Numerical modelling of hydraulic fracturing

Remij, E.W.; Remmers, J.J.C.; Huyghe, J.M.R.J.; Smeulders, D.M.J.

Published in:Computer Methods and Recent Advances in Geomechanics - Proceedings of the 14th Int. Conference ofInternational Association for Computer Methods and Recent Advances in Geomechanics, IACMAG 2014

Published: 01/01/2015

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Numerical modelling of hydraulic fracturing

E.W. Remij & J.J.C. Remmers & J.M. Huyghe & D.M.J. SmeuldersDepartment of Mechanical EngineeringEindhoven University of Technology, PO BOX 513, 5600 MB Eindhoven

ABSTRACT: In this paper we present a numerical model for hydraulic fracturing purposes. The rockformation is modelled as a poroelastic material based on Biot’s Theory. A fracture is represented in adiscrete manner using the eXtended Finite Element Method (X-FEM). The fluid flow is governed by alocal mass balance. This means that there is an equilibrium between the opening of the fracture, thetangential fluid flow, and the fluid leakage. The mass balance in the fracture is solved with a separateequation by including an additional degree of freedom for the pressure in the fracture. The fracture cangrow in arbitrary directions by using an average stress criterion. We show a result of hydraulic fracturepropagation for a 2D circular borehole. The fracture direction is consistent with the expected direction.

1 INTRODUCTION

Hydraulic fracturing is the process in which a frac-ture propagates by applying a high pressure insidethe fracture. In geo-mechanics this process is usedto stimulate oil and gas reservoirs by fracturingthe underground rock formation. The induced frac-tures remain open, due to the addition of a prop-pant to the fracturing fluid, and therefore greatlyenhance the permeability in the formation. Numer-ical models can be used to obtain more insight inthe fracture process and may eventually be usedto optimize the fracture process.Boone & Ingraffea (1990) developed a numerical

model based on the finite element method (FEM)for the poroelastic material where a cohesive zonedescription was used for the fracture. Using a meshadaptation scheme in a poroelastic FEM, Schref-fler and co-workers (Schrefler et al. 2006, Secchiet al. 2007, Secchi and Schrefler 2012) simulatedpropagating hydraulic fractures in a arbitrary di-rections. Carrier & Granet (2012) used a similarapproach with a priori placed interface elementsthat contained an additional degree of freedom forthe pressure in the fracture.

The eXtended Finite Element Method (X-FEM)was used by Mohammadnejad & Khoei (2012) forcohesive crack growth in multiphase porous ma-terials. They also successfully applied their modelfor hydraulic fracturing simulations (Mohammad-nejad & Khoei 2013).X-FEM is a common technique in solid mechan-

ics and has as an important advantage comparedto the previously mentioned fracture models; afracture can grow in arbitrary directions withoutthe need to remesh (Belytschko and Black 1999,Moes et al. 1999). By exploiting the partition ofunity property of finite element shape functions(Melenk & Babuska 1996), a fracture can be in-serted in a finite element mesh by adding addi-tional degrees of freedom to the nodes surround-ing the fracture (Figure 1). In this contributionwe present a similar X-FEM model as was usedby Mohammadnejad & Khoei (2013) but we alsoinclude the pressure in the fracture as a separatedegree of freedom. Such a formulation ensures thatall the pumping fluid goes into the fracture. Thenew model is refereed to as the Enhanced LocalPressure (ELP) model.

Enhanced node

Regular node

Ω+Ω−

Figure 1: A two dimensional finite element meshcrossed by a fracture represented by the black line.The black nodes surrounding the fracture are en-hanced with additional degrees of freedom. Thegrey elements therefore contain additional termsin the stiffness matrix and the force vector.

2 KINEMATIC RELATIONS

The total displacement field of the solid skeletoncan, at any time t, be described by a continuousdisplacement field u(x, t) and a discontinuous dis-placement field u(x, t) (Belytschko & Black 1999,Moes, Dolbow, & Belytschko 1999)

u(x, t) = u(x, t) +HΓd(x)u(x, t), (1)

We can also write the pressure field as a sum of thecontinuous field p(x, t) and a discontinuous fieldp(x, t).

p(x, t) = p(x, t) +HsΓd(x)p(x, t). (2)

The functions HΓdand Hs

Γdare Heaviside step

functions defined over the fracture as

HΓd=

1 if x ∈ Ω+

0 if x ∈ Ω−,(3)

HsΓd

=

12

if x ∈ Ω+

−12

if x ∈ Ω−.(4)

With these formulations the pressure inside thefracture is undetermined. We therefore introduce athird pressure pd, representing the pressure in thefracture. The continuous fields are present in allthe finite element nodes. The discontinuous fieldsand the pressure in the fracture are only includedin the enhanced nodes (Figure 1).

The pressure in the fracture is a one dimensionalvariable that can be used to describe the fluid massbalance in the fracture. Integrating the local massbalance gives

q+Γd

· nd − q−Γd

· nd + [u]n + un〈δus

δs〉− (5)

unδ

δs(kd

δpdδs

) +un

Mf

pd = 0,

with q+Γd

and q−Γd

being the fluid flow from thefracture into formation for the fracture side of theΩ+ and the Ω− domain, respectively, [u]n denotingthe time derivative of the normal opening of thefracture, us being the shear opening of the fracture,and kd being the permeability in the fracture.Assuming a Couette flow, the latter is given by

kd =u2n

12µ, (6)

where µ is the viscosity of the fluid, and un theopening of the fracture. The derivation of thisequation can be found in Irzal et al. (2013).Using the cohesive zone approach, the damaging

of the material is described by a traction acting onthe fracture. This traction is coupled to the hydro-static pressure in the crack. Assuming continuityof stress from the formation to the fracture, we canwrite the local momentum balance as

σ · nd = td − pdnd, (7)

3 NUMERICAL FORMULATION

The time discretization is performed with an im-plicit Euler scheme. The non-linear system of equa-tions is finally solved by using a Newton-Raphsoniterative procedure. We consider the porous mate-rial to be fully saturated with a fluid and subjectedto small variations in the displacement gradient.The bulk poroelasticity is based on Biot theory.The propagation time and the propagation di-

rection of the fracture are based on an averagestress criterion. The stress values surrounding thetip of the cohesive zone are weighted with a Gaus-sian function (Jirasek 1998). The average stress isthen the weighted sum of stresses in these integra-tion points.

x

y

~n ~s

θ

η

Figure 2: Schematic representation of a materialwith at the crack tip the global x-y coordinatesystem and the local coordinate system, describedwith a normal unit vector n and a tangential unitvector s.

σav =

nint∑i=1

wi

wtot

σe,i with wtot =

nint∑j=1

wj. (8)

Here nint is the number of integration points in thedomain, σe,i is the current effective stress state inintegration point i which has a weight factor wi

defined as

wi =(2π)

23

l3ae

−r2i2l2a , (9)

with ri being the distance between the tip of thecohesive zone and the integration point ni, and labeing a length scale parameter defining how fastthe weight factor decays as a function of the dis-tance between the integration points and the co-hesive zone tip. Based on the average stress, anequivalent traction is calculated (Camacho & Or-tiz 1996)

teq(θ) =

√< tn >2 +

1

βt2s , (10)

with

< tn >=

0 if tn ≤ 0tn if tn > 0

, (11)

where tn and ts respectively are the normal andshear traction

tn = nTσavn ts = sTσavn. (12)

Here n is the normal vector and s is the tangentvector to an axis η which is rotated by an angle θwith respect to the x-axis (Figure 2). If the maxi-mum equivalent traction exceeds the yield strengthτult of the material the fracture is extended in thedirection of angle θ through one element. The frac-ture is assumed to open as a cleavage crack. Thetraction separation relation is given by (Camacho& Ortiz 1996)

tn = tn0

(1− vn

vncr

), (13)

ts = ts0

(1− vn

vncr

)sgn(vs).

Here vn and vs are respectively the normal andsliding the displacement, sgn(·) is the Signum func-tion. The parameter vncr is the length of the fullydeveloped traction-free crack. This parameter de-pends on the fracture toughness Gc and the initialnormal traction tn0

-1

0

1

t n/t

n0

0 1vn/vn,cr

Figure 3: The normalized tractions across the frac-ture as a function of the displacement jump in.

vncr =2Gc

tn0

. (14)

The initial tractions, tn0 and ts0 , are taken equal tothe normal and shear tractions at the time of prop-agation, respectively. Self-contact of the fracture issimulated by using a penalty stiffness method. Adetailed description of the numerical implementa-tion can be found in the work (Remmers, de Borst,& Needleman 2008) and the usage of X-FEM inporous materials is given in (Kraaijeveld & Huyghe2011, Rethore, Borst, & Abellan 2007).

Numerical solutionAnalytical solution

-3

-2.5

-2

-1.5

-1

-0.5

0

0 2 4 6 8 10 12 14 16

σrr[M

Pa]

x [m]

(a) Radial stress

Numerical solutionAnalytical solution

-1.75-1.7

-1.65-1.6

-1.55-1.5

-1.45-1.4

-1.35-1.3

-1.25

0 2 4 6 8 10 12 14 16

σθθ[M

Pa]

x [m]

(b) Circumferential stress

Figure 4: The results of the numerical solution andanalytical solution for the Kirsch problem plottedover the x-axis.

4 EXAMPLES

The performance of the numerical model is ad-dressed by considering a 2D borehole problem(Figure 5). The rock formation is assumed to be asquared specimen with a dimension of 10.0m. Thecircular hole has a radius of 1.0m. The confiningstress is taken as σ0=5.0[Mpa] in y-direction, andis taken twice as large in the x-direction. The ma-terial parameters are given in Table 1.Applying the confining stress leads to consolida-

tion effects in the bulk material due to a poroelas-tic response. In order to obtain an initial equili-brated state we perform one time step with a pre-scribed zero pressure in all the nodes. After thisfirst step we release the pressure, place the initialfractures and start the fluid inflow q0.The initial stress field is compared with Kirsch

analytical solution for a circular hole in an infi-nite elastic solid (Kirsch 1898). The radial stressand the circular stress over the x-axis are shownin Figure 4.

Inintialfractures

σ0

σ0

2σ02σ0

p = 0

p = 0

y

x

Figure 5: Scheme of the borehole fracture problem.

Table 1: Model Parameters used in the boreholefracture problem.

Name Symbol ValueYoung’s modulus E 17.0 GPaPoisson’s ratio ν 0.2Ultimate strength τult 1.25 MPaFracture toughness Gc 120.0 Nm−1

Fluid viscosity µ 1.0e−4 Pa · sFluid injection rate q0 0.05 ms−1

Bulk modulus solid Ks 36.0 GPaBulk modulus fluid Kf 3.0 GPaBiot coefficient α 0.75

The preferred propagation direction of a hy-draulic fracture is perpendicular to the minimumconfining stress (Hubbert & David 1972). In or-der to investigate if the numerical model can pre-dict this behaviour we also perform the simulationwith only the initial fracture in the y-direction.The time evolution of this fracture is shown inFigure 6a. If we consider both initial fractures, in-deed only the fracture that is perpendicular to theminimum confining stress propagates, as shown inFigure 6b.

5 CONCLUSIONS

The performance of the ELP model is addressedin a 2D circular borehole problem. In the first caseonly the initial fracture perpendicular to the high-est confining stress is included in the simulation.This is the unfavourable direction of fracture prop-agation.

a b

Figure 6: Contour plots of the hydrostatic pressurefor the borehole problem at 3 different times. Thedeformed mesh is magnified 100 times. (a) A prop-agating fracture turns in the direction of the min-imum confining stress. (b) Only the fracture per-pendicular to the minimum confining stress propa-gates. The second fracture in the y-direction is notvisible because it is closed.

The propagated fracture indeed turns in the direc-tion of the highest confining stress. In the secondcase also the fracture that is already in the di-rection of the highest confining stress is included.Now only the second fracture grows while the frac-ture that is initially perpendicular to the highestconfining stress remains closed. This behaviour isconsistent with findings in the literature (Hubbert& David 1972).

ACKNOWLEDGEMENT

This research project is supported by the DutchTKI Gas Foundation.

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