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Gas-well water breakthrough time prediction model for high-sulfur gas reservoirs considering sulfur deposition
Journal of Petroleum Science and Engineering 157 (2017) 999–1006
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Gas-well water breakthrough time prediction model for high-sulfur gas reservoirs considering sulfur deposition Guo Xiao a, Wang Peng a, *, Liu Jin a, Song Ge b, Dang Hailong c, Gao Tao c a b c
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitations, Southwest Petroleum University, Chengdu 610500, China Northeastern Sichuan Gas Recovery Plant, Sinopec Southwest Petroleum Company, Langzhong, 637400, China Research Institute of Yanchang Petroleum (Group) Co. LTD., Xi'an 710075, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Gas-well water breakthrough time (GWWBT) High-sulfur gas reservoirs (HSGRs) Sulfur deposition Non-Darcy seepage
The sulfur-solubility decreases as a result of the decrease in gas reservoir pressure, leading to the solid-phase sulfur deposition. As a consequence, both the reservoir porosity and permeability decrease, subsequently influencing gas well water breakthrough time (GWWBT) in high sulfur gas reservoirs (HSGRs) with edge/bottom water. To acquire the value of GWWBT, a GWWBT prediction model for HSGRs should be established while taking into account sulfur deposition. Accordingly, based on gas Non-Darcy seepage law and sulfur deposition theory in porous media, a novel GWWBT in high sulfur gas reservoir with bottom water was developed in this study. The effect of dynamic factors (e.g., sulfur deposition, gas Non-Darcy flow, irreducible water saturation, and residual gas saturation) on GWWBT was involved in this model. The GWWBT was calculated via the proposed method and three classical models in five field basic parameters, and was compared with five filed values, respectively. This result indicates that the calculation of proposed method is in closer agreement with the field production data, and illustrates that the new proposed model is more reliable. In addition, the influence of dynamic factors was further discussed in detail by this proposed model.
1. Introduction Active edge/bottom water gas reservoir accounts for approximately 40–50% of water drive gas reservoirs. Most existing gas reservoirs of China have varying degrees of water drive; most of China's high-sulfur gas reservoirs (HSGRs) are close to water bodies of bottom water (Wang et al., 2011). The water body flows to the gas reservoir during HSGR development, forming two-phase gas-water seepage with decreasing pressure. This results in a decrease in the gas-well recovery rate (Wang et al., 2011; Zeng et al., 2013; Li, 2014; Liu et al., 2015; Yu et al., 2016) which can be mitigated using an effective bottom water coning profile and by controlling the breakthrough time accordingly. Previous scholars have proposed many prediction models for gas-well water breakthrough time (GWWBT) after investigating bottom water coning problems in conventional bottom water reservoirs. Sobocinski and Cornelius (1965), for example, studied the relation between bottom water coning and time based on the physical model of sandstone, but which was built form the static water-oil contact to breakthrough conditions. Kuo and Desbrisay (1983) modeled the dynamic relationship between bottom water coning and time numerically, which is without
* Corresponding author. E-mail address: [email protected] (W. Peng). http://dx.doi.org/10.1016/j.petrol.2017.08.020 Received 28 February 2017; Received in revised form 20 July 2017; Accepted 4 August 2017 Available online 10 August 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.
considering residual oil saturation, irreducible water saturation. Shi et al. (1992) used the system identification method and a one-dimensional differential equation for filtering flow to establish a GWWBT prediction method, the application of this model is limited known mechanism model and huge amounts of oilfield statistical data. Li (2001) and Tang (2003) obtained water breakthrough prediction time equations for bottom water reservoirs in which plane radial flow and spherical flow are taken into account, while residual oil saturation, irreducible water saturation, and oil/water viscosity are neglected. Zhang et al. (2004) proposed a water breakthrough model for condensate reservoirs with bottom water based on a relatively simple water coning model in which gas condensing effects were taken into account, but this model is only applicable for condensate reservoirs. Zhao and Zhu (2012) derived a water breakthrough time prediction equation for low-permeability bottom water reservoirs with barriers by applying the material balance principle and non-Darcy flow theory, which takes the hemispherical radial flow below the water coning barrier and plane radial flow above the barrier into consideration. However, this equation considering startup pressure gradient is only applicable for low-permeability gas reservoirs. Xiong et al. (2014) proposed a formula for water coning
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
breakthrough time in bottom water oil reservoirs in which oil-water mobility ratio, original irreducible water saturation, and residual oil saturation are taken into account, this formula is only applicable for oil reservoirs. Li et al. (2015) improved upon this model by adding oil-water contact and water coning profile factors but disregarding the impact of non-Darcy effects. Huang et al. (2016) deduced bottom-water gas reservoir water coning time based on the theory of percolation flow in porous media; their model also includes gas non-Darcy effects, skin factor, degree of opening, and daily gas production, without irreducible water saturation and residual gas saturation. Sulfur deposition is missing from these models, however, rendering them inapplicable to HSGRs. In bottom-water HSGRs, sulfur solubility decreases as gas reservoir pressure decreases, which leads to solid-phase sulfur deposition (Roberts, 1997; Zeng et al., 2005; Yang et al., 2004; Du et al., 2006; Guo et al., 2009). This further causes a decrease in reservoir porosity and permeability which affects GWWBT. Extant prediction models cannot yield accurate results without considering sulfur deposition. In this paper, a novel GWWBT model was established with consideration of sulfur deposition for HSGRs.
(2)
Hz sin φ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r þ ðH zÞ2
(3)
V¼
Qsc 2π r 2 þ ðH zÞ2
(4)
Combining Eqs. (1)–(4) yields:
V1v ¼
Qsc Hz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2π r 2 þ ðH zÞ ϕ 1 Swi Sgr Ss 2 r þ ðH zÞ2
(5)
The upward migration distance of water coning is dz in dt can be obtained by equation (5), it is as follows: t
Qsc ðH zÞ 32 dt 2 2πϕ r þ ðH zÞ2 1 Swi Sgr Ss =
z
∫ 0 dz ¼ ∫ 0
2. Gas-well water breakthrough time
(6)
Equation (6) can be rewritten in integrated form as:
8 9 12 2 > > 2 > > > H þr > > > > > > > > > > > 1 2 > > 2 2 > > þ r r H > > > > > þr ln > > > 2 < = 3 H 3 1 2 1 2 2 H þ r 2 2 ðH zÞ þ r 2 þ r 2 1 > ðH zÞ2 þ r 2 2 > 3 3 > > > > > > > > > > > > 1 > > 2 2 2 > > > > þ r r ðH zÞ > > > r ln > > > > > 2 : ; ðH zÞ
The physical bottom water coning model is shown in Fig. 1. Assumptions: (1) Capillary forces, gravity, reservoir anisotropy, stress sensitivity, slippage, and skin factor effects during the displacement process are ignored; (2) Imperforated formation is assumed for the bottom of the distal radial flow and bottom hemispherical centripetal flow, while perforated formation is assumed for the plane radial flow. The existence of a shaft axis z with homogeneous distribution in the original gas water interface (r axis) was also assumed (Fig. 1). According to the water quality point seepage law, there is a mass point A at the initial gas-water interface with a radial flow for t to point A (z, r) and fluid velocity at point A (V) in the porous media seepage. Using the reservoir of porous medium porosity (ϕ) and irreducible water saturation (Swi ), residual gas saturation (Sgr ), sulfur saturation (Ss ), and the actual seepage velocity of water points V1 , the upward seepage velocity of water points is V1v can be calculated as follows:
V V1 ¼ ϕ 1 Swi Sgr Ss
V1v ¼ V1 sin φ
¼
Qsc t 2πϕ 1 Swi Sgr Ss (7)
When the water cone breaks through to the gas well, assuming r ¼ 0, z ¼ H, t ¼ tp (bottom water coning breakthrough critical height) in Eq. (7), then the novel GWWBT for HSGRs is:
tp ¼
2πϕ 1 Swi Sgr Ss 3 H 3Qsc
(1)
Fig. 1. Schematic diagram of water coning in bottom-water reservoir. 1000
(8)
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
3. Calculating several model parameters
K ¼ Ka exp
3.1. Sulfur saturation and effective gas reservoir permeability
As shown in Fig. 1, imperforated reservoir bottom gas flow is divided into hemispherical centripetal flow and planar radial flow based on actual simulation of reservoir and different seepage characteristics. According to the basic law of gas-water two-phase fluid flow, the hemispherical centripetal flow equation is:
(9)
q 2 dp μg q ¼ þ βρ g dr K 2πr 2 2πr 2
The sour gas reservoir pore volume can be calculated as follows:
dVs ¼
2πrhϕð1 Swi Þdrdc ρs
(10)
q 2 dp μg q ¼ þ βρg dr K 2πrh 2πrh (11)
β¼
(12)
ρg ¼
Bg ¼
S
p
(14)
Kpdp ¼ (16)
Kpdp ¼
(17)
(26)
Qsc psc ZTμg 1 3:33 106 Q2sc p2sc TZγ g 1 dr þ r2 r4 2πTsc 4π 2 Tsc2 Rϕ
(27)
Qsc psc ZTμg 1 3:33 106 Q2sc p2sc TZγ g 1 þ dr 2 2 2 2 r 2πhTsc r 4π h Tsc Rϕ
(28)
To simplify the calculation, μg and Z are constant. Substituting and integrating Eq. (20) into Eqs. (21) and (28) yields the following hemispherical centripetal flow equation:
Qsc psc ZTμg 1 αA 4 p r ∫ pw Ka exp pi p4 pdp ¼ ∫ rw 4 r2 2πTsc
(18)
þ
In the establishment of a high-sulfur gas reservoir seepage equation, the precipitation of sulfur damages gas reservoir permeability. The effective permeability is typically characterized by absolute permeability and relative permeability (Guo and Zhou, 2015; Hu et al., 2013):
K ¼ Ka Krg
Psc ZT Zsc Tsc p
Substituting Eqs. (24)–(27) into Eq. (23) yields the differential form of the plane radial flow yield:
Elemental sulfur may precipitate when pressure drops below a critical pressure. Given the effects of deposited sulfur on reservoir permeability, an empirical equation for relative permeability and sulfur saturation is useful (Roberts, 1997):
Krg ¼ expðαSs Þ
(25)
Substituting Eqs. (23)–(26) into Eq. (21) yields the differential form of the hemispherical centripetal flow yield:
(15)
Equation (16) can be changed into Eq. (17), a novel formula about sulfur saturation is as follows:
A Ss ¼ p4i p4 4
(24)
Psc ZT Zsc Tsc p
q ¼ Qsc Bg ¼ Qsc
where (assuming that Z and γ g are constant):
4 ð1 Swi Þ Ma γ g 4666 4:5711 A¼ 4 exp ρs T ZRT
Ma γ g p ZRT
The production under the geological conditions can be transformed into production under ground conditions:
Equation (14) can be changed to:
∫ 0 s dSs ¼ ∫ pi Ap3 dp
(23)
and the gas volume factor is:
(13)
Combining Eqs. (12) and (13) yields:
4 dSs ð1 Swi Þ Ma γ g 4666 4:5711 p3 ¼ 4 exp ρs T dp ZRT
1:15 107 Kϕ
Gas density under geological conditions is:
Based on the principles of associative law and entropy, an empirical equation was obtained utilizing Brunner's experimental data (Roberts, 1997; Brunner and Woll, 1980):
Ma γ g 4 dc 4666 ¼4 4:5711 p3 exp dp T ZRT
(22)
where β is the turbulence coefficient (1/m) and can be calculated as follows (Li and Engler, 2001):
Equation (11) can be rewritten as follows:
dSs ð1 Swi Þ dc ¼ ρs dp dp
(21)
The planar radial flow equation is:
Elemental sulfur saturation is:
ð1 Swi Þdc dVs ¼ dSs ¼ ρs 2πrhϕdr
(20)
3.2. HSGR production equation considering deposited sulfur
This model assumes that elemental sulfur is not precipitated under the original formation pressure and that sulfur deposition migration does not occur in the channel. The quality of elemental sulfur in the gas changes due to the pressure drop in solubility (dc) at r (radial distance). The precipitated sulfur quality can be expressed as follows:
dm ¼ 2πrhϕð1 Swi Þdrdc
αA 4 pi p4 4
3:33 106 Q2sc p2sc TZγ g 1 dr r4 4π 2 Tsc2 Rϕ
(29)
and planar radial flow equation:
Qsc psc ZTμg 1 αA 4 p r ∫ pe Ka exp pi p4 pdp ¼ ∫ re 4 2πhTsc r
(19)
þ
Combined with Eqs. (17)–(19), we obtained the new reservoir effective permeability as follows: 1001
3:33 106 Q2sc p2sc TZγ g 1 dr r2 4π 2 h2 Tsc2 Rϕ
(30)
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
Assuming:
where:
αA 4 pi p4 pdp φðpÞ ¼ ∫ exp 4
a¼
(31)
The following were derived by combining Eqs. (29)–(31) to rewrite the hemispherical centripetal flow equation:
e¼
The planar radial flow equation was altered as well:
Qsc psc ZTμg ðln re ln rÞ 2πhKa Tsc 3:33 106 Q2sc p2sc TZγ g 1 1 þ 2 2 2 r re 4π h Tsc Ka Rϕ
Qsc ¼
ðφðpe Þ φðpÞÞ 2:15105 Qsc γ g 1 1 TZ μg ln rrae þ ra re HRϕ
1549:2Ka ðφðpÞ φðpw ÞÞ 7:16104 Qsc γ g 1 1 TZ μg r1w r1a þ 3 3 Rϕ r r w
γ g 2:15 105 1 1 1 1 þ 7:16 104 3 1:5H re rw 3:375H 3 Rϕ H
(39)
(40)
Equation (37) can be changed to:
aeQ2sc þ abQsc ðφðpe Þ φðpw ÞÞ ¼ 0
(41)
Equation (41) uses the extract roots formula:
Qsc ¼
ab þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 þ 4aeðφðpe Þ φðpw ÞÞ 2ae
(42)
4. Model simplification To analyze the influence of specific parameters on the GWWBT for HSGR, the initial GWWBT equation (8) was simplified by substituting Eqs. (17) and (42) into (8). The GWWBT for HSGR is then expressed as follows:
(34)
In the second part, the bottom hole proximal radial distance (ra ) is in hemispherical centripetal flow to the bottom radial distance (rw ), and the corresponding gas-well productivity equation is (Wang et al., 2007):
Qsc ¼
1 re 1 1 ln þ H 1:5H rw 1:5H
(33)
The imperforated reservoir bottom gas flow is divided into the bottom reservoir distal radial distance (re ) to the bottom proximal radial distance (ra ) in the proximal plane radial flow, as well as into the bottom proximal radial distance (ra ) to the bottom hole radial distance (rw ) in the hemispherical centripetal flow (Fig. 1). Because it can simulate really the seepage process of gas flow and water coning. In the first part, the bottom reservoir distal radial distance (re ) is in planar radial flow to the bottom proximal radial distance (ra ) and the corresponding gas-well productivity equation is (Wang et al., 2007):
1549:2HKa
(38)
b ¼ μg
Qsc psc ZTμg 1 1 3:33 106 Q2sc p2sc TZγ g 1 1 φðpÞ φðpw Þ ¼ þ rw r rw3 r 3 2πKa Tsc 12π 2 Tsc2 Ka Rϕ (32)
φðpe Þ φðpÞ ¼
TZ 1549:2Ka
tp ¼
4acπϕ 1 Swi Sgr Ss pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH 3 3abBg þ 3Bg a2 b2 4acðφðpe Þ φðpw ÞÞ
(43)
5. Case calculation and analysis
(35)
The GWWBT values calculated using different prediction methods based on five gas wells of the Puguang Gas Field were compared against field data to validate the proposed model. The parameters of the field are shown in Table 1; the comparison results are shown
a
Both stresses at critical point are identical when gas from a planar radial flow transforms into a hemispherical centripetal flow. Combining Eqs. (34) and (35) yields:
9 8 1 re 1 1 > > > > ln þ μ > > g > > = < H r r r a w a Qsc TZ φðpe Þ φðpw Þ ¼ > 5 1549:2Ka > > > Qsc γ g 2:15 10 1 1 1 1 > > > þ 7:16 104 3 3 > ; :þ ra re Rϕ H rw ra
According to the electrolytic test (Wattenbarger, 1968), ra ¼ 1.5 H is the critical point where planar radial flow transforms into hemispherical centripetal flow. When ra < 1.5 H, the gas near the bottom hole only receives hemispheric centripetal flow, resulting in the corresponding gaswell productivity equation:
(36)
in Table 2. As shown in Table 2, Kuo's model shows the greatest deviation from the field value; Huang's model is slightly more precise. Li's model is more accurate than both, but still results in substantial deviation. This maybe because the Kuo's model was established by numerical simulation just
9 8 1 re 1 1 > > > > ln þ μg > > > > = < H r 1:5H 1:5H w Qsc TZ φðpe Þ φðpw Þ ¼ > > Qsc γ 2:15 105 1549:2Ka > > 1 1 1 1 g > > > >þ þ 7:16 104 3 ; : 1:5H re Rϕ H rw 3:375H 3 1002
(37)
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
Table 1 Basic parameters of five high-sulfur gas wells. Basic parameters
Value
Reservoir temperature (K) Reservoir initial pressure (MPa) Reservoir porosity (%) Reservoir absolute permeability (μm2) Gas relative density Gas average viscosity (Pa⋅s) Irreducible water saturation Residual gas saturation Z-factor Gas reservoir radius (m) Well radius (m)
P102-1
P105-1H
P106-2
P203-1
P304-1
351.75 48.33 11.80 15.72 103 0.75 2.48 105 0.18 0.12 0.97 1500 0.1
351.75 48.33 8.60 9.33 103 0.75 2.48 105 0.20 0.20 0.97 1500 0.1
351.75 48.33 8.20 5.46 103 0.75 2.48 105 0.21 0.16 0.97 1500 0.1
427.15 68.50 12.50 12.05 103 0.82 3.52 105 0.26 0.32 1.37 1000 0.1
378.55 31.45 8.95 2.37 103 0.72 2.32 105 0.23 0.26 0.92 1000 0.1
Table 2 Comparison results of different prediction methods against field GWWBT. Well No.
GWWBT (d)
P102-1 P105-1H P106-2 P203-1 P304-1
Relative deviation (%)
Field data
Proposed method
1013 1366 1057 812 1889
1334 1676 1404 1271 2400
Proposed method 1564 1588 1322 1053 2312
1957 2297 1899 1589 2856
6.38 5.47 8.01 9.48 5.17
23.29 15.99 22.19 41.69 20.48
16.82 9.90 15.06 17.39 16.06
80.87 58.96 65.27 77.15 43.37
1082 1445 1149 897 1992
Next, the effects of several sensitive parameters were analyzed via P105-1H in to further explore the feasibility and effectiveness of the proposed model.
considering known production data and ignoring the effect of gas nonDarcy flow, irreducible water saturation, residual gas saturation; Huang's model also considers gas non-Darcy flow but irreducible water saturation and residual gas saturation. However, irreducible water and residual gas but non-Darcy is considered in Li's model. In addition, above three classical models are the negligence of sulfur deposition. Their model makes calculated velocity of water coning relatively low, therefore, they overpredicts GWWBT in HSGRs. The proposed model considering non-Darcy flow, irreducible water saturation, residual gas saturation and sulfur deposition underpredicts barely GWWBT in HSGRs but yields the most accurate result in comparison to the field data. Because sulfur-solubility of empirical model may lead to an under prediction of the proposed model. Result given from model which was developed by different methods or considering different factors has prodigious difference (Table 2). Meanwhile, Eq. (43) is a function of pseudo-pressure and the effects of some factors on result (GWWBT) cannot directly show in it. Therefore, the effects of some factors on GWWBT should be discussed so that we understand the impact of various factors on GWWBT in HSGRs.
5.1. Effect of sulfur deposition on GWWBT Fig. 2 shows the relationship between final sulfur deposition amount and BHP (Eq. (17)). Sulfur deposition mainly occurred at the early stage of pressure drop, but did not markedly increase as pressure continued to decrease. The model without considering sulfur deposition is suitable for conventional gas reservoirs, where Ss ¼ 0 and the productivity equation does not need to consider the effect of sulfur deposition on permeability (Eq. (43)). Fig. 2 also shows the relationship between GWWBT and BHP considering sulfur deposition. This GWWBT value was higher than that considering sulfur deposition for constant BHP, suggesting that sulfur deposition can promote GWWBT. Because sulfur precipitation not only occupies the pore channels, but also damages reservoir permeability, 4000
3000
0.6 without considering sulfur deposition considering sulfur deposition
Sgr=0.1 Sgr=0.2 Sgr=0.3 Swi=0.1 Swi=0.2 Swi=0.3
3500 3000 0.4
2500
1000
GWWBT (d)
GWWBT (d)
sulfur saturation
2000
2000 1500
0.2
1000 500 0 10
15
20
25 30 BHP (MPa)
35
40
0 0.1
0 45
Fig. 2. Effect of sulfur deposition on gas-well water breakthrough time.
0.15
0.2
0.25 sulfur saturation
0.3
0.35
0.4
Fig. 3. Effect of irreducible water and residual gas on gas-well water breakthrough time. 1003
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3500
10000 H=10m H=20m H=30m
9000 8000
3000
2500
7000 6000
GWWBT (d)
GWWBT (d)
Non-Darcy Darcy
5000 4000 3000
2000
1500
1000
2000
500
1000 0 0.1
0.15
0.2
0.25 sulfur saturation
0.3
0.35
0 0.1
0.4
0.15
0.2
0.25 sulfur saturation
0.3
0.35
0.4
Fig. 4. Effect of imperforated reservoir thickness on gas-well water breakthrough time.
Fig. 5. Effect of different flow states on gas-well water breakthrough time.
water flow velocity increased and thus the GWWBT increased. The results also suggest that effect of sulfur deposition on GWWBT increases earlier and decreases later as BHP decreases. The reason for this is that GWWBT is mainly affected by sulfur deposition at the early stage of pressure drop and is mainly affected by BHP at the later stage. During early gas reservoir development, higher sulfur deposition rate shows in higher BHP, results in a marked decrease in GWWBT. Later on in the gas reservoir exploitation process, conversely, the BHP is smaller, sulfur deposition rate slows down, and GWWBT is more stable.
governed by Darcy's law – but when the flow rate is very high (e.g., near the wellbore), Darcy's law is inadequate (Li and Engler, 2001). The nonDarcy flow effect is significant in terms of water coning around vertical and horizontal wells (Civan and Evans, 1991). The GWWBT discussed in this paper is located near the wellbore, so the effect of non-Darcy flow is not negligible in the HSGR. 6. Conclusion (1) In this paper, a novel prediction model for bottom water breakthrough time in high-sulfur gas reservoirs is developed based on the theory of Non-Darcy fluids flow in porous media. The analytical expression is related to the sulfur deposition. The validity of the proposed model is verified by comparing the model predictions with other three classical models predictions and the gas-field production value. The present results show that the proposed model that takes sulfur deposition into consideration can effectively evaluate more GWWBT in HSGRs than conventional models, which fail to take sulfur deposition into consideration. Owing to sulfur occupying the pore channels and damaging reservoir permeability, sulfur precipitation can significantly accelerate water coning breakthrough. GWWBT decreases with precipitated sulfur increases, so sulfur deposition is a key factor in GWWBT in HSGRs. (2) The influence of gas Non-Darcy flow, irreducible water saturation, residual gas saturation, BHP, and imperforated reservoir thickness on GWWBT was further discussed in detail by this proposed model. We find that GWWBT also increases as imperforated reservoir thickness and BHP increase; GWWBT also decreases as irreducible water and residual gas increase. This is in view of the fact that the larger the imperforated reservoir thickness is, the longer the distance of water coning is; the bigger BHP is, the bigger the resistance of water coning is; the more irreducible water and residual gas is, the bigger the velocity of water coning is. To ensure a constant GWWBT value, Non-Darcy flow requires less sulfur saturation than Darcy flow because the velocity of water coning in non-Darcy flow is larger than that in Darcy flow.
5.2. Irreducible water and residual gas effects on GWWBT Fig. 3 shows the relationship between GWWBT and residual gas saturation (or irreducible water) when imperforated reservoir thickness is 20 m and BHP is 25 MPa. Greater residual gas saturation (irreducible water saturation) results in faster effective water cone advancing speed and shorter GWWBT. As sulfur saturation increases, GWWBT decreases. Fig. 3 also shows where the effects of irreducible water and residual gas on GWWBT differ slightly, as irreducible water affects both sulfur saturation and reservoir permeability while residual gas does not (Eq. (16)). 5.3. Effect of imperforated reservoir thickness on GWWBT Fig. 4 shows the relationship between imperforated reservoir thickness and GWWBT when BHP is 25 MPa, irreducible water is 0.2, and residual gas is 0.2. The thicker the imperforated reservoir, the longer the GWWBT when sulfur saturation is constant because a longer advancing distance requires more time to traverse. When sulfur saturation is 0.15 and imperforated reservoir thickness increases from 10 m to 20 m, GWWBT increases by 615.68 (1767.85%). In other words, imperforated reservoir thickness severely affects the GWWBT value. 5.4. Effect of flow state on GWWBT Fig. 5 shows the relationship between flow state and GWWBT when irreducible water is 0.2, residual gas is 0.2, imperforated reservoir thickness is 20 m, and BHP is 25 MPa; GWWBT increases as sulfur saturation increases. When sulfur saturation is 0.15, GWWBT in Darcy flow is higher 6.69% than that in non-Darcy flow. To achieve a constant GWWBT value, the required sulfur non-Darcy saturation is smaller than that with Darcy flow because high-speed non-Darcy flow increases the pressure drop, which increases sulfur precipitation and water flow velocity. Darcy's law does not accurately describe the fluid flow when the flow rate is high. In most cases in the recovery process, fluid flow is
Acknowledgment This work was supported by National Science and Technology Major Project (2016ZX05017-005). The authors recognize the support of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitations (Southwest Petroleum University) for the permission to publish this article. 1004
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Nomenclature
English letters Bg gas volume factor c sulfur solubility, kg/m3 h gas reservoir thickness, m H unperforated reservoir thickness, m K effective formation permeability, μm2 Ka absolute formation permeability, μm2 Krg relative gas phase permeability m solid sulfur quality, kg Ma dry air molecular weight, 28.97 kg=kmol p gas reservoir pressure, MPa pi original gas reservoir pressure, MPa Psc standard state pressure, 0.101325 MPa pe boundary pressure, MPa pw bottom hole pressure, MPa q subsurface production rate, m3/d Qsc standard state production, m3/d r radial distance, m ra bottom hole proximal radial distance, m re drainage radius, m rw well radius, m R conventional gas constant, 0.008314 MPa⋅m3/(kmol⋅K) Ss sulfur saturation Swi irreducible water saturation Sgr residual gas saturation t production time, d tp cone bottom water breakthrough time, d T gas reservoir temperature, K Tsc standard state temperature, 293 K Vs solid sulfur volume, m3 z water point higher level, m Z deviation factor of natural gas Zsc standard state gas deviation factor, 1 Greek letters α empirical constant, 6.22 ϕ gas reservoir porosity ρs solid sulfur density, 2070 kg/m3 ρg subsurface gas density, kg/m3 γg relative density of natural gas gas fluid viscosity, Pa⋅s μg β turbulence coefficient, m1
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References Brunner, E., Woll, W., 1980. Solubility of sulfur in hydrogen sulfide and sour gases. SPEJ 20 (5), 377–384. Oct. Civan, F., Evans, R.D., 1991. Non-darcy flow coefficients and relative permeabilities for gas/brine systems. In: SPE Gas Technology Symposium. Du, Z.M., Guo, X., Zhang, Y., 2006. Gas-liquid-solid coupled flow modelling Modeling in fractured carbonate gas reservoir with high H2S content. In: First International Oil Conference and Exhibition in Mexico. Guo, X., Du, Z.M., Yang, X., Fu, D.K., 2009. Sulfur deposition in sour gas reservoir: laboratory and simulation study. Petrol. Sci. 6 (4), 405–414. Guo, X., Zhou, X.F., 2015. A prediction model of sulfur saturation in the near-wellbore areas of high sulfur gas wells considering the effects of non-Darcy flow. Nat. Gas. Ind. 35 (4), 40–44. Hu, J.H., He, S., Zhao, J.Z., Li, Y.M., 2013. Modeling of sulfur plugging in a sour gas reservoir. J. Nat. Gas Sci. Eng. 11 (11), 18–22. Huang, Q.H., Lu, Y., Lu, C., 2016. Bottom water reservoir with partition prediction method of water breakthrough time. Lithol. Reserv. 28 (4), 82–87. Kuo, M.C., Desbrisay, C.L., 1983. A simplified method for water coning predictions. In: SPE 12067, SPE Annual Technical Conference, pp. 5–8. Dallas, Texas, September.
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006 Yu, Q.K., Su, Y.X., Li, Z.H., 2016. Establishment of water invasion identification standard for Puguang fracture pore type gas reservoir. Petrochem. Technol. 23 (5), 191. Zeng, P., Zhao, J.Z., Zhou, H.B., 2005. Formation damage from elemental sulfur deposition in sour gas reservoir. Pet. Explor. Dev. 32 (6), 113–115. Zeng, D.Q., Peng, X.L., Liu, Z.Y., 2013. Haracterization methods of reef-beach facies reservoirs in the Puguang gas field. Nat. Gas. Ind. 31 (3), 9–13. Zhang, L.H., Li, Y., Li, X.P., 2004. New method to predict condensing effect on water breakthrough in condensate reservoirs with bottom water. Nat. Gas. Ind. 24, 74–75. Zhao, X., Zhu, S., 2012. Prediction of water breakthrough time for oil wells in lowpermeability bottom water reservoirs with barrier. Petroleum Explor. Dev. 39 (4), 504–507.
Wang, H.Q., Li, X.Q., Yang, Q., et al., 2007. A prediction of water breakthrough time in gas reservoirs with bottom water. Xinjiang Pet. Geol. 28 (1), 92–93. Wang, S.P., Kong, F.Q., Peng, X.L., Zhang, S., 2011. Optimization of the development indexes of the Puguang gas field. Nat. Gas. Ind. 31 (3), 5–8. Wattenbarger, R.A., 1968. Gas well testing with turbulence, damage and wellbore storage. J. Petroleum Technol. 20 (8), 877–887. Xiong, X., Li, Y., Zhang, J., Wang, Y., 2014. A new method to predict water coning performance and water breakthrough time in bottom water oil reservoir. Fault-Block Oil Gas Field 11–15. Yang, M.P., Peng, C.Z., Li, C.N., 2004. Study of sulfur deposition model and its application of sour gas fields. J. Southwest Pet. Univ. 26 (6), 54–59.
1006
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Gas-well water breakthrough time prediction model for high-sulfur gas reservoirs considering sulfur deposition Guo Xiao a, Wang Peng a, *, Liu Jin a, Song Ge b, Dang Hailong c, Gao Tao c a b c
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitations, Southwest Petroleum University, Chengdu 610500, China Northeastern Sichuan Gas Recovery Plant, Sinopec Southwest Petroleum Company, Langzhong, 637400, China Research Institute of Yanchang Petroleum (Group) Co. LTD., Xi'an 710075, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Gas-well water breakthrough time (GWWBT) High-sulfur gas reservoirs (HSGRs) Sulfur deposition Non-Darcy seepage
The sulfur-solubility decreases as a result of the decrease in gas reservoir pressure, leading to the solid-phase sulfur deposition. As a consequence, both the reservoir porosity and permeability decrease, subsequently influencing gas well water breakthrough time (GWWBT) in high sulfur gas reservoirs (HSGRs) with edge/bottom water. To acquire the value of GWWBT, a GWWBT prediction model for HSGRs should be established while taking into account sulfur deposition. Accordingly, based on gas Non-Darcy seepage law and sulfur deposition theory in porous media, a novel GWWBT in high sulfur gas reservoir with bottom water was developed in this study. The effect of dynamic factors (e.g., sulfur deposition, gas Non-Darcy flow, irreducible water saturation, and residual gas saturation) on GWWBT was involved in this model. The GWWBT was calculated via the proposed method and three classical models in five field basic parameters, and was compared with five filed values, respectively. This result indicates that the calculation of proposed method is in closer agreement with the field production data, and illustrates that the new proposed model is more reliable. In addition, the influence of dynamic factors was further discussed in detail by this proposed model.
1. Introduction Active edge/bottom water gas reservoir accounts for approximately 40–50% of water drive gas reservoirs. Most existing gas reservoirs of China have varying degrees of water drive; most of China's high-sulfur gas reservoirs (HSGRs) are close to water bodies of bottom water (Wang et al., 2011). The water body flows to the gas reservoir during HSGR development, forming two-phase gas-water seepage with decreasing pressure. This results in a decrease in the gas-well recovery rate (Wang et al., 2011; Zeng et al., 2013; Li, 2014; Liu et al., 2015; Yu et al., 2016) which can be mitigated using an effective bottom water coning profile and by controlling the breakthrough time accordingly. Previous scholars have proposed many prediction models for gas-well water breakthrough time (GWWBT) after investigating bottom water coning problems in conventional bottom water reservoirs. Sobocinski and Cornelius (1965), for example, studied the relation between bottom water coning and time based on the physical model of sandstone, but which was built form the static water-oil contact to breakthrough conditions. Kuo and Desbrisay (1983) modeled the dynamic relationship between bottom water coning and time numerically, which is without
* Corresponding author. E-mail address: [email protected] (W. Peng). http://dx.doi.org/10.1016/j.petrol.2017.08.020 Received 28 February 2017; Received in revised form 20 July 2017; Accepted 4 August 2017 Available online 10 August 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.
considering residual oil saturation, irreducible water saturation. Shi et al. (1992) used the system identification method and a one-dimensional differential equation for filtering flow to establish a GWWBT prediction method, the application of this model is limited known mechanism model and huge amounts of oilfield statistical data. Li (2001) and Tang (2003) obtained water breakthrough prediction time equations for bottom water reservoirs in which plane radial flow and spherical flow are taken into account, while residual oil saturation, irreducible water saturation, and oil/water viscosity are neglected. Zhang et al. (2004) proposed a water breakthrough model for condensate reservoirs with bottom water based on a relatively simple water coning model in which gas condensing effects were taken into account, but this model is only applicable for condensate reservoirs. Zhao and Zhu (2012) derived a water breakthrough time prediction equation for low-permeability bottom water reservoirs with barriers by applying the material balance principle and non-Darcy flow theory, which takes the hemispherical radial flow below the water coning barrier and plane radial flow above the barrier into consideration. However, this equation considering startup pressure gradient is only applicable for low-permeability gas reservoirs. Xiong et al. (2014) proposed a formula for water coning
G. Xiao et al.
Journal of Petroleum Science and Engineering 157 (2017) 999–1006
breakthrough time in bottom water oil reservoirs in which oil-water mobility ratio, original irreducible water saturation, and residual oil saturation are taken into account, this formula is only applicable for oil reservoirs. Li et al. (2015) improved upon this model by adding oil-water contact and water coning profile factors but disregarding the impact of non-Darcy effects. Huang et al. (2016) deduced bottom-water gas reservoir water coning time based on the theory of percolation flow in porous media; their model also includes gas non-Darcy effects, skin factor, degree of opening, and daily gas production, without irreducible water saturation and residual gas saturation. Sulfur deposition is missing from these models, however, rendering them inapplicable to HSGRs. In bottom-water HSGRs, sulfur solubility decreases as gas reservoir pressure decreases, which leads to solid-phase sulfur deposition (Roberts, 1997; Zeng et al., 2005; Yang et al., 2004; Du et al., 2006; Guo et al., 2009). This further causes a decrease in reservoir porosity and permeability which affects GWWBT. Extant prediction models cannot yield accurate results without considering sulfur deposition. In this paper, a novel GWWBT model was established with consideration of sulfur deposition for HSGRs.
(2)
Hz sin φ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r þ ðH zÞ2
(3)
V¼
Qsc 2π r 2 þ ðH zÞ2
(4)
Combining Eqs. (1)–(4) yields:
V1v ¼
Qsc Hz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2π r 2 þ ðH zÞ ϕ 1 Swi Sgr Ss 2 r þ ðH zÞ2
(5)
The upward migration distance of water coning is dz in dt can be obtained by equation (5), it is as follows: t
Qsc ðH zÞ 32 dt 2 2πϕ r þ ðH zÞ2 1 Swi Sgr Ss =
z
∫ 0 dz ¼ ∫ 0
2. Gas-well water breakthrough time
(6)
Equation (6) can be rewritten in integrated form as:
8 9 12 2 > > 2 > > > H þr > > > > > > > > > > > 1 2 > > 2 2 > > þ r r H > > > > > þr ln > > > 2 < = 3 H 3 1 2 1 2 2 H þ r 2 2 ðH zÞ þ r 2 þ r 2 1 > ðH zÞ2 þ r 2 2 > 3 3 > > > > > > > > > > > > 1 > > 2 2 2 > > > > þ r r ðH zÞ > > > r ln > > > > > 2 : ; ðH zÞ
The physical bottom water coning model is shown in Fig. 1. Assumptions: (1) Capillary forces, gravity, reservoir anisotropy, stress sensitivity, slippage, and skin factor effects during the displacement process are ignored; (2) Imperforated formation is assumed for the bottom of the distal radial flow and bottom hemispherical centripetal flow, while perforated formation is assumed for the plane radial flow. The existence of a shaft axis z with homogeneous distribution in the original gas water interface (r axis) was also assumed (Fig. 1). According to the water quality point seepage law, there is a mass point A at the initial gas-water interface with a radial flow for t to point A (z, r) and fluid velocity at point A (V) in the porous media seepage. Using the reservoir of porous medium porosity (ϕ) and irreducible water saturation (Swi ), residual gas saturation (Sgr ), sulfur saturation (Ss ), and the actual seepage velocity of water points V1 , the upward seepage velocity of water points is V1v can be calculated as follows:
V V1 ¼ ϕ 1 Swi Sgr Ss
V1v ¼ V1 sin φ
¼
Qsc t 2πϕ 1 Swi Sgr Ss (7)
When the water cone breaks through to the gas well, assuming r ¼ 0, z ¼ H, t ¼ tp (bottom water coning breakthrough critical height) in Eq. (7), then the novel GWWBT for HSGRs is:
tp ¼
2πϕ 1 Swi Sgr Ss 3 H 3Qsc
(1)
Fig. 1. Schematic diagram of water coning in bottom-water reservoir. 1000
(8)
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
3. Calculating several model parameters
K ¼ Ka exp
3.1. Sulfur saturation and effective gas reservoir permeability
As shown in Fig. 1, imperforated reservoir bottom gas flow is divided into hemispherical centripetal flow and planar radial flow based on actual simulation of reservoir and different seepage characteristics. According to the basic law of gas-water two-phase fluid flow, the hemispherical centripetal flow equation is:
(9)
q 2 dp μg q ¼ þ βρ g dr K 2πr 2 2πr 2
The sour gas reservoir pore volume can be calculated as follows:
dVs ¼
2πrhϕð1 Swi Þdrdc ρs
(10)
q 2 dp μg q ¼ þ βρg dr K 2πrh 2πrh (11)
β¼
(12)
ρg ¼
Bg ¼
S
p
(14)
Kpdp ¼ (16)
Kpdp ¼
(17)
(26)
Qsc psc ZTμg 1 3:33 106 Q2sc p2sc TZγ g 1 dr þ r2 r4 2πTsc 4π 2 Tsc2 Rϕ
(27)
Qsc psc ZTμg 1 3:33 106 Q2sc p2sc TZγ g 1 þ dr 2 2 2 2 r 2πhTsc r 4π h Tsc Rϕ
(28)
To simplify the calculation, μg and Z are constant. Substituting and integrating Eq. (20) into Eqs. (21) and (28) yields the following hemispherical centripetal flow equation:
Qsc psc ZTμg 1 αA 4 p r ∫ pw Ka exp pi p4 pdp ¼ ∫ rw 4 r2 2πTsc
(18)
þ
In the establishment of a high-sulfur gas reservoir seepage equation, the precipitation of sulfur damages gas reservoir permeability. The effective permeability is typically characterized by absolute permeability and relative permeability (Guo and Zhou, 2015; Hu et al., 2013):
K ¼ Ka Krg
Psc ZT Zsc Tsc p
Substituting Eqs. (24)–(27) into Eq. (23) yields the differential form of the plane radial flow yield:
Elemental sulfur may precipitate when pressure drops below a critical pressure. Given the effects of deposited sulfur on reservoir permeability, an empirical equation for relative permeability and sulfur saturation is useful (Roberts, 1997):
Krg ¼ expðαSs Þ
(25)
Substituting Eqs. (23)–(26) into Eq. (21) yields the differential form of the hemispherical centripetal flow yield:
(15)
Equation (16) can be changed into Eq. (17), a novel formula about sulfur saturation is as follows:
A Ss ¼ p4i p4 4
(24)
Psc ZT Zsc Tsc p
q ¼ Qsc Bg ¼ Qsc
where (assuming that Z and γ g are constant):
4 ð1 Swi Þ Ma γ g 4666 4:5711 A¼ 4 exp ρs T ZRT
Ma γ g p ZRT
The production under the geological conditions can be transformed into production under ground conditions:
Equation (14) can be changed to:
∫ 0 s dSs ¼ ∫ pi Ap3 dp
(23)
and the gas volume factor is:
(13)
Combining Eqs. (12) and (13) yields:
4 dSs ð1 Swi Þ Ma γ g 4666 4:5711 p3 ¼ 4 exp ρs T dp ZRT
1:15 107 Kϕ
Gas density under geological conditions is:
Based on the principles of associative law and entropy, an empirical equation was obtained utilizing Brunner's experimental data (Roberts, 1997; Brunner and Woll, 1980):
Ma γ g 4 dc 4666 ¼4 4:5711 p3 exp dp T ZRT
(22)
where β is the turbulence coefficient (1/m) and can be calculated as follows (Li and Engler, 2001):
Equation (11) can be rewritten as follows:
dSs ð1 Swi Þ dc ¼ ρs dp dp
(21)
The planar radial flow equation is:
Elemental sulfur saturation is:
ð1 Swi Þdc dVs ¼ dSs ¼ ρs 2πrhϕdr
(20)
3.2. HSGR production equation considering deposited sulfur
This model assumes that elemental sulfur is not precipitated under the original formation pressure and that sulfur deposition migration does not occur in the channel. The quality of elemental sulfur in the gas changes due to the pressure drop in solubility (dc) at r (radial distance). The precipitated sulfur quality can be expressed as follows:
dm ¼ 2πrhϕð1 Swi Þdrdc
αA 4 pi p4 4
3:33 106 Q2sc p2sc TZγ g 1 dr r4 4π 2 Tsc2 Rϕ
(29)
and planar radial flow equation:
Qsc psc ZTμg 1 αA 4 p r ∫ pe Ka exp pi p4 pdp ¼ ∫ re 4 2πhTsc r
(19)
þ
Combined with Eqs. (17)–(19), we obtained the new reservoir effective permeability as follows: 1001
3:33 106 Q2sc p2sc TZγ g 1 dr r2 4π 2 h2 Tsc2 Rϕ
(30)
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
Assuming:
where:
αA 4 pi p4 pdp φðpÞ ¼ ∫ exp 4
a¼
(31)
The following were derived by combining Eqs. (29)–(31) to rewrite the hemispherical centripetal flow equation:
e¼
The planar radial flow equation was altered as well:
Qsc psc ZTμg ðln re ln rÞ 2πhKa Tsc 3:33 106 Q2sc p2sc TZγ g 1 1 þ 2 2 2 r re 4π h Tsc Ka Rϕ
Qsc ¼
ðφðpe Þ φðpÞÞ 2:15105 Qsc γ g 1 1 TZ μg ln rrae þ ra re HRϕ
1549:2Ka ðφðpÞ φðpw ÞÞ 7:16104 Qsc γ g 1 1 TZ μg r1w r1a þ 3 3 Rϕ r r w
γ g 2:15 105 1 1 1 1 þ 7:16 104 3 1:5H re rw 3:375H 3 Rϕ H
(39)
(40)
Equation (37) can be changed to:
aeQ2sc þ abQsc ðφðpe Þ φðpw ÞÞ ¼ 0
(41)
Equation (41) uses the extract roots formula:
Qsc ¼
ab þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 þ 4aeðφðpe Þ φðpw ÞÞ 2ae
(42)
4. Model simplification To analyze the influence of specific parameters on the GWWBT for HSGR, the initial GWWBT equation (8) was simplified by substituting Eqs. (17) and (42) into (8). The GWWBT for HSGR is then expressed as follows:
(34)
In the second part, the bottom hole proximal radial distance (ra ) is in hemispherical centripetal flow to the bottom radial distance (rw ), and the corresponding gas-well productivity equation is (Wang et al., 2007):
Qsc ¼
1 re 1 1 ln þ H 1:5H rw 1:5H
(33)
The imperforated reservoir bottom gas flow is divided into the bottom reservoir distal radial distance (re ) to the bottom proximal radial distance (ra ) in the proximal plane radial flow, as well as into the bottom proximal radial distance (ra ) to the bottom hole radial distance (rw ) in the hemispherical centripetal flow (Fig. 1). Because it can simulate really the seepage process of gas flow and water coning. In the first part, the bottom reservoir distal radial distance (re ) is in planar radial flow to the bottom proximal radial distance (ra ) and the corresponding gas-well productivity equation is (Wang et al., 2007):
1549:2HKa
(38)
b ¼ μg
Qsc psc ZTμg 1 1 3:33 106 Q2sc p2sc TZγ g 1 1 φðpÞ φðpw Þ ¼ þ rw r rw3 r 3 2πKa Tsc 12π 2 Tsc2 Ka Rϕ (32)
φðpe Þ φðpÞ ¼
TZ 1549:2Ka
tp ¼
4acπϕ 1 Swi Sgr Ss pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH 3 3abBg þ 3Bg a2 b2 4acðφðpe Þ φðpw ÞÞ
(43)
5. Case calculation and analysis
(35)
The GWWBT values calculated using different prediction methods based on five gas wells of the Puguang Gas Field were compared against field data to validate the proposed model. The parameters of the field are shown in Table 1; the comparison results are shown
a
Both stresses at critical point are identical when gas from a planar radial flow transforms into a hemispherical centripetal flow. Combining Eqs. (34) and (35) yields:
9 8 1 re 1 1 > > > > ln þ μ > > g > > = < H r r r a w a Qsc TZ φðpe Þ φðpw Þ ¼ > 5 1549:2Ka > > > Qsc γ g 2:15 10 1 1 1 1 > > > þ 7:16 104 3 3 > ; :þ ra re Rϕ H rw ra
According to the electrolytic test (Wattenbarger, 1968), ra ¼ 1.5 H is the critical point where planar radial flow transforms into hemispherical centripetal flow. When ra < 1.5 H, the gas near the bottom hole only receives hemispheric centripetal flow, resulting in the corresponding gaswell productivity equation:
(36)
in Table 2. As shown in Table 2, Kuo's model shows the greatest deviation from the field value; Huang's model is slightly more precise. Li's model is more accurate than both, but still results in substantial deviation. This maybe because the Kuo's model was established by numerical simulation just
9 8 1 re 1 1 > > > > ln þ μg > > > > = < H r 1:5H 1:5H w Qsc TZ φðpe Þ φðpw Þ ¼ > > Qsc γ 2:15 105 1549:2Ka > > 1 1 1 1 g > > > >þ þ 7:16 104 3 ; : 1:5H re Rϕ H rw 3:375H 3 1002
(37)
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
Table 1 Basic parameters of five high-sulfur gas wells. Basic parameters
Value
Reservoir temperature (K) Reservoir initial pressure (MPa) Reservoir porosity (%) Reservoir absolute permeability (μm2) Gas relative density Gas average viscosity (Pa⋅s) Irreducible water saturation Residual gas saturation Z-factor Gas reservoir radius (m) Well radius (m)
P102-1
P105-1H
P106-2
P203-1
P304-1
351.75 48.33 11.80 15.72 103 0.75 2.48 105 0.18 0.12 0.97 1500 0.1
351.75 48.33 8.60 9.33 103 0.75 2.48 105 0.20 0.20 0.97 1500 0.1
351.75 48.33 8.20 5.46 103 0.75 2.48 105 0.21 0.16 0.97 1500 0.1
427.15 68.50 12.50 12.05 103 0.82 3.52 105 0.26 0.32 1.37 1000 0.1
378.55 31.45 8.95 2.37 103 0.72 2.32 105 0.23 0.26 0.92 1000 0.1
Table 2 Comparison results of different prediction methods against field GWWBT. Well No.
GWWBT (d)
P102-1 P105-1H P106-2 P203-1 P304-1
Relative deviation (%)
Field data
Proposed method
1013 1366 1057 812 1889
1334 1676 1404 1271 2400
Proposed method 1564 1588 1322 1053 2312
1957 2297 1899 1589 2856
6.38 5.47 8.01 9.48 5.17
23.29 15.99 22.19 41.69 20.48
16.82 9.90 15.06 17.39 16.06
80.87 58.96 65.27 77.15 43.37
1082 1445 1149 897 1992
Next, the effects of several sensitive parameters were analyzed via P105-1H in to further explore the feasibility and effectiveness of the proposed model.
considering known production data and ignoring the effect of gas nonDarcy flow, irreducible water saturation, residual gas saturation; Huang's model also considers gas non-Darcy flow but irreducible water saturation and residual gas saturation. However, irreducible water and residual gas but non-Darcy is considered in Li's model. In addition, above three classical models are the negligence of sulfur deposition. Their model makes calculated velocity of water coning relatively low, therefore, they overpredicts GWWBT in HSGRs. The proposed model considering non-Darcy flow, irreducible water saturation, residual gas saturation and sulfur deposition underpredicts barely GWWBT in HSGRs but yields the most accurate result in comparison to the field data. Because sulfur-solubility of empirical model may lead to an under prediction of the proposed model. Result given from model which was developed by different methods or considering different factors has prodigious difference (Table 2). Meanwhile, Eq. (43) is a function of pseudo-pressure and the effects of some factors on result (GWWBT) cannot directly show in it. Therefore, the effects of some factors on GWWBT should be discussed so that we understand the impact of various factors on GWWBT in HSGRs.
5.1. Effect of sulfur deposition on GWWBT Fig. 2 shows the relationship between final sulfur deposition amount and BHP (Eq. (17)). Sulfur deposition mainly occurred at the early stage of pressure drop, but did not markedly increase as pressure continued to decrease. The model without considering sulfur deposition is suitable for conventional gas reservoirs, where Ss ¼ 0 and the productivity equation does not need to consider the effect of sulfur deposition on permeability (Eq. (43)). Fig. 2 also shows the relationship between GWWBT and BHP considering sulfur deposition. This GWWBT value was higher than that considering sulfur deposition for constant BHP, suggesting that sulfur deposition can promote GWWBT. Because sulfur precipitation not only occupies the pore channels, but also damages reservoir permeability, 4000
3000
0.6 without considering sulfur deposition considering sulfur deposition
Sgr=0.1 Sgr=0.2 Sgr=0.3 Swi=0.1 Swi=0.2 Swi=0.3
3500 3000 0.4
2500
1000
GWWBT (d)
GWWBT (d)
sulfur saturation
2000
2000 1500
0.2
1000 500 0 10
15
20
25 30 BHP (MPa)
35
40
0 0.1
0 45
Fig. 2. Effect of sulfur deposition on gas-well water breakthrough time.
0.15
0.2
0.25 sulfur saturation
0.3
0.35
0.4
Fig. 3. Effect of irreducible water and residual gas on gas-well water breakthrough time. 1003
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Journal of Petroleum Science and Engineering 157 (2017) 999–1006
3500
10000 H=10m H=20m H=30m
9000 8000
3000
2500
7000 6000
GWWBT (d)
GWWBT (d)
Non-Darcy Darcy
5000 4000 3000
2000
1500
1000
2000
500
1000 0 0.1
0.15
0.2
0.25 sulfur saturation
0.3
0.35
0 0.1
0.4
0.15
0.2
0.25 sulfur saturation
0.3
0.35
0.4
Fig. 4. Effect of imperforated reservoir thickness on gas-well water breakthrough time.
Fig. 5. Effect of different flow states on gas-well water breakthrough time.
water flow velocity increased and thus the GWWBT increased. The results also suggest that effect of sulfur deposition on GWWBT increases earlier and decreases later as BHP decreases. The reason for this is that GWWBT is mainly affected by sulfur deposition at the early stage of pressure drop and is mainly affected by BHP at the later stage. During early gas reservoir development, higher sulfur deposition rate shows in higher BHP, results in a marked decrease in GWWBT. Later on in the gas reservoir exploitation process, conversely, the BHP is smaller, sulfur deposition rate slows down, and GWWBT is more stable.
governed by Darcy's law – but when the flow rate is very high (e.g., near the wellbore), Darcy's law is inadequate (Li and Engler, 2001). The nonDarcy flow effect is significant in terms of water coning around vertical and horizontal wells (Civan and Evans, 1991). The GWWBT discussed in this paper is located near the wellbore, so the effect of non-Darcy flow is not negligible in the HSGR. 6. Conclusion (1) In this paper, a novel prediction model for bottom water breakthrough time in high-sulfur gas reservoirs is developed based on the theory of Non-Darcy fluids flow in porous media. The analytical expression is related to the sulfur deposition. The validity of the proposed model is verified by comparing the model predictions with other three classical models predictions and the gas-field production value. The present results show that the proposed model that takes sulfur deposition into consideration can effectively evaluate more GWWBT in HSGRs than conventional models, which fail to take sulfur deposition into consideration. Owing to sulfur occupying the pore channels and damaging reservoir permeability, sulfur precipitation can significantly accelerate water coning breakthrough. GWWBT decreases with precipitated sulfur increases, so sulfur deposition is a key factor in GWWBT in HSGRs. (2) The influence of gas Non-Darcy flow, irreducible water saturation, residual gas saturation, BHP, and imperforated reservoir thickness on GWWBT was further discussed in detail by this proposed model. We find that GWWBT also increases as imperforated reservoir thickness and BHP increase; GWWBT also decreases as irreducible water and residual gas increase. This is in view of the fact that the larger the imperforated reservoir thickness is, the longer the distance of water coning is; the bigger BHP is, the bigger the resistance of water coning is; the more irreducible water and residual gas is, the bigger the velocity of water coning is. To ensure a constant GWWBT value, Non-Darcy flow requires less sulfur saturation than Darcy flow because the velocity of water coning in non-Darcy flow is larger than that in Darcy flow.
5.2. Irreducible water and residual gas effects on GWWBT Fig. 3 shows the relationship between GWWBT and residual gas saturation (or irreducible water) when imperforated reservoir thickness is 20 m and BHP is 25 MPa. Greater residual gas saturation (irreducible water saturation) results in faster effective water cone advancing speed and shorter GWWBT. As sulfur saturation increases, GWWBT decreases. Fig. 3 also shows where the effects of irreducible water and residual gas on GWWBT differ slightly, as irreducible water affects both sulfur saturation and reservoir permeability while residual gas does not (Eq. (16)). 5.3. Effect of imperforated reservoir thickness on GWWBT Fig. 4 shows the relationship between imperforated reservoir thickness and GWWBT when BHP is 25 MPa, irreducible water is 0.2, and residual gas is 0.2. The thicker the imperforated reservoir, the longer the GWWBT when sulfur saturation is constant because a longer advancing distance requires more time to traverse. When sulfur saturation is 0.15 and imperforated reservoir thickness increases from 10 m to 20 m, GWWBT increases by 615.68 (1767.85%). In other words, imperforated reservoir thickness severely affects the GWWBT value. 5.4. Effect of flow state on GWWBT Fig. 5 shows the relationship between flow state and GWWBT when irreducible water is 0.2, residual gas is 0.2, imperforated reservoir thickness is 20 m, and BHP is 25 MPa; GWWBT increases as sulfur saturation increases. When sulfur saturation is 0.15, GWWBT in Darcy flow is higher 6.69% than that in non-Darcy flow. To achieve a constant GWWBT value, the required sulfur non-Darcy saturation is smaller than that with Darcy flow because high-speed non-Darcy flow increases the pressure drop, which increases sulfur precipitation and water flow velocity. Darcy's law does not accurately describe the fluid flow when the flow rate is high. In most cases in the recovery process, fluid flow is
Acknowledgment This work was supported by National Science and Technology Major Project (2016ZX05017-005). The authors recognize the support of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitations (Southwest Petroleum University) for the permission to publish this article. 1004
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Nomenclature
English letters Bg gas volume factor c sulfur solubility, kg/m3 h gas reservoir thickness, m H unperforated reservoir thickness, m K effective formation permeability, μm2 Ka absolute formation permeability, μm2 Krg relative gas phase permeability m solid sulfur quality, kg Ma dry air molecular weight, 28.97 kg=kmol p gas reservoir pressure, MPa pi original gas reservoir pressure, MPa Psc standard state pressure, 0.101325 MPa pe boundary pressure, MPa pw bottom hole pressure, MPa q subsurface production rate, m3/d Qsc standard state production, m3/d r radial distance, m ra bottom hole proximal radial distance, m re drainage radius, m rw well radius, m R conventional gas constant, 0.008314 MPa⋅m3/(kmol⋅K) Ss sulfur saturation Swi irreducible water saturation Sgr residual gas saturation t production time, d tp cone bottom water breakthrough time, d T gas reservoir temperature, K Tsc standard state temperature, 293 K Vs solid sulfur volume, m3 z water point higher level, m Z deviation factor of natural gas Zsc standard state gas deviation factor, 1 Greek letters α empirical constant, 6.22 ϕ gas reservoir porosity ρs solid sulfur density, 2070 kg/m3 ρg subsurface gas density, kg/m3 γg relative density of natural gas gas fluid viscosity, Pa⋅s μg β turbulence coefficient, m1
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