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Mechanism and prediction analysis of sustained casing pressure in “A” annulus of CO2 injection well
Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Mechanism and prediction analysis of sustained casing pressure in ‘‘A’’ annulus of CO2 injection well Hongjun Zhu a,n, Yuanhua Lin a, Dezhi Zeng b, Deping Zhang c, Hao Chen b, Wei Wang b a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), Chengdu, Sichuan 610500, China CNPC Key Lab for Tubular Goods Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, China c Jilin Oil Field Company, PetroChina, Songyuan, Jilin 138000, China b
a r t i c l e i n f o
abstract
Article history: Received 22 July 2011 Accepted 2 June 2012 Available online 15 June 2012
High value of sustained casing pressure (SCP) threatens the wellbore essential safety and environmental protection seriously. Considering the special and complex characteristic of SCP in ‘‘A’’ annulus of CO2 injection well, a prediction model of buildup of casing pressure has been proposed in this paper. Then finite difference method was used to solve the discrete equations in order to predict the SCP variation with time. The value is closely related to size and position of leak point, leakage rate, height of annular protective liquid, length of gas column at wellhead and the state of CO2. The buildup of SCP in one well in Jilin Oilfield is obtained by this model, which is in good agreement with the actual result, verifying the reliability of coupled model and calculation method. & 2012 Elsevier B.V. All rights reserved.
Keywords: sustained casing pressure CO2 injection well gas migration gas–liquid two-phase flow supercritical CO2
1. Introduction After well completion, measuring pressure in all of the casing strings should be zero if the well is flowing at steady state conditions, and a small volume of fluid caused by thermal expansion effects has to be bled through a needle valve in order to decrease the casing pressure to atmospheric pressure (Bourgoyne et al., 1999). If the casing pressure builds up when the needle valve is closed, the casing generally exhibits sustained casing pressure (SCP). High casing pressure may induce blowout, explosion and other serious accidents, reducing the productive life of oil and gas wells and resulting in huge economic losses. SCP in gas producing wells is very common, such as 11,498 casing strings in 8122 wells in the Gulf of Mexico (Bourgoyne et al., 1999). Thus, gas migration in gas producing well has been studied by some researchers (Somei, 1992; Xu and Wojtanowicz, 2001; Xu, 2002). Among them, Xu’s work is the most detailed one. In Xu’s (2002) dissertation, she had built a mathematical model of casing pressure in ‘‘B’’ or ‘‘C’’ annulus wellhead of gas producing well and had given the numerical solution. However, a well usually has several annuli. According to the location, the annulus from inside to outside can be named ‘‘A’’ annulus, ‘‘B’’ annulus, ‘‘C’’ annulus, and so forth (Anders et al., 2006). As shown in Fig. 1, ‘‘A’’ annulus is the annulus between the tube and production casing. ‘‘B’’ annulus is the annulus between production casing and adjacent intermediate
n
Corresponding author. Tel.: þ86 28 83032206. E-mail address: [email protected] (H. Zhu).
0920-4105/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2012.06.013
casing. The rest can be obtained by analogy. The gas migration path in ‘‘A’’ annulus is not the same as that in ‘‘B’’ or ‘‘C’’ annulus. In ‘‘A’’ annulus, gas may flow through annular protective liquid and accumulate at the gas chamber in wellhead. While in ‘‘B’’ or ‘‘C’’ annulus, gas may sequentially flow through cement and mud column to reach the gas chamber. Moreover, annular protective liquid is water added by corrosion inhibitor, while mud is usually a non-Newtonian liquid–solid two-phase fluid. Therefore, Xu’s model cannot be directly applied in ‘‘A’’ annulus. Recently, with the spread of CO2 enhanced oil recovery (Haynes and Alston, 1990; Hargrove, 2004; Keeling, 1984; Tanner and Baxley, 1992), SCP in CO2 injection well has become a salient problem, which has been found in all oil fields using CO2-EOR in China. Especially, SCP in ‘‘A’’ annulus is the most serious challenge. In Jilin Oilfield, the maximum casing pressure in ‘‘A’’ annulus of CO2 injection well is up to 17.5 MPa. In some injection wells, obvious casing pressure appears just one week after operating, and the pressure builds up quickly and approaches the tubing pressure after pressure relief. Therefore, it is urgent to understand the mechanism of SCP in ‘‘A’’ annulus of CO2 injection well and predict the value of SCP. However, there are few reports about the mechanism of SCP and gas migration in CO2 injection well, which is a highly complex problem due to its own special characteristics such as acidic property and supercritical state of CO2. In addition, unlike gas producing well, fluid flows in the opposite direction in tubing of gas injection well. It is another distinctiveness of SCP in CO2 injection well. The work presented here, therefore, focuses on the mechanism and prediction analysis of SCP in ‘‘A’’ annulus of CO2 injection
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H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
tubing hanger tubing head
A annulus monitor B annulus monitor C annulus monitor
casing head C annulus
gas chamber
B annulus mud A annulus gas bubble
surface casing intermediate casing
cement gas
production casing production tubing formation Fig. 1. Annulus schematic drawing of gas producing well.
well, which may provide some reference for the following security evaluation and solution measures of SCP.
2. Field data analysis SCP is a universal problem in CO2 injection wells in Jilin Oilfield, which is the first Oilfield using CO2-EOR in China. The field data are ‘‘A’’ annulus casing pressure records provided by various operators from six injection wells in one block in Jilin Oilfield as shown in Fig. 2. According to Xu’s classification (Xu and Wojtanowicz, 2001; Xu, 2002), three typical response patterns could be concluded from the field data, which are incomplete casing pressure buildup pattern, S-shape casing pressure buildup pattern and stable casing pressure pattern. Incomplete casing pressure buildup pattern is noted when casing pressure increase at early time is relatively low and no pressure stabilization is apparent in the recording time, such as A well and B well. Comparatively, the rate of pressure rise in A well is faster than that in B well. There are two possible reasons. One is that A well is in a quick buildup period soon after pressure relief. The other is that there is a more serious leakage in A well. Pressure stabilization occurs at an early time due to stable gas channeling. If the diameters of leaks in tubing or casing expand or new leaks arise, the casing pressure would increase gradually and finally stabilize at a new level. It is the S-shape casing pressure buildup pattern, which takes place in C well and D well. The amplitude of leaks expansion or increment of leaks in D well is greater than that in C well. If the casing pressure has reached stabilization before recording, stable casing pressure pattern would be observed in the recording time, as seen in E well and F well. Disabled tubing
Fig. 2. Casing pressure records of six CO2 injection wells.
hanger of F well was replaced with a new one during the recording time, so the curve has two sections. However, the later pressure stabilization indicates that leaks still exist in the well after the replacement of tubing hanger.
3. Mechanism of SCP In CO2 injection well, tubing, casing, packer and even cement may be corroded by sour CO2 under appropriate humidity and
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
pressure. Then pitting, perforation or even rupture of tubing and casing (Crow et al., 2010), sealing failure of packer and cracks of cement occur after corrosion. All of these provide the annulus upward channel for CO2 (Vignes and Enoksen, 2010). For instance, blowouts of CO2 injection wells in America are mostly due to mechanical failure of tubing hangers, threaded adapters, valves and so on (Duncan et al., 2009; Skinner, 2003). In addition, CO2 is usually injected as liquid, giving rise to unavoidable phase transition in wellbore. Intense phase transition in wellbore would induce sealing failure of tubing connector. Especially, as shown in Fig. 3, supercritical CO2 develops with temperature higher than 31 1C and pressure more than 7.1 MPa. The density of supercritical CO2 is close to liquid, but the viscosity is like gas, and the diffusion coefficient is about 100 times higher than that of liquid. So supercritical CO2 has better dissolving capacity and mass transfer characteristics, eroding the rubber elastic seal components strongly, which is the main reason of sealing failure of packer. In Jilin Oilfield, the majority of CO2 injection wells have been reconstructed from old producing wells. The degradation of casing and packer aggravates the formation of gas channels. It is thus clear that there are many potential leak points and leakage paths in typical CO2 injection wells as shown in Fig. 4. Once CO2 sneaks into annulus from the leak points of tubing or packer, it will accumulate at wellhead after migrating through annular protective liquid or cement and mud column, leading to SCP. As shown in Fig. 4, there is no cement column in ‘‘A’’ annulus, and the annulus is always not filled with liquid, including a certain length of gas column at wellhead. Compared to tubing and casing, cement has a lower corrosion rate. So corrosion leaks are more likely to occur on tubing and casing. Even though there is channeling path in cement column, gas migration rate in interstices and cracks of cement is relatively slow. And due to the high viscosity of mud, gas rising velocity in mud is also relatively slow. So the buildup of casing pressure in ‘‘B’’ annulus is slow. In contrast, once there is corrosion leak in tubing, CO2 will sneak into ‘‘A’’ annulus rapidly. And gas rising velocity is fast in annular
protective liquid because of its low viscosity. Then wellhead pressure increases quickly followed by CO2 accumulation at wellhead. When the pressure rises to a relatively high value, tubing and casing in contact with the gas column at wellhead will be in a more harsh corrosion environment because of high CO2 partial pressure. If there is a leak in this location, CO2 will sneak into wellhead gas column directly, leading to more serious SCP. These are the main reasons why SCP in ‘‘A’’ annulus is always found first and more serious than that in ‘‘B’’ annulus. So we focus
CO2 gas column normal route of injected CO2 gas bubble
CO2 channeling from leak of packer
B annulus
CO2 channeling from leak of tubing
mud
Pressure (bar-a)
CO2 channeling from upper leak of tubing due to high CO2 partial pressure
A annulus
CO2 channeling from leak of tubing hanger
tubing leak
annular protective liquid
CO2 channeling from crack of cement
packer
packer failure formation
Fig. 4. Possible gas channeling path in typical CO2 injection well which has two annuli.
Liquid state Solid state
3
Supercritical state
solid-liquid equilibrium
73.6 bar-a critical point
gas-liquid equilibrium
5.2 bar-a triple point
Gas state
gas-solid equilibrium
-56.6 Gas state
31.0 Temperature ( ) Supercritical state
Liquid state Fig. 3. (a) CO2 P–T phase map and (b) supercritical CO2 phase transformation.
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H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
tube
A annulus
casing
cement
formation Z
tubing
corrosion leak
annular fluid
seal element hanger failure
Tci Tf
packer failure connector leak
valve failure
Prediction of SCP in ‘‘A’’ annulus of CO2 injection well is a highly complex problem because casing pressure relates to many factors such as size and position of leak point, leakage rate, height of annular protective liquid, length of gas column at wellhead, the state of CO2 and so on. Due to too much unknown conditions, we should simplify this problem and give some assumed conditions. In this paper, we have devised a calculation program. Firstly, a temperature and pressure calculation model of CO2 in tubing is formed. Secondly, based on pressure balance principle, a recorded stable value of casing pressure is needed to calculate the position of leak point. Then a coupled mathematical model of CO2 gas migration in annular protective liquid and accumulation at wellhead is developed. Finally, the prediction of buildup of SCP and the equivalent-size of leak point are obtained by iterative computations.
Tco
ΔZ
Tti
Fig. 5. The possible way leading to SCP in ‘‘A’’ annulus of CO2 injection well.
4. Prediction model of SCP
Te
Tto
rti rto
on the mechanism of SCP in ‘‘A’’ annulus. Based on the above analysis, we summarized the possible causes leading to SCP in ‘‘A’’ annulus as shown in Fig. 5.
Th
Z+ΔZ rci rco rh
Fig. 6. Schematic diagram of heat transfer in infinitesimal section of wellbore.
where transient heat transfer function (TD) can be described using the approximate formula recommended by Hasan and Kabir (1994): ( pffiffiffiffiffi pffiffiffiffiffi 1010 rt D r 1:5 1:1281 t D ð10:3 t D Þ, TD ¼ ð3Þ ð0:4063 þ 0:5 lnt D Þð1 þ0:6=t D Þ, t D 4 1:5 t D ¼ xt=r 2h
ð4Þ
and the formation temperature (Te) meets the increasing linear law: T e ¼ T head þg T z
ð5Þ
where W is the mass flux (kg/s), ke is formation thermal conductivity (W/(m 1C)), Th is the temperature outer cement ring (1C), x is the formation thermal diffusivity (m2/s), t is the injection length (s), rh is the outer radius of cement ring (m), Thead is the wellhead temperature (1C) and gT is the geothermal gradient (1C/m). Heat flux from cement to fluid in tubing can be described as dQ 2pr to U to ¼ ðT h T f Þ dz W
ð6Þ
where, the overall heat transfer coefficient (Uto) is 4.1. Calculation model of temperature distribution U to ¼
An infinitesimal length (dz) of a vertical CO2 injection well was taken for analysis, shown in Fig. 6. Usually, the injection of CO2 is steady even if a slight leak has occurred. So it is assumed that CO2 is flowing in one-dimensional steady-state, and temperature and pressure are equal in the same depth. Heat transfer is considered as one-dimensional radial one without vertical transfer and it is steady from tubing to cement, while non-steady from cement to formation. According to energy conservation equation (Hasan and Kabir, 1991a, 1991b), we get dT f dp 1 dH dp 1 dQ dvt þ ¼ CJ þ gvt ¼ CJ dz C pm dz dz C pm dz dz dz
ð1Þ
where p is pressure (Pa), z is the vertical position (m), g is gravitational acceleration (m/s2), vt is the flow velocity in tubing (m/s), Tf is the fluid temperature (1C), CJ is the Joule Thomson coefficient (J/(kg 1C)) and Q is the heat exchange capacity per unit mass (J/kg). Heat flux from formation to cement can be expressed as dQ 2pke ¼ ðT e T h Þ dz WT D
ð2Þ
r to rto lnðr to =rti Þ 1 r to lnðrco =rci Þ rto lnðr h =rco Þ þ þ þ þ kt hc þhr kcas kcem r ti hto
1
ð7Þ
where rto is the outer radius of tubing (m), rti is the inner radius of tubing (m), hto is the convective heat transfer coefficient of fluid in tubing (W/(m2 1C)), kt is the thermal conductivity of tubing (W/(m 1C)), hc is the convective heat transfer coefficient of fluid in annulus (W/(m2 1C)), hr is the radiation heat transfer coefficient of fluid in annulus (W/(m2 1C)), rci and rco are the inner and the outer radii of casing respectively (m), and kcas and kcem are the thermal conductivities of casing and cement respectively (W/(m 1C)). Th can be eliminated combining Eqs. (2) and (6) as follows: dQ 2p r to U to ke ¼ ð8Þ ðT e T f Þ dz W ke þ r to U to T D According to Eqs. (8) and (1), the temperature distribution of fluid in tubing can be obtained. dT f dp 2p r to U to ke g vt dvt þ ¼ CJ ð9Þ ðT e T f Þ dz WC pm ke þ r to U to T D C pm C pm dz dz Then the function of temperature vs. depth is dp g vt dvt g T T f ¼ T head þg T z þ Bð1ez=B Þ C J dz C pm C pm dz þez=B ðT f in T head g T zÞ
ð10Þ
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
where, WC pm ke þr to U to T D B¼ 2p r to U to ke
ð11Þ
4.2. Calculation model of pressure distribution and leak depth After packer failure or tubing failure at one point, CO2 will cross through the leak point into ‘‘A’’ annulus, and then rise through annular protective liquid to accumulate at wellhead. Annulus wellhead pressure increases with the accumulation of gas, and reaches a stable value after a period of time. Then the pressure inner and outer leak point achieves balance and channeling comes to a stop. So the prediction model of leak depth could be developed using U-tube pressure balance principle (Chen, 2005). Leak position of packer failure is very clear, but the location of tubing failure is difficult to determine. Sometimes tubing failure may produce more than one leak. But sustained casing pressure will be present as long as there is emergence of one leak. In order to facilitate the calculation, it is assumed that there is an equivalent-size leak. All escaping CO2 should cross through this leak into annulus. Thus, the whole process from CO2 injection to accumulation at annulus wellhead includes four steps, namely, flowing in tubing, channeling in leak, rising in liquid and accumulating at wellhead, seen from Fig. 7. According to the momentum conservation law (Caetano et al., 1992a, 1992b), the steady CO2 flow in tubing of straight well meets the following equation: dp 2f rvt 2 dvt ¼ rg rvt dz dt dz
ð12Þ
where the friction coefficient, f, is (Paterson et al., 2008) 2 e=dt 5:0452 lgL f ¼ 2 lg Re 3:7065
L¼
ðe=dt Þ1:1098 7:149 0:8981 Re 2:8257
ð13Þ
ð14Þ
CO2 gas chamber
5
where r is the fluid density (kg/m3), dt is the inner diameter of tubing (m), e is the tubing wall roughness (m) and Re is the Reynolds number. As the pressure loss caused by acceleration is much smaller than that caused by gravity or friction, it can be ignored. CO2 is often injected as low-temperature and high-pressure liquid. And the injection pressure is usually higher than the critical pressure. So CO2 will directly transform into supercritical fluid at a certain depth (hl). The pressure loss caused by gravity and friction is divided into two parts described in Eq. (15). 8 2f rl,co vl 2 > 2 < dp ¼ rl,co2 g , z r hl dt dz ð15Þ 2 2f r v sc > : dp ¼ rsc,co g sc,co2 , z 4 hl dt dz 2 where hl is the depth of CO2 transforming from liquid into supercritical state (m), rl,co2 is the density of liquid CO2 (kg/m3), rsc,co2 is the density of supercritical CO2 (kg/m3), vl is the velocity of liquid CO2 in tubing (m/s) and vsc is the velocity of supercritical CO2 in tubing (m/s). Integrating Eq. (15), we get pto ¼ pin þ rl,CO2 ghl þ rsc,CO2 ghsc
2f hl rl,CO2 v2l 2f hsc rsc,CO2 v2sc dt dt ð16Þ
where pto is upstream pressure of leakage (Pa), pin is the injection pressure in wellhead (Pa) and hsc is the length of supercritical CO2 above the leak point (m). Channeling of CO2 in leak point includes two processes, sudden contraction and sudden expansion. The resulting pressure loss includes the two local losses as follows: A v2 A v2 ð17Þ Dpo ¼ pto pco ¼ 0:5 1 2 o þ 1 2 o A1 2g A3 2g where Dpo is the pressure loss of CO2 crossing through leak (Pa), pco is the downstream pressure of leak (Pa), vo is the velocity in leak (m/s), A1 is the cross-sectional area of flow in tubing (m2), A2 is the cross-sectional area of flow in leak (m2) and A3 is the cross-sectional area of flow in annulus (m2). Since the cross-sectional area of flow in leak is smaller than that in tubing or annulus, Eq. (17) can be described as
Dpo ¼ 1:5
v2o 2g
ð18Þ
CO2 gas migration in annular protective liquid can be modeled as gas–liquid two-phase flow (Hasan and Kabir, 1992), meeting the following equation:
liquid CO2 ht hl annular liquid
@ðrm vm Þ @ðrm vm 2 Þ @p f þ þ þ rm g þ r v2 ¼ 0 @t @z @z 2dh m m
ð19Þ
where
hal supercritical CO2
dh ¼ do di
ð20Þ
rm ¼ rg a þ rl ð1aÞ
ð21Þ
tube casing
hsc leak
axis Fig. 7. The whole path of CO2 migration and computational domain.
in which rm is the mixture density (kg/m3), rg is the gas density (kg/m3), rl is the liquid density (kg/m3), vm is the mixture velocity (m/s), dh is the hydraulic diameter of annulus (m), di is the outer diameter of tubing (m), do is the inner diameter of casing (m). After casing pressure reaches a stable value, the pressure inner and outer leak point will achieve a balance. Then CO2 channeling reduces greatly, namely vo E0 and Wpo E0 and bubbles in liquid are few. Annular protective liquid can be taken as stationary. So Eq. (19) is integrated as pco ¼ pwhs þ rl ghal
ð22Þ
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H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
where pwhs is the annulus wellhead pressure (Pa) and hal is the length of annular protective liquid above the leak point (m). According to U-tube pressure balance principle and combining Eqs. (16) and (22), we can obtain pin þ rl,CO2 ghl þ rsc,CO2 ghsc
2f l hl rl,CO2 v2l 2f sc hsc rsc,CO2 v2sc dt dt
¼ pwhs þ rl ghal
@ðarg Þ ð23Þ
In Eq. (23), injection pressure and rate are known operation parameters. Stable wellhead pressure can be measured by pressure gage. And liquid surface position can be measured by the acoustic detectors. Thus, the key to calculating leak depth is to determine the interface position of liquid and supercritical CO2, which can be determined by coupling calculation of temperature and pressure. Because density, viscosity and specific heat of CO2 are changing with temperature and pressure, their expressions thus should be added to achieve coupling calculation of temperature and pressure. CO2 density can be calculated using the high precision PR equation (Tong, 1996): p¼
ð24Þ
þ
@ðarg vg Þ @z
¼0
@½ð1aÞrl @½ð1aÞrl vl þ ¼0 @t @z
2
a ¼ 0:457235
ðRT C Þ 2 ½1 þ ð0:37464 þ1:54226$0:26992$2 Þð1T 0:5 r Þ pc
ð25Þ b ¼ 0:077796
RT c pc
ð26Þ
in which Tc is the critical temperature (1C), pc is the critical pressure (Pa), o is the eccentric factor and Tr is the reduced temperature. Using the viscosity model based on the PR state equation (Guo et al., 1999), viscosity can be expressed as 0
0
2
T m3 þð2bTb Tr 0 pÞm2 ð2bb T þTb þ 2r 0 bpaÞm 0 2
2
ð32Þ
ð33Þ
where vg ¼ vs þC 0 vm
ð34Þ
in which C0 is the distribution factor, vs is the gas slip velocity (m/s) and a is the gas void fraction. For bubble flow, according to the equilibrium of buoyancy force and flow drag (Caetano et al., 1992b; Harmathy, 1960), slip velocity can be calculated as " #1=4 ðrl rg Þg s vs ¼ 1:53 ð1aÞ1=2 ð35Þ 2 where
s ¼ 0:07275½10:002ðT18Þ
where
0
þðTb b þ r 0 pb ab Þ ¼ 0
The value of distribution factor C0 is related to the distribution density and velocity of gas bubbles, which is equal to 2.0 for bubble laminar flow. And for bubble turbulent flow or slug flow, the value is equal to 1.2 (Hasan and Kabir, 1988). Fanning friction factor is related to flow regime and flow channel geometry (Caetano et al., 1992b; Xu, 2002). For laminar flow in concentric annulus, Fanning friction factor is f¼
pffiffiffiffi pffiffiffiffi 2 0 b ¼ be½Q 2 ð T r 1Þ þ Q 3 ð pr 1Þ
mc T c pffiffiffiffiffiffiffiffiffi 2 pc Z c 1 þ Q 1 pr T r 1
ð28Þ ð29Þ
T 1=6 c
ð30Þ
ð31Þ
where R is the gas constant, 8.314 J/(mol 1C). 4.3. Coupled mathematical model of CO2 gas migration According to leakage rate, bubble flow (gas void fraction value less than 0.2) or slug flow (gas void fraction value more than 0.2)
ð38Þ
where K¼
di do
ð39Þ
rm vm dh mm
mm ¼ mg a þ ml ð1aÞ
in which m is the kinetic viscosity of fluid (Pa s), pr is the reduced pressure and Zc is the critical compressibility factor. The values of Q1, Q2 and Q3 relate with the eccentric factor. For CO2, the eccentric factor is 0.225. So the corresponding Q1, Q2 and Q3 are 0.871, 1.368 and 0.460. Heat capacity can be calculated using the continuous model based on the PR state (Wu et al., 2009): C pm ¼ 4:728 þ 0:01754T2:338 105 T 2 þ4:079 109 T 3 0:6766p þ ZRR þ ZRT 2 þ 29:7903 106 pT
16 ð1KÞ2 Rem ½ðð1K 4 Þ=ð1K 2 ÞÞðð1K 2 Þ=ðlnð1=KÞÞÞ
Rem ¼
2=3
51:076pc
ð36Þ
in which s is the surface tension of liquid (N/m). For slug flow, the Taylor bubble rise velocity in vertical annulus is given by (Hasan and Kabir, 1992) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gdo ðrl rg Þ 0:1di vs ¼ 0:345 þ ð37Þ do rl
ð27Þ
where
mc ¼
@t
rl
rRT ar2 1rb 1þ 2rbr2 b2
r0 ¼
may occur (Caetano et al., 1992a) after CO2 sneaking into annular protective liquid. Besides momentum equation (Eq. (19)), continuity equations for the continuous (annular protective liquid) and the dispersed phase (CO2) constitute controlling equations of two-phase flow (Xu, 2002), which are
ð40Þ ð41Þ
in which K is the consistency coefficient, ml is the liquid viscosity (Pa s), mg is the gas viscosity (Pa s), mm is the mixture viscosity (Pa s) and Rem is the Reynolds number of two-phase flow. For turbulent flow in concentric annulus, Fanning friction factor is given by (Caetano et al., 1992b; Xu, 2002) 1 n o1=2 6 2 2 f ½ð1K þð1 þK ÞlnKÞ=ðð1K 2 ÞlnKÞ0:45 exp½ðRem 3000Þ=10 8 " #0:225 exp½ðRem 3000Þ=106 9 < = 2 2 1=2 1K þð1 þK ÞlnK ¼ 4 lg Rem f 0:4 2 : ; ð1K ÞlnK ð42Þ As gas is accumulating at the wellhead, casing pressure and gas volume are constantly changing with time. Assuming gas column at wellhead only receives gas below the liquid surface and has no other channel to escape, then after Dt, the gas volume at
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
wellhead can be expressed as V 0wh ¼ V wh þ V R
ð43Þ
where VR ¼
p 4
2
2
ðdo di Þvg Dt
ð44Þ 3
in which Vwh is the volume of the gas column (m ) and VR is the volume of the gas received below the liquid surface (m3). After Dt, the mole number of CO2 in gas column is n0 ¼ n þ Dn
ð45Þ
where,
Dn ¼
prg vg Dt 4Mg
2
2
ðdo di Þ
ð46Þ
in which Mg is the mole mass of gas (kg/mol). For temperature at wellhead essentially unchanged, casing pressure after Dt can be described according to the state equation (Sterner and Pitzer, 1994): pwh f ðrwh Þ f ðrwh Þ 0
p0wh ¼
ð47Þ
where f ðrwh Þ ¼ rwh þ a1 rwh 2 rwh 2
a3 þ 2a4 rwh þ 3a5 rwh 2 þ 4a6 rwh 3 ða2 þ a3 rwh þ a4 rwh 2 þ a5 rwh 3 þ a6 rwh 4 Þ2
þa7 rwh 2 ea8 rwh þa9 rwh 2 ea10 rwh
rwh ¼
!
ð48Þ ð49Þ
ð52Þ
1 þ1 pni1=2 ðrm vm 2 Þniþ 1=2 ðrm vm 2 Þni1=2 pni þþ1=2 ðrm vm Þni þ 1 ðrm vm Þni þ þ Dt Dz Dz n f þ ðrm gÞn þ r v2 ¼ 0 ð53Þ 2dh m m
In the computational cells shown in Fig. 8, Eqs. (43), (45) and (47) can be given by þ1 V jwh ¼ V jwh þ
nj þ 1 ¼ nj þ
þ1 ¼ pjwh
p 4
2
2
ðdo di Þðvg Þjn þ 1=2 Dt
ðrg vg Þjn þ 1=2 Dt p 4M g
2
ð54Þ
2
ðdo di Þ
ð55Þ
þ1 pjwh f ðrjwh Þ
ð56Þ
f ðrjwh Þ
In the progress of CO2 migration in annular protective liquid, we also consider the solubility of CO2 in water. So the volume fraction of CO2 is set as (Weiss, 1974) ! K o M g rl pg ai þ 1 ¼ ai ð57Þ
rg
i
þ ci,5 ðT þ273:15Þ þ ci,6 ðT þ 273:15Þ2
ar
Dt
þ
n nþ1 g Þi þ 1=2 ðvg Þi þ 1=2 ð
ðar
n nþ1 g Þi1=2 ðvg Þi1=2
ar
Dz
in which Ko is the solubility of CO2 (mol/(kg Pa)) and pg is the CO2 partial pressure (Pa). At initial time, annulus has no CO2 gas, and the casing pressure at wellhead is atmospheric pressure. Thus, the initial and the boundary conditions used in calculation are as follows: p0wh ¼ pa
¼0
ð59Þ
i1 p0i1=2 ¼ p0wh þ rl ghal 1 n
1 ri rn
ð60Þ
a0i ¼ 0 1r i rn
ð61Þ
ðrg Þ0i ¼ 0
ð62Þ
1 ri r n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u j
ðqg Þj1=2 ð51Þ
pd2c u t2ðpc p1=2 Þ ¼ 4 rgc
ð63Þ
where pc is the leak point pressure (Pa).
Table 1 Coefficients ci,j of equation of state for CO2. i 1 2 3 4 5 6 7 8 9 10
ci,1 – – – – – – 0.3934 1012 – – –
ci,2 – – – – – – 0.9092 108 – 0.2300 108 –
2
ð58Þ
ð50Þ
in which rwh is the density of CO2 at wellhead (kg/m3) and values of coefficients ci,j are given in Table 1. Flowing parameters of CO2 gas channeling are changing with time and space constantly. So we choose iterative method to predict the variation of SCP using computational mesh shown in Fig. 8. Computational domain within tubing is divided into n grids. And to account for gas migration in annulus, we also discretize the annular protective liquid into n cells, leaving gas column at wellhead as a separate grid, denoted by nþ1 (Xu, 2002). A staggered grid assignment method is used in calculation. Pressure and velocity are stored at cell surfaces and scalars, such as density, temperature and volume fraction, are stored at the center of the cell. Using semi-implicit central difference method (Xu, 2002), Eqs. (32), (33) and (19) are discreted as n g Þi
Ko ¼ e60:217 þ 93:452½100=ðT þ 273:15Þ þ 23:359ln½ðT þ 273:15Þ=1000:024½ðT þ 273:15Þ=100 þ 0:005½ðT þ 273:15Þ=100
ai ¼ ci,1 ðT þ 273:15Þ4 þci,2 ðT þ 273:15Þ2 þci,3 ðT þ 273:15Þ1 þ ci,4
ðar
½ð1aÞrl ni þ 1 ½ð1aÞrl ni Dt 1 þ1 ½ð1aÞrl niþ 1=2 ðvl Þni þþ1=2 ½ð1aÞrl ni1=2 ðvl Þni1=2 ¼0 þ Dz
where
nM g V wh
nþ1 ð g Þi
7
ci,3
ci,4 7
0.1826 10 – – 1.3270 0.1246 – 0.4278 106 0.4028 103 0.7897 105 0.9503 105
2
0.7922 10 0.6656 10 4 0.5996 10 2 0.1521 4.9045 0.7552 0.2235 102 0.1197 103 0.6338 102 0.1804 102
ci,5
ci,6
– 0.5715 10 5 0.7170 10 4 0.5365 10 3 0.9822 10 2 – – – – –
– 0.3022 10 9 0.6242 10 8 0.7112 10 7 0.5596 10 5 – – – – –
8
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
Define recorded stable casing pressure pwhs, length of gas chamber ht, injection pressure pin, injection mass flux Win and injection temperature Tin
CO2 liquid CO2 hl
1
Assume the leak depth h*
2
Calculate hal and Pco
.
i=0
.
Pi=Pin, Ti=Tin, hi=0
.
Pi>7.1MPa and Ti>31ºC
i-1 supercritical CO2
i=i+1
i+1 . . .
hsc
Guess Pi* Calculate Ti Calculate Pi
n-1
|Pi-Pi*|<ε Y hi=hi+Δ z
n
leak point
hl=hi
N Calculate Cpmi, ρi and μi
i gridding
Y
Pi*=(Pi+Pi*)/2 N
N
hi>h* Y N |Pi-Pco|<ε Y Calculate the leak point pressure pc
z
Z
Assume the diameter of leak point dc
gr idding h al
Z=0
Pc
t=0
n+1 gas
ht
n n-1 . . . i+1 i i-1 . . . 2 1 0
Define the initial and boundary conditions
i+1
t=t+Δt
i+1/2
P i+1/2 vgi+1/2
Guess pwh*
out
i=1
i
ρi T i αi
Calculate pi+½* Calculate the average pressure pi, and calculate the corresponding density ρi
in P i-1/2 vgi-1/2
i-1/2
Calculate αi, (vg)i using Eq.52, Eq.53 and Eq.54
i-1
i=i+1 i≥n Y Calculate pwh
Fig. 8. Computational grid of gas channeling. (a) Computational grid in tubing and (b) computational grid in annulus.
pwh*=0.5(pwh*+pwh)
In iterative process, firstly assume that the pressure meets the static pressure distribution: i1 1 ri r n ð64Þ pji1=2 ¼ pjwh þ rl ghal 1 n Substituting Eq. (51) into (64) gives the coupling model of CO2 gas channeling in ‘‘A’’ annulus. The calculation accuracy depends on the meshing elements and the size of the interval values. The whole coupling calculation process is shown in Fig. 9.
5. Illustration One CO2 injection well in Jilin Oilfield is selected for analysis. The detailed information about this well is listed in Table 2. Significant casing pressure is present after this well has been put into production in less than one year. The temperature and stable casing pressure at wellhead are measured before pressure relief. Then liquid surface position is measured by acoustic detector and relief valve is closed after pressure relief. Based on coupling
N
N
|pwh-pwh*|<ε Y Caculate p½ |p½-pc|<ε Y |pwh-pwhs|<ε Y Stop
N N
Fig. 9. Flow chart of coupling calculation.
calculation process mentioned above, ‘‘A’’ annulus casing pressure build-up curve (the casing pressure variation with time) of this well was obtained, as shown in Fig. 10. It is observed from Fig. 10 that the calculated value basically coincided with the measured value. The pressure build-up curve shows an ‘‘S’’ shape, which can be divided into three parts: the initial smooth section, a sharp rise section and the later stable section. After closing the relief valve, a certain time of about 2 h is required for CO2 gas migration to reach the fluid surface. Then the gas reaches the wellhead and accumulates at the wellhead,
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
9
Table 2 Detailed information of the illustration well. Wellbore configuration
Category
Value
Diameter of tubing Material of tubing Diameter of casing Material of casing Formation pressure Pressure coefficient Formation temperature Geothermal gradient Injection pressure Injection mass flux Injection temperature Temperature measured at wellhead Recorded stable casing pressure Liquid surface apart from wellhead
F73 5.51 mm N80 F139.7 6.2 mm J55 24.3 MPa 0.99 98.9 1C 3.97 1C/100 m 16.4 MPa 0.463 kg/s 6 1C 25 1C 14.5 MPa 120 m
migration in annular protective liquid and accumulation at wellhead. Finally, by continuous assumption, the buildup of SCP and the equivalent-size of leak point are obtained until the calculated stable casing pressure is approximately equal to the recorded stable casing pressure. (3) Calculation result of illustration fits well with the actual monitoring data, verifying the reliability of coupled model and calculation method.
Acknowledgment Research work was co-financed by China National Natural Science foundation (No. 51074135) and Key Project of Sichuan Provincial Education Department (No. 12ZA189). Without their support, this work would not have been possible.
Fig. 10. ‘‘A’’ annulus casing pressure build-up curve of illustration well.
leading to the rise of casing pressure. Due to the large initial pressure difference, the casing pressure recovers rapidly, corresponding to 2–640 h shown in Fig. 10. Finally, the casing pressure remains stable after achieving 14.5 MPa.
6. Conclusions
(1) Special and complex characteristic of SCP in ‘‘A’’ annulus of CO2 injection well has been proposed in this paper, such as liquid in annulus, flowing direction, supercritical state of CO2, solubility of CO2 and so on. And the possible causes leading to SCP are summarized. (2) A prediction model of buildup of casing pressure has been proposed in this paper. Firstly, temperature and pressure distributions of CO2 in tubing are obtained by the calculation model. Secondly, based on pressure balance principle, the position of leak point is calculated with a recorded stable value of casing pressure. Then assuming the equivalentdiameter of leak point, casing pressure variation with time is calculated by a coupled mathematical model of CO2 gas
References Anders, J., Rossberg, S., Dube, H., Engel, H., Andrews D., 2006. Well integrity operations at Prudhoe Bay, Alaska. In: Proceedings of the SPE Annual Technical Conference and Exhibition, Paper SPE 102524, San Antonio, Texas, September 24–27, 2007. Bourgoyne Jr., A.T., Scott, S.L., Regg, J.B., 1999. Sustained casing pressure in offshore producing wells. In: Proceedings of the Offshore Technology Conference, Paper OTC 11029, Houston, May 3–6, 1999. Caetano, E.F., Shoham, O., Brill, J.P., 1992a. Upward vertical two-phase flow through an annulus, part I: single-phase friction factor, Taylor bubble rise velocity and flow pattern prediction. J. Energy Resour. Technol. 114 (1), 1–13. Caetano, E.F., Shoham, O., Brill, J.P., 1992b. Upward vertical two-phase flow through an annulus, part II: modeling bubble, slug, and annular flow. J. Energy Resour. Technol. 114 (1), 14–30. Chen, P., 2005. Drilling and Completion Engineering. Publishing House of Oil Industry, China, Beijing. Crow, W., Carey, J.W., Gasda, S., Williams, D.B., Celia, M., 2010. Wellbore integrity analysis of a natural CO2 producer. Int. J. Greenhouse Gas Control 4 (2), 186–197. Duncan, I.J., Nicot, J.P., Choi, J.W., 2009. Risk assessment for future CO2 sequestration projects based CO2 enhanced oil recovery in the US. Energy Procedia 2 (1), 2037–2042. Guo, X.Q., Rong, S.X., Yang, J.T., Guo, T.M., 1999. The viscosity model based on PR equation of state. Acta Petrolei Sin. 20 (3), 56–61. Hargrove, B., 2004. Upcoming U.S. CO2 EOR projects. In: Proceedings of the 4th Annual EOR Carbon Management Workshop, the Woodlands Waterway Marriott Hotel and Conference Center, Houston, Texas, December 4–5, 2004. Harmathy, T.Z., 1960. Velocity of large drops and bubbles in media of infinite or restricted extent. AIChE J. 6 (2), 281–288. Hasan, A.R., Kabir, C.S., 1988. A study of multiphase flow behavior in vertical wells. SPE Prod. Eng. 3 (2), 263–272.
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Hasan, A.R., Kabir, C.S., 1991a. Heat transfer during two-phase flow in wellbores: part I: formation temperature. In: Proceedings of the the SPE Annual Technical Conference and Exhibition, Paper SPE 22866, Dallas, Texas, October 6–9, 1991. Hasan, A.R., Kabir, C.S., 1991b. Heat transfer during two-phase flow in wellbores: part II: wellbore fluid temperature. In: Proceedings of the SPE Annual Technical Conference and Exhibition, Paper SPE 22948, Dallas, Texas, October 6–9, 1991. Hasan, A.R., Kabir, C.S., 1992. Two-phase flow in vertical and inclined annuli. Int. J. Multiphase Flow 18 (2), 279–293. Hasan, A.R., Kabir, C.S., 1994. Aspects of wellbore heat transfer during two-phase flow. SPE Prod. Facilities 9 (3), 211–216. Haynes Jr., S., Alston, R.B., 1990. Study of the mechanisms of carbon dioxide flooding and applications to more efficient eor projects. In: Proceedings of the Enhanced Oil Recovery Symposium, Paper SPE 20190-MS, SPE/DOE, Tulsa, Oklahoma, October 22–25, 1990. Keeling, R.J., 1984. CO2 miscible flooding evaluation of the south Welch unit, Welch San Andres Field. In: Proceedings of the SPE Enhanced Oil Recovery Symposium, Paper SPE 12664-MS, Tulsa, Oklahoma, April 15–19, 1984. Paterson, L., Meng, L., Luke, D.C., Jonathan, E.K., 2008. Numerical modeling of pressure and temperature profiles including phase transitions in carbon dioxide wells. In: Proceedings of the SPE Annual Technical Conference and Exhibition, Paper SPE 115946, Denver, Colorado, September 21–24, 2008. Skinner, L., 2003. CO2 blowouts an emerging problem. World Oil 224 (1) (pp. 38, 40–42).
Somei, N., 1992. Mechanisms of Gas Migration after Cement Placement and Control of Sustained Casing Pressure. Master Thesis. Louisiana State University. Sterner, S.M., Pitzer, K.S., 1994. An equation of state for carbon dioxide valid from zero to exterme pressures. Contrib. Mineral. Petrol. 117 (4), 362–374. Tanner, C.S., Baxley, P.T., 1992. Production performance of the Wasson Denver unit CO2 flood. In: Proceedings of the SPE Enhanced Oil Recovery Symposium, Paper SPE 24256-MS, Tulsa, Oklahoma, April 22–24, 1992. Tong, J.S., 1996. Thermophysical Properties of Fluid. Publishing House of Petrochemical Industry, China, Beijing. Vignes, B., Enoksen, A.M., 2010. CO2 injection and storage-qualification of well barriers in adjacent wells. In: Proceedings of the Offshore Technology Conference, Paper OTC 20724, Houston, Texas, May 3–6, 2010. Weiss, R., 1974. Carbon dioxide in water and seawater: the solubility of a nonideal gas. Mar. Chem. 2 (3), 203–215. Wu, X.D., Wang, Q., He, Y.F., 2009. Temperature-pressure field coupling calculation model considering phase behavior change in CO2 injection well borehole. J. China Univ. Pet. 33 (1), 73–77. Xu, R., Wojtanowicz, A.K., 2001. Diagnosis of sustained casing pressure from bleedoff/buildup testing patterns. In: Proceedings of the SPE Production and Operations Symposium, Paper SPE 67194, Tulsa, Oklaho, March 24–27, 2001. Xu, R., 2002. Analysis of Diagnostic Testing of Sustained Casing Pressure in Wells. Ph.D. Thesis. Louisiana State University.
Contents lists available at SciVerse ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Mechanism and prediction analysis of sustained casing pressure in ‘‘A’’ annulus of CO2 injection well Hongjun Zhu a,n, Yuanhua Lin a, Dezhi Zeng b, Deping Zhang c, Hao Chen b, Wei Wang b a
State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), Chengdu, Sichuan 610500, China CNPC Key Lab for Tubular Goods Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, China c Jilin Oil Field Company, PetroChina, Songyuan, Jilin 138000, China b
a r t i c l e i n f o
abstract
Article history: Received 22 July 2011 Accepted 2 June 2012 Available online 15 June 2012
High value of sustained casing pressure (SCP) threatens the wellbore essential safety and environmental protection seriously. Considering the special and complex characteristic of SCP in ‘‘A’’ annulus of CO2 injection well, a prediction model of buildup of casing pressure has been proposed in this paper. Then finite difference method was used to solve the discrete equations in order to predict the SCP variation with time. The value is closely related to size and position of leak point, leakage rate, height of annular protective liquid, length of gas column at wellhead and the state of CO2. The buildup of SCP in one well in Jilin Oilfield is obtained by this model, which is in good agreement with the actual result, verifying the reliability of coupled model and calculation method. & 2012 Elsevier B.V. All rights reserved.
Keywords: sustained casing pressure CO2 injection well gas migration gas–liquid two-phase flow supercritical CO2
1. Introduction After well completion, measuring pressure in all of the casing strings should be zero if the well is flowing at steady state conditions, and a small volume of fluid caused by thermal expansion effects has to be bled through a needle valve in order to decrease the casing pressure to atmospheric pressure (Bourgoyne et al., 1999). If the casing pressure builds up when the needle valve is closed, the casing generally exhibits sustained casing pressure (SCP). High casing pressure may induce blowout, explosion and other serious accidents, reducing the productive life of oil and gas wells and resulting in huge economic losses. SCP in gas producing wells is very common, such as 11,498 casing strings in 8122 wells in the Gulf of Mexico (Bourgoyne et al., 1999). Thus, gas migration in gas producing well has been studied by some researchers (Somei, 1992; Xu and Wojtanowicz, 2001; Xu, 2002). Among them, Xu’s work is the most detailed one. In Xu’s (2002) dissertation, she had built a mathematical model of casing pressure in ‘‘B’’ or ‘‘C’’ annulus wellhead of gas producing well and had given the numerical solution. However, a well usually has several annuli. According to the location, the annulus from inside to outside can be named ‘‘A’’ annulus, ‘‘B’’ annulus, ‘‘C’’ annulus, and so forth (Anders et al., 2006). As shown in Fig. 1, ‘‘A’’ annulus is the annulus between the tube and production casing. ‘‘B’’ annulus is the annulus between production casing and adjacent intermediate
n
Corresponding author. Tel.: þ86 28 83032206. E-mail address: [email protected] (H. Zhu).
0920-4105/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2012.06.013
casing. The rest can be obtained by analogy. The gas migration path in ‘‘A’’ annulus is not the same as that in ‘‘B’’ or ‘‘C’’ annulus. In ‘‘A’’ annulus, gas may flow through annular protective liquid and accumulate at the gas chamber in wellhead. While in ‘‘B’’ or ‘‘C’’ annulus, gas may sequentially flow through cement and mud column to reach the gas chamber. Moreover, annular protective liquid is water added by corrosion inhibitor, while mud is usually a non-Newtonian liquid–solid two-phase fluid. Therefore, Xu’s model cannot be directly applied in ‘‘A’’ annulus. Recently, with the spread of CO2 enhanced oil recovery (Haynes and Alston, 1990; Hargrove, 2004; Keeling, 1984; Tanner and Baxley, 1992), SCP in CO2 injection well has become a salient problem, which has been found in all oil fields using CO2-EOR in China. Especially, SCP in ‘‘A’’ annulus is the most serious challenge. In Jilin Oilfield, the maximum casing pressure in ‘‘A’’ annulus of CO2 injection well is up to 17.5 MPa. In some injection wells, obvious casing pressure appears just one week after operating, and the pressure builds up quickly and approaches the tubing pressure after pressure relief. Therefore, it is urgent to understand the mechanism of SCP in ‘‘A’’ annulus of CO2 injection well and predict the value of SCP. However, there are few reports about the mechanism of SCP and gas migration in CO2 injection well, which is a highly complex problem due to its own special characteristics such as acidic property and supercritical state of CO2. In addition, unlike gas producing well, fluid flows in the opposite direction in tubing of gas injection well. It is another distinctiveness of SCP in CO2 injection well. The work presented here, therefore, focuses on the mechanism and prediction analysis of SCP in ‘‘A’’ annulus of CO2 injection
2
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
tubing hanger tubing head
A annulus monitor B annulus monitor C annulus monitor
casing head C annulus
gas chamber
B annulus mud A annulus gas bubble
surface casing intermediate casing
cement gas
production casing production tubing formation Fig. 1. Annulus schematic drawing of gas producing well.
well, which may provide some reference for the following security evaluation and solution measures of SCP.
2. Field data analysis SCP is a universal problem in CO2 injection wells in Jilin Oilfield, which is the first Oilfield using CO2-EOR in China. The field data are ‘‘A’’ annulus casing pressure records provided by various operators from six injection wells in one block in Jilin Oilfield as shown in Fig. 2. According to Xu’s classification (Xu and Wojtanowicz, 2001; Xu, 2002), three typical response patterns could be concluded from the field data, which are incomplete casing pressure buildup pattern, S-shape casing pressure buildup pattern and stable casing pressure pattern. Incomplete casing pressure buildup pattern is noted when casing pressure increase at early time is relatively low and no pressure stabilization is apparent in the recording time, such as A well and B well. Comparatively, the rate of pressure rise in A well is faster than that in B well. There are two possible reasons. One is that A well is in a quick buildup period soon after pressure relief. The other is that there is a more serious leakage in A well. Pressure stabilization occurs at an early time due to stable gas channeling. If the diameters of leaks in tubing or casing expand or new leaks arise, the casing pressure would increase gradually and finally stabilize at a new level. It is the S-shape casing pressure buildup pattern, which takes place in C well and D well. The amplitude of leaks expansion or increment of leaks in D well is greater than that in C well. If the casing pressure has reached stabilization before recording, stable casing pressure pattern would be observed in the recording time, as seen in E well and F well. Disabled tubing
Fig. 2. Casing pressure records of six CO2 injection wells.
hanger of F well was replaced with a new one during the recording time, so the curve has two sections. However, the later pressure stabilization indicates that leaks still exist in the well after the replacement of tubing hanger.
3. Mechanism of SCP In CO2 injection well, tubing, casing, packer and even cement may be corroded by sour CO2 under appropriate humidity and
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
pressure. Then pitting, perforation or even rupture of tubing and casing (Crow et al., 2010), sealing failure of packer and cracks of cement occur after corrosion. All of these provide the annulus upward channel for CO2 (Vignes and Enoksen, 2010). For instance, blowouts of CO2 injection wells in America are mostly due to mechanical failure of tubing hangers, threaded adapters, valves and so on (Duncan et al., 2009; Skinner, 2003). In addition, CO2 is usually injected as liquid, giving rise to unavoidable phase transition in wellbore. Intense phase transition in wellbore would induce sealing failure of tubing connector. Especially, as shown in Fig. 3, supercritical CO2 develops with temperature higher than 31 1C and pressure more than 7.1 MPa. The density of supercritical CO2 is close to liquid, but the viscosity is like gas, and the diffusion coefficient is about 100 times higher than that of liquid. So supercritical CO2 has better dissolving capacity and mass transfer characteristics, eroding the rubber elastic seal components strongly, which is the main reason of sealing failure of packer. In Jilin Oilfield, the majority of CO2 injection wells have been reconstructed from old producing wells. The degradation of casing and packer aggravates the formation of gas channels. It is thus clear that there are many potential leak points and leakage paths in typical CO2 injection wells as shown in Fig. 4. Once CO2 sneaks into annulus from the leak points of tubing or packer, it will accumulate at wellhead after migrating through annular protective liquid or cement and mud column, leading to SCP. As shown in Fig. 4, there is no cement column in ‘‘A’’ annulus, and the annulus is always not filled with liquid, including a certain length of gas column at wellhead. Compared to tubing and casing, cement has a lower corrosion rate. So corrosion leaks are more likely to occur on tubing and casing. Even though there is channeling path in cement column, gas migration rate in interstices and cracks of cement is relatively slow. And due to the high viscosity of mud, gas rising velocity in mud is also relatively slow. So the buildup of casing pressure in ‘‘B’’ annulus is slow. In contrast, once there is corrosion leak in tubing, CO2 will sneak into ‘‘A’’ annulus rapidly. And gas rising velocity is fast in annular
protective liquid because of its low viscosity. Then wellhead pressure increases quickly followed by CO2 accumulation at wellhead. When the pressure rises to a relatively high value, tubing and casing in contact with the gas column at wellhead will be in a more harsh corrosion environment because of high CO2 partial pressure. If there is a leak in this location, CO2 will sneak into wellhead gas column directly, leading to more serious SCP. These are the main reasons why SCP in ‘‘A’’ annulus is always found first and more serious than that in ‘‘B’’ annulus. So we focus
CO2 gas column normal route of injected CO2 gas bubble
CO2 channeling from leak of packer
B annulus
CO2 channeling from leak of tubing
mud
Pressure (bar-a)
CO2 channeling from upper leak of tubing due to high CO2 partial pressure
A annulus
CO2 channeling from leak of tubing hanger
tubing leak
annular protective liquid
CO2 channeling from crack of cement
packer
packer failure formation
Fig. 4. Possible gas channeling path in typical CO2 injection well which has two annuli.
Liquid state Solid state
3
Supercritical state
solid-liquid equilibrium
73.6 bar-a critical point
gas-liquid equilibrium
5.2 bar-a triple point
Gas state
gas-solid equilibrium
-56.6 Gas state
31.0 Temperature ( ) Supercritical state
Liquid state Fig. 3. (a) CO2 P–T phase map and (b) supercritical CO2 phase transformation.
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H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
tube
A annulus
casing
cement
formation Z
tubing
corrosion leak
annular fluid
seal element hanger failure
Tci Tf
packer failure connector leak
valve failure
Prediction of SCP in ‘‘A’’ annulus of CO2 injection well is a highly complex problem because casing pressure relates to many factors such as size and position of leak point, leakage rate, height of annular protective liquid, length of gas column at wellhead, the state of CO2 and so on. Due to too much unknown conditions, we should simplify this problem and give some assumed conditions. In this paper, we have devised a calculation program. Firstly, a temperature and pressure calculation model of CO2 in tubing is formed. Secondly, based on pressure balance principle, a recorded stable value of casing pressure is needed to calculate the position of leak point. Then a coupled mathematical model of CO2 gas migration in annular protective liquid and accumulation at wellhead is developed. Finally, the prediction of buildup of SCP and the equivalent-size of leak point are obtained by iterative computations.
Tco
ΔZ
Tti
Fig. 5. The possible way leading to SCP in ‘‘A’’ annulus of CO2 injection well.
4. Prediction model of SCP
Te
Tto
rti rto
on the mechanism of SCP in ‘‘A’’ annulus. Based on the above analysis, we summarized the possible causes leading to SCP in ‘‘A’’ annulus as shown in Fig. 5.
Th
Z+ΔZ rci rco rh
Fig. 6. Schematic diagram of heat transfer in infinitesimal section of wellbore.
where transient heat transfer function (TD) can be described using the approximate formula recommended by Hasan and Kabir (1994): ( pffiffiffiffiffi pffiffiffiffiffi 1010 rt D r 1:5 1:1281 t D ð10:3 t D Þ, TD ¼ ð3Þ ð0:4063 þ 0:5 lnt D Þð1 þ0:6=t D Þ, t D 4 1:5 t D ¼ xt=r 2h
ð4Þ
and the formation temperature (Te) meets the increasing linear law: T e ¼ T head þg T z
ð5Þ
where W is the mass flux (kg/s), ke is formation thermal conductivity (W/(m 1C)), Th is the temperature outer cement ring (1C), x is the formation thermal diffusivity (m2/s), t is the injection length (s), rh is the outer radius of cement ring (m), Thead is the wellhead temperature (1C) and gT is the geothermal gradient (1C/m). Heat flux from cement to fluid in tubing can be described as dQ 2pr to U to ¼ ðT h T f Þ dz W
ð6Þ
where, the overall heat transfer coefficient (Uto) is 4.1. Calculation model of temperature distribution U to ¼
An infinitesimal length (dz) of a vertical CO2 injection well was taken for analysis, shown in Fig. 6. Usually, the injection of CO2 is steady even if a slight leak has occurred. So it is assumed that CO2 is flowing in one-dimensional steady-state, and temperature and pressure are equal in the same depth. Heat transfer is considered as one-dimensional radial one without vertical transfer and it is steady from tubing to cement, while non-steady from cement to formation. According to energy conservation equation (Hasan and Kabir, 1991a, 1991b), we get dT f dp 1 dH dp 1 dQ dvt þ ¼ CJ þ gvt ¼ CJ dz C pm dz dz C pm dz dz dz
ð1Þ
where p is pressure (Pa), z is the vertical position (m), g is gravitational acceleration (m/s2), vt is the flow velocity in tubing (m/s), Tf is the fluid temperature (1C), CJ is the Joule Thomson coefficient (J/(kg 1C)) and Q is the heat exchange capacity per unit mass (J/kg). Heat flux from formation to cement can be expressed as dQ 2pke ¼ ðT e T h Þ dz WT D
ð2Þ
r to rto lnðr to =rti Þ 1 r to lnðrco =rci Þ rto lnðr h =rco Þ þ þ þ þ kt hc þhr kcas kcem r ti hto
1
ð7Þ
where rto is the outer radius of tubing (m), rti is the inner radius of tubing (m), hto is the convective heat transfer coefficient of fluid in tubing (W/(m2 1C)), kt is the thermal conductivity of tubing (W/(m 1C)), hc is the convective heat transfer coefficient of fluid in annulus (W/(m2 1C)), hr is the radiation heat transfer coefficient of fluid in annulus (W/(m2 1C)), rci and rco are the inner and the outer radii of casing respectively (m), and kcas and kcem are the thermal conductivities of casing and cement respectively (W/(m 1C)). Th can be eliminated combining Eqs. (2) and (6) as follows: dQ 2p r to U to ke ¼ ð8Þ ðT e T f Þ dz W ke þ r to U to T D According to Eqs. (8) and (1), the temperature distribution of fluid in tubing can be obtained. dT f dp 2p r to U to ke g vt dvt þ ¼ CJ ð9Þ ðT e T f Þ dz WC pm ke þ r to U to T D C pm C pm dz dz Then the function of temperature vs. depth is dp g vt dvt g T T f ¼ T head þg T z þ Bð1ez=B Þ C J dz C pm C pm dz þez=B ðT f in T head g T zÞ
ð10Þ
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
where, WC pm ke þr to U to T D B¼ 2p r to U to ke
ð11Þ
4.2. Calculation model of pressure distribution and leak depth After packer failure or tubing failure at one point, CO2 will cross through the leak point into ‘‘A’’ annulus, and then rise through annular protective liquid to accumulate at wellhead. Annulus wellhead pressure increases with the accumulation of gas, and reaches a stable value after a period of time. Then the pressure inner and outer leak point achieves balance and channeling comes to a stop. So the prediction model of leak depth could be developed using U-tube pressure balance principle (Chen, 2005). Leak position of packer failure is very clear, but the location of tubing failure is difficult to determine. Sometimes tubing failure may produce more than one leak. But sustained casing pressure will be present as long as there is emergence of one leak. In order to facilitate the calculation, it is assumed that there is an equivalent-size leak. All escaping CO2 should cross through this leak into annulus. Thus, the whole process from CO2 injection to accumulation at annulus wellhead includes four steps, namely, flowing in tubing, channeling in leak, rising in liquid and accumulating at wellhead, seen from Fig. 7. According to the momentum conservation law (Caetano et al., 1992a, 1992b), the steady CO2 flow in tubing of straight well meets the following equation: dp 2f rvt 2 dvt ¼ rg rvt dz dt dz
ð12Þ
where the friction coefficient, f, is (Paterson et al., 2008) 2 e=dt 5:0452 lgL f ¼ 2 lg Re 3:7065
L¼
ðe=dt Þ1:1098 7:149 0:8981 Re 2:8257
ð13Þ
ð14Þ
CO2 gas chamber
5
where r is the fluid density (kg/m3), dt is the inner diameter of tubing (m), e is the tubing wall roughness (m) and Re is the Reynolds number. As the pressure loss caused by acceleration is much smaller than that caused by gravity or friction, it can be ignored. CO2 is often injected as low-temperature and high-pressure liquid. And the injection pressure is usually higher than the critical pressure. So CO2 will directly transform into supercritical fluid at a certain depth (hl). The pressure loss caused by gravity and friction is divided into two parts described in Eq. (15). 8 2f rl,co vl 2 > 2 < dp ¼ rl,co2 g , z r hl dt dz ð15Þ 2 2f r v sc > : dp ¼ rsc,co g sc,co2 , z 4 hl dt dz 2 where hl is the depth of CO2 transforming from liquid into supercritical state (m), rl,co2 is the density of liquid CO2 (kg/m3), rsc,co2 is the density of supercritical CO2 (kg/m3), vl is the velocity of liquid CO2 in tubing (m/s) and vsc is the velocity of supercritical CO2 in tubing (m/s). Integrating Eq. (15), we get pto ¼ pin þ rl,CO2 ghl þ rsc,CO2 ghsc
2f hl rl,CO2 v2l 2f hsc rsc,CO2 v2sc dt dt ð16Þ
where pto is upstream pressure of leakage (Pa), pin is the injection pressure in wellhead (Pa) and hsc is the length of supercritical CO2 above the leak point (m). Channeling of CO2 in leak point includes two processes, sudden contraction and sudden expansion. The resulting pressure loss includes the two local losses as follows: A v2 A v2 ð17Þ Dpo ¼ pto pco ¼ 0:5 1 2 o þ 1 2 o A1 2g A3 2g where Dpo is the pressure loss of CO2 crossing through leak (Pa), pco is the downstream pressure of leak (Pa), vo is the velocity in leak (m/s), A1 is the cross-sectional area of flow in tubing (m2), A2 is the cross-sectional area of flow in leak (m2) and A3 is the cross-sectional area of flow in annulus (m2). Since the cross-sectional area of flow in leak is smaller than that in tubing or annulus, Eq. (17) can be described as
Dpo ¼ 1:5
v2o 2g
ð18Þ
CO2 gas migration in annular protective liquid can be modeled as gas–liquid two-phase flow (Hasan and Kabir, 1992), meeting the following equation:
liquid CO2 ht hl annular liquid
@ðrm vm Þ @ðrm vm 2 Þ @p f þ þ þ rm g þ r v2 ¼ 0 @t @z @z 2dh m m
ð19Þ
where
hal supercritical CO2
dh ¼ do di
ð20Þ
rm ¼ rg a þ rl ð1aÞ
ð21Þ
tube casing
hsc leak
axis Fig. 7. The whole path of CO2 migration and computational domain.
in which rm is the mixture density (kg/m3), rg is the gas density (kg/m3), rl is the liquid density (kg/m3), vm is the mixture velocity (m/s), dh is the hydraulic diameter of annulus (m), di is the outer diameter of tubing (m), do is the inner diameter of casing (m). After casing pressure reaches a stable value, the pressure inner and outer leak point will achieve a balance. Then CO2 channeling reduces greatly, namely vo E0 and Wpo E0 and bubbles in liquid are few. Annular protective liquid can be taken as stationary. So Eq. (19) is integrated as pco ¼ pwhs þ rl ghal
ð22Þ
6
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
where pwhs is the annulus wellhead pressure (Pa) and hal is the length of annular protective liquid above the leak point (m). According to U-tube pressure balance principle and combining Eqs. (16) and (22), we can obtain pin þ rl,CO2 ghl þ rsc,CO2 ghsc
2f l hl rl,CO2 v2l 2f sc hsc rsc,CO2 v2sc dt dt
¼ pwhs þ rl ghal
@ðarg Þ ð23Þ
In Eq. (23), injection pressure and rate are known operation parameters. Stable wellhead pressure can be measured by pressure gage. And liquid surface position can be measured by the acoustic detectors. Thus, the key to calculating leak depth is to determine the interface position of liquid and supercritical CO2, which can be determined by coupling calculation of temperature and pressure. Because density, viscosity and specific heat of CO2 are changing with temperature and pressure, their expressions thus should be added to achieve coupling calculation of temperature and pressure. CO2 density can be calculated using the high precision PR equation (Tong, 1996): p¼
ð24Þ
þ
@ðarg vg Þ @z
¼0
@½ð1aÞrl @½ð1aÞrl vl þ ¼0 @t @z
2
a ¼ 0:457235
ðRT C Þ 2 ½1 þ ð0:37464 þ1:54226$0:26992$2 Þð1T 0:5 r Þ pc
ð25Þ b ¼ 0:077796
RT c pc
ð26Þ
in which Tc is the critical temperature (1C), pc is the critical pressure (Pa), o is the eccentric factor and Tr is the reduced temperature. Using the viscosity model based on the PR state equation (Guo et al., 1999), viscosity can be expressed as 0
0
2
T m3 þð2bTb Tr 0 pÞm2 ð2bb T þTb þ 2r 0 bpaÞm 0 2
2
ð32Þ
ð33Þ
where vg ¼ vs þC 0 vm
ð34Þ
in which C0 is the distribution factor, vs is the gas slip velocity (m/s) and a is the gas void fraction. For bubble flow, according to the equilibrium of buoyancy force and flow drag (Caetano et al., 1992b; Harmathy, 1960), slip velocity can be calculated as " #1=4 ðrl rg Þg s vs ¼ 1:53 ð1aÞ1=2 ð35Þ 2 where
s ¼ 0:07275½10:002ðT18Þ
where
0
þðTb b þ r 0 pb ab Þ ¼ 0
The value of distribution factor C0 is related to the distribution density and velocity of gas bubbles, which is equal to 2.0 for bubble laminar flow. And for bubble turbulent flow or slug flow, the value is equal to 1.2 (Hasan and Kabir, 1988). Fanning friction factor is related to flow regime and flow channel geometry (Caetano et al., 1992b; Xu, 2002). For laminar flow in concentric annulus, Fanning friction factor is f¼
pffiffiffiffi pffiffiffiffi 2 0 b ¼ be½Q 2 ð T r 1Þ þ Q 3 ð pr 1Þ
mc T c pffiffiffiffiffiffiffiffiffi 2 pc Z c 1 þ Q 1 pr T r 1
ð28Þ ð29Þ
T 1=6 c
ð30Þ
ð31Þ
where R is the gas constant, 8.314 J/(mol 1C). 4.3. Coupled mathematical model of CO2 gas migration According to leakage rate, bubble flow (gas void fraction value less than 0.2) or slug flow (gas void fraction value more than 0.2)
ð38Þ
where K¼
di do
ð39Þ
rm vm dh mm
mm ¼ mg a þ ml ð1aÞ
in which m is the kinetic viscosity of fluid (Pa s), pr is the reduced pressure and Zc is the critical compressibility factor. The values of Q1, Q2 and Q3 relate with the eccentric factor. For CO2, the eccentric factor is 0.225. So the corresponding Q1, Q2 and Q3 are 0.871, 1.368 and 0.460. Heat capacity can be calculated using the continuous model based on the PR state (Wu et al., 2009): C pm ¼ 4:728 þ 0:01754T2:338 105 T 2 þ4:079 109 T 3 0:6766p þ ZRR þ ZRT 2 þ 29:7903 106 pT
16 ð1KÞ2 Rem ½ðð1K 4 Þ=ð1K 2 ÞÞðð1K 2 Þ=ðlnð1=KÞÞÞ
Rem ¼
2=3
51:076pc
ð36Þ
in which s is the surface tension of liquid (N/m). For slug flow, the Taylor bubble rise velocity in vertical annulus is given by (Hasan and Kabir, 1992) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gdo ðrl rg Þ 0:1di vs ¼ 0:345 þ ð37Þ do rl
ð27Þ
where
mc ¼
@t
rl
rRT ar2 1rb 1þ 2rbr2 b2
r0 ¼
may occur (Caetano et al., 1992a) after CO2 sneaking into annular protective liquid. Besides momentum equation (Eq. (19)), continuity equations for the continuous (annular protective liquid) and the dispersed phase (CO2) constitute controlling equations of two-phase flow (Xu, 2002), which are
ð40Þ ð41Þ
in which K is the consistency coefficient, ml is the liquid viscosity (Pa s), mg is the gas viscosity (Pa s), mm is the mixture viscosity (Pa s) and Rem is the Reynolds number of two-phase flow. For turbulent flow in concentric annulus, Fanning friction factor is given by (Caetano et al., 1992b; Xu, 2002) 1 n o1=2 6 2 2 f ½ð1K þð1 þK ÞlnKÞ=ðð1K 2 ÞlnKÞ0:45 exp½ðRem 3000Þ=10 8 " #0:225 exp½ðRem 3000Þ=106 9 < = 2 2 1=2 1K þð1 þK ÞlnK ¼ 4 lg Rem f 0:4 2 : ; ð1K ÞlnK ð42Þ As gas is accumulating at the wellhead, casing pressure and gas volume are constantly changing with time. Assuming gas column at wellhead only receives gas below the liquid surface and has no other channel to escape, then after Dt, the gas volume at
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
wellhead can be expressed as V 0wh ¼ V wh þ V R
ð43Þ
where VR ¼
p 4
2
2
ðdo di Þvg Dt
ð44Þ 3
in which Vwh is the volume of the gas column (m ) and VR is the volume of the gas received below the liquid surface (m3). After Dt, the mole number of CO2 in gas column is n0 ¼ n þ Dn
ð45Þ
where,
Dn ¼
prg vg Dt 4Mg
2
2
ðdo di Þ
ð46Þ
in which Mg is the mole mass of gas (kg/mol). For temperature at wellhead essentially unchanged, casing pressure after Dt can be described according to the state equation (Sterner and Pitzer, 1994): pwh f ðrwh Þ f ðrwh Þ 0
p0wh ¼
ð47Þ
where f ðrwh Þ ¼ rwh þ a1 rwh 2 rwh 2
a3 þ 2a4 rwh þ 3a5 rwh 2 þ 4a6 rwh 3 ða2 þ a3 rwh þ a4 rwh 2 þ a5 rwh 3 þ a6 rwh 4 Þ2
þa7 rwh 2 ea8 rwh þa9 rwh 2 ea10 rwh
rwh ¼
!
ð48Þ ð49Þ
ð52Þ
1 þ1 pni1=2 ðrm vm 2 Þniþ 1=2 ðrm vm 2 Þni1=2 pni þþ1=2 ðrm vm Þni þ 1 ðrm vm Þni þ þ Dt Dz Dz n f þ ðrm gÞn þ r v2 ¼ 0 ð53Þ 2dh m m
In the computational cells shown in Fig. 8, Eqs. (43), (45) and (47) can be given by þ1 V jwh ¼ V jwh þ
nj þ 1 ¼ nj þ
þ1 ¼ pjwh
p 4
2
2
ðdo di Þðvg Þjn þ 1=2 Dt
ðrg vg Þjn þ 1=2 Dt p 4M g
2
ð54Þ
2
ðdo di Þ
ð55Þ
þ1 pjwh f ðrjwh Þ
ð56Þ
f ðrjwh Þ
In the progress of CO2 migration in annular protective liquid, we also consider the solubility of CO2 in water. So the volume fraction of CO2 is set as (Weiss, 1974) ! K o M g rl pg ai þ 1 ¼ ai ð57Þ
rg
i
þ ci,5 ðT þ273:15Þ þ ci,6 ðT þ 273:15Þ2
ar
Dt
þ
n nþ1 g Þi þ 1=2 ðvg Þi þ 1=2 ð
ðar
n nþ1 g Þi1=2 ðvg Þi1=2
ar
Dz
in which Ko is the solubility of CO2 (mol/(kg Pa)) and pg is the CO2 partial pressure (Pa). At initial time, annulus has no CO2 gas, and the casing pressure at wellhead is atmospheric pressure. Thus, the initial and the boundary conditions used in calculation are as follows: p0wh ¼ pa
¼0
ð59Þ
i1 p0i1=2 ¼ p0wh þ rl ghal 1 n
1 ri rn
ð60Þ
a0i ¼ 0 1r i rn
ð61Þ
ðrg Þ0i ¼ 0
ð62Þ
1 ri r n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u j
ðqg Þj1=2 ð51Þ
pd2c u t2ðpc p1=2 Þ ¼ 4 rgc
ð63Þ
where pc is the leak point pressure (Pa).
Table 1 Coefficients ci,j of equation of state for CO2. i 1 2 3 4 5 6 7 8 9 10
ci,1 – – – – – – 0.3934 1012 – – –
ci,2 – – – – – – 0.9092 108 – 0.2300 108 –
2
ð58Þ
ð50Þ
in which rwh is the density of CO2 at wellhead (kg/m3) and values of coefficients ci,j are given in Table 1. Flowing parameters of CO2 gas channeling are changing with time and space constantly. So we choose iterative method to predict the variation of SCP using computational mesh shown in Fig. 8. Computational domain within tubing is divided into n grids. And to account for gas migration in annulus, we also discretize the annular protective liquid into n cells, leaving gas column at wellhead as a separate grid, denoted by nþ1 (Xu, 2002). A staggered grid assignment method is used in calculation. Pressure and velocity are stored at cell surfaces and scalars, such as density, temperature and volume fraction, are stored at the center of the cell. Using semi-implicit central difference method (Xu, 2002), Eqs. (32), (33) and (19) are discreted as n g Þi
Ko ¼ e60:217 þ 93:452½100=ðT þ 273:15Þ þ 23:359ln½ðT þ 273:15Þ=1000:024½ðT þ 273:15Þ=100 þ 0:005½ðT þ 273:15Þ=100
ai ¼ ci,1 ðT þ 273:15Þ4 þci,2 ðT þ 273:15Þ2 þci,3 ðT þ 273:15Þ1 þ ci,4
ðar
½ð1aÞrl ni þ 1 ½ð1aÞrl ni Dt 1 þ1 ½ð1aÞrl niþ 1=2 ðvl Þni þþ1=2 ½ð1aÞrl ni1=2 ðvl Þni1=2 ¼0 þ Dz
where
nM g V wh
nþ1 ð g Þi
7
ci,3
ci,4 7
0.1826 10 – – 1.3270 0.1246 – 0.4278 106 0.4028 103 0.7897 105 0.9503 105
2
0.7922 10 0.6656 10 4 0.5996 10 2 0.1521 4.9045 0.7552 0.2235 102 0.1197 103 0.6338 102 0.1804 102
ci,5
ci,6
– 0.5715 10 5 0.7170 10 4 0.5365 10 3 0.9822 10 2 – – – – –
– 0.3022 10 9 0.6242 10 8 0.7112 10 7 0.5596 10 5 – – – – –
8
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
Define recorded stable casing pressure pwhs, length of gas chamber ht, injection pressure pin, injection mass flux Win and injection temperature Tin
CO2 liquid CO2 hl
1
Assume the leak depth h*
2
Calculate hal and Pco
.
i=0
.
Pi=Pin, Ti=Tin, hi=0
.
Pi>7.1MPa and Ti>31ºC
i-1 supercritical CO2
i=i+1
i+1 . . .
hsc
Guess Pi* Calculate Ti Calculate Pi
n-1
|Pi-Pi*|<ε Y hi=hi+Δ z
n
leak point
hl=hi
N Calculate Cpmi, ρi and μi
i gridding
Y
Pi*=(Pi+Pi*)/2 N
N
hi>h* Y N |Pi-Pco|<ε Y Calculate the leak point pressure pc
z
Z
Assume the diameter of leak point dc
gr idding h al
Z=0
Pc
t=0
n+1 gas
ht
n n-1 . . . i+1 i i-1 . . . 2 1 0
Define the initial and boundary conditions
i+1
t=t+Δt
i+1/2
P i+1/2 vgi+1/2
Guess pwh*
out
i=1
i
ρi T i αi
Calculate pi+½* Calculate the average pressure pi, and calculate the corresponding density ρi
in P i-1/2 vgi-1/2
i-1/2
Calculate αi, (vg)i using Eq.52, Eq.53 and Eq.54
i-1
i=i+1 i≥n Y Calculate pwh
Fig. 8. Computational grid of gas channeling. (a) Computational grid in tubing and (b) computational grid in annulus.
pwh*=0.5(pwh*+pwh)
In iterative process, firstly assume that the pressure meets the static pressure distribution: i1 1 ri r n ð64Þ pji1=2 ¼ pjwh þ rl ghal 1 n Substituting Eq. (51) into (64) gives the coupling model of CO2 gas channeling in ‘‘A’’ annulus. The calculation accuracy depends on the meshing elements and the size of the interval values. The whole coupling calculation process is shown in Fig. 9.
5. Illustration One CO2 injection well in Jilin Oilfield is selected for analysis. The detailed information about this well is listed in Table 2. Significant casing pressure is present after this well has been put into production in less than one year. The temperature and stable casing pressure at wellhead are measured before pressure relief. Then liquid surface position is measured by acoustic detector and relief valve is closed after pressure relief. Based on coupling
N
N
|pwh-pwh*|<ε Y Caculate p½ |p½-pc|<ε Y |pwh-pwhs|<ε Y Stop
N N
Fig. 9. Flow chart of coupling calculation.
calculation process mentioned above, ‘‘A’’ annulus casing pressure build-up curve (the casing pressure variation with time) of this well was obtained, as shown in Fig. 10. It is observed from Fig. 10 that the calculated value basically coincided with the measured value. The pressure build-up curve shows an ‘‘S’’ shape, which can be divided into three parts: the initial smooth section, a sharp rise section and the later stable section. After closing the relief valve, a certain time of about 2 h is required for CO2 gas migration to reach the fluid surface. Then the gas reaches the wellhead and accumulates at the wellhead,
H. Zhu et al. / Journal of Petroleum Science and Engineering 92–93 (2012) 1–10
9
Table 2 Detailed information of the illustration well. Wellbore configuration
Category
Value
Diameter of tubing Material of tubing Diameter of casing Material of casing Formation pressure Pressure coefficient Formation temperature Geothermal gradient Injection pressure Injection mass flux Injection temperature Temperature measured at wellhead Recorded stable casing pressure Liquid surface apart from wellhead
F73 5.51 mm N80 F139.7 6.2 mm J55 24.3 MPa 0.99 98.9 1C 3.97 1C/100 m 16.4 MPa 0.463 kg/s 6 1C 25 1C 14.5 MPa 120 m
migration in annular protective liquid and accumulation at wellhead. Finally, by continuous assumption, the buildup of SCP and the equivalent-size of leak point are obtained until the calculated stable casing pressure is approximately equal to the recorded stable casing pressure. (3) Calculation result of illustration fits well with the actual monitoring data, verifying the reliability of coupled model and calculation method.
Acknowledgment Research work was co-financed by China National Natural Science foundation (No. 51074135) and Key Project of Sichuan Provincial Education Department (No. 12ZA189). Without their support, this work would not have been possible.
Fig. 10. ‘‘A’’ annulus casing pressure build-up curve of illustration well.
leading to the rise of casing pressure. Due to the large initial pressure difference, the casing pressure recovers rapidly, corresponding to 2–640 h shown in Fig. 10. Finally, the casing pressure remains stable after achieving 14.5 MPa.
6. Conclusions
(1) Special and complex characteristic of SCP in ‘‘A’’ annulus of CO2 injection well has been proposed in this paper, such as liquid in annulus, flowing direction, supercritical state of CO2, solubility of CO2 and so on. And the possible causes leading to SCP are summarized. (2) A prediction model of buildup of casing pressure has been proposed in this paper. Firstly, temperature and pressure distributions of CO2 in tubing are obtained by the calculation model. Secondly, based on pressure balance principle, the position of leak point is calculated with a recorded stable value of casing pressure. Then assuming the equivalentdiameter of leak point, casing pressure variation with time is calculated by a coupled mathematical model of CO2 gas
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