Wellbore modeling for hybrid steam-solvent processes

Wellbore modeling for hybrid steam-solvent processes

Fuel 188 (2017) 50–60 Contents lists available at ScienceDirect Fuel journal homepage: www.elsevier.com/locate/fuel Full Length Article Wellbore m...

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Fuel 188 (2017) 50–60

Contents lists available at ScienceDirect

Fuel journal homepage: www.elsevier.com/locate/fuel

Full Length Article

Wellbore modeling for hybrid steam-solvent processes Rafael V. de Almeida, Hamid Rahnema ⇑, Marcia D. McMillan Petroleum and Natural Gas Engineering, New Mexico Tech, 801 Leroy Place, Socorro, NM 87801, United States

a r t i c l e

i n f o

Article history: Received 10 July 2016 Received in revised form 27 August 2016 Accepted 3 October 2016

Keywords: Steam/solvent Injection Wellbore Numerical modeling Thermal Compositional

a b s t r a c t The addition of hydrocarbon solvents to the cyclic steam stimulation (CSS) or steam assisted gravity drainage (SAGD) processes has recently gained significant interest from the petroleum industry. In these processes, a proper selection of solvent is critical: injected solvent must be in the vapor phase at the injection point in order to propagate inside the steam chamber and condense at the steam/oil interface to effectively reduce oil viscosity. Therefore, the wellbore have to deliver vaporized solvent near its dew point at perforation intervals. This work provides a detailed numerical formulation to predict steam and solvent qualities, temperature, and pressure profiles along the wellbore. Four phases were considered: hydrocarbon liquid and vapor phases, and aqueous liquid and vapor phases. The mass, energy and momentum balance equations are integrated with drift-flux model and discretized over the wellbore domain. Unknowns and governing equations are divided into the sets of primary and secondary equation and unknowns are solved sequentially. The model was compared against previously published models and field data. The data from two steam injection wells and two gas condensate production wells were used for validation. Also, case studies are presented to investigate the temperature and condensation behavior of the solvent-steam mixture. The use of this model will assist the industry in proper wellbore design and the engineering of injection constraints in hybrid steam/solvent injection processes. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction In hybrid steam-solvent injection processes, both steam and solvent are co-injected in different proportions to reduce the viscosity of heavy oil or bitumen by heating and dilution, which improves oil displacement, drainage, and consequently the production rate. To have a proper solvent placement at the edge of the steam chamber, the injected solvent must be in the vapor phase at the injection point in order to properly propagate inside the steam chamber. Also, the solvent needs to condense at the steam/oil interface to reduce oil viscosity effectively. Therefore, wellbore must deliver a vaporized solvent near its dew point to the perforation intervals. Predicting steam and solvent thermodynamic properties in the flowing wellbore can significantly improve the design of surface facilities, wellbore system, solvent selection, and injection constraints. When the solvent-steam mixture is injected into the wellbore, both the pressure and temperature of the injected fluid vary by time and depth. Heat exchange between the wellbore system and the lower temperature formation, change ⇑ Corresponding author. E-mail address: [email protected] (H. Rahnema). http://dx.doi.org/10.1016/j.fuel.2016.10.013 0016-2361/Ó 2016 Elsevier Ltd. All rights reserved.

in hydrostatic pressure, phase velocities, and friction determines the steam and solvent thermodynamic conditions at the sand face. Thermal modeling of wellbore heat transmission dates back to the late 1950’s, where Lesem et al. [1] presented an analytical model for calculation of bottom hole temperature in gas production wells. Similarly, Moss et al. [2] developed the solution to a system of equations predicting the temperature profile of the injected water in the wellbore based on some simplifying assumptions. Later on, Ramey [3] improved upon the previous studies on wellbore heat transmission. He developed an approximate solution considering a steady-state, non-compressible, and single phase fluid flow in the wellbore In his model, kinetic energy, and frictional pressure drop were neglected. To define an overall heat transfer coefficient, Ramey considered steady-state axial heat convection through the wellbore with transient heat conduction into the formation. This coefficient was assumed to be constant along the wellbore. Later, Willhite [4] improved the accuracy of the Ramey’s model by considering depth-dependence overall heat transfer coefficient. Satter [5] extended the Ramey model to wet steam flow in the wellbore. In all of these models, it is assumed that there are no pressure changes with respect to the depth and only temperature and steam quality varies along the wellbore.

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51

Nomenclature A Ap C0 dt f f c;p f ðt D Þ g gc hf hPi hPo hrc;an H_ p hp Jc nc n_ c;p n_ h;c n_ h;p NG n_ h;p q_ p Q_ f r Ea r ci r co ri r ins ro rw R Rh t Ta Tb T EA tD up Phc Pw P zw;g

tubing inside area (ft2) area occupied by phase p (ft2) distribution coefficient (dimensionless) inner diameter of tubing (ft) friction factor (dimensionless) fugacity of component c in the phase p (psia) time conduction function (dimensionless) acceleration due to gravity (32.17 ft/s2) unit conversion factor (32.17 ft-lbm/lbf-s2) film coefficient of heat transfer between fluid inside pipe and the pipe (Btu/ft2-day-°F) coefficient of heat transfer across any deposits of scale or dirt at the inside wall of the pipe (Btu/ft2-day-°F) coefficient of heat transfer across the contact between pipe and insulation (Btu/ft2-day-°F) radiation and convection coefficient of heat transfer in the annulus (Btu/ft2-day-°F) enthalpy rate per bulk volume of phase p (Btu/day-ft3) enthalpies per unit mass of phase p (Btu/lbm) unit conversion factor (778 ft-lbf/Btu) number of hydrocarbon components molar flux of component c in phase p (lb-mole/day) total hydrocarbon molar flux in phase p (lb-mole/day) in-situ molar flux of hydrocarbon in phase p (lb-mole/day) total number of wellbore grid blocks in-situ molar flux of hydrocarbon in phase p (lb-mole/day) molar rate per unit volume of phase p (ft/day) heat loss rate per unit length (Btu/day-ft) radius of altered zone in the formation near the wellbore (ft) inner radius of casing (ft) outer radius of casing (ft) inner radius of tubing (ft) insulation radius (ft) outer radius of tubing (ft) well radius (ft) universal gas constant (psi-ft3/(lbmol-°R)) average thermal resistance per unit length (ft-day-°F/Btu) time (days) ambient temperature (°F) bulk temperature of the fluid flowing in the tubing (°F) surrounding formation temperature (°F) dimensionless time (dimensionless) internal energy per unit mass of phase p (Btu/lbm) hydrocarbon partial pressure (psia) water partial pressure (psia) wellbore Pressure (psia) compressibility factor of steam (dimensionless)

Holst et al. [6] added the effects of frictional pressure drop in heat loss calculations to the model. In 1969, Earlougher [7] employed two–phase flow pressure drop calculations (Hagedorn and Brown [8]) to account for the pressure changes for wet steam injection. In his calculations, slippage between phases was neglected. In 1981, Fontanilla and Aziz [9] presented a method to calculate steam quality and steam pressure by considering two-phase flow and slippage using the Beggs and Brill correlation [10]. Later, Farouq Ali [11,12] and Wooley [13] developed numerical models which were able to integrate a wider range of formation and well complexities. These models con-

zh;g T

vd vl vm vp v p;i v sg v sl v sh;g v sh;p v sw;g v sw;p

xc,p

xw,g xw,w z zc

compressibility factor of hydrocarbon (dimensionless) temperature (°F) drift velocity of gas in liquid (water) (ft/day) liquid (water and oil) velocity (ft/s) average velocity of the mixture (ft/day) superficial velocity of phase p (ft/day) velocity of phase p at the grid block i (ft/day) gas phase superficial velocity (ft/day) liquid phase superficial velocity (ft/day) hydrocarbon superficial velocity in gas phase (ft/day) superficial velocity of hydrocarbon in phase p (ft/day) superficial velocity of water in gas phase (ft/day) superficial velocity of water in phase p (ft/day) hydrocarbon component mole fractions in the phase p (fraction) mole fractions of water in gas phase (fraction) mole fractions of water in water phase (fraction) elevation or depth (ft) overall hydrocarbon component mole fraction (fraction)

Greek letters ap in-situ volume fraction of phase p (dimensionless) al in-situ volume fraction of liquid phase (ao þ aw Þ (dimensionless) aE thermal diffusivity of the earth (ft2/day) h inclination of segment from horizon (radian) kins thermal conductivity of insulation (Btu/ft-day-°F) kP thermal conductivity of pipe (Btu/ft-day-°F) kcem thermal conductivity of cement (Btu/ft-day-°F) thermal conductivity of earth (Btu/ft-day-°F) kE kEa thermal conductivity of altered earth zone (Btu/ft-day-°F) q density (lb/ft3) u_ c molar flux of component c at the wellhead (lb-mole/day) u_ hc total hydrocarbon Molar flux at the wellhead (lb-mole/day) Acronyms MSCF thousand standard cubic feet STB stock tank oil barrel Superscripts c component g gas i grid block index m mixture o oil p phase w water

sider directional heat transmission in the formation with different well operational constraints. Their studies showed the importance of vapor phase slippage and flow pattern for predicting the temperature, pressure drop, and steam quality during the downward wet steam injection. Sagar et al. [14] calculated the temperature of saturated steam in deviated wells using modified Ramey model. They also included the Joule-Thomson effect due to pressure drops in the tubing. Stone et al. [15,16] extended the wellbore thermal models to integrate both the wellbore and the reservoir. In their formulation, fluid flow through the reservoir was approximated with

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single-phase radial flow and axial representation of the wellbore. Energy, momentum, and mass conservation equations were fully coupled in a numerical approach. Stone’s model was further improved by Livescu et al. [17–19] where they coupled reservoir multi-phase flow and wellbore thermal model using a numerical method. Bahonar et al. [20] presented a semi-unsteady state wellbore model to simulate the downward flow of the steam and water mixture, accounting for slippage between the phases and considering axial and radial heat transfer in the formation. Recently, Xiong et al. [21] developed a thermal wellbore simulator focused on the improvement of the heat loss calculations along the horizontal section of the wellbore. You et al. [22,23] provided nudmerical solution for wellbore heat transmission and showed that at early time, Ramey model may produced significant error in downhole tempertaure calculation. Apart from wellbore thermal and pressure modeling, efforts have been made to investigate the effect of non-condensable gases (i.e. N2 and CO2), as flowing with wet steam inside the wellbore. Using numerical methods, Barelli et al. [24] studied the effects of CO2 on the temperature and quality of produced steam in geothermal wells. They showed that the presence of CO2, even in very low concentrations, can significantly shift the condensation depth and the steam quality profile. Inspired by earlier works on geothermal wells, Fidan [25] presented the calculations of heat transmission, pressure drops, and steam quality during the co-injection of steam with the non-condensable gas additive. Despite significant studies, there is still no model that simultaneously predicts compositional variation, heat losses, and pressure changes along the surface pipeline and wellbore when steam and solvent are co-injected. This paper moves one step forward to develop a semi-transient model for thermal and compositional wellbore simulation of hybrid steam-solvent injection. 2. Model formulation This work covers two-phase flow pressure changes due to gravity, friction loss, and kinetic energy; enthalpy and temperature variation caused by heat losses; and the compositional variation of the injected fluid. The numerical formulation is based on the physical model illustrated in Fig. 1: a vertical injection wellbore, cased down to the top of injection intervals. Injected fluid is flowing through the tubing with a constant diameter. The formation has homogenous thermal and physical properties. It is assumed that at the wellhead, injection pressure, temperature, steam quality, steam-solvent ratio, injection rates, and geothermal gradient are known. The major assumptions and approximations are given below;

" # " # " # X @ X @ X q ap xc;p þ q ap v p xc;p þ xc;p q_ p ¼ 0 @t p p @z p p p c : 1; . . . ; nc

p : o; g

ð1Þ

X @ X @ X q ap xw;p þ q ap v p xw;p þ xw;p q_ p ¼ 0 p : w; g @t p p @z p p p

ð2Þ

where p refers to the three phases of oil, gas and water, xc;p is the hydrocarbon component mole fractions in the phase p,ap is volume A fraction of phase p (ap ¼ Ap ), q_ p is the phase molar flux per unit vol-

ume between the well and the reservoir (source/sink term), v p is phase velocity and qp is molar density of the phase p. Hydrocarbon components does not present in the water phase and water component only present in water and gas phases. Thus, xw;w is equal to one. In Eq. (1), xw;g is defined as ratio of molar flux of water in gas phase (n_ w;g Þ to the hydrocarbon molar flux in gas phase (n_ h;g Þ:

n_ w;g n_ w;g qw v sw;g xw;g ¼ Pnc ¼ ¼ _ qg v sh;g n_ h;g c nc;g

ð3Þ

In Eq. (3), v sw;g is superficial velocity of water and v sh;g is the hydrocarbon superficial velocity, both in the gas phase. The presence of hydrocarbon components in gas phase decreases the partial pressure of the steam. Total pressure of system (P) is summation of the water partial pressure (P w ) and hydrocarbon partial pressure (Phc ) (Dalton’s law):

P ¼ Pw þ Phc

ð4Þ

The following equation relates the partial pressure of the steam to the total pressure of the system [26–28]. zw;g n_ w;g RT

Pw Pw V ag ¼ ¼ P Pw þ P hc zhg n_ h;g RT þ zw;g n_ w;g RT V ag

ð5Þ

V ag

or

Pw ¼

zw;g xw;g P zh;g þ zw;g xw;g

ð6Þ

where V is the unit control volume, zhg and zw;g are the hydrocarbon and water gas compressibility factor, respectively. In a situation where water component appears in both liquid and gas phases (saturated water), the partial pressure of the water is coupled with the temperature of the system, assuming there is no temperature gradient between phases.

Pw ¼ Psat ðTÞ

ð7Þ

2.2. Momentum Conservation Equation  Fluid flow in the tubing is steady sate with constant mass flow rate.  Axial heat conduction in the wellbore and vertical heat conduction in the formation are neglected.  Heat transmission to the formation is transient and in the radial direction.  Slippage between liquid solvent and liquid water is ignored, meaning that they flow with the same velocity.  Injected solvent is insoluble in the aqueous phase, and aqueous phase does not impact the phase behavior of the solvent. 2.1. Mass conservation equation The mass conservation equations for each hydrocarbon component and for water, respectively, are as follow:

Drift-flux model was employed to determine the slippage and to calculate phase velocities [29].

v g ¼ C0v m  v d

ð8Þ

where v d is the gas phase drift velocity respect to liquid phase, C0 is the distribution coefficient and v m is average velocity of the mixture (v m ¼ v sl þ v sg Þ. In Drift-flux model, slippage between gas and liquid phases is expressed based on the uneven distribution of gas phase through the pipe cross section and buoyancy. These two effects were integrated to calculate in-situ gas phase velocity as summation of C0 v m and v d respectively. The values of C 0 and v d can be calculated based on the different flow patterns [29]. In this work, slippage between oil and water (liquid phases) was neglected, assuming these two phases flow with same velocity (v l ¼ v o ¼ v w Þ.

53

Temperature

R.V. de Almeida et al. / Fuel 188 (2017) 50–60

Surrounding Formaon Cement Casing Annulus Insulaon Tubing

Tb

T Ea

(a)

(b)

Fig. 1. Schematic of wellbore cross section and temperature profile. Wellbore includes tubing, insulation, annulus, casing and cement zone, all with different thermal resistant.

Gas and liquid in-situ volume fraction can be calculated using the following equations [17,30,31]:

ag ¼

vl ¼

v sg

ð9Þ

C0v m  v d 1  ag C 0 a vm þ g vd 1  ag 1  ag

ð10Þ

al ¼ ao þ aw ¼ 1  ag

ð11Þ

There are two main approaches in the literature to compute momentum balance along the fluid conduit. In one approach (Two-fluid model) momentum and mass are balanced for each phase (e.g., gas and liquid) including the interaction between the two phases. Thus, viscous stress, turbulent stress, diffusion stress and Interphase mass transfer terms appear in the basic equations. This model gives the most detailed description of two-phase flow transient momentum balance. However, the stability problems and challenges in describing the interphase interaction are the major drawbacks [32–35]. The alternative and relatively simpler method is to lump the constitutive equations of the phases using averaging techniques and reduce the basic equations to one momentum balance relation for entire fluid (mixture model). In this work, momentum balance is based on mixture model and drift-flux approach was used to account for velocity differences. The transient momentum balance for multi-phase flow inside the tubing can be expressed as follow [35–38]:

@p @ @ X  qm g sinh þ ðqm v m Þ þ 144g c q ag v 2p @z @t @z p p

!

f q v2 þ m m¼0 2dt ð12Þ

where @p is the pressure gradient along the tubing, g is the [email protected] tional acceleration, h is the angle of the wellbore segment with the horizon, dt is the inner diameter of the tubing, and f is the friction factor, calculated based on the Beggs and Brill two-phase flow friction factor correlation [10]. The mixture density (qm Þ and the mixture velocity (v m Þ are defined by Eqs. (13) and (14). v m represents the center of mass velocity (mixture velocity).

qm ¼

X

ap qp

ð13Þ

ap qp v p qm

ð14Þ

p

vm ¼

P p

One of the important parameter for the pressure gradient computation is the interfacial tension between phases. This property is calculated using Katz et al. correlation [39]. 2.3. Energy conservation equation The temperature of the gas and liquid phases flowing in the wellbore is assumed to be equal. Thus, the transient energy balance equation can be expressed as follow [18,40]

( " !# ) ( " !#) @ X 1 v 2p @ X 1 v 2p ap qp up þ qp ap v p hp þ þ @t 2 Jc g c @z p 2 Jc g c p þ

X qp ap v p g sin h p

788g c

þ

Q_ f X _ Hp ¼ A p

p : o; g; w

ð15Þ

where v p is phase velocity, up and hp are internal energy and the enthalpy of phase p respectively, Q_ f represents heat transfer rate

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R.V. de Almeida et al. / Fuel 188 (2017) 50–60

between wellbore and surrounding formation, A is the inner cross section area of the tubing and H_ p is enthalpy sink/source term of phase p. On the right side, the first term represents the energy accumulation and other two terms represent convective energy flux and gravitational force work. The conductive heat transfer along the wellbore and work rate done by the viscous force is ignored. The enthalpy calculation procedure for the hydrocarbons solvents was retrieved from data provided by Reid et al. [26], where the gas base model was used to calculate enthalpies. The steam liquid and gas enthalpies were calculated from International Association for the Properties of Water and Steam [41].

where hrc;an is radiation and convection coefficient of heat transfer in the annulus, kcem is the heat conduction coefficient of cement and kEa and kE are the heat conduction coefficient of altered and unaltered part of the surrounding formation, respectively. The calculation procedure for hrc;an has been discussed in details by Prats [43]. Also, Wilhite [4] provided values of f ðtD Þ as function of dimensionless time, tD :

tD ¼

aE t

ð20Þ

r 2w

where aE is the thermal diffusivity of the surrounding formation, r w is the well radius and t is the injection time.

2.4. Phase equilibrium 3. Numerical implementation It is assumed an immediate local thermodynamic equilibrium in the wellbore. This condition requires phase equilibrium and equality of chemical potentials (partial molar Gibbs free energies), which can be expressed in terms of fugacity. Equality of phase fugacities allows relating mole fraction of each component in different phases. Since hydrocarbon components exist in oil and gas phases, Eq. (16) is describing equilibrium condition between the two phase of oil and gas:

f c;o  f c;g ¼ 0;

c : 1; . . . ; nc

ð16Þ

where f c;o , and f c;g , are the fugacities of hydrocarbon component c in the oil and gas phases. Fugacity of hydrocarbon component in each phase is only function of hydrocarbon partial pressure, temperature and phase composition f c;p ¼ f ðP hc ; T; xc;p Þ. In addition to Eq. (11), there is a composition constraint for each phase that needs to be satisfied in flash calculations. nc X

xc;p ¼ 1;

p ¼ o; g

ð17Þ

2.5. Wellbore heat transmission Thermal modeling of the wellbore was based on the earlier works done by Ramey [3] and Willhite [4]. In this calculation, heat transmission from the tubing to the formation sand face is considered to be steady state. This means that any temperature changes in the tubing can be felt immediately at the cement behind the casing. This is a valid approximation, as noted by Ramey [3], if we assume that heat transmission through the wellbore is rapid compared to heat flow through the formation. The heat transfer into the formation occurs under transient condition. Fig. 1 shows different thermal resistances between the inside of the tubing and the formation sand face, including heat resistances of scale deposited inside the tubing, tubing wall, insulations, casing and cement. The basic equation used to calculate heat loss per unit length of pipe (Q_ loss Þ is [3,42]:

wellbore temperature profile and consequently, impact the other wellbore properties. With these assumptions, Eq. (1) can be simplified to:

ð21Þ

1

−1

     1 1 1 1 ro 1 1 r ins þ þ þ ln þ ln 2p hf r i hPi r i kP hPo ro kins ri ro        1 1 r co 1 rw 1 rEa f ðtD Þ þ þ ln þ ð19Þ þ ln ln þ hrc;an r ins kP kcem kEa kE r ci r co ro

,

,

ð18Þ

where T b is the bulk temperature of the fluid flowing inside of the tubbing, T EA is the surrounding formation temperature (linear function of depth) and Rh is average thermal resistance, which is calculated by Eq. (19). The wellbore heat loss doesn’t reach steady state condition. The rate of heat loss is monotonically decreasing as a function of time [3,42]:

Rh ¼

time, the governing equations are steady state (no accumulation for mass, momentum or energy) and all other parameters are fixed including Q_ f . As time advances, Q_ f changes and this will alter the

i @ h qp ap v p xc;p ¼ 0 @x

c

ðT b  T EA Þ Q_ f ¼ Rh

The wellbore is discretized into a number of staggered grids. The temperature, pressure, and phase properties are calculated at the center of the grid with the mixture velocity defined at the upper (exit) boundary of the cell, as shown in Fig. 2. The use of staggered grids helps to avoid interpolation of pressure in momentum and of phase velocities in mass conservation equations [44]. In this work, wellbore flow is considered steady state, while heat transfer to the formation is unsteady state radial conduction. This assumption leads to a semi-transient condition. Transient heat conduction to the surrounding formation leaves a time dependent parameter in the energy balance equation (i.e., Q_ f ). At any given

+1

,

,

,

,

, +1

Fig. 2. Schematic of the wellbore discretization with staggered grid blocks.

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R.V. de Almeida et al. / Fuel 188 (2017) 50–60

Hydrocarbon components are only present in oil and gas phases, therefore:

q o a o v o;i xc;o þ q g a g v g;i xc;g  u_ c ¼ 0

ð22Þ

_ c is the molar flux of component c at the wellhead. The where u overall hydrocarbon component mole fractions at the wellhead can be expressed as:

u_ c zc ¼ Pnc c

u_ c

X

u_ c ¼ q o a o v o;i þ q g a g v g;i  u_ hc ¼ 0

ð24Þ

Or

q o vsh;o þ q g vsh;g  u_ hc ¼ 0

ð25Þ

q o v sh;o ¼ n_ h;o

ð26Þ

q g v sh;g ¼ n_ h;g

ð27Þ

_ hc is the total hydrocarbon molar flux, v sh;o and v sh;g are where u superficial velocity of hydrocarbon, n_ h;o and n_ h;g are the in-situ molar flux of hydrocarbon in oil and gas phase. Since water component presents at water and gas phases, Eq. (2) can be reduced to:

q w v sw;w þ q g v sw;g  u_ w ¼ 0

ð28Þ

The discretized forms of the momentum and energy balance equations are given by [17,45]:

144g c ðP i  Pi1 Þ  qm g sinh  i h i P h  g v 2p  pa q p a g v 2p  q p f q v2 iþ1 i þ þ m m¼0 Dz 2dt ( " !# 1 X 1 v 2p q p a p v p hp þ 2 Jc g c Dz p þ

O il

ð23Þ

The combination of Eq. (22) and (23) imply that the overall mole fraction of hydrocarbon components at any location along the wellbore is constant and equal to the overall mole fraction at the wellhead. It is convenient to sum up the Eq. (22) for all the hydrocarbon components;

q o a o v o;i þ q g a g v g;i 

Gas

X qp ap v p g sinh Q_ f þ ¼0 Jc g c A p



iþ1

" X

q p a p v p

p

ð29Þ 1 v 2p hp þ 2 Jc g c

!# ) i

ð30Þ

 p and q  p are calculated at the boundary from the variables where a of the neighboring grids using standard Godunov first order implicit upwinding scheme [44]:

iþ1 ¼ aiþ1 b þ ð1  aiþ1 Þb b i iþ1 b is either ap or qp and the value of aiþ1 is equal 1 (if equal 0 (if v p;iþ1 < 0Þ.

ð31Þ

v p;iþ1 > 0Þ or

3.1. Primary variables and equations According to Fig. 3, In order to determine both intensive and extensive state of the system, the number of variables will be 2nc þ 8:

fX 1;g ; . . . ; X nc;g ; X w;g ; X 1;o ; . . . ; X nc;o ; P; Phc ; Pw ; T; ao ; ag ; aw g To improve the computation time, the total number of variables is divided into primary (state variables) and secondary variables. Two independent variables ðP; TÞ were selected as primary variables that are calculated from momentum and energy conservation equations for each grid block. The secondary variables are dependent and calculated as a function of primary variables. Total governing and constrain equations were divided into two groups,

Water

P, T

(2)

x1,g ,…, xnc,g , xw,g , αg , Phc

(nc+3)

x1,o ,…, xnc,o , αo

(nc+1)

xw,w =1 , αw , Pw

(2)

Total variables: (2 nc + 8) Fig. 3. 2nc þ 8 variables need to be determined to fix both intensive and extensive state of each grid block.

primary and secondary equations. The momentum and energy (Eqs. (29) and (30)) as primary equations and the rest of equations as secondary. In the following, the procedure of calculating secondly variables are explained for the condition where three phase of water, oil and gas are present: 1. Knowing T and P of the system; Partial pressure of the water phase (Pw Þ is determined by Eq. (7). Intensive properties of water and steam are calculated using International Association for the Properties of Water and Steam (IAPWS) [41] correlations. Phc is calculated from Eq. (4). 2. Hydrocarbon composition in oil and gas phases, as well as their molar fluxes (i.e., n_ ho and n_ hg ) are computed using flash calculations (SRK EOS) [46]. 3. Superficial velocities of hydrocarbon in gas and oil phases (v sh;o ,v sh;g ) are calculated with Eqs. (26) and (27). 4. Eqs. (6) and (3) are used to find xw;g and v sw;g respectively. Also, v sw;w is be calculated from Eq. (28) 5. Liquid and gas superficial velocities, including water component, are obtained using the following relations:

v sl ¼ v sw;w þ v sh;o

ð32Þ

v sg ¼ v sw;g þ v sh;g

ð33Þ

Knowing vsg and vsl , phase volume fractions and in-situ velocities fag ; ao ; aw ; vg ; vo ; vw g are calculated based on the drift-flux model [29]. Note that vo and vw are equal since slippage between oil and water phases was ignored. An iterative algorithm was implemented to calculate pressure and temperature of each grid block using momentum and energy balance equations (Eqs. (29) and (30)). The iterative algorithm consists of two iterative methods of Newton–Raphson and Bisection, starting from the inlet boundary (e.g., wellhead) to the outlet boundary (e.g., perforated intervals) until convergence is achieved for the entire grid system. In the case of multicomponent hydrocarbon mixture and water flow in the wellbore, the type of primary variables fP; Tg do not depend on the appearance or disappearance of phases. However, the secondary variables will be different. Thus, in each iteration the appearance and disappearance of the grid block phases will be checked and secondary variables is switched if phases change status. For two-phase flow of single-component hydrocarbon or water, pressure and temperature of the system are coupled. Therefore, primary variables was changed to fP; v sl g. 4. Results and discussion There is no field or experimental data available in the literature about the two-phase flow of the solvent-steam mixture. Therefore,

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R.V. de Almeida et al. / Fuel 188 (2017) 50–60

4.1. Steam injection wells

Table 1 Field data parameters for Sallie Lee and Martha Bigpond wells [47]. Parameters

Sallie lee

Martha Bigpond

Units

Well diameter Inner diameter of tubing Outer diameter of tubing Insulation diameter Inner diameter of casing Outer diameter of casing Geothermal gradient Thermal conductivity of earth, ke Thermal diffusivity of earth, ae Thermal conductivity of cement, kcem Thermal conductivity of pipe, kp Injection rate Steam quality Injection pressure Surface temperature Well depth Injection time

14.4 1.99 2.37 – 4.00 4.50 0.02 24 0.69 4.8 600 2800 0.8 520 71 1700 308

14.4 2.12 2.50 – 4.00 4.50 0.02 24 0.69 4.8 600 4850 0.8 250 50 1600 117

in. in. in. – in. in. °F/ft Btu/ft-day-°F ft2/day Btu/ft-day-°F Btu/ft-day-°F lbm/h – psia °F ft. h

the model was compared with data from steam injection and gas condensate production wells.

The field data are 14-W Sallie Lee and 61-0 Martha Bigpond steam injection tests, both located in Oklahoma and first reported by Bleakley [47]. These data frequently is used for model validation in the literature [9,20,48,49]. Table 1 summarizes the field data for these two cases. The surrounding formation temperature is defined by a geothermal gradient (0:020 F=ft), increasing from surface temperature (Ta) at the surface to reservoir temperature at the bottom of the well. For the 14-W Sara Lee well, with 1600 ft. depth, wellbore pressure was measured at different depth after 308 h of steam injection. Injection rate was recorded about 2800 lbm/h with the injection pressure of 520 psia. For the Martha Bigpond well, the parameters were measured after 117 h of steam injection with injection rate of 4850 lbm/h and injection pressure of 250 psia. In both cases, the steam quality at the surface is measured to be around 80%. Figs. 4 and 5 show the comparison of the predicted wellbore pressure by this study against the field data and other models (Fontanilla & Aziz [9], Bahonar et al. [20]). There is very good agreement between field data and simulated pressure profile for both cases. In Fig. 5, a negative pressure gradient has been observed, where the pressure of steam at sandface is lower than at the wellhead. This is due to high fluid velocity that causes

Pressure (psi) 515 0

520

525

530

535

540

545

550

555

14-W Sallie Lee 200

Depth (ft)

400 600 800 1000 1200

Fonanilla and Aziz

1400

This Study Bahonar 2009

1600

Field Data

1800 515

520

525

530

535

540

545

550

555

Fig. 4. Comparison of predicted steam pressure with 14-W Sallie Lee well data after 308 h of steam injection.

Pressure (psi) 0

0

50

100

150

200

250

300

150

200

250

300

61-0 Martha Bigpond 200 400

Depth (ft)

600 800 1000 1200

Fonanilla and Aziz

1400

Bahonar 2009 This Study

1600

Field Data

1800 0

50

100

Fig. 5. Comparison of predicted steam pressure with 61-0 Martha Bigpond well data after 117 h of steam injection.

57

R.V. de Almeida et al. / Fuel 188 (2017) 50–60

numerical predictions. For the Sallie Lee case (Fig. 6), the prediction of this study is almost the same as Bahonar’s results, while Fontanilla and Aziz’s model calculated lower value of steam quality. In the case of Martha Bigpond well (Fig. 7), this model shows a

friction pressure loss overcomes the hydrostatic pressure increase and creates a negative pressure gradient along the wellbore. Because the reported field data did not include steam quality measurements along the wellbore, the model is compared with other

Steam Quality (fraction) 0.4 0

0.5

0.6

0.7

0.8

0.9

14-W Sallie Lee 200 400

Depth (ft)

600 800 1000 1200 Bahonar 2009

1400

This Study

1600

Fonanilla and Aziz

1800 0.4

0.5

0.6

0.7

0.8

0.9

Fig. 6. Comparison of predicted steam quality with other numerical models after 308 h of steam injection for the 14-W Sallie Lee well.

Steam Quality (fraction) 0.65 0

0.7

0.75

0.8

0.85

61-0 Martha Bigpond 200 400

Depth (ft)

600 800 1000 1200 Bahonar 2009

1400

This Study

1600

Fonanilla and Aziz

1800 0.65

0.7

0.75

0.8

0.85

Fig. 7. Comparison of the predicted the steam quality with other numerical models after 71 h of steam injection for the 61-0 Martha Bigpond well.

0 200

Depth (ft)

400 600 800 1000

Field Data

1200

340ft 100ft

1400 1600 510

50ft 25ft 520

530

540

550

560

Pressure (psi) Fig. 8. Resolution study of pressure predictions on the 14-W Sallie Lee well when using gird blocks of 340 ft., 100 ft., 50 ft. and 25 ft.

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difference of 1% between Bahonar’s prediction and a 4% difference between those of Fontanilla and Aziz model. Since there is no available field data, it is not possible to determine which model is more accurate for steam quality prediction. 4.2. Resolution study In order to determine the best grid block size, a resolution study was performed by comparing the pressure predictions on the 14-W

Table 2 Composition of produced fluid of well GF-00031 and GF-00085. Component

GF-00031

GF-00085

C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 N2 CO2 C7+

76.34 9.19 5.14 0.79 1.73 0.65 0.66 1.06 0.57 0.62 3.26

74.99 9.69 4.11 0.7 1.34 0.44 0.59 0.59 1.51 4.22 1.82

MWC7+

130

128

Table 3 Well and flow parameters of well GF-00031 and GF-00085. GF-00031

GF-00085

Unit

Oil production Gas production Molar rate Well depth Well diameter Surface pressure Bottomhole pressure Surface temperature Bottomhole pressure

774 9210 11,900 8677 0.20 2214 3363 104 176

441 11,600 26,300 7439 0.25 1057 1642 75 170

STB/day MSCF/day lb-mole/day ft. ft. psia psia °F °F

100

150

(b)

Pressure (psia)

200

0

0

0

1000

1000

2000

2000

3000

3000

4000

4000

Depth ()

Depth ()

50

These data was presented by Govier and Fogarasi [50] in 1975. They provided the pressure and temperature field measurements at the surface and downhole for a set of 102 gas condensate wells. No measurement along the wellbores was reported. From this set of data, two wells were selected for validation of this model (i.e., wells GF-00085 and GF-00085). Table 2 shows the well stream composition for the two wells. Since the original data set did not contain information about downhole water flow, the selection criterion for these two wells was zero water production, making these wells the best choice for the 100% hydrocarbon flow in the wellbore. Small modifications in the model was done to take into account the flow direction, which is from bottom to top. Table 3 shows the reported well and flow parameters. The calculated temperature and pressure profiles are presented in Fig. 9. Simulated results are very close to the field data. For well GF-00031, the difference between predicted and measured pressures is about 1%, and for well GF-00085 is less than 0.9%.

To test our wellbore model for steam and solvent co-injection, a series of sensitivity studies were performed. Five different singlecomponent solvents were selected: butane (C4H10), pentane (C5H12), hexane (C6H14), heptane (C7H16) and octane (C8H18). Type of solvent, concentration and injection constraints are among the critical parameters in the engineering of steamsolvent processes. The utilization of butane was successfully tested in EnCana Senlac and Christina Lake projects [51] and in the Cold Lake project by ExxonMobil [52]. The majority of research studies

Temperature (°F) 0

4.2. Gas condensate wells

4.3. Steam-solvent injection

Composition

(a)

Sallie Lee well when using grid blocks of 340 ft., 100 ft., 50 ft. and 25 ft. Fig. 8 illustrates the comparisons of the predictions and the field data. The pressure at the sand face (1700 ft.) for different grid block sizes is compared. The difference of the error between the grid blocks of 100 ft. and 50 ft. is only 0.1%.

5000 6000

1000

2000

3000

4000

5000 6000

7000

7000 Field Data-GF0086

8000 9000

This Work-GF0086 Field Data-GF0031

8000 9000

This Work-GF0031

10000

10000

Fig. 9. Comparison of calculated temperature and pressure profile with wellhead and bottom hole measured values for two gas condensate wells.

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Table 4 Well parameters used for sensitivity runs of steam-solvent co-injection. Parameters

Value

Units

Well depth Well diameter Inner diameter of tubing Outer diameter of tubing Inner diameter of casing Outer diameter of casing Formation thermal conductivity, ke Formation thermal diffusivity, ae Thermal conductivity of cement, kcem Thermal conductivity of pipe, kp Surface temperature

2200 12 1/4 2.44 2 7/8 8.84 9 5/8 24 0.69 4.8 600 53

ft. in. in. in. in. in. Btu/ft-day-°F ft2/day Btu/ft-day-°F Btu/ft-day-°F °F

Pressure (psia) 530 0

550

570

590

610

630

Depth (ft)

500

1000 Steam + C8 Steam + C7

1500

Steam Quality (fraction) 0.50 0

0.55

0.60

0.65

0.70

0.75

0.80

Depth (ft)

500

1000 Steam + C8 Steam + C7

1500

Steam + C6 Steam + C5 2000

Steam + C4 Steam Only

2500 0.50

0.55

0.60

0.65

0.70

0.75

0.80

Fig. 11. Predicted steam quality profile for different single component hydrocarbon co-injected with steam compared to steam injection case.

Solvent Quality (fraction) 0.90 0

0.95

1.00

1.05

1.10 Steam + C8 Steam + C7

500

Steam + C6

Depth (ft)

support medium weighted solvents (C4H10- C7H16) as the best additive for solvent aided steam injection. The strong support for this range of hydrocarbons solvents are based on experimental, and simulation studies in porous media. These studies analysed the effectiveness of the injected solvent to mix and reduce insitu heavy oil viscosity in the presence of steam. However, it is also important to understand the behavior of these solvents along the wellbore to assure that the solvent will reach the perforation intervals in vapor phase in order to propagate through the steam chamber and condense at the steam-oil contact. To study the condensation behavior of different solvents when co-injected with steam, the well parameters of one of the injector (i.e., AO2I02) of the Encana’s Solvent Aided Process (SAP) Pilot at the Christina Lake Project was used (see Table 4). The geothermal gradient was taken from Deroo and Powell [53] (Institute of Sedimentary and Petroleum Geology of Canada). Besides the parameters that provided in Table 4, injection pressure, temperature and mass flow rate are extracted from Cold Lake field pilot [52,54] reports, where 25 m3/day (about 20% of steam injection rate) of solvent was injected at an average pressure of 600 psia and temperature of 458°F. Steam quality at the wellhead is fixed at 80%. Wellbore profiles were calculated after one day of steam-solvent injection. Fig. 10 shows the pressure profile for different cases. Results show higher pressure drop along the wellbore for all the cases because the pressure gradient due to friction is bigger than hydrostatic. For heavier hydrocarbons, the pressure drop is bigger because of the higher density of the mixture and consequently higher friction pressure loss. Steam quality is plotted versus depth for different hydrocarbon additives in Fig. 11.

Steam + C5

1000

Steam + C4 1500

2000

2500 0.90

0.95

1.00

1.05

1.10

Fig. 12. Predicted solvent quality profile for different single component hydrocarbon co-injected with steam compared to steam injection case, Octane starts to condense at around 1450 ft. depth.

Steam quality is higher for solvent-steam co-injection cases compared to the situation where only steam is injected. This is due to extra energy that was carried by hydrocarbon additive. Fig. 12 shows the solvent quality, defined as the ratio of moles of hydrocarbon in the vapor phase to the total moles of hydrocarbon, profile along the wellbore. All the hydrocarbons additives, except octane, remains in the vapor phase. The octane started to condense at around 1450 ft. depth and its quality drops to 0.91 at the well bottom hole. Typically, heavier components are more efficient in reducing the in-situ heavy crude oils. The challenge remains, however, in the transportation of hydrocarbon solvent from the surface to the edge of the steam chamber. Heavier components are more likely to condense in the wellbore.

Steam + C6 Steam + C5 2000

5. Conclusions

Steam + C4 Steam Only

2500 530

550

570

590

610

630

Fig. 10. Predicted pressure profile for different single component hydrocarbon coinjected with steam compared to steam injection case, Heavy weighted solvent will cause higher pressure drops.

In this work, a detailed formulation of multi-phase thermal and compositional wellbore modeling was presented. The wellbore volume is divided into a set of axial staggered grids. The mass, energy and momentum balance equations are discretized over the wellbore domain. Phase slippage is captured using a driftflux model and the wellbore heat loss to the formation is calculated

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by an overall heat transfer coefficient. Following the assumption of instantaneous thermal and thermodynamic equilibrium in the wellbore, two-phase flash calculations are used to predict quality and properties of each phase. The developed model was tested and validated against data from steam injection and gas condensate production wells. This comparison demonstrated the accuracy of the applicability of this model. An application to a real well data from a Canadian heavy oil pilot is presented. Simulation of thermal, multiphase flow of steam and hydrocarbon in the wellbore shows that solvents with heavier components have higher chance to condense along the wellbore compared to lighter components. Acknowledgment The authors would like to thank the Brazilian Ministry of Education, through CAPES Foundation (‘‘Coordination for the Improvement of Higher Education Personnel”) and the Brazilian Ministry of Science and Technology, through CNPq (‘‘National Counsel of Technological and Scientific Development”) for the financial support of this study. References [1] Lesem LB, Greytok F, Marotta F, McKetta Jr., JJ. A method of calculating the distribution of temperature in flowing gas wells. Paper presented at Ninth Oil Recovery Conference at Texas A & M College; 1957. [2] Moss JT, White PD. How to calculate temperature profiles in a water-injection well. Oil Gas J 1959;57(11):174. [3] Ramey Jr HJ. Wellbore heat transmission. J Petrol Technol 1962;14 (04):427–35. [4] Willhite GP. Over-all heat transfer coefficients in steam and hot water injection wells. J Petrol Technol 1967;19(05):607–15. [5] Satter A. Heat losses during flow of steam down a wellbore. J Petrol Technol 1965;17(07):845–51. [6] Holst PH, Flock DL. Wellbore behaviour during saturated steam injection. J Can Pet Technol 1966;5(04):184–93. [7] Earlougher Jr RC. Some practical considerations in the design of steam injection wells. J Petrol Technol 1969;21(01):79–86. [8] Hagedorn AR, Brown KE. Experimental study of pressure gradients occurring during continuous two-phase flow in small-diameter vertical conduits. J Petrol Technol 1965;17(04):475–84. [9] Fontanilla JP, Aziz K. Prediction of bottom-hole conditions for wet steam injection wells. J Can Pet Technol 1982;21(02). [10] Beggs DH, Brill JP. A study of two-phase flow in inclined pipes. J Petrol Technol 1973;25(05):607–17. [11] Ali SM. A comprehensive wellbore stream/water flow model for steam injection and geothermal applications. Soc Petrol Eng J 1981;21(05):527–34. [12] Pacheco EF, Ali SM. Wellbore heat losses and pressure drop in steam injection. J Petrol Technol 1972;24(02):139–44. [13] Wooley GR. Computing downhole temperatures in circulation, injection, and production wells. J Petrol Technol 1980;32(09):1–509. [14] Sagar R, Doty DR, Schmidt Z. Predicting temperature profiles in a flowing well. SPE Prod Eng 1991;6(04):441–8. [15] Stone TW, Edmunds NR, Kristoff BJ. A comprehensive wellbore/reservoir simulator. Houston, Texas: Paper Presented at theSPE Symposium on Reservoir Simulation; 1989. [16] Stone TW, Bennett J, Law DS, Holmes JA. Thermal simulation with multisegment wells. Houston, Texas: Paper Presented at thethe SPE Reservoir Simulation Symposium; 2001. [17] Livescu S, Durlofsky LJ, Aziz K, Ginestra JC. A fully-coupled thermal multiphase wellbore flow model for use in reservoir simulation. J Petrol Sci Eng 2010;71 (3):138–46. [18] Livescu S, Aziz K, Durlofsky LJ. Development and application of a fully-coupled thermal compositional wellbore flow model. In: Paper Presented at theSPE Western Regional Meeting, San Jose, California, 2009. [19] Livescu S, Durlofsky L, Aziz K. A semianalytical thermal multiphase wellboreflow model for use in reservoir simulation. SPE J 2010;15(03):794–804. [20] Bahonar M, Azaiez J, Chen Z. A semi-unsteady-state wellbore steam/water flow model for prediction of sandface conditions in steam injection wells. J Can Pet Technol 2010;49(09):13–21. [21] Xiong W, Bahonar M, Chen ZJ. Development of a thermal wellbore simulator with focus on improving heat-loss calculations for steam-assisted-gravitydrainage steam injection. SPE Reserv Eval Eng 2016. [22] You J, Rahnema H, McMillan MD. Numerical modeling of unsteady-state wellbore heat transmission. J Nat Gas Sci Eng 2016;34:1062–76.

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