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A Finite Element Model for Three Dimensional

Hydraulic Fracturing

Philippe R.B. Devloo FEC-Unicamp

Paulo Dore Fernandes PETROBRAS

Snia M. GomesIMECCUnicamp

Cedric A. Bravo FEC-Unicamp

Renato G. Damas FEC-Unicamp

A Finite Element Model for Three Dimensional Hydraulic Fracturing p.1/1


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Motivation

Hydraulic fracturing is a method extensively used in oil industry

To enhance petroleum recovery, fractures are artificially created

inside the porous media by applying hydraulic pressure produced

by an injected fluid

Usually, numerical simulations use simple models (2d or

pseudo3d) which are not suited for high permeability and/or low

efficiency fluids (e.g.water).

The purpose is to obtain a more sofisticated numerical model,

including the fluid flow inside the fracture, the elastic response of

the porous media and leak-off. The simulation is part of a research project between Unicamp

and PETROBRAS



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Summary

Introduction

Model Construction

Mathematical Model

Finite Element Model

Algorithm for Temporal Evolution

Numerical Simulation

Conclusions



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Mathematical Model

Fluid flow inside the fracture: conservation of massLaw relating the fluid flow with the variation the

fracture opening

Elastic response of the porous media

Formula relating the fracture opening and the pressure

on the fracture walls

Fluid filtration to the porous media



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Numerical Model

For the fluid flow inside the fracturefinite element Galerkin method + finite difference time

integration

For the relation between the pressure and fracture

opening

finite element approximation + analytical formulas

Computational implementation: PZ enviroment

Based on finite element techniques for the numerical

solution of PDE

Object oriented philosophy



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Geometric Aspects

Axisx: main propagation direction

Axisz: fracture hight

Axisy: fracture opening width

|y| w(x,z,t)

2 ,



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Section orthogonal to the propagation direction

: in-situ stress distribution

h/2: frature hight Hr =b: reservoir hight



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Matemathical Model

State variable: fracture openingw =w(x,z,t) Opening displacement equation (Bui, 1977)

p(x,z,t)

(x, z) =

G

4(1 ) Z

x

1

r

w

xl

+

z

1

r

w

zl dxldzl

r =p

(x xl)2 + (z zl)2 G, shear modulus and Poissons ration : computational region on the planex

z

Leak-off: Carters model (1957)

ql(t) =A

vsp(t ) +

2

t

Fluid loss, per time unit, on areaAexposed during time period



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Fluid Flow: Conservation of mass

dx

q qqq

leakoff

porous media

fluid inside the fracture

div(q) + wt

+Ql = 0.



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Fluid flow models

Newtonian fluid

q = w3

12p

Non Newtonian fluids (power law): 0<


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Numerical Approximation Schemes

For the conservation law Galerkin Method

Spatial discretization by bilinear finite elements

w(x,z,tn) =

i

wnii(x, z)

Computational region and boundary conditions

q=0

q=0

q=0

q0

q=



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Temporal discretization

Implicit Euler scheme + mass lumping

Zi.q n+1dxdy+ i

wn+1i wnit

+ i

Z tn+1tn

Qlidt+qi| = 0

For the opening-preassure integral equation

R*j

R*jp = constant on staggered cells

pnj = j + G

4(1 )1

mj

ZRj

Z

xl

1

r

w

x +

yl

1

r

w

z

dxdzdxldzl

= j + G

4(1 )

1

m

jXi

Tijwni ,



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Opening-preassure connection matrix

p = T w + conection matrix

Tij =

ZRj

x

Z

1

r

i

xldxl dzl

dx dz+

ZRj

z

Z

1

r

i

yldxl dzl

dx dz.

2

4

6

8

10

2

4

6

8

10

-0.01

0

0.01

0.02

0.03

2

4

6

8

10



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Algorithm for temporal integration

(wn

,pn

) (wn+1

,pn+1

)First step: Algorithm for fracture evolution

Point classification

8>>>:

open w(xi, yi, tn)>0

candidate for open w(xi, yi, tn) = 0 butResi >

closed: otherwise

candidate for open

open point

closed point

zero flux: If some of the four nodes of the cell boundary is closed



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Second step: Newtons method+ GMRES

Res(m) = G(wn+1,(m),pn+1,(m)) +Fn+1n

Gi(w,p) = tZiq dxdy+ iwi

Fn+1n

i

= iwni + tiq n+1 0@

Kw Kp

T I

1A0@wn+1,(m+1) wn+1,(m)

pn+1,(m+1) pn+1,(m)

1A =

0@

Resm

0

1A

Third step: Leak-off and pos-processing


Numerical results


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Numerical results


Conclusions and Next Steps


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Conclusions and Next Steps Demontration of the numerical viability of hydraulic

fracture simulation by combining fluid flow inside

fractures, elastic response of the porous media and leak-off;

Pseudo3D simulations indicate some advantages in

including the leakoff term during Newtons iterations. Forthe 3D model, this would require a more elaborated control

of fracture opening;

There is a great interest in PETROBRAS to consider

horizontal wells simulations;

The next step of the project is to extend the formulation to

consider 3D non-planar fracture propagation.


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