manuscript submitted to J. Petr. Sci. Eng. Gravity Segregation in Steady-State Horizontal Flow in Homogeneous Reservoirs W. R. Rossen Department of Petroleum and Geosystems Engineering The University of Texas at Austin Austin, TX 78712-1061 U.S.A. email: [email protected]fax: 1-512-471-9605 to whom correspondence should be addressed and C. J. van Duijn Department of Mathematics and Computer Science Eindhoven University of Technology P. O. Box 513 5600 MB Eindhoven The Netherlands email: [email protected]fax: 31-40-247-2855 Keywords: gravity segregation, gas injection, IOR, gravity override, fractional-flow method, foam
36
Embed
Gravity Segregation in Steady-State Horizontal Flow in ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
manuscript submitted to J. Petr. Sci. Eng.
Gravity Segregation in Steady-State Horizontal Flow
in Homogeneous Reservoirs
W. R. Rossen Department of Petroleum and Geosystems Engineering
The University of Texas at Austin Austin, TX 78712-1061
Persoff, P., Radke, C. J., Pruess, K., Benson, S. M., Witherspoon, P. A., 1991. A
laboratory investigation of foam flow in sandstone at elevated pressure, SPE
Reservoir Eng. 6, 365-372.
Rossen, W. R., Zeilinger, S. C., Shi, J.-X., Lim, M. T., 1999. Simplified mechanistic
simulation of foam processes in porous media, SPE J. 4, 279-287.
Rossen, W. R., Zhou, Z. H., 1995. Modeling foam mobility at the limiting capillary
pressure, SPE Adv. Technol. 3, 146-152.
Shan, D., Rossen, W. R.,. 2002. Optimal injection strategies for foam IOR, SPE 75180,
SPE/DOE Symposium on Improved Oil Recovery, Tulsa, OK, April 13-17.
Shi, J.-X., Rossen, W. R., 1998a. Improved surfactant-alternating-gas foam process to
control gravity override, SPE 39653, SPE/DOE Improved Oil Recovery
Symposium, Tulsa, April 19-22.
Shi, J.-X., Rossen, W, R., 1998b. Simulation of gravity override in foam processes in
porous media, SPE Reservoir Eval. and Eng. 1, 148-154.
Siddiqui, F. I., Lake, L. W., 1992. A dynamic theory of hydrocarbon migration, Math.
Geol. 24, 305-325.
Stone, H. L., 1982. Vertical conformance in an alternating water-miscible gas flood, SPE
11140, 1982 SPE Annual Tech. Conf. and Exhibition, New Orleans, LA, Sept.
26-29.
Zhou, Z. H., Rossen, W. R.,1995. Applying fractional-flow theory to foam processes at
the 'limiting capillary pressure', SPE Adv. Technol. 3, 154-162.
Appendix A: Model Parameters Based on Data of Persoff et al. (1991) The gas and water relative-permeability data of Persoff et al. (1991) in the
absence of foam can be fit by the functions
19
3.1)1( .940 Skrg −= (A1)
2.4 .20 Skrw = . (A2)
with Swr = Sgr = 0.2 (Eq. (4)). We assume µw = 0.001 Pa s and µg = 2 x 10-5 Pa s, ρw =
1000 kg/m3, ρg = 153 kg/m3., which corresponds roughly to N2 gas at 2000 psi and 300K.
In Figs. 2 and 8 we assume that the injected water fractional flow fJ is 0.2.
The equations of Stone and Jenkins use two factors computed from these
parameters: Mgm is the ratio of the mobility of gas in the mixed zone to the mobility of
gas in the override zone, and Mgw is ratio of the mobility of gas in the override zone to the
mobility of water in the underrride zone. The override zone is at Sw = Swr = 0.2; the
underride zone is at Sw = 1 (S = 1), with krw = 1, since it is assumed gas has never entered
there. To calculate the mobilities in the mixed zone it is necessary to calculate Sw there
from the injected fractional flow fw:
fw = 0.2 =
)()(
1
1
wrw
w
g
wrg
SkSk µ
µ+
(A4)
which leads to Sw = 0.777, Mgm = 69.24, and Mgw = 47.
Both gas relative permeability and viscosity are altered by foam, but for
simplicity here we account for all effects for foam by altering the gas relative
permeability (Rossen et al., 1999). The data of Persoff et al. in the presence of foam are
fit by retaining the functions above for Sw < 0.37. For Sw > 0.37, krg is reduced by a factor
of 18,500 (Zhou and Rossen, 1995). For Sw = 0.37, krg is not a unique function of Sw, but
must be determined from fw. For instance, for injected fw = 0.2, Sw = 0.37 and (cf. Eq.
(A4))
fw = 0.2 =
3
3
5 1010
102)(
1
1
−
−
−⋅+ wrg Sk
(A5)
which gives krg = 8⋅10-5, Mgw = 47 and Mgm = 11750.
20
Appendix B: Free-Boundary Problem The curves separating the phases in the reservoir (in the original x,z coordinates)
are determined by the solution of the stream-function equation (Eq. (16)) subject to Eqs.
(20), (30) and (31) across the a priori-unknown phase boundaries. This is a free-
boundary problem which we pose here for completeness.
We begin with a non-dimensionalization and some notation. Setting
x: = x/H, z = z/H and Lg* = Lg/H , (B1)
let
Ω = (x,z) : 0 < ∞, 0 < z < 1 (B2)
denote the semi-infinite scaled reservoir in which we identify the regions of mixed flow
(Ωm) and the gas override (Ωg) and water underride zones (Ωw) as in Fig. B1. The
corresponding phase boundaries are denoted by Cmg, Cmw and Cgw. We assume that they
have the horizontal parameterizations
Cmg = (x,z) : 0 ≤ x ≤ Lg*, z = hmg(x) , (B3)
Cmw = (x,z) : 0 ≤ x ≤ Lg*, z = hmw(x) , (B4)
Cgw = (x,z) : Lg* ≤ x ≤ ∞, z = hgw(x) . (B5)
Next we set
α : =fJ (B6)
γw : = gw
w
ρααρρ
)1( −+ , Kw : = w
t
mt
λλ
(B7)
γg : = gw
g
ρααρρ
)1( −+ , Kg : = g
t
mt
λλ
(B7)
and we nondimensionalize ψ and Q according to
21
(ψ,Q) = (ψ,Q) / H kmtλ v (αρw + (1-α)ρg) g . (B8)
Then for ψ results the equation
∇•(K T ∇ψ + γ xe ) = 0 in Ω (B9)
where
=
h
v
kkT 001
(B10)
and where
K = γ = (B11)
gg
ww
m
in ΩK in ΩK
in Ω1
gg
ww
m
in Ωγ in Ωγ
in Ω1
The boundary conditions for ψ are
(BC) (B12)
∞≤≤−=∞≤≤==
xforzQzxforxQx
0 ),1(),0(0 0)1,( ,)0,(
ψψψ
and the values along the phase boundaries
ψ|Cmg = (1 - α) (γw - γg) x for 0 ≤ x ≤ Lg* (B13)
ψ|Cmw = Q - α (γw - γg) x for 0 ≤ x ≤ Lg* (B14)
ψ|Cgw = (1 - α) Q for Lg* ≤ x < ∞ (B15)
where
Lg* = Q / (γw - γg) . (B16)
22
The free-boundary problem now reads: Given 0 < α < 1, 0 < ρg < ρw (specifying
γg and γw), Q > 0 and Kw, Kg > 0, find ψ: Ω! R satisfying
)()()(2 ΩCΩLΩH loc ∩∩∈ ∞ψ (B17)
and find
hmg, hmw : [0,Lg*] ! [0,1] (B18)
hgw : [Lg*,∞) ! [0,1] (B18)
satisfying
hmw(x) < hmg(x) for 0 ≤ x < Lg* (B19)
and
hmw(Lg*) = hmg(Lg*) = hgw(Lg*) (B20)
such that
i) Eq. (B9) is satisfied weakly in Ω (B21)
ii) ψ satisfies boundary conditions (B12) (B22)
iii) ψ satisfies Eqs. (B13) to (B15) along the phase boundaries . (B23)
Note that the free-boundary conditions do not involve the parameters Kw and Kg;
however, the location of the free boundaries will strongly depend on their values. This is
illustrated in Fig. 2, where Kw = 0.68, Kg = 0.014, and in Fig. 3, where Kw = 4⋅10-3, Kg =
8.5⋅10-5.
23
List of Figures Figure 1. Schematic of three uniform zones in model of Stone and Jenkins.
Figure 2. Predictions of model of Stone and Jenkins for gas-water flow without foam.
Parameter values are based on data of Persoff et al. (1991) for N2-water flow in
Boise sandstone (cf. Appendix A).
Figure 3. Example of three zones in reservoir predicted by model of Stone and Jenkins
for foam injection using parameters based on data of Persoff et al. (1991); cf.
Appendix A.
Figure 4. Example of gravity segregation in finite-difference simulation of continuous
foam injection into rectangular reservoir, from Shan (2001). Gray scale indicates
water saturation: white = override zone, gray = mixed (foam) zone, black =
underride zone. In this case complete gravity segregation occurs at Lg ≈ 0.5. The
foam model used here is not identical to that in Fig. 3.
Figure 5. Schematic of assumption of Stone (1982) that a moving vertical fluid element
within reservoir maps segregation problem in horizontal flow on to segregation
problem without horizontal flow.
Figure 6. Function F(S) that governs gravity segregation without horizontal flow
(Siddiqui and Lake, 1992). Parameters values are those for N2 and water from
Persoff et al. (cf. Appendix A). If reservoir is initially at S = 0.8 (water saturation
Sw = 0.68), there is a shock front moving from the bottom of the reservoir, which
has saturation S = 1 (Sw = .8) (dotted line), but a spreading wave moving down
from the top at S = 0 (Sw = 0.2).
Figure 7. Schematic of boundary conditions in terms of x and either z or ψ.
Figure 8. Function F( f~ ) = F(f) that governs gravity segregation in horizontal flow for
N2 and water. Parameters values are from Persoff et al. (1991) (Appendix A).
Dotted lines indicate shock fronts between mixed zone (f = fJ = 0.2) and top of
reservoir (f = 0) and bottom of reservoir (f = 1) for foam injected at fJ = 0.2.
24
Figure 9. F(f) function for mobility functions of Persoff et al. (1991) (Appendix A),
where gas mobility decreases abruptly for Sw > Sw* = 0.37 (S* = 0.283). The
portion of the curve for S < S* matches that in Fig. 9 (note change of scale).
Figure 10. Schematic of F(f) for more general foam behavior, where foam collapses over
a small range of values of S near S*; cf. Fig. 9. Note points on curve for S near
S* fall on fan of lines originating at (1,0).
Figure B1. Schematic of regions and boundaries between them in free-boundary
problem.
25
mixed zone; gasand water flowing
underride zone;only water flowing
override zone;only gas flowing
x
z
Figure 1. Schematic of three uniform zones in model of Stone and Jenkins.
26
total flow rate through mixe d zone
0
1
0 0.5 1x/Lg
Q/Q
(x=0
)
thre e zone s in re s e rvoir
0
1
0 1x/Lg
z/H
total volume tric flux in mixe d zone
0
1
0 1x/Lg
ut/u
t(x=0
)
pre s s ure gradie nt in mixe d zone
0
1
0 1x/Lg
p/
p(x=
0)
mixed zone underride zone
override zone
∇∇
Figure 2. Predictions of model of Stone and Jenkins for gas-water flow without foam.
Parameter values are based on data of Persoff et al. (1991) for N2-water flow in Boise sandstone (cf. Appendix A).
27
thre e zone s in re s e rvoir
0
1
0 1x/Lgz/
H
Figure 3. Example of three zones in reservoir predicted by model of Stone and Jenkins
for foam injection using parameters based on data of Persoff et al. (1991); cf. Appendix A.
28
0.00 0.25 0.50 0.75 1.00
0.00
0.25
0.50
0.75
1.00
X
Z
0.40 0.55 0.70 0.85 1.00 Figure 4. Example of gravity segregation in finite-difference simulation of continuous
foam injection into rectangular reservoir, from Shan (2001). Gray scale indicates water saturation: white = override zone, gray = mixed (foam) zone, black = underride zone. In this case complete gravity segregation occurs at Lg ≈ 0.5. The foam model used here is not identical to that in Fig. 3.
29
=
Figure 5. Schematic of assumption of Stone (1982) that a moving vertical fluid element
within reservoir maps segregation problem in horizontal flow on to segregation problem without horizontal flow.
30
-1.6E-06
-1.4E-06
-1.2E-06
-1.0E-06
-8.0E-07
-6.0E-07
-4.0E-07
-2.0E-07
0.0E+000 0.2 0.4 0.6 0.8 1
S
F(S)
Figure 6. Function F(S) that governs gravity segregation without horizontal flow
(Siddiqui and Lake, 1992). Parameters values are those for N2 and water from Persoff et al.(cf. Appendix A). If reservoir is initially at S = 0.8 (water saturation Sw = 0.68), there is a shock front moving from the bottom of the reservoir, which has saturation S = 1 (Sw = .8) (dotted line), but a spreading wave moving down from the top at S = 0 (Sw = 0.2).
31
S = 0, fw = 0, ψ = 0
S = 1, fW = 1, ψ = Q
f w =
fJ
x0 L
0
H
z
Q
0
ψ
Figure 7. Schematic of boundary conditions in terms of x and either z or ψ.
32
0.E+00
1.E-06
2.E-06
0 0.2 0.4 0.6 0.8 1
f(S)
F(S)
fJ = 0.2
Figure 8. Function F(f) that governs gravity segregation in horizontal flow for N2 and
water. Parameters values are from Persoff et al. (1991) (Appendix A). Dotted lines indicate shock fronts between mixed zone (f = fJ) and top of reservoir (f = 0) and bottom of reservoir (f = 1) for foam injected at fJ = 0.2.
33
0.E+00
1.E-08
2.E-08
3.E-08
4.E-08
0 0.2 0.4 0.6 0.8 1
f(S)
F(S)
S = S*
S > S*S < S*
Figure 9. F(f) function for mobility functions of Persoff et al. (1991) (Appendix A),
where gas mobility changes abruptly at Sw = Sw* = 0.37 (S* = 0.283). The portion of the curve for S < S* matches that in Fig. 8 (note change of scale).
34
0.E+00
1.E-08
2.E-08
3.E-08
0 0.2 0.4 0.6 0.8 1
f(S)
F(S)
S ~ S*
S > S*
S < S*
Figure 10. Schematic of F(f) for more general foam behavior, where foam collapses over
a small range of values of S near S*; cf. Fig. 9. Note points on curve for S near S* fall on fan of lines originating at (1,0).
35
x
z
0
1
ΩmΩw
Ωg
Lg*
Cgw
Cmw
Cmg
Figure B1. Schematic of regions and boundaries between them in free-boundary