Heat Mass Transfer (2016) 52:1071–1080 DOI 10.1007/s00231-015-1626-z
ORIGINAL
Production of light oil by injection of hot inert gas Bidhan C. Ruidas1 · Somenath Ganguly2
Received: 24 June 2014 / Accepted: 6 July 2015 / Published online: 14 July 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract Hot inert gas, when injected into an oil reservoir is capable of generating a vaporization–condensation drive and as a consequence, a preferential movement of the lighter components to the production well. This form of displacement is an important unit mechanism in hot fluegas injection, or in thermal recovery from a watered-out oil reservoir. This article presents the movement of heat front vis-à-vis the changes in the saturation profile, and the gasphase composition. The plateau in the temperature profile due to the exchange of latent heat, and the formation of water bank at the downstream are elaborated. The broadening of the vaporization–condensation zone with continued progression is discussed. The effect of inert gas temperature on the cumulative production of oil is reviewed. The results provide insight to the vaporization–condensation drive as a stand-alone mechanism. The paper underscores the relative importance of this mechanism, when operated in tandem with other processes in improved oil recovery and CO2 sequestration. List of symbols Cp Heat capacity (J kg−1 K−1) Csd Sutherland constant, 240 K dSG Specific gravity K Phase equilibrium constant k Permeability l Length of volume element (m)
* Somenath Ganguly
[email protected] 1
Department of Chemical Engineering and Technology, Birla Institute of Technology, Mesra, Ranchi, India
2
Department of Chemical Engineering, Indian Institute of Technology, Kharagpur, Kharagpur, India
L Latent heat of vaporization (J kg−1) p Pressure (Pa) R Ideal gas constant S Saturation T Temperature (K) Tsd Reference temperature in Sutherland equation of gas viscosity, 293.15 K t Time (s) v Velocity of various phases (m s−1) x Length (m) y Mole fraction of various components in gaseous phase Z Compressibility factor Greek symbols ∂ Differential operator λ Thermal conductivity (W m−1 K−1) μ Viscosity (kg m−1 s−1 or Pa s) μsd Sutherland coefficient of viscosity, 14.8×10−6 Pa s ρ Density (kg m−3) φ Void fraction in the reservoir (dimensionless) Subscripts avg Average ct Critical (K) g Gaseous phase i Phase index ig Inert gas o Oil phase or Residual oil r Relative rc Reduced s Solid phase tot Total (m2) vp Vaporization
13
1072
w Water phase wr Residual water
1 Introduction Thermal recovery is a class of enhanced oil recovery scheme that utilizes steam injection, hot water injection, or in situ combustion to raise the temperature of the reservoir. Higher temperature reduces the viscosity of oil. Also, vaporization of oil and water, and flow of the vapor with the injected gas produces an additional mechanism for the movement of lighter components of oil. Generally, a thermal recovery process is simulated by considering multiple mechanisms that are involved in the process [1–9]. Initial models were mostly analytical [10]. Subsequent treatments involve extensive computation on commercial simulator, and even parallel computation with data communication between different processors through message passing interface library [11]. The purpose of the simulation studies has been mostly to match the production history and predict future trends for various operating parameters. In some cases, computer simulations have helped analyzing the experimental data in combustion tubes [1, 12]. Similar heat and mass transfer based computer simulations have been undertaken in the recovery of natural gas [13–15]. The exothermic reaction with oxygen in injected air has been considered a primary component of in situ combustion [16]. In the more recent simulations, the vaporization and the downstream condensation of water and lighter components of oil are included. With interest on in situ combustion in mature light oil reservoirs [17–19], the vaporization–condensation mechanism becomes more important as a unit process in the thermal drive. This article analyzes the thermal recovery of oil from injection of hot inert gas. The response to a step change in temperature at the inlet was tracked as the inert gas continued to flow through a porous matrix, containing oil, water and gas. The computer simulation of the responses is analyzed in this article. Recently, there has been an interest in understanding a multiple displacement mechanism through the unit mechanisms. In these analyses, the cooperation of unit mechanisms forms the basis for deriving the work done in the transmission process [20]. The stand-alone treatment of one mechanism exclusive of others may explain whether the mechanism acts parallel to other mechanisms or has some cross effects. In the present article, the flow of hot inert gas is treated as a stand-alone mechanism that does not include the thermal and the pressure shock from in situ combustion. However, the vaporization–condensation drive and the thermal effects on flow parameters that are also the components of in situ combustion are included here. Thus, the simulation provides a base line for in situ combustion process.
13
Heat Mass Transfer (2016) 52:1071–1080
Similarly, the injection of flue gas into the oil-bearing reservoir at a pressure below the minimum miscibility pressure may serve the purpose of sequestration. The analysis in this article presents a baseline for that process as well. In the algorithm, used for the analysis, the mass continuity equations for the phases and the heat balance equation were discretized in space using finite difference. The duration of flow was broken down into small time steps. For each time step, the set of ODEs from mass continuity was solved simultaneously using an auto-adaptive implicit algorithm. With the update on phase flow rates and saturations for the given timestep, the heat balance equation was solved subsequently in implicit manner (Crank Nicolson method), and the iterations were repeated. The evolution of the vaporization–condensation zone, the constituting elements of the zone, the progression of the zone through the bed, and the bank formation beyond the condensation zone were analyzed through temperature, pressure, saturations, gas density, and gas phase mass fraction profiles. The effect of the inlet temperature is discussed, and the enhancement in the oil production is reviewed.
2 Methods The mathematical model considers the following aspects. 1. Presence of oil, water, and gas in the porous medium. 2. Injection of hot inert gas at constant mass flow rate against the reservoir pressure. 3. Movement of oil, water, and gas in accordance with the relative permeability relations, and functional dependence of viscosity on temperature. 4. Equilibrium of the vapor phase with oil and water respectively at each axial location. 5. Evaporation and/or condensation to satisfy vapor–liquid equilibrium by exchange of latent heat. 6. Change in gas density due to temperature, pressure, evaporation and/or condensation of oil and water. The analysis is restricted to 1-D to focus on the vaporization–condensation drive. Similarly, the system was considered isolated from heat loss to the surroundings in line with the combustion tube studies. The equations are listed in the “Appendix” section. The meanings of symbols are also listed separately. The unsteady state energy balance equation assumes thermal equilibrium between the solid and the fluid phases at each point in the bed. The unsteady state mass balance equations for oil and water involve the individual phases, and the fractions that move as vapor with the gas phase. Similar equation for inert gas defines the pressure at a grid. The gas density at a grid point at any instant of time is defined by the pressure, temperature, and the mole fractions of
Heat Mass Transfer (2016) 52:1071–1080
oil-vapor and water-vapor at that grid. The equilibrium relations define the partial pressures of oil and water in the vapor phase, and are listed in the “Appendix” section. The governing equations were solved by discretizing the length-axis into grids, and iterating over small time steps. The temperature and pressure at a grid defines the equilibrium oil and water fractions in the gas phase. When the vapor pressures of oil and water cumulatively exceed the total pressure of a grid, entire oil and water evaporates from the grid. At each time step, the mass balance equations involving oil, water and inert gas were solved simultaneously. The partial differential equations were discretized using finite difference method along the length of the bed. The resulting set of ordinary differential equations was solved using the Gear algorithm [21] to obtain values at the next incremental time step. With the updated velocity profile the heat balance equation was discretized in both space and time by implicit finite difference method. The tri-diagonal system of equations was solved using Thomas algorithm [22]. The updated temperature and pressure profiles were used to compute the equilibrium composition for the next time step. The iterations continued for the entire duration (specified in Table 1) over which the inert gas was injected. The Gear algorithm is an auto-adaptive implicit algorithm, where the order changes based on the computational cost for a given accuracy. The algorithm is ideally suited for stiff equations with variables showing abrupt changes due to the dynamics at different timescales. The vapor pressure of oil and water were computed from Antoine’s equation (Eqs. 18, 19). The n-dodecane was used to represent the oil phase. The choice of n-dodecane was based on a review of true boiling point (TBP) curves for a light crude oil of API gravity around 37 (density 838.92 kg m−3). In particular, the choice was made from the molal average boiling points and the corresponding mid-boiling point in the TBP curve with a note that only 90 % of the total weight finally evaporated [23]. The rest, comprising of asphaltenes will stay away from evaporation, and burn at extreme temperatures. Accordingly, after complete evaporation, the oil saturation reaches the irreducible value of 0.088 instead of zero. The system of equation was solved using Matlab 7.1 from Mathworks Incorporate. The sensitivity of the profiles to the sizes of spatial grids and time steps were checked to choose the right size for the simulation. In the final simulation, the length was divided into 120 grids, and time was divided in 380 steps. For each time step, the gear algorithm decided the micro-step and order in auto-adaptive mode.
3 Results and discussions Table 1 lists the material properties and the operating parameters, used in the simulation. Initially, the entire bed
1073 Table 1 Input parameters for simulation Parameters
Symbol Value
Unit
Length of porous bed
l
1.83
m
Cross sectional diameter of the bed
d
0.0985
m
Porosity Permeability
φ
ktot
0.426 5
Dimensionless Darcya
Oil density
ρo
37
°APIb
Initial saturation: oil
So
54.4 %
–
Initial saturation: water
Sw
36.3 %
–
Initial saturation: gas
Sg
9.3 %
–
Residual oil saturation
Sor
8.8 %
Dimensionless
Residual water saturation
Swr
0.0 %
Dimensionless
Gas flow rate
Qin
7.167 × 10−5 m3 s−1
Thermal conductivity of porous medium
λs
2.4
Outlet pressure
pout
9.303 × 106
Pa
Initial temperature
T0
349
K
Inlet temperature
Tin
623
K
μsd
10.6 × 3600 14.8 × 10−6
s Pa s
Csd
Duration of gas injection Sutherland coefficient of viscosity Sutherland constant
W m−1 K−1
240
K
Sutherland equation param- Tsd eter Critical temperature of the Tct gas
293.15
K
304.20
K
Critical pressure of the gas
pct
7.387 × 106
Pa
Specific gravity of oil
dSG
0.839
Dimensionless
a
1 Darcy = 9.869233 × 10−13 m2
b
Density in kg m−3 = 141.5 × 103/(°API + 131.5)
was held at a temperature of 76 °C (349 K), and the oil, water and gas saturations of 0.54, 0.36, and 0.1 respectively. Figures 1, 2, 3, 4, 5, 6, 7, 8 and 9 describe the profiles of various process parameters along the length of the packed bed for the inlet temperature of 350 °C (623 K). The subsequent figures were developed considering the inlet temperature of 86 °C (359 K). Figure 1 describes the pressure at two different time instances in the bed as a function of axial distance from the inlet. The pressure was made dimensionless by dividing the pressure by the back pressure, held at the outlet of the pack. The axial distances of 0 and 1 refer to the inlet and the outlet of the pack respectively. The continuous line in blue represents the pressure profile near the end of the 10.6 h (3.816 × 104 s) of hot air injection. The dotted line in green represents the pressure profile near the end of the first half, i.e., after 5.3 h (1.9808 × 104 s) of hot gas injection. A distinct change in slope was observed in the pressure profile along the length of the bed. The change in slope
13
1074
Heat Mass Transfer (2016) 52:1071–1080 0.7 10.6 hours
0.6
5.3 hours
1.0002
Oil saturation
Dimensionless pressure
1.0003
1.0001
1
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
Dimensionless distance from the inlet
Fig. 4 Oil saturation along the length at two time instances (inlet temperature 623 K)
450 400 350 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 2 Temperature profile along the length at two time instances (inlet temperature 623 K)
0.7
90 85 80 75 70
10.6 hours 5.3 hours
65 60 55 50 45 0
0.2
0.4
0.6
0.8
1
Dimensionless distance from the inlet
Fig. 5 Density of gas along the length at two time instances (inlet temperature 623 K)
10.6 hours 5.3 hours
0.6
Density of the gas phase (kg/cubic meter)
500
0.5 0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1
Dimensionless distance from the inlet
Fig. 3 Water saturation along the length at two time instances (inlet temperature 623 K)
suggests a significant change in viscosity, arising from a phase change. In simple displacement of fluid by another fluid of different viscosity, the velocities of the two fluids have to be same. As per Darcy’s law, the pressure gradients on the two sides of the interface will be proportional
13
Mass fraction of oil in gas phase
Temperature in Kelvin
550
Dimensionless distance from the inlet
water saturation
0.2
10.6 hours 5.3 hours
600
300
0.3
0
Fig. 1 Pressure profile along the length at two time instances (inlet temperature 623 K)
10.6 hours 5.3 hours
0.4
0.1
1
Dimensionless distance from the inlet
650
0.5
0.07 0.06 10.6 hours 5.3 hours
0.05 0.04 0.03 0.02 0.01 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless distance from the inlet
Fig. 6 Mass fraction of oil in gas phase along the length at two time instances (inlet temperature 623 K)
to the viscosities of the respective phases, resulting in a discrete change in the pressure profile at the interface. Here, a continuous transition happened over a length of
Mass fraction of water in gas phase
Heat Mass Transfer (2016) 52:1071–1080
1075
0.2 10.6 hours 5.3 hours
0.15 0.1 0.05 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless distance from the inlet
Fraction of in−place oil produced
Fig. 7 Mass fraction of water in gas phase along the length at two time instances (inlet temperature 623 K)
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
0
0.5
1
1.5
2
2.5
3
3.5
4 4
Time (seconds)
x 10
Fraction of in−place water produced
Fig. 8 Cumulative flow of oil at the outlet as a function of time (inlet temperature 623 K)
−3
6
x 10
5 4 3 2 1 0 0
0.5
1
1.5
2
2.5
Time (seconds)
3
3.5
4 4
x 10
Fig. 9 Cumulative flow of water at the outlet as a function of time (inlet temperature 623 K)
the porous bed. Another noteworthy point is the choice of the duration of the simulation run. Too small a duration will not be enough to establish a discernible vaporization–condensation drive. Too long a duration will have
the vaporization–condensation zone moving beyond the length, considered in this investigation. Duration of 10.6 h was considered in this investigation. Empirically, this is the time required for a controlled in situ combustion front to travel through the same length and initial saturations of porous medium, when air is injected at a rate, same as the inert gas flow rate in the simulation [10]. Figure 2 describes the temperature profile along the length of the bed at the two time instances, referred before. Temperature at the inlet was held at 350 °C, whereas the initial temperature of the bed was 76 °C. For heat transfer simply by conduction and convection, associated with the flow of a single phase, the temperature profile would have been a curved one. However, the exchange of latent heat occurred over a length of the bed due to evaporation of oil and water. The reverse exchange of heat continued to a downstream location, where the vapors condensed. Over this axial span, the change in enthalpy was primarily due to the exchange of latent heat. The sensible heat of the system over this length did not change significantly, resulting in a plateau in the temperature profile. The vaporization– condensation process has been suggested by researchers in the context of in situ combustion [7]. However, this aspect was suppressed in the profiles due to the high heat of reaction and conduction through solid phase in case of in situ combustion. Figure 3 describes the saturation of water as a function of distance from the inlet end at the two time instances, referred before. The initial water saturation of the bed was 0.36. With injection of the inert gas, the brine saturation decreased by two mechanisms. Firstly, the pressure gradient imposed by the flow of gas displaced the oil and the water to the downstream end, resulting in the decrease of saturation at the upstream side. The second mechanism is by evaporation of oil and water at the elevated temperature. The first mechanism may reduce the water saturation to a level, referred as residual saturation. On the other hand, the second mechanism may reduce the water saturation to zero. The formation of water bank is evident from the hump in the saturation profile. The hump due to local accumulation is expected from a sudden increase in velocity over a small length of the porous bed. The condensation of vapor may lead to such accumulation. Figure 4 describes the saturation of oil along the porous bed at the two time instances, referred before. Two major differences water saturation profiles are noted here. The evaporation at elevated temperature reduced the saturation to irreducible oil saturation near the inlet end. The true boiling point curve of a light oil sample indicates that the final 10 % of oil does not get evaporated due to the presence of bituminous remnants. Accordingly, an irreducible oil saturation of 10 % was imposed in the algorithm. Secondly, the viscosity got reduced manifold at elevated temperature,
13
1076
13
temperature and the pressure profiles. In the temperature profile, the change was more discrete, and a plateau was observed due to the exchange of latent heat. In the pressure profile, the change in gradient was continuous over the entire stretch. At the upstream end of the vaporization zone, water was the major liquid phase, since oil at reduced viscosity was driven out from this zone. The downstream end of the condensation zone was evenly shared by oil and water. No major carry-over of oil and water vapors by the injected gas was observed. To emphasize the importance of vaporization–condensation drive, another set of simulations was performed where the inert gas temperature is marginally higher than the temperature of the in-place fluids. Figures 10 and 11 present the temperature and pressure profiles, obtained with the temperature at the inlet as 10 °C higher than the initial temperature of the bed. That is, the earlier plots were based on an inlet temperature of 350 °C. The new plots are based on the inlet temperature of 86 °C. For the
Temperature in Kelvin
360 358
10.6 hours 5.3 hours
356 354 352 350 348 0
0.2
0.4
0.6
0.8
1
Dimensionless distance from inlet
Fig. 10 Temperature profile with the inlet temperature reduced to 359 K
1.00014
Dimensionless pressure
unlike the viscosity of water. The viscosity of crude oil may decrease by orders of magnitude for a 300 °C increase in temperature. If the change in viscosity is the only mechanism, the oil would have moved faster than water. This would have resulted in a hump for oil, and a small complementary dip in water saturation. These were not observed in Figs. 3 and 4. The complete reversals in both the profiles signify the dominance of vaporization–condensation in progression of the saturation fronts. This is noted that the reduction in viscosity of oil and the condensation of oil vapor became dominant at different axial positions of the porous bed. The local accumulation from one mechanism could not sustain due to another accumulation at the downstream from the other mechanism. From the axial location of the banks, one may conclude that the condensation of oil and water did not happen at the same location at a time. From prior research in thermal flooding, the bank formation for oil water was generally indicated [3]. In the present simulation, the accumulation showed up in somewhat different manifestation. Figure 5 describes the gas density at different axial locations in the porous bed. The density of the gas is inversely proportional to the temperature in the Kelvin scale. Accordingly, the density increased with the distance from the inlet face. Also, the density of the gas increased in the evaporation zone due to the addition of oil and water vapor to the gas phase. The change in pressure along the length of the bed is not significant to influence a profile in density along the length of the bed. Figures 6 and 7 describe the oil and water mass fractions along the bed at the two time instances, referred before. The mass fraction is zero at the upstream side of the vaporization front, as the last trace of volatile liquid has evaporated from this region. Beyond this region, there is a sharp increase in the mass fraction, as the entire oil and water remains in the evaporated form. At further downstream, the temperature decreases significantly, and the vapor and liquid coexist. The mass fraction in this region is stipulated by the vapor liquid equilibrium at the local temperature and pressure. Therefore, the evaporation–condensation of oil and water are quite evident in these plots. The stretch of the bed over which the spikes were observed is consistent with the plateau in the profiles for temperature, pressure and saturations. Figures 8 and 9 present the cumulative flow of oil and water from the outlet with time. The sharp change in cumulative flow suggests the arrival of the fluid bank. With this understanding the evolution of the vaporization–condensation zone is revisited. This is noted that the vaporization–condensation zone was focused to a small band initially, and then it grew moderately with continued progression through the porous media. The vaporization–condensation entails a change in gradient in both the
Heat Mass Transfer (2016) 52:1071–1080
1.00012 10.6 hours 5.3 hours
1.0001 1.00008 1.00006 1.00004 1.00002 1
0
0.2
0.4
0.6
0.8
1
Dimensionless distance from inlet
Fig. 11 Pressure profile with the inlet temperature reduced to 359 K
Heat Mass Transfer (2016) 52:1071–1080
1077 Fraction of in−place oil produced
0.4
Water saturation
0.35 0.3 10.6 hours 5.3 hours
0.25 0.2 0.15 0.1 0.05 0
0.2
0.4
0.6
0.8
0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
1
0
0.5
1
Oil saturation
0.55 0.5 10.6 hours 5.3 hours
0.45 0.4 0.35 0.3 0.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Dimensionless distance from inlet
Fig. 13 Oil saturation along the length with the inlet temperature reduced to 359 K
system pressure considered here, no major evaporation is expected at this level of temperature. Accordingly, the profiles simply indicate the progression of the thermal front by exchange of sensible heat. The heat propagates by conduction through the solid phase of the bed, and by convection through the flowing phases. Also, no major change in oil viscosity is anticipated at this level of increase in temperature. Figures 12 and 13 present the saturation profiles. The hump in the profile was not observed. This clearly rules out any major flow of oil and water within the bed. The cumulative flows of oil and water from the outlet are plotted in Figs. 14 and 15. For the same volume of inert gas injection, significantly less volume of oil was recovered. Therefore, the net production of oil was enhanced four times, and the net production of water was enhanced three times due to the 260 °C increase in inlet temperature. The increase in water production was through vaporization–condensation drive. This became apparent
2.5
3
3.5
4
4
x 10
Fig. 14 Cumulative flow of oil at the outlet with the inlet temperature reduced to 359 K
Fraction of in−place water produced
0.6
2
Time (seconds)
Dimensionless distance from inlet
Fig. 12 Water saturations along the length with the inlet temperature reduced to 359 K
1.5
−3
2
x 10
1.5
1
0.5
0 0
0.5
1
1.5
2
2.5
Time (seconds)
3
3.5
4 4
x 10
Fig. 15 Cumulative flow of water at the outlet with the inlet temperature reduced to 359 K
from the hump is the saturation profile beyond the condensation zone (Fig. 3), and the high mass fraction of water vapor in the gas phase in the vaporization zone (Fig. 7). On the other hand, the increase in oil production was due to the combined effect of viscosity reduction, and the vaporization–condensation drive. If the viscosity effect had dominated, one would have clearly seen the bank formation with a complementary dip in water saturation. That had not happened. The vaporization–condensation drive was not dominating either, which was evident from the low mass fraction of oil vapor in the gas phase (Fig. 6). Therefore, it was truly a combined effect. Though for oil it was a combined drive, the two mechanisms did not apply to the same extents at an axial location. No localized accumulation could sustain under such operation. No clear hump in the saturation profile was observed (Fig. 4). However, some distortions in the saturation profile were observed.
13
1078
Heat Mass Transfer (2016) 52:1071–1080
4 Conclusions
The mass balance equation for inert gas
The progression of the thermal front arising from the flow of inert gas through oil bearing porous matrix is studied in this article. The vaporization–condensation of oil and water introduced a change in gradient or a plateau in the temperature profile. This shows the exchange of latent heat, dominating over the exchange of sensible heat in this stretch. The change in gradient is evident in the pressure and gas density plots, and corresponds well with the vapor phase composition and the liquid phase saturations along the length of the bed. The change in gradient in the pressure profile is continuous over the vaporization–condensation zone. The vaporization–condensation entails a bank formation with water. The dominance of vaporization is evident from the high mass fraction of water vapor in the gas phase. For oil, the vaporization–condensation drive occurred in tandem with the pressure driven flow at reduced viscosity. The former mechanism cannot dominate with little mass fraction of oil vapor in the gas phase in evaporation zone. If the latter mechanism had dominated, a hump in the oil saturation profile at the expense of a dip in the water saturation would have been observed. Rather, the reverse was observed, indicating the importance of both the mechanisms for the crude oil considered here. The hump was not observed in the saturation profile for oil, possibly due to the cumulative effects of two mechanisms operation over different axial positions, little away from each other. The mass fractions of oil and water in the vapor phase were of significance only within the vaporization–condensation zones. Beyond this zone on the downstream side, no major carry-over was observed. The vaporization–condensation zone was focused into a small stretch of the porous bed. At the upstream-end of the zone, water is the major liquid phase. The vapor space of the condensation zone was evenly shared by oil and water. The width of the zone broadened somewhat with progression through the porous bed. When the temperature at the inlet was less by 250 °C, the cumulative production of oil got reduced four times for the time frame, considered in this work.
−
∂ ∂ (yig vg ρg ) = φSg (yig ρg ) ∂x ∂t
(2)
The mass balance equation for hydrocarbon
−
∂ ∂ (vo ρo + yo vg ρg ) = φ (ρo So + yo Sg ρg ) ∂x ∂t
(3)
The mass balance equation for water
−
∂ ∂ (vw ρw + yw vg ρg ) = φ (ρw Sw + yw Sg ρg ) ∂x ∂t
(4)
Sum-constraint for gaseous phase (5)
yig + yo + yw = 1 Sum-constraint for saturation
(6)
So + S w + S g = 1
Compressibility factor of gas phase (Berthelot equation) [24] prc 1 − 6Trc−2 Z = 1 + 0.0703 (7) T rc
where, prc = ppct and Trc = TTct Relation for viscosity of gas (Sutherland equation) [25]
µg = µSd
TSd + CSd T + CSd
T TSd
3/
2
Specific heat capacity of gas phase [26] 5721 Cpg = 103 × 1.0 + 0.000123 T − 2 T
(8)
(9)
Specific heat capacity of water phase [26] For, 283.15 K < T ≤ 513.15 K
Cpw = 1000 × {4.182 − 1.5 × 10−4 × (T − 273.15) + 3.44 × 10−7 × (T − 273.15)2 + 4.26 × 10−8 × (T − 273.15)3 }
Appendix
For, 513.15 K < T≤573.15 K
The overall energy balance equation
∂ ∂ 2T vg ρg Cpg T + vg ρg Lvpavg − 2 ∂x ∂x ∂ − (vo ρo Cpo T + vw ρw Cpw T ) ∂x = {(1 − φ)ρs Cps
Cpw = 1000 × (11.55 − 0.64518 × (T − 273.15) +1.5087 × 10−4 × (T − 273.15)2
s
+ φ(So ρo Cpo + Sw ρ w Cpw ∂Lvpavg + φSg ρg ∂t
13
∂T + Sg ρg Cpg )} ∂t
(10)
(11)
Viscosity of oil [1] (1)
µo = 0.001 × 2.5875 × 107 × TF−3.158 where, TF is the temperature in degree Fahrenheit. Conversion factor: T = (TF + 459.67)/1.8 Specific heat capacity of oil phase [26]
(12)
Heat Mass Transfer (2016) 52:1071–1080
1079
Average heat of vaporization [3]
Cpo = 1.605 × 103 + 4.361 × (T − 273) − 4.04 × 10
−3
× (T − 273)
2
(13)
Density of gas phase
ρg =
(20)
where,
pMavg ZRT
Lvpo =
Density of oil phase [26]
ρo = 881.0252 × {1.022 − (0.000378 × TF )}
Lvpavg = yo Lvpo + yw Lvpw
(14)
Conversion factor: T = (TF + 459.67)/1.8 Relative permeability relation for oil, water, and gaseous phases [3]
4184 [194.4 − 0.162(T − 273.15)] J kg−1 dSG
(21)
and Lvpw = 2.26 × 106 J kg−1 The velocity of each phase is expressed in accordance with Darcy’s law
vi =
−ktot kri ∂p . µi ∂x
for So ≤ Sor 3 5 1 − Sg − Sw 1 + Sw − Sg − 2Swr 1 − 1 − Sw ; kro = 4 1 − Swr 1 − (1 − Sw )5
(22)
kro = 0;
krw = 0; for Sw ≤ Swr (Sw − Swr ) 4 ; for Sw > Swr krw = (1 − Swr ) krg = 0; for Sg ≤ 0.05 Sg − 0.05 krg = ; for Sg > 0.05 0.95
µw =
0.001 × 2.185 [0.04012TF + 5.1547 × 10−6 TF2 − 1.0]
Conversion factor: T = (TF + 459.67)/1.8
(15)
References (16)
(17)
The definition of Ko = yo/xo, where xo is the mole fraction of oil in liquid. Kw is similarly defined as Kw = yw/ xw, where xw is the mole fraction of water in liquid. As oil and water are immiscible, both xo and xw are taken as 1.0. This leaves yo = Ko and yw = Kw. As the two liquids are immiscible they exert the partial pressures equals to their individual vapor pressures respectively at the prevailing temperature in the gas phase. Antoine equation is adopted to calculate the vapor pressure of oil and water. Phase equilibrium coefficient, K-value for crude oil and water [3] 6738.91 1 So Ko = exp 12.12767 − p TR − 167.13 So + 10−4 (18) 1 TR Sw Kw = (19) p 116 Sw + 10−4 where, TR is the temperature in degree Rankine Conversion factor: T = TR/1.8 The viscosity of water phase is give as [3]
for So > Sor
1. Gottfried BS (1968) Combustion of crude oil in a porous medium. Combust Flame 12(1):5–13 2. Prats M (1969) The heat efficiency of thermal recovery processes. J Pet Technol 246:323–332 3. Crookston RB, Culham WE, Chen WH (1979) A numerical simulation model for thermal recovery processes. Soc Pet Eng J (AIME) 19(1):35–58 4. Yortsos YC, Gavalas GR (1982) Heat transfer ahead of moving condensation fronts in thermal oil recovery processes. Int J Heat Mass Transf 25(3):305–316 5. Oklany JSFA (1992) An in-situ combustion simulator for enhanced oil recovery. Ph.D. thesis. University of Salford, Salford 6. Bahadir AR, Ellerby FB (2002) On the performance of certain direct and iterative methods on equation arising on a two–dimensional in situ combustion simulator. Appl Math Comput 125:347–358 7. Kristensen MR, Gerritsen MG, Thomsen PG, Michelsen ML, Stenby EH (2008) An equation of state compositional in situ combustion model: a study of phase behavior sensitivity. Transp Porous Media 76(2):219–246 8. Bruining J, Marchesin D (2007) Maximal oil recovery by simultaneous condensation of alkane and steam. Phys Rev E 75:036312-1–036312-16 9. Liu Z, Jessen K, Tsotsis TT (2011) Optimization of in-situ combustion processes: a parameter space study towards reducing the CO2 emissions. Chem Eng Sci 66:2723–2733 10. Baily HR, Larkin BK (1960) Conduction–convection in underground combustion. Trans AIME 219:320–331 11. Ma Y, Chen Z (2004) Parallel computation for reservoir thermal simulation of multicomponent and multiphase fluid flow. J Comput Phys 201:224–237 12. Belgrave JDM, Moore RG (1992) A model for improved analysis of in-situ combustion tube tests. J Pet Sci Eng 8(2):75–88 13. Vinaykumar BG, Singh AP, Ganguly S (2014) Effect of heat diffusion in the burden on the dissociation of methane in a hydrate bearing formation. J Nat Gas Sci Eng 16:70–76 14. Castaldi MJ, Zhou Y, Yegulalp TM (2007) Down-hole combustion method for gas production from methane hydrates. J Pet Sci Eng 56(1–3):176–185
13
1080 15. Ruidas BC, Ganguly S (2014) Progression of a thermal front in porous media of finite length due to the injection of an inert gas. J Porous Media 17(2):179–184 16. Akkutlu IY, Yortsos YC (2003) The dynamics of in-situ combustion fronts in porous media. Combust Flame 134(3):229–247 17. Fassihi MR, Yannimaras DV, Kumar VK (1997) Estimation of recovery factors in light-oil air-injection projects. SPE Reserv Eng 12(3):173–178 18. Greaves M, Young TJ, El-Usta S, Rathbone RR, Ren SR, Xia TX (2000) Air injection into light and medium heavy oil reservoirs: combustion tube studies on west of shetlands clair oil and light Australian oil. Chem Eng Res Des 78(5):721–730 19. Turta AT, Singhal AK (2001) Reservoir engineering aspects of light-oil recovery by air injection. SPE Reserv Eval and Eng 4(4):336–343 20. Liu Y, Li YC, Cheng QL, Xiang XY, Wang ZG (2009). Exergy transfer analysis of thermal driving oil process. In: Proceeding of
13
Heat Mass Transfer (2016) 52:1071–1080 Asia-Pacific power and energy engineering conference, Wuhan, China 21. Gear GW (1971) Numerical initial value problems in ordinary differential equations. Prentice Hall, Englewood Cliffs 22. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing. Cambridge University Press, Cambridge 23. Riazi M (2005) Characterization and properties of petroleum fractions. ASTM International, West Conshohocken 24. Berthelot D (1907) Travaux et Mémoires du Bureau international des Poids et Mesures—Tome XIII. Gauthier-Villars, Paris 25. Smits AJ, Dussauge J-P (2006) Turbulent shear layers in supersonic flow. Birkhäuser, Basel, p 46. ISBN 0-387-26140-0 26. Butler RM (1991) Thermal recovery of oil and bitumen. Prentice Hall, New Jersey
