Int J Fract (2012) 178:245–258DOI 10.1007/s10704-012-9742-y

ORIGINAL PAPER

A method for 3-D hydraulic fracturing simulation

S. Secchi · B. A. Schrefler

Received: 15 February 2012 / Accepted: 14 June 2012 / Published online: 11 July 2012© Springer Science+Business Media B.V. 2012

Abstract We present a method for the simulationof 3-D hydraulic fracturing in fully saturated porousmedia. The discrete fracture(s) is driven by the fluidpressure. A cohesive fracture model is adopted wherethe fracture follows the face of the elements around thefracture tip which is closest to the normal direction ofthe maximum principal stress at the fracture tip. Nopredetermined fracture path is needed. This requirescontinuous updating of the mesh around the crack tipto take into account the evolving geometry. The updat-ing of the mesh is obtained by means of an efficientmesh generator based on Delaunay tessellation. Thegoverning equations are written in the framework ofporous media mechanics theory and are solved numer-ically in a fully coupled manner. An examples dealingwith a concrete dam is shown.

Keywords Hydraulic fracturing · Cohesive fracturemodel · 3D finite element solution

S. SecchiCNR, ISIB, Corso Stati Uniti 4, 35127 Padua, Italy

B. A. Schrefler (B)Department of Civil, Environmental and ArchitecturalEngineering, University of Padova, Via Marzolo 9,35131 Padua, Italye-mail: bas@dic.unipd.it

1 Introduction

Fluid-driven fracture propagating in geomaterials suchas rocks, soil or concrete are encountered in manyengineering problems. Hydraulic fracturing is used toenhance the recovery of hydrocarbons from under-ground reservoirs and most recently for extracting gasand oil from shales (TAMEST 2011). This is an emerg-ing field where the consequences on the environmenthave still to be assessed. On the other hand, the reservesof shale gas both in the United States (estimated around70 tr m3) and in a lesser measure in Europe (estimated14 tr m3) are enormous and justify the interest on mod-elling of hydraulic fracturing. Another application ofimportance is related to the overtopping stability anal-ysis of dams.

Contributions to the mathematical modelling offluid-driven fractures have been made continuouslysince the 1960s, beginning with Perkins and Kern(1964). These authors made various simplifyingassumptions, for instance regarding fluid flow, fractureshape and leakage velocity from the fracture. Amongother contributions, reference can be made to Riceand Cleary (1976), Cleary (1978), Huang and Russel(1985a), Huang and Russel (1985b) and Detournay andCheng (1991). All these contributions present analyti-cal solutions, in the frame of linear fracture mechanicsassuming the problem to be stationary. Further, theysuffer the limits of the analytical approach, in partic-ular the inability to represent an evolutionary problemin a domain with a real complexity. Other papers deal

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246 S. Secchi, B.A. Schrefler

with the analysis of solid and fluid behaviour near thecrack tip (Advani et al. 1997; Garagash and Detour-nay 2000). A general model has been presented byBoone and Ingraffea (1990) in the context of linearfracture mechanics. It allows for fluid leakage in themedium surrounding the fracture and assumes a mov-ing crack depending on the applied loads and materialproperties. Hence, fracture length is a natural prod-uct of the solution algorithm, not the result of previ-ous assumptions. A finite element/difference solutionis adopted in this case. Carter et al. (2000) put forwarda fully 3-D hydraulic fracture model which relies onhypotheses similar to Boone and Ingraffea (1990), inparticular assuming the positions where fractures initi-ate, but completely neglects the fluid continuity equa-tion in the medium surrounding the fracture. Schrefleret al. (2006) and Secchi et al. (2007) presented a twodimensional cohesive fracture model in a fully satu-rated porous medium where remeshing is used to fol-low the advancing fracture and the above listed limit-ing assumptions have been eliminated. Finally Réthoréet al. (2008) used the Extended Finite Element Methodfor 2-D hydraulic fracturing simulation.

As far as numerical fracture analysis in general isconcerned, mainly two approaches can be found inliterature: smeared crack analysis and discrete crackanalysis. We consider here the second one. In this con-text embedded discontinuity elements have been pro-posed by Bolzon and Corigliano (2000), Wells andSluys (2001), Oliver et al. (2001), Feist and Hofstetter(2006). Extended Finite Elements have been introducedby Moes and Belytschko (2002), a moving rosette ina fixed mesh was used by Wawrzynek and Ingraffea(1989) and a moving fracture in a fixed mesh by Cama-cho and Ortiz (1996).

We present a model for 3-D hydraulic fracturingbased on a discrete fracture approach which uses reme-shing in an unstructured mesh together with automaticmesh refinement. In our procedure the fracture followsthe face of the elements around the fracture tip which isclosest to the normal direction of the maximum princi-pal stress at the fracture tip. The solution can be appliedas it stands to realistic 3-D problems or be used as abenchmark problem for less precise methods such asthe Particle Finite Elemet Method PFEM, intdoducedby Oñate et al. (2004). Oñate and Owen (2011).

The paper is organized as follows: in Sect. 2 we sum-marise briefly the governing equations and the cohe-sive fracture model adopted, in Sect. 3 we present the

remeshing strategy for an unstructured three-dimen-sional mesh, in Sect. 4 the adopted procedure for frac-ture advancement and in Sect. 5 one examples.

2 The mathematical model and its discretization

The porous medium is considered fully saturatedthroughout the domain. Also the fracture is filled withthe same fluid. The derivation of the linear momentumbalance equations and the mass balance equations fora fully saturated porous medium can be found in text-books (Lewis and Schrefler 1989; Zienkiewicz et al.1999) and is not repeated here. The mass balance equa-tion for the fluid in the fracture (filler) is similar to thatof the fluid in the pores. Simulation of fluid driven frac-ture propagation in a porous medium requires the intro-duction of appropriate constitutive relationships for thesolid, and for the fluid defining the permeability withinthe crack and the rest of the domain. We will show in thesequel the weak form of the balance equations wherethe constitutive equations have been introduced.

2.1 Solid phase

In the framework of discrete crack models, the mechan-ical behaviour of the solid phase at a distance from theprocess zone is usually assumed as simple as possible.Let the complete domain be composed of a set of differ-ent homogeneous sub-domains. In the present formula-tion a Green-elastic or hyperelastic material is assumedfor each component, being the mechanical behaviourdependent on effective stress as

σ′ij = cijrsεrs (1)

where the elastic coefficients depend on the strainenergy function W and can be expressed in terms ofLamé constants as

cijrs = ∂2W

∂εij ∂εrs= μ

(δisδjr + δirδjs

) + λ δijδrsδij

(i,j,r,s = 1, 2, 3) (2)

For the process zone we use the cohesive fracture modelshown in Fig.1 for sake of simplicity in a 2-D setting.The relations are however generalized for a 3-D sit-uation as will be explained below. Between the realfracture apex which appears at macroscopic level andthe apex of a fictitious fracture there is the process zone

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A method for 3-D hydraulic fracturing simulation 247

Fig. 1 Definition of cohesive crack geometry and model param-eters

σσ

δσ δσδσcrδσcr

σ0 σ0

σ1

σ2

δσ1 δσ2

A

BG

0 0

(a) (b)

Fig. 2 Fracture energy (a) and loading/unloading law (b) foreach homogeneous component

where cohesive forces act. A constitutive model linksthe cohesive forces to the crack opening in the processzone. In our approach the fracture is supposed to followthe side of the finite element in the apex zone which isclosest to the normal to the plane with maximum tensilestress. Hence the following relations are valid for theside of the particular element considered in the apex.Clearly with propagating fracture this approach needscontinuous updating of the mesh.

Within a generic component of the solid skele-ton, a fracture can initiate or propagate under theassumption of mode I crack opening, provided thattangential relative displacements of the fracture lips arenegligible. The cohesive forces are hence orthogonal tofractures planes. Following the Barenblatt (1959) andDugdale (1960) model and accounting for Hilleborget al. (1976), the cohesive law when opening monoton-ically increases is (Fig. 2)

σ = σ0

(1 − δσ1

δσ cr

)(3)

σ0 being the maximum cohesive traction (closed crack),δσ the current relative displacement normal to the crack,δσcr the maximum opening with exchange of cohesivetractions and G = σ0 × δσcr/2 the fracture energy. Ifafter some opening δσ1 < δσcr the crack begins toclose, tractions obey a linear unloading as

σ = σ0

(1 − δσ1

δσcr

)δσ

δσ1(4)

When the crack reopens, Eq. (4) is reversed until theopening δσ1 is recovered, then tractions obey againEq. (3). The unloading path is also presented in Fig. 2.The individual homogeneous components differ onlydue to different values attributed to the fundamentalparameters of Eqs. (3) and (4).

When tangential relative displacements of the sidesof a fracture in the process zone can not be disregarded,mixed mode crack opening takes place. This is usuallythe case of a crack moving along an interface separatingtwo solid components. In fact, whereas the crack pathin a homogeneous medium is governed by the principalstress direction, the interface has an orientation that isusually different from the principal stress direction. Themixed cohesive mechanical model involves the simul-taneous activation of normal and tangential displace-ment discontinuity and corresponding tractions. For thepure mode II, the model presented in Fig. 3 is referredto, where the relationship between tangential tractionsand displacements is

τ = τ0δσ

δσ1

δτ

|δτ| (5)

τ0 being the maximum tangential stress (closed crack),δτ the relative displacement parallel to the crack andδσcr the limiting value opening for stress transmission.The unloading/loading from/to some opening δσ1 <

δσcr follow the same behaviour as for mode I and thetraction law is

τ = τ0

(1 − δσ1

δσcr

)δσ

δσ1

δτ

|δτ| (6)

which is valid for opening in the set [0 < δσ1 < δσcr],then the original path (Eq. 5) is followed. Figure 3presents also the unloading/reloading relation.

For the mixed mode crack propagation, the interac-tion between the two cohesive mechanisms is treatedas in Camacho and Ortiz (1996). By defining an equiv-alent or effective opening displacement δ and the scalareffective traction t as

δ =√

β2δ2τ + δ2

σ, t =√

β−2t2τ + t2

σ (7)

the resulting cohesive law is

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248 S. Secchi, B.A. Schrefler

Fig. 3 Fracture energy (a)and loading unloading lawfor the interface and mixedmode

(a) (b) ττ

τ0

τ1

τ2

−τ2

−τ1

−τ0

δτδτ1 δτ2 δτcr

−δτ1−δτ2−δτcr

δτδτcr

τ0

G

0 0

D

C

A

B

Fig. 4 Hydraulic crack domain

t = t

δ

(β2

δτ + δσ

)(8)

β being a suitable material parameter that defines theratio between the shear and the normal critical com-ponents. The cohesive law takes the same aspect as inFig. 2, by replacing displacement and traction param-eter with the corresponding effective ones.

Taking into account the cohesive forces, the effec-tive stress principle and the symbols of Fig. 4, the linearmomentum balance equation of the mixture is writtenas∫

�

δεi j ci jrsεrs d�−∫

�

δεi j αδi j p d�−∫

�

ρδui gi d�

−∫

�e

δui ti d� −∫

�′δui ci d�′ = 0 (9)

where � is the domain of the initial boundary valueproblem, �e is the external boundary and �′ the bound-

ary of the fracture and process zone. δεi j is the strainassociated with virtual displacement δui , ρ the densityof the mixture (solid plus fluid), gi the gravity accel-eration vector, ti the traction on boundary �e and ci

the cohesive tractions on the process zone as definedabove. Tractions and compressive values of the fluidpressure are taken as positive.

2.2 Liquid phase

Constant absolute permeability is assumed for the fluidfully saturated medium surrounding the fracture. As faras permeability within the crack is concerned, the valid-ity of Poiseuille or cubic law is assumed. This has beenstated for a long time in the case of laminar flow throughopen fractures and its validity has been confirmed in thecase of closed fractures where the surfaces are in con-tact and the aperture decreases under applied stress. Thecubic law has been found to be valid when the fracturesurfaces are held open or are closed under stress, with-out significant changes when passing from opening toclosing conditions. Permeability is not dependent onthe rock type or stress history, but is defined by crackaperture only. Deviation from the ideal parallel surfacesconditions cause only an apparent reduction in flow andcan be incorporated into the cubic law, which reads as(Witherspoon et al. 1980)

ki j = 1

f

w2

12(10)

w being the fracture aperture and f a coefficient in therange 1.04–1.65 depending on the solid material. In the

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A method for 3-D hydraulic fracturing simulation 249

following, this parameter will be assumed as constantand equal to 1.0. Incorporating the Poiseuille law intothe weak form of water mass balance equation of thefiller within the crack results in∫

�′δp

{n

Kw

∂p

∂t+ ∂w

∂t

}d�′

−∫

�′(δp),i

[w2

12μw

(−p, j + ρwg j)]

d�′

+∫

�′δpqwd�′ = 0 (11)

which represents the fluid flow equation along the frac-ture. It should be noted that the last term, representingthe leakage flux into the surrounding porous mediumacross the fracture borders, is of paramount importancein hydraulic fracturing techniques. This term can berepresented by means of Darcy’s law using the mediumpermeability and pressure gradient generated by theapplication of water pressure on the fracture lips. Noparticular simplifying hypotheses are hence necessaryfor this term. Equation (11) can be directly discretizedby finite elements at the same stage as the other gov-erning equations. Except for the compressibility term,this equation is also presented in (Boone and Ingraffea1990), where it is discretized by the finite differencemethod and integrated separately along a predeter-mined path by using a staggered approach to obtain thepressure along the crack. Further, particular relation-ships for the leakage term have been introduced there,for instance impermeable boundaries. This staggeredprocedure resulted in a cumbersome method, requiringseveral thousand iterations, as the authors declare, dueto the strong coupling of displacement and pressurefields, but, more importantly, it needs particular con-vergence and stability analyses (iteration convergence)to assess the numerical performance of the solution(Turska and Schrefler 1993). We solve this equationtogether with the other balance equations.

The mass balance equation for the pore water of theporous medium surrounding the fracture, becomes afterincorporating Darcy’s law

∫

�

δp

{(α − n

Ks+ n

Kw

)∂p

∂t+ αvs

i,i

}d� −

∫

�

(δp),i

[ki j

μw

(−p, j +ρwg j)]

d�+∫

�e

δpqwd�+∫

�′δpqwd�′ = 0

(12)

where δp is a continuous pressure distribution satis-fying boundary conditions, n the porosity, Kw the bulkmodulus for liquid phase, vs

i the velocity vector of thesolid phase, ki j the permeability tensor of the medium,μw the dynamic viscosity of water, ρw its density andqw the imposed flux on the external boundary. In thelast term of Eq. (12) qw represents the water leakageflux along the fracture toward the surrounding medium.This term is defined along the entire fracture, i.e. theopen part and the process zone.

It is worth mentioning that the topology of thedomain � changes with the evolution of the fracturephenomenon. In particular, the fracture path, the posi-tion of the process zone and the cohesive forces areunknown and must be regarded as products of themechanical analysis. The discretized governing equa-tions which are shown next, are solved simultaneouslyto obtain the displacement and pressure fields togetherwith the fracture path.

2.3 Discretized governing equations and solutionprocedure

Space discretization by means of the Finite ElementMethod of Equations (9) and (11, 12), adopting avector notation and incorporating the constitutive equa-tions, results in the following system of time differentialequations (dot represents time derivative) at elementlevel,

[KE −LE

−LTE SE

] [dE

pE

]+

[0 00 HE

] [dE

pE

]

=[

FE

0

]+

[0GE

](13)

We have taken advantage of the fact that Eqs. 11 and

12 have the same structure, hence only the parametershave to be changed in the appropriate element, depend-ing whether they belong to the fracture or the sur-rounding porous medium. The submatrices in Eq. (13)are

KTE =

∫

VE

BT D BdV (14a)

LTE =

∫

VE

NpT(

mT − 1

3KsmT D

)BdV (14b)

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250 S. Secchi, B.A. Schrefler

SE =∫

VE

NpT(

1 − n

Ks+ n

K f− mT Dm

(3Ks)2

)NpdV

(14c)

HE =∫

VE

(∇Np)T K∇NpdV (14d)

GE =∫

VE

NpTqE dV +

∫

�q,E

NpTq�

E d�+∑

NpTQE

(14e)

FE =∫

VE

NT fE dV +∫

�E

NT tE d�+∫

�Ecrack

NT cE d�

(14f)

where B is the strain operator, D the solid phase consti-tutive matrix, K the permeability matrix, N the matrixcontaining shape functions for the solid displacementsand Np for the pressures, tE the vector of boundary trac-tions, fE the vector of body forces and m = [1 1 1 0 0 0].Formally the only change with respect to a fully cou-pled consolidation model (Lewis and Schrefler 1989)is given in Eq. (14f) where cE represents the cohe-sive traction rate and is different from zero only if theelement has a side on the lips of the fracture �Ecrack.Given that the liquid phase is continuous over the wholedomain, leakage flux along the opened fracture lips isaccounted for through Eq. (14d) together with the fluxalong the crack. Finite elements are in fact present alongthe crack, as shown in Fig. 4, which account only forthe pressure field and have no mechanical stiffness. Inthe present formulation, non-linear terms arise throughcohesive forces in the process zone and permeabilityalong the fracture.

Global equations are assembled in usual way andcan be integrated in time by means of the generalizedtrapezoidal rule. This yields the algebraic system of dis-cretized equations, written for simplicity in a conciseform as

Axn+1 = Vn + Zn+1 (15)

being

xn+1 =[

dp

]

n+1

An+1 =[

K −L−LT S + αtH

]

n+1

Vn =[

K −L−LT S − t (1 − α) H

]

n

[dp

]

n

Zn+1 =[

F0

]

n+1−

[F0

]

n+ t (1 − α)

[0G

]

n

−tα

[0G

]

n+1

(16)

n represents the time station and α the time discret-ization parameter. Implicit integration is used in thefollowing applications (α = 0.5).

3 3-D remeshing algorithm

As a consequence of the propagation of the cracks andensuring evolution of the geometry an automatic 3-Dremeshing technique is adoped. The starting point forthe following developments is the numerical procedurefor managing generic tetrahedral elements. In partic-ular a modification of Quad-Edge structure proposedby Guibas and Stolfi (1985) is used for non-manifoldsurfaces (Weiler 1985, 1988). A useful survey on thissubject and related implementation can be found inCampagna and Karbacher (2000). Modifications to anexisting topological structure or new proposals are inany case possible, together with a definition of the perti-nent Euler functions, also depending on the structure’sfield of application. To create the finite element sub-divisions, the TMWEdge data structure that simplifiesthe Quad-Edge and is similar to the Face-Edge sug-gested by Weiler (1985) is proposed here. This structurepreserves the edge algebra and all mathematical andtopological properties stated for the Quad-Edge formu-lation as well as efficiency in handling the adjacency ofthe Face-Edge. In addition it has been equipped withsome new primitives directed to managing all the dif-ferent meshing operations. The edge of the TMWEdgeprovides a cyclic list of the faces incident and allowsmoving from one triangle to the next one according toeither a clockwise, or a counterclockwise order.

The choice of the data structure results from a com-promise between computational and storage efficiency.In the TMWEdge structure the operations needed toconstruct an incremental Delaunay triangulation dealwith creating, locating and deleting nodes, edges andtriangles and require repetitive queries of neighbour-ing vertices, edges and polygons. These operation areimplemented in the TMWEdge structure or obtained

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A method for 3-D hydraulic fracturing simulation 251

Box 1 TVertex object

Box 2 TEdge object

Box 3 TMWEdge structure

by the pertinent Euler functions. Geometrical informa-tion is preserved, hence the queries are performed ina limited time, almost independently of the number ofelements in the list. For this reason, the topologicalstructure also allows for a rapid execution of other oper-ations such as swapping of edges, mesh quality con-trol and similar tasks. Although not used in this paper,

the TMWEdge structure maintains the information forspatial operations (see Boxes 1, 2, 3, 4), in particu-lar it is suitable for representing spatial surfaces andsolids by means of triangles or polygonal surfaces ofhigher complexity and to deal with tetrahedral subdi-visions. The usual topological transformations (trans-lations, rotations, scaling) as well as more complex

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252 S. Secchi, B.A. Schrefler

Box 4 TFace object

operations in space (such as sectioning of solids andtheir spatial visualisation) are also possible by addinga small amount of extra data.

Thebasicplanetopologicalentitiesarevertices(TVer-tex object), edges (TEdge object) and faces (TFaceobject). Their union establishes the ordered complexTMWEdge structure.

The structure elements must fulfil some require-ments which make their application possible and someconditions ensuing from the topological consistency:

− each element is required to reach, by means ofinternal pointers, its corresponding neighbouringentity;

− each edge can be shared by more than two faces(non-manifold);

− each edge must possess at least one face connectedto it;

− each vertex must be connected to at least threeedges;

− each face is defined by at least three vertices;− there are no theoretical bounds to the number of

edges connected to a vertex. This is a necessaryrequirement to obtain unstructured meshes.

Other information concerns the orientation. For eachedge there is the possibility to define two orientations,the first, from the origin to the destination node (callednatural) and the opposing one.

In an object-oriented framework, the TVertex classmanages the node. It contains (Box 1) a TCoord object,which controls the coordinates, and a pointer to one ofthe edges connected to the vertex. The set of edgesjoined to each vertex forms the node ring.

The TEdge class contains and manages the informa-tion of each edge of the subdivision. The member dataare (Box 2): pVertex pointer to the origin node, pFacepointer to the first face of the ring located at the left

of the edge when observing from pVertex, and Index, ashort-type datum, i.e. a one byte char described in thefollowing.

The three main entities (TVertex, TMWEdge, TFace)of the topological structure refer to 1, n or 3 TEdgeobjects respectively. In the first case, the pointer allowsthe object to know what and how many edges are linkedto the vertex. Recognising whether the vertex belongsto the boundary of the domain without introducing newdata is straightforward. The two pointers in TMWEdgeestablish an oriented edge and, finally, the three pointersin TFace describe the edges of the triangular elementof the subdivision.

The efficiency of the structure increases if the TEdgeobject is able to recognise itself within the other entitiesin which it is contained. This means that each TEdgeobject has to know its position within the arrays con-taining it, and present in TMWEdge and TFace. Theinformation contained in the Index datum is used forthis purpose.

Depending on the way the edge operators are built,particularly for the function Sym() to be operative, itis necessary that operations which create the TEdgeobjects be exclusively performed in pairs of TMWEdgeobjects, by using dynamical memory allocation.

Box 2 lists the basic operators for the TEdge object.For instance Sym() yields the pointer to the edge havingan opposite direction to the one under consideration.By using these operators the different entities neces-sary for meshing the domain are easily identified. E.g.it is possible to identify the vertex destination of edgee, which is represented by e->Sym()->Origin(), thevertex of the left triangle not shared by e is given bye->D_Prev()->Origin().

The topological entities are completed by the struc-ture TFace, which defines a face of the tessellationusing NumEdges() edges suitably allocated in the

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A method for 3-D hydraulic fracturing simulation 253

computer memory. The functions SetConnection() andAdjustConnections() (Box 4) initialise and modify datarelated to the incidences of the analysed and neighbour-ing elements, respectively.

4 Fracture advancement

Because of the continuous variation of the domain asa consequence of the propagation of the cracks, alsothe boundary �’ and the related mechanical condi-tions change. An extension to 3-D case of the proce-dure proposed by Schrefler et al. (2006) is adopted.Note that as compared to a 2-D situation the fracturetip becomes here a curve in space (front). Along theformed crack faces and in the process zone, boundaryconditions are the direct result of the field equations.The adopted remeshing technique accounts for all thesechanges.

At each time station tn , all the necessary spatialrefinements are made, i.e. j successive front advance-ments are possible within the same time step (Fig. 5). If

a new node is created at the fracture advancement frontthe resulting elements for the filler are tetrahedral. If aninternal node along the process zone advances, a newwedge element result in the filler.

The number of advancements in general depends onthe chosen time step increment t, the adopted cracklength increment s,and the variation of the appliedloads. This requires continuous remeshing with a con-sequent transfer of nodal vector Vm (Eq. 15) fromthe old to the continuously updated mesh (Fig. 6).Projection of this nodal vector between two consec-utive meshes is obtained by using a suitable operatorVm(�m+1) = ℵ (Vm(�m)) (Secchi et al. 2007). Thesolution is then repeated with the quantities of mesh mbut re-calculated on the new mesh m+1 before advanc-ing the crack tip to preserve as far as possible energyand momentum, see Fig. 6.

Fluid lag, i.e. negative fluid pressures at the cracktip which may arise if the speed with which the cracktip advances is sufficiently high so that for a given per-meability water cannot flow in fast enough to fill thecreated space, can be obtained numerically only if an

Fig. 5 Multiple advancingfracture step at the sametime station

Fig. 6 Nodal forcesprojection algorithm. aNodal forces at time stationn on mesh m. b Nodalforces of time station n onmesh m + 1 beforeprojection. c Nodal forcesof time n on mesh m + 1referred to nodes of thelatter

Ω

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254 S. Secchi, B.A. Schrefler

Fig. 7 Problem geometryfor ICOLD benchmark andcalculated crack path with2-D analysis

Fig. 8 Fracture at the baseof the dam

element threshold number is satisfied over the processzone. This number is given by the ratio of elements overprocess zone and can be estimated in advance from theproblem at hand and the expected process zone length.Hence a sort of object oriented refinement is neededlocally (Secchi et al. 2008).

5 Example

The 3-D code has been validated with respect to con-solidation problems (Secchi et al. 2008).

The following application deals with a bench-mark proposed by ICOLD. The benchmark consists

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A method for 3-D hydraulic fracturing simulation 255

Fig. 9 Pressure field insidethe ‘filler’

in the evaluation of failure conditions as a conse-quence of an overtopping wave acting on a concretegravity dam. The geometry of the dam is shownin Fig. 7 together with initial and boundary condi-tions and an intermediate discretization. Differentlyfrom the original benchmark, the dam concrete foun-dation is also considered, which has been assumedhomogeneous with the dam body. In such a situa-tion, the crack path is unknown. On the contrary,when a rock foundation is present, the crack naturallydevelops at the interface between dam and founda-tion.

As far as initial conditions for water pressure areconcerned, it is assumed that during building opera-tions and before filling up the reservoir, pressure candissipate in all the dam body. As consequence zero ini-tial pore pressure is assumed in the simulation. A morerealistic assumption is the hypothesis of partial satu-ration of the concrete, which would require a furtherextension of the present mathematical model.

Applied loads are the dam self-weight and the hydro-static pressure due to water in the reservoir growingfrom zero to the overtopping level h (which is higherthan the dam). The material data for the concrete are

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256 S. Secchi, B.A. Schrefler

Fig. 10 Principal stressmap-contour

those assigned in the benchmark, whereas for perme-ability the value of 10−12 cm/s has been assumed. Thisvalue could suggest the hypothesis of an impermeablematerial. This limit case can be analysed by the presentmodel locating the diffusion phenomenon in restrictedareas near the wetted side of the dam and along thecrack sides. Such a condition is easily handled bythe used mesh generator, but has not been applied inthe following. The proper representation of the cohe-sive forces requires a fine mesh in the area of the pro-cess zone. This is shown in Fig. 8, which represents theprocess zone when the fracture length is about 15 mand corresponds to an intermediate step of the analysiswhen the water level is 80 m. The pressure distributioninside the fracture is shown in Fig. 9 and the contourof principal stresses in Fig. 10.

This problem has been solved as a 2-D case in Schre-fler et al. (2006). The nucleation point of the fractureand the inclination of about 80◦ during the first 5 m ofthe fracture are similar in both solutions. During furtheradvancement steps the inclination of the 3-D solutiondiminishes also as a consequence of the boundary con-ditions at the base. At the crack tip after each advance-ment appears a fluid lag as in the 2-D case. The negativepressures diminish rapidly once the fracture is stabi-lized within the considered time step because of theinflow of water. The 3-D ad 2-D solutions give compa-rable stress and pressure distributions being the plane-

strain model representative of the 3-D case; in fact thisproblem has been chosen for validation purposes. Thegeneral 3-D space-adaptive procedure involves a highernumber of dofs of the corresponding plain strain modeland requires, therefore, a higher computational effortto obtain mesh-independent solutions. In many engi-neering situations 3-D analysis is compulsory.

6 Conclusion

Some important conclusions can be drawn from thisapplication:

− the mechanical behaviour of the solid skeletonstrongly depends on the characteristic permeabil-ity of the fluid within the crack. Crack paths are infact different, as result of the different stress fields;

− the crack path cannot be forecast; hence thetraditional use of special/interface elements to sim-ulate fracture propagation in large structures is pre-vented. One alternative to the successive remeshingis the use of cumbersome discretizations of theareas interested by fracture, but also this strategyis not viable in the case of dams. Further, the usedtechnique for the analysis of the nucleation of thefracture does not require the presence of an ini-tial notch and requires a very limited amount of

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A method for 3-D hydraulic fracturing simulation 257

information to be initially defined. Another alterna-tive would be the use of X-Fem, but there are onlyfew 3-D applications around (Moes et al. 2002;Gravouil et al. 2002; Sukumar et al. 2008); none ofthem addresses hydraulic fracturing.

− numerical results show a mesh-size dependencethat can be only reduced with the adaptivity inspace considered in the analysis. A full elimina-tion would require also adaptivity in time whichwas investigated in (Secchi et al. 2008) using atime discontinuous Galerkin method;

− trial simulations have shown that results depend onthe presence of the ‘filler’ inside the crack. In par-ticular a different crack surface path and a differ-ent fracture propagation velocity has been obtainedwith and without filler.

Acknowledgments Partial support was provided by the Strate-gic Research Project ”Algorithms and Architectures for Compu-tational Science and Engineering”—AACSE (STPD08JA32—2008) of the University of Padova (Italy)

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