Model GDK

Stimulation Fracture Propagation Models

The modeling of hydraulic fractures applies three fundamental equations:

1. Continuity

2. Momentum (Fracture Fluid Flow)

3. LEFM (Linear Elastic Fracture Mechanics)

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Stimulation Fracture Propagation Models

Solution Technique • The three sets of equations need to be coupled to simulate the

propagation of the fracture. • The material balance and fluid flow are coupled using the relation

between the fracture width and fluid pressure. • The resulting deformation is modeled through LEFM. • Complex mathematical problem requires sophisticated numerical

schemes. • 2D models provide tractable solutions but are limited by assumptions • 3D and pseudo-3D are less restrictive but require computer analysis

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Stimulation Fracture Propagation Models

The following assumptions simplify the complex problem: 1. The fracture height,hf, is fixed and independent of fracture length. 2. The fracture fluid pressure is constant in the vertical cross sections perpendicular to the direction of propagation. 3. Reservoir rock stiffness, its resistance to deformation prevails in the vertical plane; i.e, 2D plane-strain deformation in the vertical plane 4. Each plane obtains an elliptic shape with maximum width in the center,

Perkins-Kern-Nordgren Model (PKN) without leakoff

Schematic representation of linearly propagating fracture with laminar fluid flow according to PKN model

G

hp

fh1

)t,x(w

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Stimulation Fracture Propagation Models

5. The fluid pressure gradient in the x-direction can be written in terms of a narrow, elliptical flow channel, 6. The fluid pressure in the fracture falls off at the tip, such that at x = L and thus p = h. 7. Flow rate is a function of the growth rate of the fracture width, 8. Combining provides a non-linear PDE in terms of w(x,t): subject to the following conditions, w(x,0) = 0 for t = 0 w(x,t) = 0 for x > L(t)

q(0,t) = qi/2 for two fracture wings

Perkins-Kern-Nordgren Model (PKN) without leakoff

fh

3w

q64

x

hp

t

w

4

fh

x

q

0t

w

2x

2w

2

fh)1(64

G

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Stimulation Fracture Propagation Models

Assumptions: 1. Fixed fracture height, hf.

2. Rock stiffness is taken into account in the horizontal plane only. 2D plane strain deformation in the horizontal plane.

3. Thus fracture width does not depend on fracture

height and is constant in the vertical direction.

4. The fluid pressure gradient is with respect to a

narrow, rectangular slit of variable width,

Geertsma-de Klerk (GDK) Model without leakoff

Schematic representation of linearly propagating fracture with laminar fluid flow according to GDK model

x

0 )t,x(3

w

dx

fh

iq12

)t,x(p)t,0(p

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Stimulation Fracture Propagation Models

Assumptions: 5. The shape of the fracture in the horizontal plane is elliptic with maximum width at

the wellbore

Geertsma-de Klerk (GDK) Model without leakoff

Schematic representation of linearly propagating fracture with laminar fluid flow according to GDK model

G

)hf

p(L)1(2)t,0(w

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Stimulation Fracture Propagation Models

Comparison

0

100

200

300

400

500

600

700

800

900

1000

0 2000 4000 6000 8000

Ne

t p

ress

ure

at

we

llbo

re, p

si

Fluid volume, gals

PKN

KGD

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Stimulation Fracture Propagation Models

Comparison

0

500

1000

1500

2000

2500

3000

0 2000 4000 6000 8000

frac

ture

len

gth

, ft

Fluid volume, gals

PKN

KGD

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Stimulation Fracture Propagation Models

Comparison

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0 2000 4000 6000 8000

max

imu

m w

idth

at

we

llbo

re, i

n

Fluid volume, gals

PKN

KGD

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Stimulation 3D Fracture Propagation Models

Applications

• Primarily for complex reservoir conditions – Multiple zones with varying elastic or leakoff properties

– Closure stress profiles indicate complex geometries

Vertical fracture profile illustrating the changes in width across the fracture

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Stimulation 3D Fracture Propagation Models

Components Assumptions 1. 3D stress distribution linear elastic behavior

propagation criterion given by fracture toughness

2. 2D fluid flow in fracture laminar flow of newtonian or non-newtonian fluid

3. 2D proppant transport

4. Heat transfer

5. Leakoff Leakoff is 1D, to fracture face

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Stimulation 3D Fracture Propagation Models

Formulation • Elliptic D.E. for elasticity

• Convective-diffusive eq. for heat transfer

• Parabolic D.E. for leakoff

Solution • Finite element method – discretization of formation to solve for stresses

and displacements

• Boundary integral method – discretization of boundary

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Stimulation 3D Fracture Propagation Models

Pseudo 3D models (P3D) • Crack height variations are approximate…dependent on position and time

• 1D fracture fluid flow

• Similar to PKN, i.e., vertical planes deform independently

2D P3D 3D

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• Comparison to validate 2D models

• Example A: Strong stress barriers, negligible leakoff

• More examples in Chapter 5 of SPE monograph Vol 12

Stimulation 3D Fracture Propagation Models

3D simulator

Stimulation Fracture Propagation Models

Dynamic Fracture Propagation Design PKN Model

• Includes effects of non-newtonian fluids and net-to-gross height

1. Initial guess of maximum wellbore width, wwb = 0.10 in.

2. Calculate the average width,

wbw

2

4w

3. Calculate the effective viscosity,

1n

2w

gh

iq842.80

K47880e

4. Calculate dimensionless time,

3/2

5

nh

gh

Gg

h5

C32

2

iq

e14

10x7737.1B

B

t

Dt

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Stimulation Fracture Propagation Models

Dynamic Fracture Propagation Design PKN Model

• Includes effects of non-newtonian fluids and net-to-gross height

5. Calculate dimensionless width,

1645.0

Dt78.0

Dw

6. Calculate the maximum wellbore width,

3/1

2

2

21162

100782.5

nh

gh

Gg

hC

iq

exe

Dew

wbw

7. Test for convergence,

TOL1n

wbw

n

wbw

YES Continue

NO Go to step 2) with updated wwb.

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Stimulation Fracture Propagation Models

Dynamic Fracture Propagation Design PKN Model

• Includes effects of non-newtonian fluids and net-to-gross height

8. Calculate the fracture length,

3/1

8

48256

512

104768.7

nh

gh

Gg

hC

iq

exa

6295.

Dt5809.0

DL

DaLL

1. Calculate the fracture volume,

12

Lg

hwV

10. Calculate the fracture pressure

min,h

4/1

31

Lei

q3

G

gh

02975.0)t,0(

fP

11. Update pumping time and repeat the procedure, starting at step 1).

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Stimulation Fracture Propagation Models

Dynamic Fracture Propagation Design GDK Model

1. Initialize the procedure by guessing wwb = 0.1 in.

2. Calculate the dimensionless fluid loss parameter and fracture length,

gh

nh

spV8

12

wew

gh

nh

tC8

L

)(

2

12

812

2

2

11168.0L

erfcL

eL

gh

nh

spVwe

w

nh

gh

iq

Cg

h

L

3. Average width,

wbw

4w

4. Calculate the effective viscosity, 1n

2w

gh

iq842.80

K47880e

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Stimulation Fracture Propagation Models

Dynamic Fracture Propagation Design GDK Model

5. Simplified expression for fracture width,

4/1

gGh

2L

iq

e)1(84

1295.0wb

w

6. Test for convergence,

TOL1n

wbw

n

wbw

YES Continue

NO Go to step 2) with updated wwb.

7. Volume of one wing of the fracture,

48

wbhLw

V

8. Bottomhole fracture pressure,

min,

4/1

231

33

2

03725.0),0(

hL

gh

eiqG

gh

tf

P

9.Update pumping time and repeat the procedure, starting at step 1).

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Stimulation Fracture Propagation Models

Dynamic Fracture Propagation Design GDK Model

5. Simplified expression for fracture width,

4/1

gGh

2L

iq

e)1(84

1295.0wb

w

6. Test for convergence,

TOL1n

wbw

n

wbw

YES Continue

NO Go to step 2) with updated wwb.

7. Volume of one wing of the fracture,

48

wbhLw

V

8. Bottomhole fracture pressure,

min,

4/1

231

33

2

03725.0),0(

hL

gh

eiqG

gh

tf

P

9.Update pumping time and repeat the procedure, starting at step 1).

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Stimulation Fracture Propagation Models

Nomenclature

a = length constant, ft.

B = time constant, min.

C = fluid loss coefficient, ft/(min)1/2

E = width constant, in.

G = shear modulus, psi

hg = gross fracture height, ft.

hn = net permeable sand thickness, ft.

K = consistency index, (lbf-secn)/ft

2

L = fracture length, ft.

LD = dimensionless fracture length

Pf = bottomhole fracture pressure, psi

qi = flow rate into single wing of fracture, bpm

t = pumping time, min.

tD = dimensionless time

V = volume of single wing, ft3

Vsp = spurt loss, ft3/ft

2

w = volumetric average fracture width, in.

wD = dimensionless fracture width

wwb = fracture width at wellbore, in.

wwe = fracture width at wellbore at end of pumping, in.

L = dimensionless fluid-loss parameter including spurt loss

e = effective fracture fluid viscosity, cp

h = horizontal, minimum stress, psi

= poisson’s ratio

a = length constant, ft.

B = time constant, min.

C = fluid loss coefficient, ft/(min)1/2

E = width constant, in.

G = shear modulus, psi

hg = gross fracture height, ft.

hn = net permeable sand thickness, ft.

K = consistency index, (lbf-secn)/ft

2

L = fracture length, ft.

LD = dimensionless fracture length

Pf = bottomhole fracture pressure, psi

qi = flow rate into single wing of fracture, bpm

t = pumping time, min.

tD = dimensionless time

V = volume of single wing, ft3

Vsp = spurt loss, ft3/ft

2

w = volumetric average fracture width, in.

wD = dimensionless fracture width

wwb = fracture width at wellbore, in.

wwe = fracture width at wellbore at end of pumping, in.

L = dimensionless fluid-loss parameter including spurt loss

e = effective fracture fluid viscosity, cp

h = horizontal, minimum stress, psi

= poisson’s ratio

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