May 4, 2007
-
Upload
nadine-brown -
Category
Documents
-
view
25 -
download
1
description
Transcript of May 4, 2007
1
Study of Pressure Front Propagation in a Reservoir from a Producing Well
by
Hsieh, B.Z., Chilingar, G.V., Lin, Z.S.
May 4, 2007
2
Outline
• Introduction• Purpose• Basic theory and simulation tool• Results and discussions• Conclusions
3
Introduction
4
Producing rate and flowing pressure at wellbore
Pwf
(psi)
p=pi
q(stb/day)
q=constant
t (hours)t=tit=0
r
rw
t (hours)t=ti
5
Pressure distribution in reservoir at t = ti
P(psi)
r (ft)
p=pi
t=ti
q(stb/day)
q=constant
t (hours)
Radius of investigation (ri)
Pressure front
Pressure disturbance area ( Drainage area )
Non-disturbed area
t=ti
r=rw
t=0
ri
rw
t=ti
r=ri
6
Pressure distribution in reservoir at various times
P(psi)
r (ft)
p=pi
t=t3
q(stb/day)
q=constant
t=t2
t=t1
t (hours)
ri 3ri 1 ri 2
t=t1 t=t2 t=t3
pressure front s
7
t1
rw
r1 r2
r3
t2
t3
Plane view of pressure fronts at various times
8
Dimensionless Variables
Dimensionless radius
Dimensionless time
Dimensionless pressure Bq
ppkhp i
D 2.141
2
000264.0
w
Drc
ktt
wD r
rr
9
Pressure distribution in a reservoir in terms of dimensionless variables
PD
rD
tD3
tD2tD1
riD1 riD2 riD3
pressure front s
10
Radius of investigation (riD) and time (tD)
• The relationship between the dimensionless radius of investigation (riD) and the dimensionless time (tD) is (Muskat, 1934; Tek et al., 1957; Jones, 1962; Van Poolen, 1964; Lee, 1982; Chandhry, 2004, etc.)
riD2 = α tD
where the radius coefficient (α) is a constant and varied in different studies, from 3.18 to 16
11
Literature on radius of investigation equation
Author (Year)
Radius of investigation
equation
Definition of radius of investigation
or method used
Muskat (1934) riD2= 4tD Material balance method
Tek et al. (1957) riD2= 18.4tD The fluid flow rate at the radius of
investigation is 1% of that flowing into the wellbore
Hurst et al. (1961) riD2= 6.97tD Pressure build-up test
Jones (1962) riD2= 16tD The pressure drop at the radius of
investigation is 1% of pressure drop at the wellbore
Van Poolen (1964) riD2= 4tD Y-function of an infinite and a finite
reservoir
Hurst (1969) riD2= 8tD The analytical van Everdingen and
Hurst solution
12
Literature on Radius of Investigation (Cont.)
Author (Year)
Radius of investigation equation
Definition of radius of investigation
or method used
Earlougher (1977) riD2= 3.18tD After van Poolen (1964)
Lee (1982) riD2= 4tD The solution of the diffusivity
equation for an instantaneous line source in an infinite medium
Kutasov and Hejri (1984) riD2= 4.12tD Constant bottom-hole pressure test
Johnson (1986) riD2= 7.89tD The radius enclosing a volume in the
reservoir accounts for a specified fraction of the cumulative production of 96.1%
Chandhry (2004) riD2= 4tD Pressure transient analysis of pressure
build-up test
13
Purposes of the study
• To estimate the propagation of the radius of investigation from a producing well by using both analytical and numerical methods, including variable flow rates case, skin factor, and wellbore storage effect.
• To estimate the starting time of transient pressure affected by the reservoir boundary to concurrently determine the radius coefficient
14
Basic theory and simulation tool
15
Analytical Solution – Ei solution
• The analytical solution of the diffusivity equation for a well (line source) producing in an infinite cylindrical reservoir is (van Everdingen and Hurst, 1949; Earlougher, 1977):
)4
(2
1 2
D
DD t
rEip
n
k
kk
x
u
i kk
xxdu
u
exExE
1
1
1 )!(
)1(ln0.5772)()(
Bq
ppkhp i
D 2.141 where
2
000264.0
w
Drc
ktt
wD r
rr
16
Numerical Solution of Diffusivity Equation
• Numerical solutions are also used in this study for the cases that no analytical is available or the comparisons are required.
• The IMEX simulator (CMG) is used in this study to generate results in numerical simulation.
17
Basic reservoir parameters used in this study
18
The pressure behavior check-- infinite reservoir
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
tD
PD
van Everdingen & Hurst solution
CMG calculated PD
Even the specific oil reservoir is used in this study, the pressure behavior(dimensionless pressure as function of time and radius) is checked by comparison with analytical solution that exist in the literature
19
The pressure behavior check-- bounded reservoir
7
8
9
10
11
12
13
14
1.E+06 1.E+07 1.E+08
tD
PD
re/rw=2000
re/rw=2200
re/rw=2400
re/rw=2600
re/rw=2800
re/rw=3000
re/rw=infinite
CMG re/rw=2000
CMG re/rw=2200
CMG re/rw=2400
CMG re/rw=2600
CMG re/rw=2800
CMG re/rw=3000
CMG re/rw=infinite
20
The pressure behavior check-- bounded reservoir
6
7
8
9
10
11
12
13
1.E+05 1.E+06 1.E+07tD
PD
re/rw=700
re/rw=800
re/rw=900
re/rw=1000
re/rw=1200
re/rw=1400
re/rw=1600
re/rw=1800
re/rw=infinite
CMG re/rw=700
CMG re/rw=800
CMG re/rw=900
CMG re/rw=1000
CMG re/rw=1200
CMG re/rw=1400
CMG re/rw=1600
CMG re/rw=1800
CMG re/rw=infinite
21
Results and Discussions
22
Definition of pressure front
P(psi)
r (ft)
p=pi
t=ti
Δp1 Δp2 Δp3
α1α2
α3
△p= pressure drop defined at the pressure front
α= radius coefficient
● ● ●
● ● ●
23
Definition of pressure front
PD
rD
tD5
△pD= the dimensionless pressure drop defined at the pressure front α= radius coefficient
α1 α2 α3
ΔpD1 ΔpD2 ΔpD3
24
Pressure front and radius of investigation • From Ei solution such as
• By defining or giving ΔpD (or y), the following equation can be derived
.4
2
constt
r
D
iD
)4
(2
1 2
D
iDD t
rEip
DD
iD pt
rEior 2)
4(
2
DiD tror 2
Note: The radius coefficient (α) is dependent on the criteria defined at the pressure front (the value of ΔpD).
25
Radius coefficients from analytical solution with constant flow rate in an infinite reservoir
• By defining a small dimensionless pressure value (ΔpD) at the pressure front, the value of riD
2/4tD in the Ei solution can be estimated.
26
Radius coefficients from analytical solution with constant flow rate in an infinite reservoir
0.0E+00
5.0E+10
1.0E+11
1.5E+11
2.0E+11
2.5E+11
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09
tD
r iD2
α = 4.00 (ΔpD=1.095*10-1)
α = 10.39 (ΔpD=1.095*10-2)
α = 51.22 (ΔpD=10-7)
α = 59.84 (ΔpD =10-8)
α = 71.15 (ΔpD=10-9)
α = 26.06 (ΔpD=10-4)
α = 34.28 (ΔpD=10-5)
α = 42.69 (ΔpD=10-6)
α = 17.82 (ΔpD=1.095*10-3)
27
0.0E+00
2.0E+10
4.0E+10
6.0E+10
8.0E+10
1.0E+11
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09
tD
r iD2
Δ pD=0.1095
Δ pD=0.01095
Δ pD=0.001095
Radius coefficients from numerical solution with constant flow rate in an infinite reservoir
α = 3.986 (ΔpD=1.095*10-1)
α = 10.363 (ΔpD=1.095*10-2)
α = 17.799 (ΔpD=1.095*10-3)
28
Radius investigation equation from analytical and numerical solution -- constant flow rate case
Constant flow rate test
(q=100 stb/day)
riD criteria (I)
for ΔpD=0.1095
riD criteria (II)
for ΔpD=0.01095
riD criteria (III)
for ΔpD=0.001095
Analytical solution
riD2= 4.00tD
R2=1
riD2= 10.39tD
R2=1
riD2= 17.82tD
R2=1
Numerical solution
riD2= 3.986tD
R2=0.9999
riD2= 10.363tD
R2=0.9999
riD2= 17.799tD
R2=0.9999
Different criteria for pressure front will obtain different radius coefficient (α)
The smaller the ΔpD, the larger the radius coefficient (α), i.e., the faster the pressure front propagation.
29
Results and Discussions (2)
Effect of variable flow rates
30
Ei solution with superposition – variable flow rate
)}][4
()()4
({6.70
)(1
2
21
2
1
iD
Dn
iii
D
Dtotal ttt
rEiqq
t
rEiq
kh
Bp
)}][4
()()4
({2
1
1
2
2 1
12
1
iD
Dn
i
ii
D
DD ttt
rEi
q
t
rEip
Bq
pkhp total
D 11 2.141
)(
21
1
)(000264.0][
w
iiD
rc
ttkttt
or
where
31
(a) Increasing flow rate (two-rates) test
32
Radius of investigation equations from analytical solution and numerical solution with increasing flow rate test
Increasing flow rate test
q1=100 stb/day
q2=150 stb/day
riD criteria (I)
for ΔpD=0.1095
riD criteria (II)
for ΔpD=0.01095
riD criteria (III)
for ΔpD=0.001095
Analytical solution
riD2= 4.101tD
R2=0.9991
riD2= 10.411tD
R2=0.9999
riD2= 17.819tD
R2=0.9999
Numerical solution
riD2= 4.082tD
R2=0.9991
riD2= 10.381tD
R2=0.9999
riD2= 17.804tD
R2=0.9999
Note: Radius coefficient(α) increase slightly for smaller ΔpD
33
(b) Decreasing flow rate (two-rates) test
34
Radius of investigation equations from analytical solution and numerical solution with decreasing flow rate test
Decreasing flow rate test
q1=100 stb/day
q2= 50 stb/day
riD criteria (I)
for ΔpD=0.1095
riD criteria (II)
for ΔpD=0.01095
riD criteria (III)
for ΔpD=0.001095
Analytical solution
riD2= 3.887tD
R2=0.9983
riD2= 10.374tD
R2=0.9999
riD2= 17.814tD
R2=0.9999
Numerical solution
riD2= 3.868tD
R2=0.9983
riD2= 10.344tD
R2=0.9999
riD2= 17.799tD
R2=0.9999
Note: Radius coefficient(α) decrease slightly for smaller ΔpD
35
(c) Middle flow rate increasing test
36
Radius of investigation equations from analytical solution and numerical solution with middle flow rate increasing test
Middle flow rate increasing test:
q1=100 stb/day
q2=150 stb/day
q3=100 stb/day
riD criteria (I)
for ΔpD=0.1095
riD criteria (II)
for ΔpD=0.01095
riD criteria (III)
for ΔpD=0.001095
Analytical solutionriD
2= 4.246tD
R2=0.9981
riD2= 10.494tD
R2=0.9999
riD2= 17.849tD
R2=0.9999
Numerical solutionriD
2= 4.227tD
R2=0.9981
riD2= 10.464tD
R2=0.9999
riD2= 17.833tD
R2=0.9999
Note: Radius coefficient(α) increase for smaller ΔpD
37
(d) Middle flow rate decreasing test
38
Radius of investigation equations from analytical solution and numerical solution with middle flow rate decreasing test
Middle flow rate decreasing test:
q1=100 stb/day
q2= 50 stb/day
q3=100 stb/day
riD criteria (I)
for ΔpD=0.1095
riD criteria (II)
for ΔpD=0.01095
riD criteria (III)
for ΔpD=0.001095
Analytical solutionriD
2= 3.698tD
R2=0.9956
riD2= 10.280tD
R2=0.9998
riD2= 17.782tD
R2=0.9999
Numerical solutionriD
2= 3.679tD
R2=0.9956
riD2= 10.249tD
R2=0.9998
riD2= 17.767tD
R2=0.9999
Note: Radius coefficient(α) decrease for smaller ΔpD
39
The results of the dimensionless radius of investigation at the criterion ΔpD= 0.1095
0.0E+00
2.0E+09
4.0E+09
6.0E+09
8.0E+09
1.0E+10
1.2E+10
1.4E+10
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09
tD
r iD2
q1=100 q2=150
q1=100 q2=50
q1=100 q2=150 q3=100
q1=100 q2=50 q3=100
q=100
Note: Radius coefficient(α) is affected by rate changes for larger ΔpD
40
The results of the dimensionless radius of investigation at the criterion ΔpD= 0.01095
0.0E+00
5.0E+09
1.0E+10
1.5E+10
2.0E+10
2.5E+10
3.0E+10
3.5E+10
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09
tD
r iD2
q1=100 q2=150
q1=100 q2=50
q1=100 q2=150 q3=100
q1=100 q2=50 q3=100
q=100
Note: Radius coefficient(α) is slightly affected by rate changes for small ΔpD
41
The results of the dimensionless radius of investigation at the criterion ΔpD= 0.001095
0.0E+00
1.0E+10
2.0E+10
3.0E+10
4.0E+10
5.0E+10
6.0E+10
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09
tD
r iD2
q1=100 q2=150
q1=100 q2=50
q1=100 q2=150 q3=100
q1=100 q2=50 q3=100
q=100
Note: Radius coefficient(α) is very slightly affected by rate changes for smaller ΔpD
42
Results and Discussions (3)
Effect of skin factor
43
The effect of skin factor to the radius coefficients in simulation studies (constant flow
rate test)
0.0E+00
1.0E+10
2.0E+10
3.0E+10
4.0E+10
5.0E+10
6.0E+10
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09 3.5E+09 4.0E+09 4.5E+09 5.0E+09
tD
riD2
α = 17.799 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-3)
α = 10.363 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-2)
α = 3.986 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-1)
The radius coefficient (α) is independent of skin factor
44
Results and Discussions (4)
Effect of wellbore storage volume
45
The effect of wellbore storage volume (constant flow rate test)
0.0E+00
1.0E+10
2.0E+10
3.0E+10
4.0E+10
5.0E+10
6.0E+10
0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09 3.5E+09 4.0E+09 4.5E+09 5.0E+09
tD
riD2
α = 17.799 (CD=102, 103, 104, 105
for ΔpD=1.095*10-3)
α = 10.363 (CD=102, 103, 104, 105
for ΔpD=1.095*10-2)
α = 3.986 (CD=102, 103, 104, 105
for ΔpD=1.095*10-1)
Note: Radius coefficient(α) is independent of wellbore storage volume in late time
46
The effect of wellbore storage volume (constant flow rate test)
1.0E-01
1.0E+00
1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10
tD
riD
cD=103
cD=0
cD=105
cD=104cD=102
ΔpD=0.1095
Note: Radius coefficient(α) is affected by wellbore storage volume in early time
47
Which criteria for defining pressure front is suitable in conjunction with pressure behavior affected by bounded reservoir?
48
Pressure response for a bounded reservoir
PDwf
Log (tD)
Infinite reservoir pressure response
Bounded reservoir pressure response
Dimensionless boundaryaffecting time, tD
*
Deviated point
re
49
Boundary affecting time equation
• From radius of investigation equation, such as
– When pressure front reaches boundary then back to the wellbore, i.e., pressure front propagates two-times of external boundary radius ( riD= 2reD ), is applied
*2 **4 DeD tr re(in terms of wellbore radius, rw)
riD2 = α tD
2* )/4( eDD rtor
50
(a) bounded circular reservoir with reD=3000
No-flow boundary
re = 1050 ft rw = 0.35 ft
300035.0
1050
ft
ft
r
rr
w
eeD
51
Boundary affecting time estimated from radius of investigation equation for the bounded circular reservoir with reD = 3000
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
1.0E+05 1.0E+06 1.0E+07 1.0E+08
tD
p D
infinite reservoir
reD=3000 (re=1050 ft)
( I )
( II )
( III )
The visually deviated point from type curve analysis
2* 4eDD rt
52
(b) Bounded circular reservoir with reD=1000
No-flow boundary
re = 350 ft rw = 0.35 ft
100035.0
350
ft
ft
r
rr
w
eeD
53
4
6
8
10
12
14
1.0E+04 1.0E+05 1.0E+06 1.0E+07
tD
p D
infinite reservoir
reD=1000 (re=350 ft)
Boundary affecting time estimated from radius of investigation equation for the bounded circular reservoir with reD = 1000
( I )
( II )( III )
The visually deviated point from type curve analysis
2* 4eDD rt
54
Discussions of radius of investigation equation
• The study of radius of investigation in an infinite reservoir– Using different criterion ΔpD defined at the pressure front, we obtained
different radius coefficients (α) which vary from 4 to 71.15 for the pressure front varied from 0.1095 to 10-9, respectively.
• The study of boundary effect time in a bounded reservoir– The results of boundary effect time from the radius of investigation
equation with radius coefficient (α) of 17.82 (i.e., riD2=17.82tD for
ΔpD= 0.001095) are consistent with those from the deviated point of pressure type curves of the infinite and bounded reservoirs.
• The radius of investigation equation should be riD2=17.82tD,
where the radius coefficient (α) is 17.82.
55
Conclusions
• The relationship between the square of the dimensionless radius of investigation and the dimensionless time is linear (riD
2= αtD) for a constant flow rate but not necessarily linear for variable flow rates
• The radius coefficient (α) in constant flow rate cases varies from 4 to 71.15 as the defined dimensionless pressure at the pressure front of the radius of investigation is varies from 0.1095 to 10-9 , respectively
56
Conclusions (Cont.)
• The radius of investigation equation is independent of skin factor. The wellbore storage effect affects the propagation of the radius of investigation only at an early time and depending on the size of wellbore storage volume.
• The radius coefficient (α) is 17.82 and should be used in the equation of radius of investigation.
57
Thank you for your attention
58