Stimulation
Fracture Propagation Models
The modeling of hydraulic hydraulic fractures applies three fundamental equations: 1. Continuity 2. Mome Moment ntum um (Fr (Frac actu turre Flui Fluid d Flow) Flow) 3. LEFM LEFM (Line (Linear ar Elas Elasti ticc Frac Fractur ture e Mech Mechani anics cs))
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 restrictive but require computer analysis
Stimulation
Fracture Propagation Models
Perkins-Kern-Nordgren Model (PKN) without leakoff
The following assumptions simplify the complex problem: 1. 2.
3.
4.
The fracture height,hf , is fixed and independent of fracture length. The fracture fluid pressure is constant in the vertical cross sections perpendicular to the direction of propagation. Reservoir rock stiffness, its resistance to deformation prevails in the vertical plane; i.e, 2D plane-strain deformation in the vertical plane Each plane obtains an elliptic shape with maximum width in the center,
w ( x, t )
1 h f p h G
Schematic representation of linearly propagating fracture with laminar fluid flow according to PKN model
Stimulation
Fracture Propagation Models
Perkins-Kern-Nordgren Model (PKN) without leakoff
5. The fluid pressure gradient in the x-direction can be written in terms of a narrow, elliptical flow channel,
p h 64 x
q 3 w h
f
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,
h f w q x 4 t 8. Combining provides a non-linear PDE in terms of w(x,t):
64(1 )h
2 2 w 2 x
G
f
w 0 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
Stimulation
Fracture Propagation Models
Geertsma-de Klerk (GDK) Model without leakoff
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,
12q x dx i p(0, t ) p( x, t ) 3 h f 0 w ( x, t ) Schematic representation of linearly propagating fracture with laminar fluid flow according to GDK model
Stimulation
Fracture Propagation Models
Geertsma-de Klerk (GDK) Model without leakoff
Assumptions: 5. The shape of the fracture in the horizontal plane is elliptic with maximum width at the wellbore
w (0, t )
2(1 )L( p
f
h )
G
Schematic representation of linearly propagating fracture with laminar fluid flow according to GDK model
Stimulation
Fracture Propagation Models
Comparison 1000
i s p , e r o b l l e w t a e r u s s e r p t e N
900 800 700 600
PKN
500 KGD 400 300 200 100 0 0
2000
4000
Fluid volume, gals
6000
8000
Stimulation
Fracture Propagation Models
Comparison 3000
2500
t f , 2000 h t g n e l 1500 e r u t c a r 1000 f
PKN KGD
500
0 0
2000
4000
Fluid volume, gals
6000
8000
Stimulation
Fracture Propagation Models
Comparison 0.400
n i 0.350 , e r o 0.300 b l l e w0.250 t a h t 0.200 d i w0.150 m u 0.100 m i x a m0.050
PKN KGD
0.000 0
2000
4000
Fluid volume, gals
6000
8000
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
Stimulation
Components 1.
3D stress distribution
3D Fracture Propagation Models
Assumptions linear elastic behavior propagation criterion given by fracture toughness
2.
2D fluid flow in fracture
3.
2D proppant transport
4.
Heat transfer
5.
Leakoff
laminar flow of newtonian or non-newtonian fluid
Leakoff is 1D, to fracture face
Stimulation
3D Fracture Propagation Models
Formulation •
Elliptic D.E. for elasticity
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Convective-diffusive eq. for heat transfer
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Parabolic D.E. for leakoff
Solution •
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Finite element method – discretization of formation to solve for stresses and displacements Boundary integral method – discretization of boundary
Stimulation
3D Fracture Propagation Models
Pseudo 3D models (P3D) •
Crack height variations are approximate…dependent on position and time
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1D fracture fluid flow
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Similar to PKN, i.e., vertical planes deform independently
2D
P3D
3D
Stimulation •
•
•
3D Fracture Propagation Models
Comparison to validate 2D models Example A: Strong stress barriers, negligible leakoff More examples in Chapter 5 of SPE monograph Vol 12
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, w wb = 0.10 in. 2. Calculate the average width,
2 w w 4 wb 3. Calculate the effective viscosity,
n 80.842q i e 47880K 2 h w g
1
4. Calculate dimensionless time,
2 5 h 4 1 eqi g B 1.7737x10 5 32C h G h n g t
t
2/3
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, w
0.1645
0.78t D D
6. Calculate the maximum wellbore width, 1/ 3
2 2 h 2 161 eqi g e 5.0782 x10 2 C h G hn g w ewD wb 7. Test for convergence, w
n n 1 w wb wb
Continue NO Go to step 2) with updated w wb. YES
TOL
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, 1/ 3
5 8 h 2 1 eqi g a 7.4768 x10 256C 8h4G hn g L
.6295
0.5809t D D L aL D
1. Calculate the fracture volume,
V
wh L g 12
10. Calculate the fracture pressure 1/ 4
3 0.02975 G qieL P (0, t ) f 3 h g 1
h,min
11. Update pumping time and repeat the procedure, starting at step 1).
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,
h 8C t n hg L h w we 8Vsp n 12 hg h 0.11168 g L q i 2 h n h C g
2
h 2 we 8V n L sp 12 h g
w
1 e
3. Average width, w
w wb 4
4. Calculate the effective viscosity,
80.842q i e 47880K 2
n 1
2 L
erfc ( ) L
Stimulation
Fracture Propagation Models
Dynamic Fracture Propagation Design
GDK Model
5. Simplified expression for fracture width, 1/ 4
2 84(1 )eqi L w 0.1295 wb Ghg
6. Test for convergence, w
n n 1 w wb wb
TOL
Continue NO Go to step 2) with updated w wb. YES
7. Volume of one wing of the fracture, V
hLw wb 48
Stimulation
Fracture Propagation Models
Dynamic Fracture Propagation Design
8. Bottomhole fracture pressure, 1/ 4 3 3 0.03725 G qi eh g h,min P (0, t ) f 2 2h 3 g 1 L
9.Update pumping time and repeat the procedure, starting at step 1).
GDK Model
Stimulation
Fracture Propagation Models
Nomenclature a
= length constant, ft.
V
= volume of single wing, ft
B
= time constant, min.
Vsp
= spurt loss, ft /ft
C
= fluid loss coefficient, ft/(min)1/2
w
= volumetric average fracture width, in.
E
= width constant, in.
wD
= dimensionless fracture width
G
= shear modulus, psi
wwb
= fracture width at wellbore, in.
hg
= gross fracture height, ft.
wwe
= fracture width at wellbore at end of pumping, in.
hn
= net permeable sand thickness, ft.
L
= dimensionless fluid-loss parameter including spurt loss
K
= consistency index, (lbf-sec )/ft
e
= effective fracture fluid viscosity, cp
L
= fracture length, ft.
h
= horizontal, minimum stress, psi
LD
= dimensionless fracture length
= poisson’s ratio
Pf
= bottomhole fracture pressure, psi
qi
= flow rate into single wing of fracture, bpm
t
= pumping time, min.
tD
= dimensionless time
n
2
3
3
2
