Mechanistic Rate Decline Analysis in Shale Gas Reservoirs
Dr. George Stewart Chief Reservoir Engineer
Weatherford Intl. Inc
Reservoir Engineering Technology Symposium May 11, 2012
Lfar = xe/2
W
ExternalFractures
Interior
Fractures
Dominantly
Multiple TransverseHydraulic Fractures
Horizontal WellTrajectory
xf
Approximating
Virtual No-Flow
Boundaries
N = 5frac
xe
y = We
y = W/2w
ConfiningExternal
Boundary
W = W/2far
W
ExternalFractures
Interior
Fractures
Dominantly
Multiple TransverseHydraulic Fractures
Horizontal WellTrajectory
xf
Approximating
Virtual No-Flow
Boundaries
N = 5frac
xe
y = We
y = W/2w
ConfiningExternal
Boundary
W = W/2far
Essentially ZeroPermeability
ModifiedPermeability
W
W
xf
Fracture in a Closed Channel - Image Source Solution
FirstImage
FirstImageFirst
Image
PlaneSource
VNFB LNFB
Areal View
Constant Terminal Pressure (CTP) Solution for Rate, q
q(t)
Time, t
Well produced atconstant BHP
Drawdown = p pi wf
Boundary condition of the first kind
Basis of decline curve analysis
CTP solution can be generated from the CRD solution
CRD solutions form the basis of well test analysis
CTP solutions used to analyse production data
CTP solution obtained from the CRD solution by convolution
Real time superposition method is termed forecastingLaplace space method is termed CTP convolution
pi
pwf
0 t
Dxf
3wfi
D
t4/
1
ppkh2
tqq
dwfi
D t4ln
2
1
1
ppkh2
tqq
Canonical CTP Solutions
I.-A. Radial Flow (Line Source)
I.-A. Linear Flow (Plane Source)
. . . Carslaw and Jaeger p 43
2
wt
Drc
ktt
where
2
ft
Dxfxc
ktt
where
CTP Linear Flow Plot for Gas
t
slope mT
A k clf
f t
, . 64 2881
A = hxf f
Requires knowledge of pi
Yields x kf
assuming h known
telf
0
0
S = 0fs
m(p) m(p )i wf
Q(t)
blf
fslf Skh
Tb
1422
33.0
2/Wc
kt000263679.02
t
elf
Flowing Material Balance Cartesian Plot of m(p )/Q versus t wf a
m(p)
SSS Depletion
slope, m* = 2 355. T
hA c t i b g
Rate Normalised Average Pressure Pseudotime, ta
tQ t
c Q t
p c pdta
i gi
g
t
1
0b g b g b g
pd m p Q
dta
/
Derivative of Specialised Plotb*
Intercept
VRD
t = f(A)a
straightline
Closed System
Q
Rate Normalised Pseudopressure Change
Flowing (VRD) Material Balance
Method 1: Approximate Deconvolution or equivalent constant rate (ECR)
Method 2: Material Balance Time, te
• developed by Stewart for analysing VRD data
• developed by Agarwal and Gardner following a
suggestion by Blasingame
ppcVpcVqtNVtqdN i
t
epp 0
Flowing Material Balance
Liquid material balance equation
pppwheretcVq
pie
1Rate Normalised
Average Pressure
Drop from pi
wf
tpp
qJ
wfwfi
tip
q
pp
qJ
Classical Definition of
Transient P.I.
Transient P.I.
Based on p pwf
t or t'
Analysis of Variable Rate Extended Drawdown
pwf
pwf
pwf
corr
p
p
p p m g t q t Swf i lf l l ( )
p p q m t Swf i r lf l
CRD - I.-A.
VRD-I.-A.
mh k c x
lf
t f
4 06411
2.
(field units)
ECR Method
Based on
LSTF
Approximate Deconvolution
Transformed Time, t (hr)
pwf
corr
(psia)
Approximate Deconvolution of Spanning Fracture in a Closed Reservoir
x =100 ft W = 200 ft k = 1 mdf
Exact Volume = 14,2500 bbl
Error ~ 6%
i = 1 +
p pi
iM
k
ki =
i
3
M = E1
(1 + )(1 2)
K = M
3
1 +
1
Palmer and Mansoori CBM Rock Mechanics Model
E = Young’s Modulus = Poisson’s Ratio
Constrained Axial Modulus
Bulk Modulus
Recommended by Mavor
- Based on Linear Elasticity
very sensitive to i
SPE 52607
WellborePressure
Flow-Rate, q
pw
Reservoir Pressure, pe
Locus of Roots of the Stress-Dependent Radial Flow Equation
Low Drawdown RootHigh Drawdown Roots
Only Accessible ThroughReverse Direction Integration High Drawdown Root
Fig. 18.13.5
0
500
1000
1500
2000
2500
3000
3500
4000
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
y(p)
Pressure, p (psia)
Combined Pseudopressure Function(p)
(p)
(psia)
Real Gaspseudop
pseudop includingSDPP effect
g i
i i
= 0.7 T = 150 F p = 5000 psia
= 0.01 E = 500,000 = 0.25 n = 3 k = 10 md
o
p
pi
i
i
i
b
pdp1
pk
z
p
k
1
p
zp
Palmer and MansooriModel
i
Linear Steady-State Flow with Stress Dependent Permeability
Linear Steady-State Flow with Stress Dependent Permeability
dy
dppkphxq ff
4Darcy’s Law
e
f
e
f
p
p
eeff
W p
p
f
fdp
p
pkphxdppkph
xdyq
1
1)(4)()(
4
0
Integrating:
e
f
p
pe
e
efef
dpp
pk
kphkx
qW
1
1
4i.e.
Defining:
p
pe
e
b
pdp
pk
kp
1
1SDPP
Normalised
Pseudopressure
fe
efef
pphkx
qW
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1000 1500 2000 2500 3000 3500 4000 4500 5000
kf/k
i
Pressure, pwf (psia)
Permeability Ratio, kf/ki versus pwf and i
M = 600,000 psi
pi = 5000 psia
i = 0.05
i = 0.005
i = 0.01
Choked