.2 MPaMass injection rate = 0.025 Kg/m/sec

Fracturing fluid viscosity = 0.001 Kg/m/sec

Medium permeability = 10 nD

fmax = 100Ddcr = 0.25ndamage = 0.25scr = 0 (wc = 0)

where fmax is the maximum allowable permeability insidethe fracture and ndamage is the power law coefficient. Notethat the absolute value of fmax does notmatter when simulat-ing an infinite fracture conductivity as long as it is a relativelylarge value. Table 1 reports the simulation parameters.

The domain discretization and numerical integration toconvert the governing integral equations into coupled alge-braic equations are the same as the 1D consolidation problemdescribed above. The additional equation for fracture flowtakes the following algebraic form

f [xi ] f [xi ]+ f [xi ] f [xi ] =k

j=1

{Q [xi ] x j xi

Q [x j ] xi x j } Ax j + R [xi ]+ I [xi ]. (53)The coupled governing equations are solved implicitly. Wechoose m =

x = 3 and perform a convergence test withrespect to the predicted fracture length to find an appropriatehorizon size to simulate the problem. As expected, we findthe relative difference in the fracture length predicted fromperidynamic formulation and the KGDmodel (49) decreaseswith decreasing horizon size (Fig. 11). We choose the small-est considered horizon size = 3Lx/400 and m = x = 3for performing the numerical peridynamic simulations. Wenote that the damage field does not evolve every time step asother field variables; however, using this variable in closurerelations to identify the proper flow regime has not shownany adverse effects or otherwise oscillations in the numeri-cal solution.

In Fig. 12(ac), we compare the variation in injectionpressure, fracture half-length andmaximum fracture openingwith time and in Fig. 12(d) the variation in fracturewidthwithdistance from fracture tip at t = 22.05 s. The peridynamicpredictions are in good agreement with the corresponding

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Fig. 11 convergence curve for the relative difference in predictedfracture length from peridynamics formulation and the KGDmodel form =

x = 3

results from the KGDmodel. The KGDmodel does not haveany compressive boundary stress and the model assumed thestress intensity factor at any time is the critical stress intensityfactor so the fraure initiates and propagates for any deforma-tion in the medium. However, in the peridynamic fmulation,the domain is initially compressed by far field stresses, there-fore, the fracture initiates only after enough fracture pressureis built to break the required bonds for crack initiation. This is

Fig. 13 Schematic of a 2D fracture opening of length L with internalpressure p f for the Sneddon solution

why injection pressure from peridynamics is initially higherthan the KGD model. For late time, the peridynamic predic-tion of fracture pressure is slightly lower due to formulationbeing non-local and reflection of stress waves from bound-aries of finite domain.

Note that we propose an additional equation to solve forfracture flow and the resulting fracture pressure was used inthe force density calculation for the interaction of the mater-ial points across the fracture surface. The validity of the pro-posed modification in the force density of the bonds crossingthe fracture surface can be seen through prediction of theeffective normal stress distribution in y-direction, yy that isobtained from Eq. (39). We plot in Fig. 14(a) the contours ofyy from peridynamic formulation and compare the same in

Fig. 12 Variation in a Fluid injection pressure, b fracture half length, and c maximum fracture width with time from the KGD model and theperidynamics solution, and d variation in fracture width along the fracture length at time = 35 s from the KGDmodel and the peridynamics solution

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Fig. 14 Contours of effective normal stress in y-direction from a Sneddon solution, b Peridynamics solution, c variation is effective normal stressin y-direction along y = 0, and d variation is effective normal stress in y-direction along x = 0

Fig. 14(b) with the Sneddon solution [57] that analyticallypredicts the stress field around a 2-D fracture of finite lengthwith constant internal pressure p f in an infinite medium. InFig. 14(cd), we plot the variation in yy along (y = 0) andnormal (x = 0) to fracture opening respectively from peri-dynamics and the Sneddon solution given below

yy = p f1 S(

S1 S2) 12

cos

( 1 + 2

2

)

S(S1 S

) 32

sin sin

(3

2(1 + 2)

) ,

= tan1( x

y

), 1 = tan1

( x1+ y

),

2 = tan1( x1 y

), (54)

where x , y, S1, S2, and S are the coordinates and distancesshown in Fig. 13 normalized with the fracture half length

l f /2. For comparison, time t = 6.2 s is chosen so thatthe fracture length (l f = 2.5m) is shorter in comparisonto domain length to minimize the effects of reflection ofstress waves from the boundary and simulate an infinitelylong domain as required for Sneddon solution. The fracturepressure p f at t = 6.2 s is 9.09 Mpa. The contours of yydue to fracture pressure fromperidynamics show good agree-ment with the one from Sneddon solution. The peridynamicprediction of yy is more diffused for the formulation beingnon-local; only in the limit of non-locality going to zero,the peridynamic formulation converges to the correspondinglocal solution. The agreement between the peridynamic solu-tion of yy with the Sneddon solution is more quantitative inFig. 14(cd) where we plot yy along y = 0 and x = 0.It is important to note that the average value of yy + Tyalong the fracture length in Fig. 14(c) is 9.04 Mpa. This 0.55% deviation in total stress (yy + Ty) at the fracture sur-face from fracture p f = 9.09 supports the validity of theproposed force density in Eq. (36) for bonds crossing thefracture surface. In Fig. 14(c), the prediction of yy at the

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boundary is abruptly changing for the horizon not being afull disk. The peridynamic solutions are generally known tobe affected the boundary, however, away from the boundarythe peridynamic predictions are consistent with the contin-uum formulation [54].

4 Conclusions

We have developed a new peridynamics based model to sim-ulate the growth of fluid driven cracks. Multiple, non-planar,competing fractures can now be simulated in unconventionalreservoirs more realistically and this will allow us to bet-ter design wells and hydraulic fractures to better drain thereservoir volume. Computational modeling of HF process isa challenging problem. Virtually all current approaches tohydraulic fracture modeling rely on finite difference, finiteelement (FEM) or boundary element methods (BEM) tosolve PDEs of the LEFM based fracture formulation. How-ever, computing derivatives on domains containing fracturescauses problems with such methods. In addition, PDEs ofthe classical local formulation do not have any characteristiclength scale to capture nonlocal physics around the crack tip.

With our interest in investigating the complex geometryandnetwork of fluid-driven fractures in unconventional reser-voirs, a generalized 3-D state-based peridynamic model isdeveloped by modifying the existing peridynamic formula-tion of solid mechanics for porous and fractured media andcoupling it with a previously developed peridynamic formu-lation of porous flow. The coupled poroelastic formulationproduces closematchwith the analytical solution for the clas-sical 1D consolidation problem. The coupling includes anadditional equation for flow inside created fracture space.For simplicity, we currently consider only Newtonian andslightly compressible fracturing fluid. A novel approach ispresented to impose the non-local traction boundary condi-tion and the resulting deformations for 2-D plane-strain prob-lem are in close agreement with the corresponding analyticalsolution. A previously developed energy-based criterion isused to simulate autonomous material failure and fracturepropagation. Fluid-driven fracture propagation is verified ina 2-D plane-strain setting against the corresponding classicalanalytical solution from the KGD model. In spite of the lim-itations to represent the KGD crack numerically, a close pre-diction of the fracture geometry and injection pressure fromperidynamic model supports its ability to simulate complexfracture propagation patterns.

Since the formulation solves the flow physics outside aswell as inside the fracture, unlike several previous models, itprovides an excellent means to simulate the effects of hetero-geneity (in form of mechanical properties, permeability het-erogeneity and anisotropy and natural fractures). In addition,the developed peridynamic model, being based on particle-

based discretization, overcomes the limitation of re-meshingduring fracture propagation in previous continuum mechan-ics models. In this paper, we have confirmed the validity ofthe newly developed model to simulate simple planar, fluid-driven fracture propagating in a poroelastic heterogeneousmedium that would enable us to investigate complex frac-ture geometry and created networks for fracture design andoptimized well stimulation.

Acknowledgments This work is supported by Department of Energy(DOE) Grant No. DE-FOA-0000724 and by the member companiesparticipating in the Joint Industry Program on Hydraulic Fracturing andSand Control at the University of Texas at Austin. We thank MichaelBrothers, a graduate studentworkingwith JTF, for sharing the insight onimplementation of peridynamic formulation into a computation code.

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A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approachAbstract 1 Introduction2 Mathematical model2.1 Review of the state-based peridynamic formulation for solid mechanics2.2 State-based peridynamic formulation of poroelasticity2.3 Material failure modeling in peridynamics2.4 State-based peridynamic formulation of porous flow2.5 State-based peridynamic formulation of fracture flow2.6 Imposing boundary conditions2.6.1 Imposing nonlocal displacement boundary condition2.6.2 Imposing nonlocal traction boundary condition2.6.3 Imposing nonlocal boundary conditions for porous flow

3 Results and discussion3.1 Traction boundary condition3.2 Poroelasticity3.3 Single fracture propagation

4 ConclusionsAcknowledgmentsReferences