stimulation 7
DESCRIPTION
stimulation noteTRANSCRIPT
Hydraulic Fracturing Basic Relations
Basic Relations
• Necessary to understand to apply to fracturing pressure analysis and design parameters
• Material balance – provides basic design requirements for fluid volumes and proppants
• Fluid flow in a fracture
Interaction of fluid and formation
• Rock deformation
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Material Balance expressions
At the end of pumping,
(1)
Qi
Qout
Qf
Flow rate In = Flow Rate Out + Accumulation
Lpfpi VVV
formation to
lost fluid
of volume
fracture
created
of volume
injected
volume
cumulative
Lost VLp
Stored Vfp
VLs (Dt)
Vf (Dt)
VLs (Dt)
Vprop
Pumping, tp
Closure, Dt
Hydraulic Fracturing Basic Relations
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Material Balance expressions
At any time during shutin,
(2)
At closure, Dt = Dtc
(3)
Combining (1) and (3)
(4)
Lost VLp
Stored Vfp
VLs (Dt)
Vf (Dt)
VLs (Dt)
Vprop
Pumping, tp
Closure, Dt
)()(
p tfrom
lost fluid
of volume
fracture
created
of volume
fracture
theof
volume
tLsfptf VVV
t
DD
D
propVctfV D
)(
)space poresolids(
proppant
of volume
fracture
theof
volume
)( ctLsVLpVpropViV D
Hydraulic Fracturing Basic Relations
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At the end of pumping,
Lost VLp
Stored Vfp
pii tqV *
wAV ffp *
pt
fA
pr
LC
LK2
LpV
Lpfpi VVV
formation to
lost fluid
of volume
fracture
created
of volume
injected
volume
cumulative
Af = area of one face of fracture w = average created fracture width,in CL = fluid loss coefficient, ft/(min)1/2
rp = fluid loss area to fracture area KL = fluid loss multiplier
qi = injection rate tp = injection time
Hydraulic Fracturing Basic Relations
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Proppant scheduling
Define efficiency as:
Or
time for the fracture to close defines the efficiency
Amount of pad and proppant scheduling depends on h
Pad fraction:
Lost VLp
Stored Vfp=hVi P
rop
Co
nc,
c/c
f
1
0
0 fpVi Vi
i
fp
V
Vh
hD
p
c
t
t
h
h
1
1pf
Hydraulic Fracturing Basic Relations
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Example
Hydraulic Fracturing Basic Relations
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Example
Hydraulic Fracturing Basic Relations
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pad
Pre-pad flush
Pumping
Pressure geometry
Closure
Pressure CL
Time h
Design
Model
Volume required
Proppant schedule
Pumping
Pressure geometry
Closure
Pressure CL
Time h
Design
Model
Volume required
Proppant schedule
Hydraulic Fracturing Basic Relations
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p
tp
rL
CL
K2wf
Ap
ti
q
Rock deformation • Compliance of fracture describes the ease of fracture deformation
• Principle of crack advance and stresses at the crack tip
• The strain in the formation created by hydraulic fracturing is minor.
• As a result, formation deformation is linear elastic
• Based on the linear elastic assumption, the behavior of a fracture can be modeled using Sneddon’s classical solutions: 2D crack or radial crack Both are: • 2D with one-dimension infinite in extent • Elliptic shaped cracks • Inversely proportional to E’…plain strain modulus • Proportional to a characteristic dimension and net pressure
21
EE
Hydraulic Fracturing Basic Relations
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Rock deformation Stress intensity factor, KI, - characterizes the magnitude of the stresses near the crack tip
- f (geometry of body, loading parameters) - LEFM states a fracture will advance when KI reaches a critical value.
Stress concentration near the tip of the crack. Fracture toughness – measure of the resistance of the rock to crack, i.e., propagate.
Hydraulic Fracturing Basic Relations
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Fluid flow in fracture Pressure gradient exists along the fracture Local pressure gradient is given by the fluid rheology, velocity, and fracture width. where k’ and n’ are consistency and behavior indices, respectively, for a power law
model. If k’ = m and n’ = 1, then this equation reduces to Newtonian fluid.
Pnet
xf
Pnet=0
tiprw
closureP
fractureP
netP D
n
f
i
n h
q
w
k
dx
dp
12
Hydraulic Fracturing Basic Relations
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Fluid flow in fracture: pressure gradient correction Classical fracture models assume pressure in the fracture is constant. However, the fluid flow relation indicates a gradient from pwf to pc. Thus define, where pf is average pressure within the fracture. Consequently, substitute for pf – smin in width equations with, • Includes the pressure gradient effect from flow and fluid rheology
along the fracture • Includes wellbore pressure
f
f
cw
cf
p
p
pp
pp
D
D
fff ppp DD s min
Hydraulic Fracturing Basic Relations
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Fracture compliance, cf
Proportionality between the pressure and width
With
Hydraulic Fracturing Basic Relations
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c
pw
pf
cf
pf
cw D
Radial
GDK
PKN
R
fx
fh
Efc
23/32
22
Rock deformation
Pc smin ??
• Fracture closure pressure is a global parameter which defines the fluid pressure for which the fracture effectively closes. It is the average of formation heterogeneities.
• The minimum stress is a local parameter which generally varies over the plane of the fracture.
Hydraulic Fracturing Basic Relations
© Copyright, 2011