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Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore
Journal of Natural Gas Science and Engineering xxx (2015) 1e8
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Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore* Weiqiang Song, Ruihe Wang, Hongjian Ni*, Hongjun Huo, Zhonghou Shen School of Petroleum Engineering in China University of Petroleum, Qingdao 266580, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 October 2014 Received in revised form 20 April 2015 Accepted 20 April 2015 Available online xxx
Supercritical carbon dioxide (CO2) is of great interest as a sand-flushing fluid with significant potential advantages; therefore, it is of fundamental and practical importance to investigate its sand transporting mechanism. Numerical simulation was conducted to analyse the heat transfer along the whole deviated wellbore, and the results served as boundary conditions for the multiphase flow model of CO2 transporting sand. In addition, the physical properties of CO2 as a function of temperature and pressure were considered in the numerical simulation. The multiphase flow model was solved to investigate the effects of various factors on the sand transporting efficiency of CO2, including displacement, sand production rate and annulus eccentricity, in which the sand size distribution at the annulus entrance was set based on the measurement data from the experiments. The computed results show that the sand transporting efficiency decreases with inclination abruptly at first and subsequently increases, which is consistent with the experimental results. The inclination where sand is most difficult to transport varies from 48 to 72 and depends on the sand’s diameter and production rate at the bottom hole. The efficiency can be improved by increasing displacement with a trend that is greater at first and later slows. A larger sand production rate at the bottom hole means it is more difficult to transport sand. The efficiency decreases with increasing eccentricity at first and subsequently begins to increase when the eccentricity reaches 0.8. Under the same conditions, the pressure drop of sand-flushing with CO2 is 52.3% that observed with water. The results lay an important foundation for practical application. © 2015 Elsevier B.V. All rights reserved.
Keywords: Supercritical carbon dioxide Inclination Multi-phase flow Sand size distribution Sand-transporting efficiency
1. Introduction Increasing numbers of unconventional/offshore gas reservoirs have been exploited to meet the world’s need for energy, and a portion of the need is met by uncemented horizontal wells intersecting multi-stage fractures to obtain enhanced recovery. Hydraulic fracturing sand or shallow formation sand, alone or mixed with reservoir fluids, can easily penetrate into the annulus/tubing during production. Sand can induce flow resistance and negatively affect gas production (Li et al., 2010). Hence, sand cleanout has developed into a standard practice in oilfield operations (Heinrichs and Dedora, 1995) and is generally conducted with coiled tubing
DOI of original article: http://dx.doi.org/10.1016/j.jngse.2015.04.022. This article is a reprint of a previously published article. For citation purposes, please use the original publication details; Journal of Natural Gas Science and Engineering 25 (2015) 140-147. * Corresponding author. E-mail address: [email protected] (H. Ni). *
and circulating out the fills with carrying fluid (Falk and Fraser, 1995; Ozbayoglu et al., 2005). Water or brine is most commonly used as the circulation fluid in oil reservoirs due to the low cost; however, it is limited in low bottom-hole pressure (BHP) gas reservoirs and unconventional reservoirs due to potential damage (e.g., lost circulation) (Li et al., 2010; Lage et al., 1996). Stable foams have been successfully applied in both vertical and inclined wells as sand/cutting transporting fluids since the mid-1980s (Hall and Roberts, 1984; Fraser and Moore, 1987; Falk and McDonald, 1995; Nakagawa et al., 1999; Wang et al., 2009). Foams transport the fills in high temperature and pressure conditions and may get polluted by various formation fluids penetrating into the annulus. The difficulties lie in how to keep foams stable economically (Doane et al., 1996; Negrao and Lage, 1997; Li et al., 2014). Carbon dioxide (CO2) has been used as a drilling and fracturing fluid with potential advantages and insignificant corrosion (Kolle, 2000; Gupta et al., 2005; Lillies. 1982; Gupta and Bobier, 1998), and it is now drawing more interest. CO2 changes into the
http://dx.doi.org/10.1016/j.jngse.2015.11.017 1875-5100/© 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
supercritical state in bottom hole (when the temperature is 304.2 K or higher and the pressure is above 7.38 MPa), which provides significant benefits, including increased rate of penetration (ROP) (Du et al., 2012, 2013) and productivity (by reducing formation damage and competitive adsorption with methane) (Sun et al., 2013). Thus, CO2 is one of the most promising sand cleanout fluids (Li et al., 2013). However, knowledge of cutting/sand transporting with supercritical carbon dioxide is inadequate to support field application. Former investigations mainly focused on how temperature and pressure (by changing the density and viscosity of CO2) influenced the cutting transporting efficiency in horizontal sections, and the experimental results and simulation were in good agreement (Shen et al., 2011; Li et al., 2011). However, how the engineering factors (including inclination and displacement) influence multiple phase flow is not known. Additionally, the density/viscosity change of CO2 in the flow field and the variation of the cutting diameter were neglected in the earlier simulations, which contradicted the actual engineering conditions. In hydraulics calculations, the compressibility of CO2 and heat transfer along the wellbore must be considered because liquid CO2 is pumped into tubing whose temperature is much lower than that of formation. Span and Wagner (1996) presented a new equation of state for CO2 covering the fluid region from the triple-point temperature to 1100 K at pressure up to 800 MPa. Based on the SpaneWagner equation and Vesovic model (Vesovic and Wakeham, 1990), Wang and Ni (2013) investigated the heat and pressure transferring mechanism of CO2 in vertical wellbores. The preliminary research lays the foundation for this study. With consideration of the sand size distribution and eccentricity of the annulus, this paper investigates the sand transporting mechanism with CO2 in the whole inclined wellbore. For the hydraulics calculations, we developed a closed mathematical model to fully couple the hydrostatic pressure, temperature, density and viscosity of CO2 and friction. The study offers a theoretical basis for field application.
ensure accuracy. The coupling relationship is illustrated in this section. The implicit equation of CO2 density can be expressed as (Span and Wagner, 1996)
Pðd; tÞ ¼ rRT 1 þ dFrd
where d ¼ r/rc is the dimensionless reduced density and t ¼ Tc/T is the inverse reduced temperature. Dimensionless Frd is the partial derivative of the Helmholtz energy F(d,t). After the pressure P and temperature T are obtained, the density r can be calculated by numerical algorithms. The isobaric heat capacity of CO2 can be written as
2 o 1 þ d4rd dt4rdt M$cp 2 r ¼ t 4tt 4rr þ R 1 þ 2d4rd þ d2 4rdd
hðT; rÞ ¼ h0 þ DhðT; rÞ þ Dhc ðT; rÞ
(4)
where the zero-density viscosity h0 is in units of mPa.s and temperature T is in kelvin.
h0 ðTÞ ¼
1:00697T 1=2 G*h ðT * Þ
(5)
The reduced effective cross section G*h ðT * Þ is represented by empirical equation (6), T* ¼ T/251.196 K. 4 X i ln G*h T * ¼ ai ln T *
(6)
i¼0
The excess viscosity is in the following explicit form
Dhðr; TÞ ¼ d11 r þ d21 r2 þ
d64 r6 d r8 þ d81 r8 þ 82* *3 T T
(7)
Similar to the viscosity, the thermal conductivity of CO2 is given
In the sand-cleanout process, hypothermic liquid CO2 is pumped down to the bottom and then jetted into the tubing and casing annulus. Fills are flushed and dispersed, and then CO2 carries sand and flows back to the surface via the annulus. Sand can continue to penetrate into the annulus via perforations, and heat transfers from the formation into the tubing through the cement sheath, metal casing and annulus (Fig. 1). The inside diameter of the tubing is 50 mm, the outside diameter of the tubing (ro) is 60 mm and the inside diameter of the casing (Ri) is 100 mm. The eccentricity ε is defined as
D Rr
(3)
The viscosity of CO2 can be written as (Vesovic and Wakeham, 1990; Fenghour and Wakeham, 1998)
2. Mathematical models
ε¼
(2)
(1)
2.1. Physical properties of CO2 As documented in earlier research (Kolle, 2000; Gupta, 2005), CO2 changes into the supercritical state in bottom hole when the temperature is 304.2 K or higher and the pressure is above 7.38 MPa (Fig. 2). The phase change of CO2 is reflected by the thermodynamic properties changes. Li et al. (2011) experimental results show that the cutting/sand transporting capability of CO2 can be improved by reducing temperature or increasing pressure. The coupling relationship between temperature/pressure (attributed to well depth) and the thermodynamic properties of CO2 was considered to
by
lðT; rÞ ¼ l0 þ DlðT; rÞ þ Dlc ðT; rÞ
(8)
2.2. Mathematical model for heat transfer In heat transfer, both conductive resistance and convective resistance are involved, and it is assumed that the geothermal gradient is constant. Compared with the radial temperature difference between the annulus and formation, the axial temperature difference between neighbouring infinitesimal sections is negligible. The influence of sand on the temperature distribution is also neglected due to its low volume fraction. The energy equations in an infinitesimal section (Fig. 1) are represented as
Qea Qap ¼ cp mDTa Qap ¼ cp mDTp
(9) (10)
where Qea stands for the heat transferred from the formation rocks to CO2 in the annulus, and Qap is the heat transferred from CO2 in annulus to that in pipe, W. DTp is the temperature difference of CO2 between infinitesimal sections in the tubing, and DTa is the temperature difference in the annulus, K.
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
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Fig. 1. Physical model of flow field.
With consideration of the throat effect, when CO2 flows through the jet connecting tubing and annulus, the pressure drop DPj(DPj ¼ P1 P2) and temperature drop DTj are expressed as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 2 kþ1 #ffi u u 2k P2 k P2 k m ¼ AP2 t Rs T1 ðk 1Þ P1 P1
(13)
ZP2 DT ¼
mJT dP
(14)
P1
where A is the section area of the jet, m3. P1 and P2 are the pressure of the jet inlet and outlet, Pa. T1 is the temperature of the jet inlet, K. Isentropic coefficient k ¼ 1.28, specific gas constant Rs ¼ 0.1889 kJ/ (kg K), and throttle coefficient mJT ¼ 1/CP[T(vV/vT)P V]. 2.3. Mathematical model for multiphase flow The Eulerian model (Wang and Sun, 2009) solves a set of n momentum and continuity equations for each phase and takes interference among particles into consideration, which is not negligible in the flow field of a sand bed. The relative velocity ! v qp between particle phase and primary phase is given by
Fig. 2. Phase diagram of CO2.
Qea ¼
1
e ~ h2pr
Qap ¼
a1 l
Te Ta þ 2pl1 l ln rra2 þ 2pl1 l ln rra2e a1 ca
ce
Ta Tp 1 h pdi l
_
1 ln do þ þ 2pl l d t
(11)
i
(12)
1
e ~ hpd
ol
e ~ is the convective heat transfer coefficient of casing, W/ where h 2 (m K). r stands for the radius and d for the diameter, m. lca is the thermal conductivity of the casing, lce is the thermal conductivity of the cement sheath, and lt is the thermal conductivity of the pipe, _ W/(m K). h stands for the convective heat transfer coefficient of the inner wall of the pipe, W/(m2 K).
! v qp ¼ tqp ! a
(15)
v! vm ! ! vm a ¼ g ! v m $V! vt
(16)
tqp ¼
rm rp d2p 18mq fd
(17)
where ! a is the acceleration of a particle, m/s2. tqp is the relaxation time of a particle, s. fd stands for the drag force, N. The interaction force between two phases is expressed by the Schiller and Naumann model (Schiller and Naumann, 1933). The interaction force between two sand particles is illustrated by the Syamlal-O’Briensymmetric model, and the Brown force and Saffman lifting force are
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
both neglected because the particle size is much larger than 1e10 mm. Based on the continuity equation of the primary phase, the volume fraction equation of particles is given as (Wang and Wang, 2005; Tang et al., 2003)
v ap rp þ V$ ap rp vm ¼ V$ ap rp vdr;p vt
(18)
8 1 > > Qap Tp ðm þ 1ÞðiÞ ¼ Tp ðmÞðiÞ þ > > cm > > > < Ta ðm þ 1ÞðiÞ ¼ Ta ðmÞði1Þ i ¼ 1;2;3…; m ¼ 1;2;3…;n 1 > > ðiÞ ðiÞ > P ðm þ 1Þ ¼ P ðmÞ þ DP > p p p > > > : Pa ðm þ 1ÞðiÞ ¼ Pa ðmÞðiÞ þ DPa (20)
where ap is the volume fraction of the particle. The mean quality Pn ! ak r v k velocity ! v m is defined as ! v m ¼ k¼1r k , m/s. The drift velocity m ! v is given by ! v ¼! v ! v , m/s.
The k-3 model is utilized to illustrate the turbulence and close the equations, which is appropriate for compressible fluid flow.
With consideration of both the static hydraulic differential and flow pressure loss, DPp and DPa are related to the well structure and are calculated with the Fanning equation (Huang et al., 2011). The bottom-hole parameters are modified during iteration (as follows).
dr;p
dr;p
p
Tp ðnÞ Ta ðnÞ ¼ DTB Pp ðnÞ Pa ðnÞ ¼ DPB
(21)
m
! 8 > vðrkÞ v vk vk > > þ ruj ðmþmt Þ ¼ ttij Sij rεþQk > > vxj vxj vxj < vt ! > 2 > > > vðrεÞ þ v ru ε mþ mt vε ¼ 1:45 ε t S 1:92f r ε þQε > 2 j tij ij : vt vxj k 1:3 vxj k (19) where ttij ¼ 2mt(Sij Snndij/3) 2rkdij/3, and the eddy viscosity mt ¼ 0.09furk2/ε. The near wall attenuation functions are given as 2 ðRe2t Þ , where Re ¼ rk2/mε. fu ¼ eð3:4=ð1þ0:02Ret Þ Þ and f2 ¼ 1 0:3e t pffiffiffi The wall terms are given as Qk ¼ 2mðv k=vyÞ2 and Qε ¼ 2mmt/r(v2mε/ vy2)2. Sij stands for the mean-velocity strain-rate tensor, and dij stands for the Kronecker delta.
2.4. Sand size distribution In boundary conditions, the sand diameter is set based on the experimental statistical data. Sand produced from reservoir rocks (tight sandstone) was collected to determine their diameters by mass fraction (Table 1).
2.5. Solution procedure 2.5.1. Temperature and pressure distribution along the wellbore The wellbore is divided into many flow units (every 1 m) axially, and the mathematical model is solved by numerical methods from the wellhead to the bottom hole (as follows). In every flow unit, the temperature and pressure are considered to be constant so that the properties of CO2, the heat transfer and the pressure loss can be gained. According to field application, the backpressure, inlet temperature and inlet flow rate are given, whereas the inlet pressure and outlet temperature are assumed at first and then gained after iteration reaches convergence.
(
Ta ðnÞ* ¼ Tp ðnÞ DTB Pp ðnÞ* ¼ Pa ðnÞ þ DPB
(22)
The convergence condition is given by
8
T ðnÞ* T ðnÞ
a a > > > d < Tc
>
* > > : Pp ðnÞ Pp ðnÞ
d Pc
(23)
where d ¼ 0.001%, critical temperature of CO2 Tc ¼ 304.25 K and the critical pressure of CO2 Pc ¼ 7.38 MPa. Interactions continue from the bottom hole to wellhead, with Ta(n)*,Pa(n),Tp(n),Pp(n)* as the boundary conditions, and reverse as shown in Eq. (20) until the convergence condition is met.
8 ðiÞ ði1Þ > ðm 1Þ Tp ðm 1Þ ¼ Tp > > > > > > > < TaðiÞ ðm 1Þ ¼ TaðiÞ ðmÞ þ 1 Qea Qap cm i ¼ 2;3;4…; m ¼ 2;3;…n > > ðiÞ ðiÞ > > Pp ðm 1Þ ¼ Pp ðmÞ DPp > > > > : ðiÞ ðiÞ Pa ðm 1Þ ¼ Pa ðmÞ DPa (24)
2.5.2. Multiphase flow in the wellbore The annulus model is faceted axially and radially by a structured grid dividing method (Fig. 1), and it is assumed that sand penetrates into the annulus at a constant rate. The multiphase flow model is solved with the Eulerian model as described earlier, where the pressure-linked equations for the flow field are calculated by a semi-implicit method, and the velocity-linked equations are solved with a modified SIMPLE algorithm. 3. Model validation
Table 1 Experimental data of the particle size distribution. No.
Average diameter (mm)
Mass (g)
Mass fraction
1 2 3 4 5
0.95 0.74 0.53 0.39 0.18
1.7 7.1 4.4 14.4 26.2
0.0316 0.1320 0.0818 0.2676 0.4870
The well trajectory of the example is presented in Fig. 3. Initially, liquid CO2 of 20 C was pumped into the coiled tubing with a mass flow rate of 5.37 kg/s. The geothermal gradient and sand density were set as 2.8 K/100 m and 2.65 g/cm3. In the bottom hole, sand penetrates into the annulus constantly, and the volume fraction is 1% of the multiphase flow. To compare with the experimental results, 13 points along the
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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5
Fig. 4. Influence of the inclination on the sand cleanout efficiency.
Fig. 3. Schematic diagram of the well trajectory.
wellbore were selected to investigate the sand transporting efficiency, and the average value of a 9 m well section was numerically calculated for every point. Parameters related to the sand transporting efficiency in the annulus are list in Table 2, which are then used as the boundary conditions for the multi-phase flow. As CO2 flows upward in the annulus, the formation gets cooler, and CO2 is also cooled by the tubing in which CO2 flows downward. Finally, CO2 changes from the supercritical state into the liquid state in the annulus at a depth of 400 me500 m. The sand retention rate (SRR) was selected as the standard measure of the sand transporting efficiency, and it was defined as:
SRR ¼
mr rs Vf x rs Vf x
(25)
where mr stands for the real sand mass in the annulus, kg; rs is the density of sand, kg/m3; Vf stands for the volume of the annulus, m3; x presents the sand production rate and is given by the sand volume fraction at the annulus inlet, dimensionless. A larger SRR means it is more difficult to transport sand. The results show that SRR changes little along the wellbore in the vertical section. The distribution of SRR along the inclined wellbore is presented in Fig. 4. SRR increases with inclination at first and then decreases after a certain critical point. It is most difficult to transport sand when the inclination is 48 e72 . Laboratory experiments have been conducted to
investigate the influence of the inclination on the sand transport efficiency in the annulus with supercritical CO2 as the circulation fluid (Li et al., 2011). Researchers placed a certain amount of sand in the equipment, then started the CO2 pump and increased the displacement to investigate the sand mass that was retained in the equipment after a certain time. In the results, researchers took the critical flow rate (linearly correlated with displacement) that could transport 90% of the sand out of the equipment as the measuring standard. A higher critical flow rate means it is more difficult to transport sand at different inclinations. The experimental result is presented in Fig. 4. The trend of how multiphase flow develops is consistent with the simulation result, which validates the simulation model to some extent. Due to the differences in the boundary conditions, the inclination at which it is most difficult to transport sand differs from the experimental results to the simulation results. Except for the inclination, the other boundary conditions are set to the same values as in experiment (313.2 K/8.5 MPa), which means the physical properties of CO2 are the same in the different sections. The heat transfer is also neglected in the experiment. In the simulation, the density, viscosity and velocity of CO2 were coupled with the annulus temperature and pressure, and they all changed with well depth and inclination. Therefore, the boundary conditions in the simulation are consistent with the actual engineering. As depicted in Table 2, as the temperature and pressure along with depth changed, the flow rate and physical properties of CO2 changed accordingly. For example, it is more difficult to transport sand when the inclination is 72 than when it is 54 in the experiment. In contrast, more sand is transported out when the
Table 2 Flow field variables of the test points. No.
Depth m
Vertical depth m
Inclination
1 2 3 4 5 6 7 8 9 10 11 12 13
50 500 1000 1200 1400 1600 1800 1900 2000 2100 2200 2400 2550
50.0 500.0 1000.0 1198.5 1387.5 1560.5 1709.1 1772.0 1826.5 1872.0 1907.9 1949.6 1954.9
0 0 0 12 24 36 48 54 60 66 72 84 90
Temperature K
Pressure MPa
Density kg/m3
Viscosity 105 Pa$s
Flow rate m/s
279.5 305.0 317.7 322.7 327.3 331.5 335.1 336.6 337.9 339.0 339.8 340.7 340.9
6.740 10.16 13.79 15.23 16.59 17.86 18.95 19.41 19.82 20.17 20.44 20.81 20.88
865.8 705.1 695.8 694.0 693.0 693.2 693.3 693.5 693.9 694.3 694.9 696.8 697.5
7.996 5.643 5.585 5.582 5.596 5.617 5.637 5.646 5.658 5.668 5.679 5.707 5.717
1.230 1.510 1.531 1.534 1.537 1.536 1.536 1.536 1.535 1.534 1.532 1.528 1.527
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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inclination is 72 in the simulation. The reason is that the density of CO2 increases as the inclination and depth increase; thus, the sand transporting capability of CO2 is improved. 4. Sand cleanout mechanism of CO2 Quantitative calculation was conducted to investigate the difference between water and CO2 with respect to sand transporting capability. In the calculation, the boundary conditions were the same as No. 11 in Table 2 and sand diameter were all set as 0.3 mm. The results show that the SRR when transporting sand with CO2 (22.0%) is 1.85 times that with water (11.9%) because the viscosity of water is more than 10 times larger than that of CO2, so the Reynolds of water flow is smaller and the boundary layer is thicker in the radial direction. With respect to Wang’s (2009) force model, the drag force applied on the near-wall hemisphere of sand is smaller than that on the other side; thus, sand reverses to the bottom of the boundary layer along the low-side hole. For turbulent CO2 flow, the difference of the drag force applied on the near-wall hemispheres and the other is smaller and gravity is dominant, so that sand tends to assemble at the bottom side of the annulus (Fig. 5). A higher sand volume fraction means the sand particles collide more frequently; thus, more dynamic energy is consumed and SRR finally increases. Because the viscosity of water is not sufficient to clean the wellbore, it is necessary to add viscosifier into CO2 to increase its sand transporting capability. The pressure drop of sand cleanout with CO2 is 52.3% that with water, which confirms the advantages of the mentioned field attributes. The sand size distribution was also investigated to reveal the mechanism of how the sand bed was formed at different sections. As is shown in Fig. 6, SRR do not have a simple positive correlation with the diameters. The production rate of 0.18 mm sand at the inlet is much higher than the others, and the particles collide with other particles and the wall more frequently. Finally, more dynamic energy is consumed and its SRR is the largest. The production rate of 0.74 mm sand is a little smaller than that of 0.53 mm sand, but the SRR of 0.74 mm sand is larger than that of 0.53 mm sand because of the gravity difference. The comparison of the SRR of 0.39 mm sand and 0.74 mm sand indicates that the SRR of heavier sand (0.74 mm) is larger in the lightly deviated section. In the highly deviated section, 0.39 mm sand (with a larger production rate at the bottomhole) is more difficult to transport. The section where sands of different diameters are most difficult to transport varies from 48 to 66 . The sand transporting process is influenced by gravity and momentum exchange (caused by collision), and in the lightly deviated section, sand disperses in the annulus uniformly and
Fig. 6. Distribution of sand diameters along wellbore.
gravity is dominant. As in the highly deviated section, sand assembles at the bottom side of the annulus and collides frequently; thus, momentum exchange along with dynamic energy consumption dominates the sand transporting process. 5. Analysis of the influencing factors 5.1. Displacement In the drilling field, displacement can be translated as “the CO2 volume that is pumped into the tubing/pipe in a second”. The annulus flow rate has a linear positive correlation with displacement. As shown in Fig. 7, the SRR decreases as the annulus flow rate increases with a trend that develops fast at first and then slows. The critical flow rate is approximately 3.0 m/s in this study. Sand should be transported with a flow rate no larger than the critical value in field application to control cost. MVT theory has been widely used as a standard in air drilling to transport cuttings effectively and economically. This theory has been well-documented in earlier research (Angel, 1957; Doan et al., 2003) and provided engineers a convenient method to calculate the displacement for gas drilling of different density (related to pressure and temperature). CO2 is in the supercritical state rather than gaseous state at the bottom-hole; therefore, we propose to establish similar theory to meet the need for cleanout application with CO2. In addition to the density, the viscosity of CO2 has a non-negligible influence on the sand transporting efficiency; thus, the proposed theory should also take the
Fig. 5. Comparison chart of the sand volume fraction when transporting sand with CO2 and water.
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
7
Fig. 7. Influence of annulus flow rate on sand retention rate.
Fig. 9. Influence of the eccentricity on the sand retention rate.
viscosity of CO2 (changes significantly with pressure and temperature) into consideration to ensure accuracy.
increase of the eccentricity due to the influence of the boundary layer. When the pipe is close to the casing, more sand assembles in the bilateral area of the pipe where the average velocity of CO2 is influenced less by the solid wall; thus, more sand is transported out. In field applications, engineers can search from the area near the critical eccentricity to find the sticking point.
5.2. Sand production rate The sand production rate (measuring the velocity of sand penetrating into the annulus) is given by the sand volume fraction at the bottom-hole. As presented in Fig. 8, the SRR increases with the increase of the production rate with a trend that grows increasingly fast. As more sand particles penetrate into the annulus, they assemble at the low side of the annulus and collide with each other more frequently. Therefore, more dynamic energy is consumed and more sand stagnates to form a sand bed. 5.3. Eccentricity In inclined wellbores, pipes tend to settle upon the low side of the annulus due to the influence of gravity. The influence of the eccentricity on the sand transporting efficiency is also investigated, and the result is presented in Fig. 9. As the eccentricity increases, the SRR increases at first and then decreases when the eccentricity is larger than 0.8. The mean velocity of CO2 in the low side of the annulus (MVLS) where most sand assembles has a strong influence on the sand cleanout efficiency. At first, MVLS decreases with the
6. Conclusions (1) A closed mathematical multi-phase flow model, which fully couples the hydrostatic pressure, temperature, physical properties of CO2 and friction, is solved and verified with experiments to investigate the sand transporting mechanism of CO2. The effects of various factors on CO2’s sand transporting efficiency, including displacement, the sand production rate and the annulus eccentricity, are also analysed. (2) Sand with larger diameter is more difficult to transport in vertical wellbores, whereas in highly deviated sections, sand with larger production rate is more difficult to transport. It is most difficult to transport sand when the inclination is approximately 48 e72 . A general decreasing trend in the SRR, which develops fast at first and then slows, was observed with increasing displacement. The increasing trend in the SRR with increasing sand production rate grows faster and faster. As the annulus eccentricity increases, the SRR increases at first and then begins to decrease when the eccentricity reaches 0.8. The pressure drop of sand cleanout with CO2 is 52.3% that with water, which confirms the advantages (e.g., less leakage) of the mentioned attributes. (3) The findings of this study predict the necessity of developing a new sand/cutting transporting theory to facilitate field engineers. Because the SRR when transporting sand with CO2 is 1.85 times that with water, it is advised to add viscosifier to improve the sand cleanout efficiency. Acknowledgements
Fig. 8. Influence of the production rate on the sand retention rate.
The financial support from the Major State Basic Research Development Program of China (2014CB239202), the Doctoral Fund of Ministry of Education (20120133110011 and 20130133110006) and the Fundamental Research Funds for the Central Universities (27R1402019A) is highly appreciated. We sincerely thank our colleagues at the Water Jet Research Centre in China University of Petroleum (East China) for helping with the theory research.
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Nomenclature ro Ri D ε
d t P T
h0 r cp M R T*
l0 Qea Qap
DTB DTa e ~ h
lca lce lt _
h
A P1 P2 T1 Rs ! a
tqp fd ! v qp DTj DPj
ap
! vm ! v p
SRR mr
rs
Vf x
outside diameter of the pipe, mm inside diameter of the casing, mm distance between the pipe centre and the casing centre, mm eccentricity, dimensionless reduced density, dimensionless inverse reduced temperature, dimensionless pressure, MPa temperature, K zero-density viscosity, mPa$s density, kg/m3 isobaric heat capacity, J/(kg$K) molecular weight, kg/mol universal gas constant, R ¼ 8.314 Pa m3 K1$mol1 T/251.196 K, dimensionless thermal conductivity, W/(m K) heat transferred from formation rocks to CO2 in the annulus, W heat transferred from CO2 in the annulus to that in the pipe, W temperature difference of CO2 in pipe between infinitesimal sections, K temperature difference of CO2 in the annulus between infinitesimal sections, K convective heat transfer coefficient of the casing, W/ (m2 K) thermal conductivity of the casing, W/(m K) thermal conductivity of the cement sheath, W/(m K) thermal conductivity of the pipe, W/(m K) convective heat transfer coefficient of the inner wall of the pipe, W/(m2 K) section area of the jet, m3 pressure of the jet inlet, Pa pressure of the jet outlet, Pa temperature of the jet inlet, K specific gas constant, Rs ¼ 0.1889 kJ/(kg K) acceleration of a particle, m/s2 relaxation time of a particle, s drag force, N relative velocity, m/s temperature drop of the jet pressure drop of the jet volume fraction of particles, dimensionless mean quality velocity, m/s velocity of a particle, m/s sand retention rate, dimensionless real sand mass in the annulus, kg density of sand, kg/m3 volume of the annulus, m3 sand production rate, dimensionless
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Feasibility of supercritical carbon dioxide as a drilling fluid for deep underbalanced drilling operation. SPE Annual Technical Conference and Exhibition Held in Dallas, Texas, 9e12 October, 2005 (SPE paper 96992). Gupta, D.V.S., Bobier, D. M. The history and success of liquid CO2 and CO2/N2 fracturing system. SPE Gas Technology Symposium Held in Calgary, Alberta, 15e18 March, 1998 (SPE paper 40016). Hall, D. L. and Roberts, R. D. Offshore drilling with preformed stable foam. SPE California Regional Held in Long Beach, California, 11e13 April, 1984 (SPE paper 12794). Herichis, B. D. and Dedora, O. Horizontal well sand cleanout. SPE International Heavy Oil Symposium Held in Calgary, 19e21 June, 1995 (SPE paper 30269). Huang, Y., Yin, K., Zhu, L.H., et al., 2011. Velocity and pressure model of largediameter DTH hammer pneumatic reverse circulation fluid. J. Jilin Univ. (Earth Sci. Ed. 41 (S1), 232e236 (in Chinese). Kolle, J. J. Coiled-tubing drilling with supercritical carbon dioxide. SPE/CIM International Conference on Horizontal Well Technology Held in Calgary, Alberta, 6e8 November, 2000 (SPE paper 65534). Lage, A.C.V.M., Nakagawa, E. Y., Souza, A.A.De., et al. Recent case histories of foam drilling in Brazil. SPE Latin America/Caribbean Petroleum Engineering Conference, Held in Port-of-Spain, Trinidad, 23e26 April, 1996 (SPE paper 36098). Li, G. Sh, Wang, H. Zh, Shen, Zh H., et al., 2013. Application investigation and prospects of supercritical carbon dioxide jet in petroleum engineering. J. China Univ. Petrol. Ed. Nat. Sci. 37 (5), 76e87 (in Chinese). Li, J., Pan, R., Guo, B.Y., et al., 2014. Thermal stability of brine foams for shale gas drilling. J. Nat. Gas Sci. Eng. 17, 131e135. Li, L. Ch, Wang, Zh M., Qiu, Zh S., et al., 2011. An experimental study on carrying cuttings features for supercritical carbon dioxide drilling fluid. ACTA PET. SIN. 32 (2), 355e359. Lillies, A. T. Sand fracturing with liquid carbon dioxide. SPE Annual Technical Meeting Held in Calgary, Alberta, 6e9 June, 1982 (SPE paper 11341). Li, S.Y., Li, Z.M., Lin, R.Y., et al., 2010. Modeling of sand cleanout with foam fluid for vertical well. SPE J. 15 (3), 805e811 (SPE paper 120449). Nakagawa, E. Y., Silva, V. Jr., Boas, P.R.C., et al. Comparison of aerated fluids/foam drilling hydraulics simulators against field data. SPE Asia Pacific Oil and Gas Conference and Exhibition Held in Jakarta, Indonesia, 20e22 April, 1999 (SPE paper 54319). Negrao, A. F. and Lage, A C V M. An overview of air/gas/foam drilling in Brazil. SPE/ IADC Drilling Conference Held in Amsterdam, Netherlands, 4e6 March, 1997 (SPE paper 37678). Ozbayoglu, M.E., Miska, S.Z., Reed, T., et al., 2005. Using foam in horizontal well drilling: a cuttings transport modeling approach. J. Petrol. Sci. Eng. 46, 267e282. Schiller, L., Naumann, A.Z., 1933. Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereiting. Z Ver. Deut Ing. 77, 318e320. Shen, Zh H., Wang, H. Zh, Li, G.S., 2011. Numerical simulation of the cutting-carrying ability of supercritical carbon dioxide drilling at horizontal section. Petrol. Explor. Develop. 38 (2), 233e236 (in Chinese). Span, R., Wagner, W., 1996. A new equation of state for CO2 covering the fluid region from the triple-point temperature to 1100 K at pressure up to 800 MPa. J. Phys. Chem. Ref. Data 25 (6), 1509e1596. Sun, B.J., Zhang, Y.L., Du, Q.J., et al., 2013. Property evaluation of CO2 adsorption and desorption on shale. J. China Univ. Petrol. Ed. Nat. Sci. 37 (5), 95e99 (in Chinese). Tang, X.L., Xu, Y., Wu, Y.L., 2003. Mixed model for silt-laden solid-liquid two-phase flows. TSINGHUA Sci. Technol. 8 (6), 692e696. Vesovic, A., Wakeham, W.A., 1990. The transport properties of carbon dioxide. J. Phys. Chem. Ref. Data 19 (3), 763e808. Wang, R.H., Cheng, R.C., Wang, H.G., et al., 2009. Numerical simulation of Transient cuttings transport with foam fluid in horizontal wellbore. J. Hydrodynamics 21 (4), 437e444. 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Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore* Weiqiang Song, Ruihe Wang, Hongjian Ni*, Hongjun Huo, Zhonghou Shen School of Petroleum Engineering in China University of Petroleum, Qingdao 266580, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 October 2014 Received in revised form 20 April 2015 Accepted 20 April 2015 Available online xxx
Supercritical carbon dioxide (CO2) is of great interest as a sand-flushing fluid with significant potential advantages; therefore, it is of fundamental and practical importance to investigate its sand transporting mechanism. Numerical simulation was conducted to analyse the heat transfer along the whole deviated wellbore, and the results served as boundary conditions for the multiphase flow model of CO2 transporting sand. In addition, the physical properties of CO2 as a function of temperature and pressure were considered in the numerical simulation. The multiphase flow model was solved to investigate the effects of various factors on the sand transporting efficiency of CO2, including displacement, sand production rate and annulus eccentricity, in which the sand size distribution at the annulus entrance was set based on the measurement data from the experiments. The computed results show that the sand transporting efficiency decreases with inclination abruptly at first and subsequently increases, which is consistent with the experimental results. The inclination where sand is most difficult to transport varies from 48 to 72 and depends on the sand’s diameter and production rate at the bottom hole. The efficiency can be improved by increasing displacement with a trend that is greater at first and later slows. A larger sand production rate at the bottom hole means it is more difficult to transport sand. The efficiency decreases with increasing eccentricity at first and subsequently begins to increase when the eccentricity reaches 0.8. Under the same conditions, the pressure drop of sand-flushing with CO2 is 52.3% that observed with water. The results lay an important foundation for practical application. © 2015 Elsevier B.V. All rights reserved.
Keywords: Supercritical carbon dioxide Inclination Multi-phase flow Sand size distribution Sand-transporting efficiency
1. Introduction Increasing numbers of unconventional/offshore gas reservoirs have been exploited to meet the world’s need for energy, and a portion of the need is met by uncemented horizontal wells intersecting multi-stage fractures to obtain enhanced recovery. Hydraulic fracturing sand or shallow formation sand, alone or mixed with reservoir fluids, can easily penetrate into the annulus/tubing during production. Sand can induce flow resistance and negatively affect gas production (Li et al., 2010). Hence, sand cleanout has developed into a standard practice in oilfield operations (Heinrichs and Dedora, 1995) and is generally conducted with coiled tubing
DOI of original article: http://dx.doi.org/10.1016/j.jngse.2015.04.022. This article is a reprint of a previously published article. For citation purposes, please use the original publication details; Journal of Natural Gas Science and Engineering 25 (2015) 140-147. * Corresponding author. E-mail address: [email protected] (H. Ni). *
and circulating out the fills with carrying fluid (Falk and Fraser, 1995; Ozbayoglu et al., 2005). Water or brine is most commonly used as the circulation fluid in oil reservoirs due to the low cost; however, it is limited in low bottom-hole pressure (BHP) gas reservoirs and unconventional reservoirs due to potential damage (e.g., lost circulation) (Li et al., 2010; Lage et al., 1996). Stable foams have been successfully applied in both vertical and inclined wells as sand/cutting transporting fluids since the mid-1980s (Hall and Roberts, 1984; Fraser and Moore, 1987; Falk and McDonald, 1995; Nakagawa et al., 1999; Wang et al., 2009). Foams transport the fills in high temperature and pressure conditions and may get polluted by various formation fluids penetrating into the annulus. The difficulties lie in how to keep foams stable economically (Doane et al., 1996; Negrao and Lage, 1997; Li et al., 2014). Carbon dioxide (CO2) has been used as a drilling and fracturing fluid with potential advantages and insignificant corrosion (Kolle, 2000; Gupta et al., 2005; Lillies. 1982; Gupta and Bobier, 1998), and it is now drawing more interest. CO2 changes into the
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Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
supercritical state in bottom hole (when the temperature is 304.2 K or higher and the pressure is above 7.38 MPa), which provides significant benefits, including increased rate of penetration (ROP) (Du et al., 2012, 2013) and productivity (by reducing formation damage and competitive adsorption with methane) (Sun et al., 2013). Thus, CO2 is one of the most promising sand cleanout fluids (Li et al., 2013). However, knowledge of cutting/sand transporting with supercritical carbon dioxide is inadequate to support field application. Former investigations mainly focused on how temperature and pressure (by changing the density and viscosity of CO2) influenced the cutting transporting efficiency in horizontal sections, and the experimental results and simulation were in good agreement (Shen et al., 2011; Li et al., 2011). However, how the engineering factors (including inclination and displacement) influence multiple phase flow is not known. Additionally, the density/viscosity change of CO2 in the flow field and the variation of the cutting diameter were neglected in the earlier simulations, which contradicted the actual engineering conditions. In hydraulics calculations, the compressibility of CO2 and heat transfer along the wellbore must be considered because liquid CO2 is pumped into tubing whose temperature is much lower than that of formation. Span and Wagner (1996) presented a new equation of state for CO2 covering the fluid region from the triple-point temperature to 1100 K at pressure up to 800 MPa. Based on the SpaneWagner equation and Vesovic model (Vesovic and Wakeham, 1990), Wang and Ni (2013) investigated the heat and pressure transferring mechanism of CO2 in vertical wellbores. The preliminary research lays the foundation for this study. With consideration of the sand size distribution and eccentricity of the annulus, this paper investigates the sand transporting mechanism with CO2 in the whole inclined wellbore. For the hydraulics calculations, we developed a closed mathematical model to fully couple the hydrostatic pressure, temperature, density and viscosity of CO2 and friction. The study offers a theoretical basis for field application.
ensure accuracy. The coupling relationship is illustrated in this section. The implicit equation of CO2 density can be expressed as (Span and Wagner, 1996)
Pðd; tÞ ¼ rRT 1 þ dFrd
where d ¼ r/rc is the dimensionless reduced density and t ¼ Tc/T is the inverse reduced temperature. Dimensionless Frd is the partial derivative of the Helmholtz energy F(d,t). After the pressure P and temperature T are obtained, the density r can be calculated by numerical algorithms. The isobaric heat capacity of CO2 can be written as
2 o 1 þ d4rd dt4rdt M$cp 2 r ¼ t 4tt 4rr þ R 1 þ 2d4rd þ d2 4rdd
hðT; rÞ ¼ h0 þ DhðT; rÞ þ Dhc ðT; rÞ
(4)
where the zero-density viscosity h0 is in units of mPa.s and temperature T is in kelvin.
h0 ðTÞ ¼
1:00697T 1=2 G*h ðT * Þ
(5)
The reduced effective cross section G*h ðT * Þ is represented by empirical equation (6), T* ¼ T/251.196 K. 4 X i ln G*h T * ¼ ai ln T *
(6)
i¼0
The excess viscosity is in the following explicit form
Dhðr; TÞ ¼ d11 r þ d21 r2 þ
d64 r6 d r8 þ d81 r8 þ 82* *3 T T
(7)
Similar to the viscosity, the thermal conductivity of CO2 is given
In the sand-cleanout process, hypothermic liquid CO2 is pumped down to the bottom and then jetted into the tubing and casing annulus. Fills are flushed and dispersed, and then CO2 carries sand and flows back to the surface via the annulus. Sand can continue to penetrate into the annulus via perforations, and heat transfers from the formation into the tubing through the cement sheath, metal casing and annulus (Fig. 1). The inside diameter of the tubing is 50 mm, the outside diameter of the tubing (ro) is 60 mm and the inside diameter of the casing (Ri) is 100 mm. The eccentricity ε is defined as
D Rr
(3)
The viscosity of CO2 can be written as (Vesovic and Wakeham, 1990; Fenghour and Wakeham, 1998)
2. Mathematical models
ε¼
(2)
(1)
2.1. Physical properties of CO2 As documented in earlier research (Kolle, 2000; Gupta, 2005), CO2 changes into the supercritical state in bottom hole when the temperature is 304.2 K or higher and the pressure is above 7.38 MPa (Fig. 2). The phase change of CO2 is reflected by the thermodynamic properties changes. Li et al. (2011) experimental results show that the cutting/sand transporting capability of CO2 can be improved by reducing temperature or increasing pressure. The coupling relationship between temperature/pressure (attributed to well depth) and the thermodynamic properties of CO2 was considered to
by
lðT; rÞ ¼ l0 þ DlðT; rÞ þ Dlc ðT; rÞ
(8)
2.2. Mathematical model for heat transfer In heat transfer, both conductive resistance and convective resistance are involved, and it is assumed that the geothermal gradient is constant. Compared with the radial temperature difference between the annulus and formation, the axial temperature difference between neighbouring infinitesimal sections is negligible. The influence of sand on the temperature distribution is also neglected due to its low volume fraction. The energy equations in an infinitesimal section (Fig. 1) are represented as
Qea Qap ¼ cp mDTa Qap ¼ cp mDTp
(9) (10)
where Qea stands for the heat transferred from the formation rocks to CO2 in the annulus, and Qap is the heat transferred from CO2 in annulus to that in pipe, W. DTp is the temperature difference of CO2 between infinitesimal sections in the tubing, and DTa is the temperature difference in the annulus, K.
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
3
Fig. 1. Physical model of flow field.
With consideration of the throat effect, when CO2 flows through the jet connecting tubing and annulus, the pressure drop DPj(DPj ¼ P1 P2) and temperature drop DTj are expressed as
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 2 kþ1 #ffi u u 2k P2 k P2 k m ¼ AP2 t Rs T1 ðk 1Þ P1 P1
(13)
ZP2 DT ¼
mJT dP
(14)
P1
where A is the section area of the jet, m3. P1 and P2 are the pressure of the jet inlet and outlet, Pa. T1 is the temperature of the jet inlet, K. Isentropic coefficient k ¼ 1.28, specific gas constant Rs ¼ 0.1889 kJ/ (kg K), and throttle coefficient mJT ¼ 1/CP[T(vV/vT)P V]. 2.3. Mathematical model for multiphase flow The Eulerian model (Wang and Sun, 2009) solves a set of n momentum and continuity equations for each phase and takes interference among particles into consideration, which is not negligible in the flow field of a sand bed. The relative velocity ! v qp between particle phase and primary phase is given by
Fig. 2. Phase diagram of CO2.
Qea ¼
1
e ~ h2pr
Qap ¼
a1 l
Te Ta þ 2pl1 l ln rra2 þ 2pl1 l ln rra2e a1 ca
ce
Ta Tp 1 h pdi l
_
1 ln do þ þ 2pl l d t
(11)
i
(12)
1
e ~ hpd
ol
e ~ is the convective heat transfer coefficient of casing, W/ where h 2 (m K). r stands for the radius and d for the diameter, m. lca is the thermal conductivity of the casing, lce is the thermal conductivity of the cement sheath, and lt is the thermal conductivity of the pipe, _ W/(m K). h stands for the convective heat transfer coefficient of the inner wall of the pipe, W/(m2 K).
! v qp ¼ tqp ! a
(15)
v! vm ! ! vm a ¼ g ! v m $V! vt
(16)
tqp ¼
rm rp d2p 18mq fd
(17)
where ! a is the acceleration of a particle, m/s2. tqp is the relaxation time of a particle, s. fd stands for the drag force, N. The interaction force between two phases is expressed by the Schiller and Naumann model (Schiller and Naumann, 1933). The interaction force between two sand particles is illustrated by the Syamlal-O’Briensymmetric model, and the Brown force and Saffman lifting force are
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W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
both neglected because the particle size is much larger than 1e10 mm. Based on the continuity equation of the primary phase, the volume fraction equation of particles is given as (Wang and Wang, 2005; Tang et al., 2003)
v ap rp þ V$ ap rp vm ¼ V$ ap rp vdr;p vt
(18)
8 1 > > Qap Tp ðm þ 1ÞðiÞ ¼ Tp ðmÞðiÞ þ > > cm > > > < Ta ðm þ 1ÞðiÞ ¼ Ta ðmÞði1Þ i ¼ 1;2;3…; m ¼ 1;2;3…;n 1 > > ðiÞ ðiÞ > P ðm þ 1Þ ¼ P ðmÞ þ DP > p p p > > > : Pa ðm þ 1ÞðiÞ ¼ Pa ðmÞðiÞ þ DPa (20)
where ap is the volume fraction of the particle. The mean quality Pn ! ak r v k velocity ! v m is defined as ! v m ¼ k¼1r k , m/s. The drift velocity m ! v is given by ! v ¼! v ! v , m/s.
The k-3 model is utilized to illustrate the turbulence and close the equations, which is appropriate for compressible fluid flow.
With consideration of both the static hydraulic differential and flow pressure loss, DPp and DPa are related to the well structure and are calculated with the Fanning equation (Huang et al., 2011). The bottom-hole parameters are modified during iteration (as follows).
dr;p
dr;p
p
Tp ðnÞ Ta ðnÞ ¼ DTB Pp ðnÞ Pa ðnÞ ¼ DPB
(21)
m
! 8 > vðrkÞ v vk vk > > þ ruj ðmþmt Þ ¼ ttij Sij rεþQk > > vxj vxj vxj < vt ! > 2 > > > vðrεÞ þ v ru ε mþ mt vε ¼ 1:45 ε t S 1:92f r ε þQε > 2 j tij ij : vt vxj k 1:3 vxj k (19) where ttij ¼ 2mt(Sij Snndij/3) 2rkdij/3, and the eddy viscosity mt ¼ 0.09furk2/ε. The near wall attenuation functions are given as 2 ðRe2t Þ , where Re ¼ rk2/mε. fu ¼ eð3:4=ð1þ0:02Ret Þ Þ and f2 ¼ 1 0:3e t pffiffiffi The wall terms are given as Qk ¼ 2mðv k=vyÞ2 and Qε ¼ 2mmt/r(v2mε/ vy2)2. Sij stands for the mean-velocity strain-rate tensor, and dij stands for the Kronecker delta.
2.4. Sand size distribution In boundary conditions, the sand diameter is set based on the experimental statistical data. Sand produced from reservoir rocks (tight sandstone) was collected to determine their diameters by mass fraction (Table 1).
2.5. Solution procedure 2.5.1. Temperature and pressure distribution along the wellbore The wellbore is divided into many flow units (every 1 m) axially, and the mathematical model is solved by numerical methods from the wellhead to the bottom hole (as follows). In every flow unit, the temperature and pressure are considered to be constant so that the properties of CO2, the heat transfer and the pressure loss can be gained. According to field application, the backpressure, inlet temperature and inlet flow rate are given, whereas the inlet pressure and outlet temperature are assumed at first and then gained after iteration reaches convergence.
(
Ta ðnÞ* ¼ Tp ðnÞ DTB Pp ðnÞ* ¼ Pa ðnÞ þ DPB
(22)
The convergence condition is given by
8
T ðnÞ* T ðnÞ
a a > > > d < Tc
>
* > > : Pp ðnÞ Pp ðnÞ
d Pc
(23)
where d ¼ 0.001%, critical temperature of CO2 Tc ¼ 304.25 K and the critical pressure of CO2 Pc ¼ 7.38 MPa. Interactions continue from the bottom hole to wellhead, with Ta(n)*,Pa(n),Tp(n),Pp(n)* as the boundary conditions, and reverse as shown in Eq. (20) until the convergence condition is met.
8 ðiÞ ði1Þ > ðm 1Þ Tp ðm 1Þ ¼ Tp > > > > > > > < TaðiÞ ðm 1Þ ¼ TaðiÞ ðmÞ þ 1 Qea Qap cm i ¼ 2;3;4…; m ¼ 2;3;…n > > ðiÞ ðiÞ > > Pp ðm 1Þ ¼ Pp ðmÞ DPp > > > > : ðiÞ ðiÞ Pa ðm 1Þ ¼ Pa ðmÞ DPa (24)
2.5.2. Multiphase flow in the wellbore The annulus model is faceted axially and radially by a structured grid dividing method (Fig. 1), and it is assumed that sand penetrates into the annulus at a constant rate. The multiphase flow model is solved with the Eulerian model as described earlier, where the pressure-linked equations for the flow field are calculated by a semi-implicit method, and the velocity-linked equations are solved with a modified SIMPLE algorithm. 3. Model validation
Table 1 Experimental data of the particle size distribution. No.
Average diameter (mm)
Mass (g)
Mass fraction
1 2 3 4 5
0.95 0.74 0.53 0.39 0.18
1.7 7.1 4.4 14.4 26.2
0.0316 0.1320 0.0818 0.2676 0.4870
The well trajectory of the example is presented in Fig. 3. Initially, liquid CO2 of 20 C was pumped into the coiled tubing with a mass flow rate of 5.37 kg/s. The geothermal gradient and sand density were set as 2.8 K/100 m and 2.65 g/cm3. In the bottom hole, sand penetrates into the annulus constantly, and the volume fraction is 1% of the multiphase flow. To compare with the experimental results, 13 points along the
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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5
Fig. 4. Influence of the inclination on the sand cleanout efficiency.
Fig. 3. Schematic diagram of the well trajectory.
wellbore were selected to investigate the sand transporting efficiency, and the average value of a 9 m well section was numerically calculated for every point. Parameters related to the sand transporting efficiency in the annulus are list in Table 2, which are then used as the boundary conditions for the multi-phase flow. As CO2 flows upward in the annulus, the formation gets cooler, and CO2 is also cooled by the tubing in which CO2 flows downward. Finally, CO2 changes from the supercritical state into the liquid state in the annulus at a depth of 400 me500 m. The sand retention rate (SRR) was selected as the standard measure of the sand transporting efficiency, and it was defined as:
SRR ¼
mr rs Vf x rs Vf x
(25)
where mr stands for the real sand mass in the annulus, kg; rs is the density of sand, kg/m3; Vf stands for the volume of the annulus, m3; x presents the sand production rate and is given by the sand volume fraction at the annulus inlet, dimensionless. A larger SRR means it is more difficult to transport sand. The results show that SRR changes little along the wellbore in the vertical section. The distribution of SRR along the inclined wellbore is presented in Fig. 4. SRR increases with inclination at first and then decreases after a certain critical point. It is most difficult to transport sand when the inclination is 48 e72 . Laboratory experiments have been conducted to
investigate the influence of the inclination on the sand transport efficiency in the annulus with supercritical CO2 as the circulation fluid (Li et al., 2011). Researchers placed a certain amount of sand in the equipment, then started the CO2 pump and increased the displacement to investigate the sand mass that was retained in the equipment after a certain time. In the results, researchers took the critical flow rate (linearly correlated with displacement) that could transport 90% of the sand out of the equipment as the measuring standard. A higher critical flow rate means it is more difficult to transport sand at different inclinations. The experimental result is presented in Fig. 4. The trend of how multiphase flow develops is consistent with the simulation result, which validates the simulation model to some extent. Due to the differences in the boundary conditions, the inclination at which it is most difficult to transport sand differs from the experimental results to the simulation results. Except for the inclination, the other boundary conditions are set to the same values as in experiment (313.2 K/8.5 MPa), which means the physical properties of CO2 are the same in the different sections. The heat transfer is also neglected in the experiment. In the simulation, the density, viscosity and velocity of CO2 were coupled with the annulus temperature and pressure, and they all changed with well depth and inclination. Therefore, the boundary conditions in the simulation are consistent with the actual engineering. As depicted in Table 2, as the temperature and pressure along with depth changed, the flow rate and physical properties of CO2 changed accordingly. For example, it is more difficult to transport sand when the inclination is 72 than when it is 54 in the experiment. In contrast, more sand is transported out when the
Table 2 Flow field variables of the test points. No.
Depth m
Vertical depth m
Inclination
1 2 3 4 5 6 7 8 9 10 11 12 13
50 500 1000 1200 1400 1600 1800 1900 2000 2100 2200 2400 2550
50.0 500.0 1000.0 1198.5 1387.5 1560.5 1709.1 1772.0 1826.5 1872.0 1907.9 1949.6 1954.9
0 0 0 12 24 36 48 54 60 66 72 84 90
Temperature K
Pressure MPa
Density kg/m3
Viscosity 105 Pa$s
Flow rate m/s
279.5 305.0 317.7 322.7 327.3 331.5 335.1 336.6 337.9 339.0 339.8 340.7 340.9
6.740 10.16 13.79 15.23 16.59 17.86 18.95 19.41 19.82 20.17 20.44 20.81 20.88
865.8 705.1 695.8 694.0 693.0 693.2 693.3 693.5 693.9 694.3 694.9 696.8 697.5
7.996 5.643 5.585 5.582 5.596 5.617 5.637 5.646 5.658 5.668 5.679 5.707 5.717
1.230 1.510 1.531 1.534 1.537 1.536 1.536 1.536 1.535 1.534 1.532 1.528 1.527
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inclination is 72 in the simulation. The reason is that the density of CO2 increases as the inclination and depth increase; thus, the sand transporting capability of CO2 is improved. 4. Sand cleanout mechanism of CO2 Quantitative calculation was conducted to investigate the difference between water and CO2 with respect to sand transporting capability. In the calculation, the boundary conditions were the same as No. 11 in Table 2 and sand diameter were all set as 0.3 mm. The results show that the SRR when transporting sand with CO2 (22.0%) is 1.85 times that with water (11.9%) because the viscosity of water is more than 10 times larger than that of CO2, so the Reynolds of water flow is smaller and the boundary layer is thicker in the radial direction. With respect to Wang’s (2009) force model, the drag force applied on the near-wall hemisphere of sand is smaller than that on the other side; thus, sand reverses to the bottom of the boundary layer along the low-side hole. For turbulent CO2 flow, the difference of the drag force applied on the near-wall hemispheres and the other is smaller and gravity is dominant, so that sand tends to assemble at the bottom side of the annulus (Fig. 5). A higher sand volume fraction means the sand particles collide more frequently; thus, more dynamic energy is consumed and SRR finally increases. Because the viscosity of water is not sufficient to clean the wellbore, it is necessary to add viscosifier into CO2 to increase its sand transporting capability. The pressure drop of sand cleanout with CO2 is 52.3% that with water, which confirms the advantages of the mentioned field attributes. The sand size distribution was also investigated to reveal the mechanism of how the sand bed was formed at different sections. As is shown in Fig. 6, SRR do not have a simple positive correlation with the diameters. The production rate of 0.18 mm sand at the inlet is much higher than the others, and the particles collide with other particles and the wall more frequently. Finally, more dynamic energy is consumed and its SRR is the largest. The production rate of 0.74 mm sand is a little smaller than that of 0.53 mm sand, but the SRR of 0.74 mm sand is larger than that of 0.53 mm sand because of the gravity difference. The comparison of the SRR of 0.39 mm sand and 0.74 mm sand indicates that the SRR of heavier sand (0.74 mm) is larger in the lightly deviated section. In the highly deviated section, 0.39 mm sand (with a larger production rate at the bottomhole) is more difficult to transport. The section where sands of different diameters are most difficult to transport varies from 48 to 66 . The sand transporting process is influenced by gravity and momentum exchange (caused by collision), and in the lightly deviated section, sand disperses in the annulus uniformly and
Fig. 6. Distribution of sand diameters along wellbore.
gravity is dominant. As in the highly deviated section, sand assembles at the bottom side of the annulus and collides frequently; thus, momentum exchange along with dynamic energy consumption dominates the sand transporting process. 5. Analysis of the influencing factors 5.1. Displacement In the drilling field, displacement can be translated as “the CO2 volume that is pumped into the tubing/pipe in a second”. The annulus flow rate has a linear positive correlation with displacement. As shown in Fig. 7, the SRR decreases as the annulus flow rate increases with a trend that develops fast at first and then slows. The critical flow rate is approximately 3.0 m/s in this study. Sand should be transported with a flow rate no larger than the critical value in field application to control cost. MVT theory has been widely used as a standard in air drilling to transport cuttings effectively and economically. This theory has been well-documented in earlier research (Angel, 1957; Doan et al., 2003) and provided engineers a convenient method to calculate the displacement for gas drilling of different density (related to pressure and temperature). CO2 is in the supercritical state rather than gaseous state at the bottom-hole; therefore, we propose to establish similar theory to meet the need for cleanout application with CO2. In addition to the density, the viscosity of CO2 has a non-negligible influence on the sand transporting efficiency; thus, the proposed theory should also take the
Fig. 5. Comparison chart of the sand volume fraction when transporting sand with CO2 and water.
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
7
Fig. 7. Influence of annulus flow rate on sand retention rate.
Fig. 9. Influence of the eccentricity on the sand retention rate.
viscosity of CO2 (changes significantly with pressure and temperature) into consideration to ensure accuracy.
increase of the eccentricity due to the influence of the boundary layer. When the pipe is close to the casing, more sand assembles in the bilateral area of the pipe where the average velocity of CO2 is influenced less by the solid wall; thus, more sand is transported out. In field applications, engineers can search from the area near the critical eccentricity to find the sticking point.
5.2. Sand production rate The sand production rate (measuring the velocity of sand penetrating into the annulus) is given by the sand volume fraction at the bottom-hole. As presented in Fig. 8, the SRR increases with the increase of the production rate with a trend that grows increasingly fast. As more sand particles penetrate into the annulus, they assemble at the low side of the annulus and collide with each other more frequently. Therefore, more dynamic energy is consumed and more sand stagnates to form a sand bed. 5.3. Eccentricity In inclined wellbores, pipes tend to settle upon the low side of the annulus due to the influence of gravity. The influence of the eccentricity on the sand transporting efficiency is also investigated, and the result is presented in Fig. 9. As the eccentricity increases, the SRR increases at first and then decreases when the eccentricity is larger than 0.8. The mean velocity of CO2 in the low side of the annulus (MVLS) where most sand assembles has a strong influence on the sand cleanout efficiency. At first, MVLS decreases with the
6. Conclusions (1) A closed mathematical multi-phase flow model, which fully couples the hydrostatic pressure, temperature, physical properties of CO2 and friction, is solved and verified with experiments to investigate the sand transporting mechanism of CO2. The effects of various factors on CO2’s sand transporting efficiency, including displacement, the sand production rate and the annulus eccentricity, are also analysed. (2) Sand with larger diameter is more difficult to transport in vertical wellbores, whereas in highly deviated sections, sand with larger production rate is more difficult to transport. It is most difficult to transport sand when the inclination is approximately 48 e72 . A general decreasing trend in the SRR, which develops fast at first and then slows, was observed with increasing displacement. The increasing trend in the SRR with increasing sand production rate grows faster and faster. As the annulus eccentricity increases, the SRR increases at first and then begins to decrease when the eccentricity reaches 0.8. The pressure drop of sand cleanout with CO2 is 52.3% that with water, which confirms the advantages (e.g., less leakage) of the mentioned attributes. (3) The findings of this study predict the necessity of developing a new sand/cutting transporting theory to facilitate field engineers. Because the SRR when transporting sand with CO2 is 1.85 times that with water, it is advised to add viscosifier to improve the sand cleanout efficiency. Acknowledgements
Fig. 8. Influence of the production rate on the sand retention rate.
The financial support from the Major State Basic Research Development Program of China (2014CB239202), the Doctoral Fund of Ministry of Education (20120133110011 and 20130133110006) and the Fundamental Research Funds for the Central Universities (27R1402019A) is highly appreciated. We sincerely thank our colleagues at the Water Jet Research Centre in China University of Petroleum (East China) for helping with the theory research.
Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017
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W. Song et al. / Journal of Natural Gas Science and Engineering xxx (2015) 1e8
Nomenclature ro Ri D ε
d t P T
h0 r cp M R T*
l0 Qea Qap
DTB DTa e ~ h
lca lce lt _
h
A P1 P2 T1 Rs ! a
tqp fd ! v qp DTj DPj
ap
! vm ! v p
SRR mr
rs
Vf x
outside diameter of the pipe, mm inside diameter of the casing, mm distance between the pipe centre and the casing centre, mm eccentricity, dimensionless reduced density, dimensionless inverse reduced temperature, dimensionless pressure, MPa temperature, K zero-density viscosity, mPa$s density, kg/m3 isobaric heat capacity, J/(kg$K) molecular weight, kg/mol universal gas constant, R ¼ 8.314 Pa m3 K1$mol1 T/251.196 K, dimensionless thermal conductivity, W/(m K) heat transferred from formation rocks to CO2 in the annulus, W heat transferred from CO2 in the annulus to that in the pipe, W temperature difference of CO2 in pipe between infinitesimal sections, K temperature difference of CO2 in the annulus between infinitesimal sections, K convective heat transfer coefficient of the casing, W/ (m2 K) thermal conductivity of the casing, W/(m K) thermal conductivity of the cement sheath, W/(m K) thermal conductivity of the pipe, W/(m K) convective heat transfer coefficient of the inner wall of the pipe, W/(m2 K) section area of the jet, m3 pressure of the jet inlet, Pa pressure of the jet outlet, Pa temperature of the jet inlet, K specific gas constant, Rs ¼ 0.1889 kJ/(kg K) acceleration of a particle, m/s2 relaxation time of a particle, s drag force, N relative velocity, m/s temperature drop of the jet pressure drop of the jet volume fraction of particles, dimensionless mean quality velocity, m/s velocity of a particle, m/s sand retention rate, dimensionless real sand mass in the annulus, kg density of sand, kg/m3 volume of the annulus, m3 sand production rate, dimensionless
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Please cite this article in press as: Song, W., et al., Reprint of: Multiphase flow mechanism of sand cleanout with supercritical carbon dioxide in a deviated wellbore, Journal of Natural Gas Science and Engineering (2015), http://dx.doi.org/10.1016/j.jngse.2015.11.017