A new analytical model of ultimate water cut for light oil reservoirs with bottom-water
Samir Prasun,1 Sayantan Ghosh2
1Louisiana State University, Baton Rouge, LA, USA 2University of Oklahoma, Norman, OK, USA
Received: September 11, 2018 | Published: September 17, 2018
Correspondence: Samir Prasun, Louisiana State University, Baton Rouge, LA, USA, Email
prasoonsamir@gmail.com
Citation: Prasun S, Ghosh S. A new analytical model of ultimate water cut for light oil reservoirs with bottom-water. J Oil Gas Petrochem Sci. (2018);1(3): 74–81. DOI: 10.30881/jogps.00015
Abstract
Ultimate water cut (WCult) defines well’s maximum water production for uncontained oil pay with bottom-water. The WCult is important to determine if the reservoir development is economical. Since presently-used WCult formula derives from simplifying assumption ignoring the effect of non-radial inflow, the formula needs to be redefined. A new analytical formula of WCult is developed by considering the inflow of oil and water into separate completions at the top of oil-zone and aquifer respectively. Then the formula is verified using the design of 46 simulated experiments representing wide variety of reservoir-bottomwater systems. It was found out that the for light-oil reservoirs, the presently-used theoretical formula may significantly diverge from the proposed formula which closely matches the simulated data and is more physics driven. Hence the proposed formula should be preferred. However, for the viscous oil reservoirs, the presently used formula conforms to the proposed formula, which is also proved mathematically.
Ultimate water-cut is a maximum stabilized water cut in an oil-pay affected by water coning. The scenario is physically modeled by setting a balanced-oil-rate (BOR) boundary of the well’s drainage area by replacing the produced oil at the the drainage boundary. After the water break-through time, there is an initial rapid increase of water-cut representing the water cone development stage, followed by the stabilization period until the WC value becomes constant, WCult.
Kuo and Desbrisay1 introduced the concept and formula of ultimate water-cut2:
(1)
Shirman and Wojtanowicz3 showed that WCult in DWS wells is always lower than that in conventional wells. Their experimental results revealed that it is possible to completely reduce WCult to zero at high drainage rates. Other authors3–5 showed the dependence of ultimate water-cut on production rate. For production rates slightly higher than critical rates (maximum possible production rate without water breakthrough), water-cut would stabilize at value lower than that in Eq. (1). After conducting laboratory experiments, Shirman and Wojtanowicz3 found out that the water-cut stabilization value may not predict the Kuo and Desbrisay1 model at low production rate. They modified Eq. (1) by including the effect of production-rate as,
(2)
Both Eqs. (1) and (2) assume the radial flow in the oil-zone and aquifer having a BOR boundary depicted in Figure 1, and thereby ignores any nonradial distorted inflows (in oil-zone and aquifer) to a partially penetrating well. Prasun and Wojtanowciz6,7 attempted to include the effect of partial-penetration in the closed-boundary reservoirs. However, they found that the new modified WCult formula reduces back to the original formula (Eq. (1)); thus disapproving any effect of partial-penetration on ultimate water-cut in these reservoirs. Apparently, they verified the effect of partial penetration by comparing the formula with the results from the wide variety of NFRs. However, they failed to understand that the generalized consideration of all attributes of reservoirs while verification, may conceal the partial-penetration effects for certain types of reservoirs. So, this study derives a new model of ultimate water-cut for the BOR systems considering the non-radial inflow to a partial-penetrating well, and then verifies it with particular types of reservoirs classified as light oil and viscous oil reservoirs. A good match for the particular reservoir, would justify the relevance of the partial penetration effects for this reservoir.
Figure 1 Oil and water horizontal flow in their respective zones.
Modified analytical formula of ultimate water-cut
In derivation of a new ultimate water-cut model for a partially penetrating well in BOR system, we consider the following assumptions:
There is a piston-like displacement of oil by coned water flowing into the well. So, the rising water cone development covers larger area of oil completion before final stabilization. Eventually, the ratio of well completion producing oil and water becomes equal to the ratio of oil and water zone thickness, when ultimate water-cut is reached.3
In a piston-like displacement, there is almost no mixing between the flow regions of oil and water. Assumption 1 follows that the partially penetrating oil completion region (producing only oil) is at the top of oil-zone, whereas, for simplicity, we assume the partially penetrating water completion region (producing only water) is displaced from the oil-zone to the top of aquifer as shown in Figure 2. This assumption ignores the additional skin due to the water inflow from aquifer to the completion in oil-zone.
Darcy-law flow-rate equations of oil (
) (
) and water (
) (
) well-inflow (into their respective completions) during ultimate water-cut stage, at surface conditions, can be given by (Appendix A),
(3)
(4)
where,
is the radial size of reservoir, ft;
is the skin factor due to oil-inflow defined by Eq. (A-4);
is the skin factor due to water-inflow defined by Eq. (A-7);
is the well radius, ft. Now, after incorporating the above formulas into the ultimate water-cut equation (as shown in Appendix A), a new model of ultimate water-cut is developed, given by,
Figure 2 Equivalence of oil and water inflow schematic between combined and separate systems.
(5)
Validation of the proposed models using experiments
For simulation experiments, a 2-D radial-cylindrical model is built with IMEX simulation model depicted in Figure 3 using the base case reservoir properties, PVT and simulation grid data presented in Appendix C. In the model, transition zone is neglected and the produced oil and water is injected back to the oil drainage boundary and aquifer respectively at the constant pressure boundary (representing BOR boundary). The production well is completed in 50% of the total oil-zone thickness.
Figure 3 Radial model of oil with bottom water.
We compare the ultimate water-cut values from Eq. (2) and Eq. (5) with the the design of simulated experiments shown in Table 2 representing wide variety of reservoir/bottom-water systems. For creating matrix of experiments, we use the 3-level Box-Behnken design8,9 to consider any non-linearity of the factors in the design. Three-levels (low, intermediate and high) of the reservoir parameters are chosed based on the practical field range values of reservoir properties: Mobility, horizontal permeability, aquifer thickness, penetration ratio and anisotropy ratio, as shown in Table 1. For 5 parameters chosen in this study, the design stipulates 46 number of runs (reservoir systems). Critical-rate values,
, for different reservoir systems used in Eq. (5) are estimated using Eq. A-12.
Levels
Mobility (
)
Aquifer thickness (
)
Horizontal permeability (
)
Penetration ratio (
)
Anisotropy ratio (
)
Low (-1)
1
20
50
0.2
0.01
Intermediate (0)
3
75
100
0.5
0.1
High (+1)
10
500
500
0.8
1
Table 1 Three-level values of different reservoir/aquifer system parameters
Using the pressure drawdown simulation data for different runs, oil and water production-rates were calculated using Eqs. (3) and (4) as shown in Table 2, which were then subsequently compared with their simulated data (from Table 2) shown in Figures 4 and 5. Near unit-slope correlation plot and high R2 value close to 1, approve the validity of underlying assumptions of these proposed models (Eqs. (3) and (4)) to a larger extent. The slight discrepancy is due to the assumptions of 1) piston-like displacement process and 2) displaced water completion as shown in Figure 2 that neglects the additional skin due to water inflow from aquifer to the oil-zone. Further, the comparison plot between the predicted values of WCult from Eqs. (2) and (5) and the simulated values (from Table 2) is shown in Figure 6.
Figure 4 Simulated vs. predicted oil production rate (Eq. 3).
Figure 5 Simulated vs. predicted water production rate (Eq. 4).
Figure 6 Simulated vs predicted ultimate water-cut with Eq. (2) and Eq. (5).
Reservoir-system #
Mobility(
(
Aquifer thickness, (
)
Horizontal perm. (
)
Penetration ratio (
)
Anisotropy ratio,
Simulated WCult
WCult (From Eq. 2)
WCult (From Eq. 5)
Abs. Discrepancy (Eq. 2 and 5)
Pressure drawdown (
)
Simulated oil-rate
Simulated water-rate
Predicted Oil-rate (From Eq. 3)
Predicted water-rate (From Eq. 4)
1
10
75
100
0.5
1.0
0.968
0.967
0.958
0.010
609
64
1936
70
1940
2
1
75
100
0.5
0.0
0.720
0.745
0.713
0.046
680
560
1440
480
1460
3
10
20
100
0.5
0.1
0.902
0.888
0.891
0.003
1178
196
1804
182
1800
4
10
75
500
0.5
0.1
0.959
0.966
0.958
0.007
152
82
1918
67
1950
5
1
75
50
0.5
0.1
0.720
0.748
0.708
0.057
1147
560
1440
501
1470
6
3
75
50
0.8
0.1
0.905
0.900
0.884
0.018
962
190
1810
196
1790
7
3
75
100
0.5
0.1
0.903
0.899
0.879
0.023
702
194
1806
205
1800
8
1
20
100
0.5
0.1
0.465
0.442
0.450
0.017
629
1070
930
970
960
9
10
500
100
0.5
0.1
0.974
0.995
0.989
0.006
625
52
1948
19
1990
10
3
500
100
0.5
1.0
0.965
0.982
0.946
0.038
480
70
1930
90
1940
11
3
75
100
0.5
0.1
0.903
0.899
0.879
0.023
710
194
1806
207
1820
12
3
75
100
0.8
1.0
0.909
0.899
0.880
0.022
410
182
1818
206
1820
13
3
75
50
0.5
1.0
0.916
0.899
0.873
0.030
1137
168
1832
218
1810
14
10
75
100
0.2
0.1
0.968
0.967
0.957
0.011
1535
64
1936
72
1940
15
3
20
500
0.5
0.1
0.726
0.701
0.707
0.009
194
548
1452
498
1480
16
10
75
100
0.8
0.1
0.968
0.968
0.962
0.006
524
64
1936
64
1950
17
3
500
100
0.5
0.0
0.920
0.982
0.968
0.014
716
160
1840
48
1880
18
10
75
50
0.5
0.1
0.963
0.968
0.960
0.007
1490
74
1926
65
1910
19
3
75
500
0.5
1.0
0.898
0.894
0.868
0.030
114
204
1796
218
1810
20
3
75
50
0.5
0.0
0.908
0.899
0.883
0.018
1696
184
1816
200
1820
21
3
20
100
0.8
0.1
0.753
0.705
0.710
0.006
731
494
1506
522
1535
22
3
75
100
0.2
0.0
0.887
0.898
0.876
0.025
1805
226
1774
209
1810
23
3
75
100
0.8
0.0
0.887
0.899
0.886
0.015
565
226
1774
193
1810
24
3
20
50
0.5
0.1
0.768
0.705
0.712
0.009
2043
464
1536
525
1560
25
1
500
100
0.5
0.1
0.904
0.948
0.895
0.059
575
192
1808
170
1830
26
3
75
500
0.5
0.0
0.865
0.891
0.874
0.018
166
270
1730
195
1780
27
3
75
500
0.8
0.1
0.891
0.897
0.881
0.018
97
218
1782
198
1810
28
1
75
100
0.8
0.1
0.720
0.748
0.716
0.045
395
560
1440
483
1470
29
3
75
100
0.5
0.1
0.903
0.899
0.879
0.023
714
194
1806
208
1830
30
3
20
100
0.5
1.0
0.755
0.705
0.712
0.011
846
490
1510
515
1540
31
10
75
100
0.5
0.0
0.944
0.967
0.961
0.006
899
112
1888
63
1930
32
3
20
100
0.2
0.1
0.753
0.705
0.715
0.014
1890
494
1506
507
1535
33
3
75
50
0.2
0.1
0.905
0.899
0.870
0.033
2921
190
1810
227
1845
34
3
500
100
0.2
0.1
0.946
0.982
0.949
0.034
1244
108
1892
82
1910
35
1
75
100
0.2
0.1
0.700
0.746
0.689
0.083
1132
600
1400
528
1430
36
3
75
100
0.5
0.1
0.903
0.899
0.879
0.023
714
194
1806
208
1830
37
3
75
100
0.5
0.1
0.903
0.899
0.879
0.023
718
194
1806
209
1840
38
3
500
50
0.5
0.1
0.947
0.983
0.963
0.020
1218
106
1894
60
1940
39
1
75
100
0.5
1.0
0.710
0.747
0.695
0.075
456
580
1420
523
1450
40
3
75
100
0.5
0.1
0.903
0.899
0.879
0.023
714
194
1806
208
1830
41
3
500
500
0.5
0.1
0.938
0.976
0.957
0.020
121
124
1876
60
1920
42
3
20
100
0.5
0.0
0.755
0.704
0.710
0.008
1163
490
1510
517
1530
43
3
500
100
0.8
0.1
0.946
0.983
0.967
0.016
413
108
1892
54
1940
44
1
75
500
0.5
0.1
0.700
0.733
0.693
0.057
112
600
1400
488
1430
45
3
75
100
0.2
1.0
0.909
0.898
0.855
0.051
1077
182
1818
256
1840
46
3
75
500
0.2
0.1
0.891
0.891
0.863
0.033
287
218
1782
224
1815
Table 2Simulated and predicted data (WCult, oil-rate and water-rate) for an experimental matrix:
;
It is clear from the unit-slope correlation plot (Figure 6) that both the formulas give practically the same result. This infers that though the formula 2 ignores the inevitable non-radial flow to a partially penetration well, it still manages to conform to a more realistic physics-based formula 5 and hence predict the simulated WCult value.
Figure 7a shows the average absolute discrepancy (error), in percentage between the presently-used formula 2 and the proposed formula 5 using the data from Table 2. Also, Figure 7b shows the discrepancy between the formulas Eq. (2) and Eq. (5) for light oil reservoirs (M<3). From these two figures, it can be inferred that for the light oil reservoirs (when the mobility ratio is <3), the theoretical formula 2 may significantly deviate from the better (physically accurate) formula 5 for some cases (Figure 7a) with discrepancy as high as 8% (Figure 7b), which may not be reflected in Figure 6 due to considerable wide variety of sample size. In this study, any discrepancy exceeding the limit of 5% would be considered significant. This implies that for the light oil reservoir, the simplified assumptions of formula 2 may no longer allow it to better predict the actual WCult values, for which the formula 5 can serve better. This can be also be justified by the mathematical proof in Appendix B. So, in practice, formula 5 should be preferred for general use.
Figure 7a Average absolute discrepancy, in % between formulas 5 and 2.
Figure 7b Absolute Discrepancy, in % between formulas 5 and 2 for runs having M<3.
On the other hand, for moderate to high mobility ratio reservoirs (M 3), Figure 7a shows that the average discrepancy between the formulas is less than 5%, which is insignificant. This implies that in those conditions, formula (5) can be reduced to formula (2), which is also shown mathematically in Appendix B. So, Eq. (2), being simpler than Eq. (5), suffices to predict WCult for viscous oil reservoirs (M≥3).
Conclusions
Results of the study are summarized in the following conclusions:
A new analytical formula for WCult has been proposed including the physical effect ignored in the presently-used formula: partial penetration of oil zone, and aquifer. The formula utilizes the new models of oil and water production-rates during the ultimate water-cut stage. The derivation of models considers the piston-like displacement process and the inflow of oil and water into separate completions at the top of oil-zone and aquifer respectively.
The proposed formulas are systematically verified for wide variety of reservoir systems using design of simulated experiments (IMEX). High R2 value for the plot between the simulated and the predicted oil and water production-rates approves the validity of the proposed model’s underlying assumptions to a large extent. However, sight discrepancy can be attributed to the above assumptions.
In general, both the formulas (proposed and presently-used) of WCult predicts almost the same results which matches the simulated WCult values. However, for the light oil reservoirs (mobility ratio<3), simulations showed that the theoretical presently used-formula may significantly deviate from the (physically accurate) proposed formula. This is also confirmed by mathematical proof, so in practice, proposed formula should be preferred for the possible avoidance of errors.
On the other hand, for viscous oil reservoirs (Mobility ratio≥3), comparison of the simulations with the predicted values showed that the presently-used formula suffices to predict the WCult values. This fact that the proposed formula reduces to presently-used formula for the above reservoirs, can be justified mathematically.
Nomenclature
= viscosity of oil, cp
= viscosity of water, cp
= density difference between water and oil, lb/ft3
= oil formation volume factor, bbl/stb
= water formation volume factor, bbl/stb
BOR = balanced-oil-rate
= oil-zone thickness, ft
= perforated length, ft
= length of well-completion occupied by oil during WCult stage, ft
= length of well-completion occupied by water during WCult stage, ft
= aquifer thickness, ft
= horizontal permeability, md
= effective permeability of oil, md
= relative permeability of oil
= relative permeability of water
=Anisotropy ratio, fraction
= effective permeability of water, md
= mobility ratio between water and oil, fraction
= reservoir pressure, psi
= well-bottomhole pressure, psi
=critical oil rate, bbl/day
= oil flow rate, bbl/day
= water flow rate, bbl/day
= Total production rate, bbl/day
= wellbore radius, ft
= reservoir radius, ft
= Partial penetration skin due to oil-inflow
= Partial penetration skin due to water-inflow
T = Ratio of aquifer thickness to oil-zone thickness
WC = water-cut, fraction
= Ultimate water cut, fraction
Appendix A: Derivation of new analytical WCult formula
Assuming piston-like displacement process, the rise of water cone before final stabilization covers larger area of oil completion. Eventually, the ratio of well completion producing oil and water becomes equal to the ratio of oil and water zone thickness, when ultimate water-cut is reached.3 So, the length of well-completion occupied by oil during WCult stage:
(A-1)
And, the length of well-completion occupied by water during WCult stage:
(A-2)
This follows that the well completion system during water cone stabilization stage can be assumed to be the combination of the oil completion (producing only oil) at the top of oil-zone and the displaced water completion (producing only water) at the top of aquifer (Figure 2). So, oil inflow rate due to partial penetration in oil-zone (producing only oil) is given by,
Since,k_o=k_h k_ro , we get:
(A-3)
Where,
is the skin factor10 due to oil-inflow and is given by,
(A-4)
(From Eq. (A-1)) (A-5)
;
;
Now, again water inflow rate due to partial penetration in an aquifer (producing only water) is given by,
Since,k_w=k_h k_rw , we get:
(A-6)
So, the skin factor,
due to water-inflow can be represented by10:
(A-7)
(From Eq. (A-2)) (A-8)
;
;
From Eqs. (A-5) and (A-8), we get:
(A-9)
Ultimate Water-cut, during water-cut stabilization stage3 is given by:
(A-10)
Substituting
and
from Eqs. (A-3) and (A-6) in (A-10), we get:
(A-11)
Where,
Critical rate,
in above Eq. (A-11) can be substituted by the following formula11:
(A-12)
Where, all the parameters are in field units.
Appendix B: Mathematical convergence of new formula to presently-used formula
Using Eqs. (A-4), (A-7) and (A-9), Eq. 5 can be rewritten as:
(B-1)
Substituting
, and
in Eq. (B-1), we get:
(B-2) Figure B-1 clearly shows the maximum value of
is 0.37. Subsequently, the approximate maximum possible value of
is 0.15 for the practical field operating range values of
(between 0.1 and 1) and for practical value of T (>0.8). Minimum possible value of
tends to 0 for infinite thick aquifers.
Figure B-1 Pattern graph of log(T)/T vs. T; (T=ratio of aquifer thickness to oil-zone thickness).
Now, assuming 5% maximum possible error is permissible in predicted WCult value given by Eq. (B-2); for viscous reservoirs (when mobility ratio ≥ 3), any value of
would lie withing this error margin of Eq. (B-2) and hence, the part ‘
’ can be ignored. So, Eq. (B-2) or Eq. (5) can be rewritten as:
(B-3)
Above derivation mathematically proves that Eq. (5) reduces to Eq. (2) in case of viscous oil reservoirs. However, for mobility ratio<3, Eq. (5) may or may not reduce to Eq. (2) depending upon the ratio of aquifer to oil-zone thickness.
Appendix C: Complete Reservoir Simulation Input Data
Parameter
Unit
Value
Datum depth
ft
5000
Thickness of oil zone
ft
25
Depth of WOC
ft
5025
Thickness of water zone
ft
75, varied
Reservoir pressure at datum depth
psi
6000
Position of top completion from formation top
ft
0
Perforated length
ft
12, varied
Horizontal permeability
md
100, varied
Anisotropy ratio
md
0.1, varied
Porosity
fraction
0.3
Well radius
ft
0.25
Outer radius of oil-zone
ft
1000
Outer radius of water zone
ft
1000
Total liquid Production rate
bpd
2000
Table 1 Reservoir and Well Input data
Property
Unit
Value
Reference pressure
psi
6000
Formation oil volume factor
rb/stb
1.2
Relative oil permeability at connate water saturation