production data analysis of shale gas

1. Mechanistic Rate Decline Analysis in Shale
Gas Reservoirs
Dr. George Stewart
Chief Reservoir Engineer
Weatherford Intl. Inc
Reservoir Engineering Technology Symposium
May 11, 2012

2. Lfar = xe/2

3. Dominantly
Multiple Transverse
Hydraulic Fractures Interior
W Fractures
Confining
External
Boundary
xf
xe
Wfar = W/2
Horizontal Well
Trajectory
External
Fractures Approximating
Virtual No-Flow
ye = W Boundaries
yw = W/2
Nfrac = 5

5. Dominantly Essentially Zero
Multiple Transverse Permeability
Hydraulic Fractures Interior
W Fractures
Confining
Modified External
Permeability Boundary
xf
xe
Wfar = W/2
Horizontal Well
Trajectory
External
Fractures Approximating
Virtual No-Flow
ye = W Boundaries
yw = W/2
Nfrac = 5

6. Fracture in a Closed Channel - Image Source Solution
First Plane First
Image Source Image
xf
W Areal View
W
VNFB LNFB

7. Constant Terminal Pressure (CTP) Solution for Rate, q
pi Well produced at
q(t) constant BHP
pwf
0 t
Drawdown = pi  pwf
Time, t
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 forecasting
Laplace space method is termed CTP convolution

8. Canonical CTP Solutions
I.-A. Radial Flow (Line Source)
qt  1
qD   tD 
kt
2kh p i  p wf  1 ln 4t d
where
 c t rw
2
2 
I.-A. Linear Flow (Plane Source)
qt  1
qD   kt
 / 4 t
t Dxf 
2kh pi  p wf 
where
3  c t x f2
Dxf
. . . Carslaw and Jaeger p 43

9. CTP Linear Flow Plot for Gas
1422T telf
m(pi)  m(pwf) blf  S fs
kh
Q(t)
T 1
slope, mlf  64.288 
Af k  c t
b0
lf
Af = hxf
Requires knowledge of pi
0
t
0.000263679  kt elf
 0.33 Yields xf k
Sfs 0 t W / 2
= c 2
assuming h known

10. Flowing Material Balance Cartesian Plot of m(pwf)/Q versus ta
Rate Normalised Pseudopressure Change
Closed System SSS Depletion
2.355T
slope, m* =
m(p)
b g
hA c t i
Q straight
line
Derivative of Specialised Plot
b*
d m p / Q
Intercept p  
1
t
 i cgi Q t  dt a
ta  dt 
Qtb g b p g c b p g
0 g
VRD
Rate Normalised Average Pressure Pseudotime, ta
t a = f(A)

12. Flowing (VRD) Material Balance
Method 1: Approximate Deconvolution or equivalent constant rate (ECR)
• developed by Stewart for analysing VRD data
Method 2: Material Balance Time, te
• developed by Agarwal and Gardner following a
suggestion by Blasingame

13. Flowing Material Balance
t
N p   qdt  V  N p  qte  cVp  cV  pi  p 
0 Liquid material balance equation
Rate Normalised p 1
Average Pressure  te where p  pi  p
Drop from pi q cV
q q
Classical Definition of J ti  
Transient P.I. pi  pwf pwf
q
Transient P.I.
Jt 
Based on p  pwf p  pwf

15. Analysis of Variable Rate Extended Drawdown
p wf  p i  mlf g l ( t )  q t Sl
VRD-I.-A.
p
pwf
p p wf  p i  q r mlf t   Sl
pwf CRD - I.-A.
corr
Based on
p wf
LSTF
t or t'
1 
ECR Method
mlf  4.0641 (field units)
h kc t x 2
f
Approximate Deconvolution

16. Approximate Deconvolution of Spanning Fracture in a Closed Reservoir
Exact Volume = 14,2500 bbl
corr
pwf Error ~ 6%
(psia)
Transformed Time, t (hr)
xf =100 ft W = 200 ft k = 1 md

17. Palmer and Mansoori CBM Rock Mechanics Model
- Based on Linear Elasticity
Recommended by
 p  pi
  Mavor
= 1 +
i iM
very sensitive to i
k   3
=  
ki i 
Constrained Axial
1 
M = E Modulus
(1 + )(1  2)
M 1 +  Bulk Modulus
K = 3  
1 
SPE
E = Young’s Modulus  = Poisson’s Ratio 52607

18. Locus of Roots of the Stress-Dependent Radial Flow Equation
Reservoir Pressure, pe
Wellbore
Pressure
pw
High Drawdown Roots
Only Accessible Through Low Drawdown Root
Reverse Direction Integration High Drawdown Root
Flow-Rate, q Fig. 18.13.5

19. Combined Pseudopressure Function (p)
4000
Real Gas
3500 g = 0.7 T = 150 oF pi = 5000 psia pseudop
(p) i = 0.01 E = 500,000  = 0.25 n = 3 ki = 10 md
3000
(psia)
2500
z ii 1  i p k p
p
) p    z 1  pdp
p2000
( pi k i pb
y
1500
pseudop including
Palmer and Mansoori SDPP effect
Model
1000
500
0
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Pressure, p (psia)

20. Linear Steady-State Flow with Stress Dependent Permeability

21. Linear Steady-State Flow with Stress Dependent Permeability
k  p  dp
Darcy’s Law q  4x f h f  p
 dy
Integrating:
W
4x f pe
4 x f h f ( pe )1  e  pe k  p 
q  dy   h f ( p)k ( p)dp   1    p  dp
0
 pf  pf
qW 1  e k  p
pe
i.e.
4 x f keh f  pe 

ke  1    p  dp
pf
1  e k  p SDPP
p
Defining:   p   1    p  dp Normalised
ke pb Pseudopressure
qW
   pe    p f 
4 x f kehef

24. Permeability Ratio, kf/ki versus pwf and i
1
0.9
0.8
i = 0.05
0.7
0.6
kf/ki
0.5 M = 600,000 psi
pi = 5000 psia i = 0.01
0.4
0.3
0.2 i = 0.005
0.1
Choked
0
1000 1500 2000 2500 3000 3500 4000 4500 5000
Pressure, pwf (psia)