Gas Generation during Electrical Heating of Oil ... - ACS Publications

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Gas Generation during Electrical Heating of Oil Sands Hassan Hassanzadeh,*,† Thomas G. Harding,‡ R. Gordon Moore,† Sudarshan A. Mehta,† and Matthew G. Ursenbach† †

Department of Chemical & Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada ‡ Nexen Energy ULC, Calgary, Alberta T2P 3P7, Canada ABSTRACT: Electrical heating of oil sand formations involves high temperatures, which can lead to in-situ gas generation as a result of aquathermolysis and thermal cracking of bitumen. The thermal cracking of bitumen can also lead to formation of coke at higher temperatures. Prediction of the volume of the produced gas and coke is very important in design and implementation of thermal processes dealing with high temperatures such as electrical heating and steam assisted gravity drainage with addition of oxygen. In this work a reaction kinetics model has been developed based on the experimental data of in-situ gas generation from an Athabasca bitumen sample. The kinetic parameters were estimated using the experimental data in the temperature range 360− 420 °C. The results show that coke formation can be significant at higher temperatures. An important observation was that a plateau to H2S production is not expected at higher temperatures. In addition, the results show that operation at 370 °C produces a gaseous composition that minimizes the volume of the produced gas at the surface. A simple scaling analysis is presented that allows clarification of the scatter that has been observed in the reported produced hydrogen sulfide versus the operation temperature. The developed model provides a useful tool for the estimation of produced gas composition during thermal recovery processes.

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INTRODUCTION In-situ extraction of heavy oil and bitumen from oil sands normally involves injection of steam and other hot fluids into subsurface formations. While considerable attention has been paid to the effects of high temperatures on the physical properties of reservoir fluids, such as oil viscosity reduction, the chemical interactions between steam and bitumen have often been ignored. Evolution of gases such as H2S from bitumen production operations is a concern in thermal recovery of oil sands because H2S poses a hazard to health, affects the integrity of wellbore tubulars, and also impacts surface facility design consideration. Also, the heating value of the recovered vent gases from the oil sand operations depends on the composition of the noncondensable produced gases, including H2S and CO2. In addition, the buildup of the noncondensable gases may adversely affect the thermal recovery process performance. Most often, gas production predictions made using thermal reservoir simulators are poor compared to the actual field data. This is partly because an appropriate kinetic model capable of incorporating the complex chemistry of steam/bitumen interaction is lacking. The chemical reactions resulting from injection of steam in steam stimulation have been addressed by Hyne1 in the late 70s. The term aquathermolysis was first coined by Hyne to describe the chemical processes that occur when oil sands are contacted with high temperature steam or water. Hyne believed that features of aquathermolysis have a profound effect on evolution of H2S, heavy oil viscosity reduction, and maturation of the insitu bitumen in the steamed zone. Hyne defined the concept of aquathermolysis window as the temperature range at which the aquathermolysis reactions are predominant. This window was further described as the temperature interval between slow maturation reaction (below 200 °C) and thermal cracking (above 300 °C). In addition, aquathermolysis was differentiated © 2016 American Chemical Society

from thermal cracking based on conversion of the liquid phase to coke.1 In aquathermolysis, little or no conversion to coke takes place and conversion to gas phase is minimal, as compared to the generated gas phase during the thermal cracking, which occurs above 300 °C.1 Furthermore, Hyne has shown that acid polymerization enhanced by CO2 generation from carbonate minerals is the source of viscosity increase at lower aquathermolysis temperatures or early times. However, such an unfavorable viscosity increase is followed by molecular cleavage reaction of the polymerized materials that reverses the viscosity trend with time or at higher temperatures.1 Many reactions, such as pyrolysis, hydrogenation, ring opening, ring closing, and desulfurization in relation to C−S, C−N, C−O, C−C, CS, CO, CN, and CS bonds, are involved in aquathermolysis, and these reactions result in viscosity reduction and upgrading.2 Significant viscosity reduction of heavy oils in the presence of catalysts has been reported in the literature.2,3 An understanding of the gas evolution process and viscosity variations as a result of aquathermolysis may lead to useful applications in oil sand thermal operations.1 At 300 °C, water has a density and polarity analogous to those of acetone at room temperature associated with a major drop in water dielectric constant at 300 °C.4 Therefore, water provides a considerably more favorable reaction environment for nonpolar compounds at high temperatures.5 The overall chemistry of aquathermolysis involves several interdependent chemical reactions in a complex way where water plays an important role as catalyst, reactant, and solvent.5 The similarity of the produced gas composition (CO2, H2, CH4, H2S) as a result of Received: May 22, 2016 Revised: July 17, 2016 Published: August 18, 2016 7001

DOI: 10.1021/acs.energyfuels.6b01227 Energy Fuels 2016, 30, 7001−7013

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Energy & Fuels

sample gradually increased by aging (16 days), suggesting the viscosity reduction by aquathermolysis is partially reversible. In a recent study Ayache et al.17 extended the Barroux et al.18 sulfur conversion scheme to model aquathermolysis experiments reported by Lamoureux and Barroux.19 The kinetic model was calibrated with experimental data and then implemented in a thermal reservoir simulator. Kapadia et al.20 also reported an aquathermolysis model for bitumen where the effect of the water gas shift reaction was also included. In addition, they included a decay rate to account for the decline in the reaction rate. Montgomery et al.21 conducted aquathermolysis experiments over a temperature range of 150−325 °C and concluded that H2S generation can be avoided if the temperatures in the reservoir are not allowed to exceed 200 °C. Detailed reviews of catalytic and noncatalytic aquathermolysis of bitumen have been recently reported by Tumanyan et al.22 and Muraza and Galadima23 where challenges in application of catalytic aquathermolysis for upgrading heavy oils and natural bitumen have been thoroughly discussed. While several studies have addressed the gas evolution as a result of aquathermolysis and thermolysis of bitumen using experimental and modeling approaches, there is still a lack of comprehensive predictive models and appropriate simulation data sets applicable to thermal stimulation of oil sands. Thermal stimulation of oil sand formations covers a wide range of temperature. In steam assisted gravity drainage (SAGD) and cyclic steam stimulation the steam chamber temperature can be as high as ∼250 °C while the thermal wave propagates away from the chamber by thermal conduction and heats the reservoir oil from its initial temperature of about 10 °C. In electrical heating the reservoir temperature in the vicinity of the wellbore can be as high as ∼500 °C, resulting in a wide range of temperature (∼10− 500 °C) during the operations. Another phenomenon of note is that field evidence has shown that the bitumen produced from the reservoir located between SAGD well pairs is less viscous than the oil extracted from the SAGD well pairs themselves, which suggests that chemical alteration of the virgin bitumen is occurring as a result of longer residence times in the reservoir. Currently, there is need for development of a comprehensive kinetic model of coking and aquathermolysis reactions that cover the wide range of temperature encountered in thermal stimulation of oil sand formations. In this study a kinetic model for prediction of in-situ gas evolution during electrical heating of oil sand is presented. The developed model is calibrated with the experimental data of an Athabasca oil sand sample at a temperature range of 360−420 °C, and the reaction parameters are estimated. The effects of the stimulation temperature and the asphaltene content of the bitumen on the in-situ gas evolution are studied. In addition, a simple scaling analysis is presented that can be used to relate the lab scale experiments to the field scale applications. The kinetic model and parameter estimation are presented first followed by the application of the developed model to electrical heating of oil sands.

aquathermolysis of organosulphur compounds (e.g., thiolane, thiophene) and those of oil sands has led to the idea that organosulphur compounds in the oil sands are the main initiators of H2S evolution.6−8 Comprehensive experimental data on steam stimulation of oil sands are very scarce. As pioneers, Hyne, and his co-workers conducted experimental studies of aquathermolysis of different oil sands from Canada and Venezuela.1 The experimental data reported by Hyne indicated that aquathermolysis activity of oils is an important parameter in design of thermal operations, and experimental results of an oil sample cannot be generalized. Monin and Audlbert9 studied thermal alteration of heavy oil samples in the absence and presence of water and minerals in an autoclave at 350 °C. They have reported significant modification of the oil composition and formation of insoluble organic materials and hydrocarbon gases (mainly methane). Generation of CO2 was observed to be a function of the minerals. In addition, it has been reported that water played the role of reactant in oxygen-containing compounds in the experiments and the amount and composition of products was determined to be highly dependent on the initial oil composition. A kinetic model of gas generation has been developed by Belgrave et al.10 from experiments conducted over the temperature range 360−420 °C on five heavy oil core samples.10,11 It was shown that core mineralogy played an important role in the generation of CO2 and that the volume of released H2S was dependent on oil composition, mineralogy, reaction temperature, and time. Gas production was found to be mainly related to heavy oil and asphaltene conversion. Thimm12 derived a simple equation for H2S generation by assuming the H2S evolution rate as a pseudo-zero-order reaction, which may be used as a first-order estimate of H2S production. A review of the kinetics of gas evolution during thermal recovery processes of heavy oil and bitumen has been reported by Zheng and Huang,13 where they highlighted the need for a proper kinetic model and implementation of gas evolution in thermal reservoir simulators. Lamoureux and Lorant14 have reported aquathermolysis experiments on Athabasca oil sands, and the amount of produced H2S versus time was measured at 10 MPa and at five temperatures (240−320 °C). The experiments were conducted in gold tubes filled with equal volume fractions of oil and demineralized water. They characterized the sample by quantifying the sulfur distribution over the oil sand sample SARA fractions as well the insoluble fraction before and after experiments. These data were then incorporated into a kinetic model to estimate the kinetics parameters of sulfur conversion, but the estimated parameters were not reported. Goicetty15 reported aquathermolysis experiments of 6 h duration on two heavy oil samples in a continuously stirred reactor at three temperatures of 225, 250, and 275 °C using the water/oil ratio 3. The asphaltene contents of the two samples used in the experiments were 20 and 23 wt %, and the molar masses were 2446 and 3312 g/mol, which (the molar masses) seem too high. The sample with the lower asphaltene content demonstrated a nonmonotonic H 2S production with time. The evolution of H2S was found to be specific to each sample. Song et al.16 conducted aquathermolysis experiments of 72 h duration on two oil samples at different temperatures ranging from 240 to 300 °C in an autoclave. A water−oil mass ratio of 0.3 was used in the experiments. Viscosity reduction was observed for both samples below 48 h, following which the oil viscosity stabilized, indicating achievement of an equilibrium condition. An interesting observation was that the viscosity of the reacted oil

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REACTION KINETIC MODEL The kinetic model presented here is adapted from the work of Belgrave et al.10 The kinetic model comprises 11 components with the only solid component being coke. The liquid fractions are asphaltene (ASPH), heavy oil (HO, BP > 300 °C), and light oil (LO, BP < 300 °C). The hydrocarbon gaseous species are methane (C1), ethane and ethylene (C2), and propane to hexane (C3+). The non-hydrocarbon species are hydrogen (H2), carbon monoxide (CO), carbon dioxide (CO2), and hydrogen sulfide 7002


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Energy & Fuels (H2S). Belgrave et al.10 incorporated aquathermolysis and thermal cracking reactions into three main reactions, which are more adaptable for use in reservoir simulators. The combined aquathermolysis and thermal cracking model was described using the following reactions:

dωi = kjϕω i ASPH dt

where (i, j) = (H2,3), (H2S,4), (CO,5), (CO2,6), dωC1

k1*

dωC2H 6

+ a1,6CO + a1,7CO2 + a1,8C2H6 + a1,9C3+

dt (1)

dωC3+

k 2*

HO → a 2,1Coke + a 2,2LO + a 2,3CH4 + a 2,4C2H6 + a 2,5C3

= k12ωHO + k10ωASPH + k19ωLO

dt

ASPH → a1,1Coke + a1,2HO + a1,3LO + a1,4H2 + a1,5CH4

+ a1,10H2S

+

dt

= k13ωHO + k 9ωASPH + k18ωLO

= k15ωHO + k 7ωASPH + k16ωLO

dωCoke = k14ωHO + k 8ωASPH + k17ωLO dt

(2)

k 3*

LO → a3,1Coke + a3,2CH4 + a3,3C2H6 + a3,4C3+

(7)

(8)

(9)

(10)

(11)

where ω is the mass fraction, ϕ is the reaction deactivation function, ki = Aie−Ei/RT are the reaction rate constants (1/h), A is the pre-exponential factor (1/h), and E is the activation energy (J/mol). During the processes of cracking and aquathermolysis, the reaction rate may decrease as the reactions progress. Experimental data10,11 have shown a peak in evolution of species such as nonhydrocarbon gases (H2S, CO2, H2, and CO), which may imply consumption of these species by sequential reactions. Core mineralogy also has been reported to play a key role in the generation of CO2 and the amount of H2S released.10,11 The composition of the produced gas can also be affected by water gas shift reaction (WGSR), hydrogenation of heavy oil, and coking. Including all of these mechanisms in a reaction model is very difficult, if not impossible, and so these mechanisms have not been resolved in detail in the model described above. Instead, these mechanisms are partly taken care of by incorporation of the deactivation of reaction rates into the described model. An empirical exponential deactivation function is found satisfactory in matching the experimental data as given by

(3)

where k1*and aij values are the reaction rate constants and mass fraction of component j among the products of component i. In this work, the three main reactions have been divided into subreactions to allow additional flexibility and accuracy in estimation of the reaction parameters. Figure 1 shows the general

ϕi(t ) = αi(e−κit − e−(κi + λ1)t )

(12)

where i = H2, H2S, CO, CO2, and κ and α are the temperaturedependent parameters of the deactivation rate function and parameter λ1 is given by 10

λ1 = ∑ ki i=1

Figure 1. Reaction scheme used for the combined coking and aquathermolysis kinetic model.

Generation of carbon dioxide, and the amount of hydrogen sulfide evolved is a complex function of oil composition and mineralogy of the core materials. While the physics behind the maximum observed in experimental data is quite complicated, the empirical deactivation function allows simulation of the experimental data. The reaction scheme in matrix form can be written as ω̇ = [A][ω], where ω = [ASPH, HO, LO, H2, H2S, CO, CO2, CH4, C2H6, C+3 , Coke]T, ω̇ = dω/dt, and A is a sparse matrix of the rate constants. The details of the equations are given in the Appendix at the end of this paper. The system of equations ω̇ = [A][ω], subject to the initial condition ω0 = [ASPH0, HO0, LO0, 0, 0, 0, 0, 0, 0, 0, 0]T, can be solved numerically. For a case with ϕi = 1, the system of equations can be solved analytically. The detailed analytical solutions to the system of equations are given in the Appendix at the end of this paper. The analytical and numerical solutions

reaction pathways used in this formulation. The governing differential equations of the reaction scheme shown in Figure 1 are given by dωASPH = −(k1 + k 2 + k 7 + k 8 + k 9 + k10 + ϕH 2k 3 dt + ϕH 2Sk4 + ϕCOk5 + ϕCO2k6)ωASPH dωHO = k1ωASPH − dt

(4)

15

∑ kiωHO

(5)

i = 11

dωLO = k 2ωASPH + k11ωHO − dt

19

∑ kiωLO i = 16

(6) 7003


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contents to 100 °C. The evolved gas was extracted and measured using a gas syringe, and the gas composition was determined using a gas chromatograph. The reactor was then heated to 200 °C and the volatiles extracted under a vacuum into an ice water trap and a liquid nitrogen trap. The collected liquid was analyzed using simulated distillation. The liquid fractions with boiling below and above 300 °C were defined as light oil (LO) and heavy oil (HO), respectively. The reactor was then washed with toluene to extract the remaining materials. The extracted sand fraction was desiccated and the coke content determined by mass loss on burning. The toluene soluble material was distilled in a rotary evaporator to remove the toluene. The residual material after removal of toluene was weighed and diluted with pentane, and the asphaltenes were extracted by filtration. The asphaltenes mass was obtained by direct weighing following removal of the pentane. The reaction model described in the previous section forms an unconstrained multivariable problem. Estimation of unknown parameters is possible using the unconstrained nonlinear optimization toolbox in MATLAB. Unconstrained nonlinear optimization uses the simplex search method of Langarias et al.24 The model described earlier was used as a forward model, and the experimental data were used to estimate the parameters by minimization of an objective function defined as the sum of the absolute relative errors between the experimental data and the results of the forward model. The estimated reaction parameters are shown in Table 2. Figure 2 also shows the Arrhenius plots for the reaction rate constants. The results of the parameter estimation show that all reactions follow Arrhenius type temperature dependence with a coefficient of determination (R2) close to unity. The asphaltene is essentially converted to form coke and gas. The heavy oil fraction also converted to form light oil, coke, and gaseous species. The light

serve as a forward model for estimation of the reaction parameters, and these are described in the next section.

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ESTIMATION OF REACTION PARAMETERS Gas evolution experiments were conducted on a core sample extracted from a bitumen deposit in Alberta to study the gas evolution over the temperature range 360−420 °C.10,11 Table 1 Table 1. Composition and Properties of the Bitumen Sample Oil composition (mass %) Asphaltenes (ASPH) Heavy oil (HO) Light oil (LO) Oil Density: Sand properties Surface area (m2/g) Quartz (mass %) Feldspar (mass %) Kaolinite (mass %) Illite (mass %) Type of water:

Molar mass (g/mol) 20.7 74.2 5.1

1092.8 903.2 192.3 1005.6 (kg/m3) 0.43 0.78 17 3 trace Distilled water

shows the composition and properties of the test sample. In these experiments 200 g of premixed extracted reservoir sand and oil along with distilled water were introduced into a quartz glass tube, which was then placed in a stainless steel reaction vessel equipped with a nominal 1/32-in., type K thermocouple, a pressure transducer, and a vapor product removal line. The reactor was then sealed, evacuated rapidly and charged with helium, and placed in an oven for the anticipated test period. At the end of each test, the reactor was removed from the oven and placed into an ice water quench bath for about 10 min to cool the Table 2. Estimated Reaction Kinetic Parameters Temperature (°C) Kinetic parameter k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12 k13 k14 k15 k16 k17 k18 k19

360

397 −8

9.824274 × 10 1.819523 × 10−2 1.027969 × 10−4 9.000000 × 10−3 1.095196 × 10−4 1.097706 × 10−2 8.728642 × 10−9 1.292745 × 10−13 3.483652 × 10−14 1.006397 × 10−3 2.184805 × 10−3 5.483422 × 10−9 7.411783 × 10−5 1.242863 × 10−3 1.088318 × 10−4 6.155315 × 10−3 4.807006 × 10−14 2.941919 × 10−3 2.635454 × 10−3

420 −6

2.596930 × 10 6.382963 × 10−2 2.642200 × 10−4 2.891396 × 10−2 7.288200 × 10−4 1.350000 × 10−2 2.811090 × 10−8 1.652580 × 10−8 8.693720 × 10−8 1.107702 × 10−2 3.347312 × 10−2 3.573370 × 10−8 2.203741 × 10−3 2.524507 × 10−2 3.002987 × 10−3 2.500000 × 10−3 1.330000 × 10−9 8.696330 × 10−9 2.054930 × 10−9

−5

3.233255 × 10 1.265417 × 10−1 6.983233 × 10−4 4.500921 × 10−2 1.712647 × 10−3 1.442142 × 10−2 4.878407 × 10−8 1.614591 × 10−5 5.645156 × 10−4 3.711877 × 10−2 5.030637 × 10−2 1.023576 × 10−7 7.962916 × 10−3 8.147883 × 10−2 1.415496 × 10−2 8.027997 × 10−4 1.769668 × 10−7 1.639482 × 10−11 3.440312 × 10−13

R2

Activation Energy E (J/mol)

Pre-exponential factor A (1/h)

0.9931 0.9999 0.9726 0.9907 0.9967 0.9827 0.9978 0.9999 0.9998 0.9991 0.9407 0.9999 0.9907 0.9918 0.9976 0.9505 0.9999 0.9987 0.9995

347907.8 118139.3 114106.8 99423.6 168765.7 16954.72 105434.9 1132338 1426834 220464.6 198697.5 178077.8 288883.8 258128.9 298370.6 −119536 1132338 −1162436 −1379193

4.47593 × 1021 1.01951 × 108 2.49447 × 105 1.48214 × 106 9.48521 × 109 2.77066 × 10−1 4.43177 3.26478 × 1080 1.66721 × 10104 1.58919 × 1015 6.40061 × 1013 5.483422 × 10−9 5.55603 × 1019 2.65772 × 1018 4.71288 × 1020 9.29620 × 10−13 3.26478 × 1080 3.20889 × 10−99 4.86689 × 10−117

ϕi(t) = αi(e−κti − e−(κi+λ1)t), λ1 = ∑i = 1 ki 10

ϕH2 ϕH2S ϕCO ϕCO2

−2

κ = 0.5, α = 3.509796 × 107 e−2.172505×10 T, T in K κ = e−1140.1923−0.31072639T−207.21845 ln(1/T), α = 7968356.633e−0.018819824 × 10−2T −2 κ = 1.015459 × 10−2T − 5.562899,α = 2.640342 × 1019e−6.302377×10 T −2 κ = 2.022015 × 10 T − 11.81013, α = 75 7004


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Figure 3. Comparison of experimental data (markers) and model predictions (lines) at 360 °C.

Figure 2. Arrhenius plots for the reaction rate constants.

oil was converted to coke and gaseous species. However, a net increase in the light oil content was eventually observed. The main source of gas evolution was found to be the asphaltenes. Conversion of light oil to gaseous hydrocarbons (methane, ethane, propane plus fraction) follows an Arrhenius relation but shows a negative activation energy implying that the reaction slows down with increasing temperature. From a physical viewpoint this may be an indication of a complicated reaction mechanism or multistep mechanism of surface catalyzed reactions, which have not been resolved in the proposed scheme. In addition, the experiments were conducted using the premixed extracted reservoir sand containing quartz, feldspar, kaolinite, and Illite with large surface areas. The kinetics of a surface catalyzed chemical reaction can be a strong function of the enthalpy of adsorption of reactants.25 The apparent activation energy can be expressed as Eapp = E + ΔHads.25 The enthalpy of adsorption is almost always negative. Therefore, depending on the magnitudes of E and ΔHads, the apparent activation energy can be either positive or negative.25 For instance, the cracking of larger alkanes (C18−C20) over H-ZSM-5, an acidic zeolite, has been shown to demonstrate negative apparent activation energy.25,26 It is worth noting that the results presented here show that the cracking of light oil (LO, a cut with boiling point below 300 °C) to gaseous hydrocarbons (C1, C2, C3+) has shown negative apparent activation energy. Note that the boiling point of n-C17 is 302 °C. Figures 3−5 show the comparison of the experimental data and the model predictions based on the estimated parameters for three temperatures of 360, 397, and 420 °C. The results show that the model predictions are in agreement with the experimental data. The model, however, was not able to predict the produced CO as accurately. Experimental data has shown a peak in production of CO. This is likely because CO was consumed in the water gas shift reaction, which has not been


taken into account in the developed model. In fact, in some of the experiments the concentration of produced CO has shown fluctuations and may be below the resolution of the analytical tools used to measure the gas phase composition. Evolution of non-hydrocarbon gaseous species demonstrates some degree of deactivation as a reaction takes place. Figure 6 shows the mass fraction of the nonhydrocarbon gaseous components (H2, H2S, CO, CO2) versus time predicted using the developed model with and without deactivation as compared with the experimental data. The results shown in Figure 6 reveal that the deactivation of H2 generation diminishes with temperature such that no appreciable deactivation was observed at 420 °C. Deactivation of H2S generation was observed at all temperatures and shows a nonmonotonic behavior with temperature as shown in Figure 6. In addition, the results presented in Figure 6 suggest that the deactivation of CO2 7005


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generation shows a temperature dependency particularly at lower temperature. In the following section, the described model calibrated with the experimental data will be used to predict the gas evolution during thermal stimulation of the bitumen used in the described experiments.

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APPLICATION TO ELECTRICAL HEATING OF A BITUMEN BEARING FORMATION Electrical Heating. A new recovery process called In-Situ Reflux (ISR) has been suggested recently for bitumen recovery.27,28 A schematic of the processes is depicted in Figure 7. This process aims at exclusion of surface steam generation and injection facilities and more efficient heat delivery to the formation, which can prominently cut the capital and operating costs of oil sand development projects and also minimize the environmental footprint of bitumen production. The In-Situ Reflux (ISR) process relies on electrical heating supplemented by makeup fluids (water or solvents) instead of continuously injecting steam to the oil sand formations and thus improving the energy efficiency of the recovery process. In this process, analogous to SAGD, two long parallel horizontal wells (about 1


Figure 6. Comparison of experimental data (markers), and model predictions with (lines) and without (dashed lines) deactivation at 360 °C (top), 397 °C (middle), and 420 °C (bottom). 7006


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wellbore tubulars and also impacts surface facility design considerations. In addition, thermal cracking leads to coke formation, which may hinder the process. It is important to determine the rate of evolution of the gaseous species and the coke. In order to make this assessment, it is necessary to relate the time scale of the reaction kinetics and the time scale of the gravity drainage, which is the dominant mechanism in bitumen drainage, which will be described later. Composition of Reaction Products. The model described in the previous section was used to predict the overall composition of the reaction products. The overall product composition determines the mass (volume) of gas generated per unit mass of the converted oil while the latter defines the concentration of various species (e.g., H2S) in the produced gas stream. Figure 8 shows the overall product composition against

Figure 7. Schematic of the In-Situ Reflux (ISR) process proposed by Nexen Energy ULC:27,28 (a) cross-sectional view; (b) side view.

km in length), located 4 to 6 m one above the other, are drilled. Electrical heaters are installed along the upper well to evenly heat the oil sand formation, evaporate the connate water, and mobilize the highly viscous oil. A steam chamber is formed above the top well similar to SAGD, but in the in-situ reflux process, the steam chamber results from the evaporation of the in-situ connate water in the formation or the injected makeup fluids. The generated steam loses its latent heat to the cold oil sands and condenses. The condensed steam and the mobilized oil drain under gravitational force, and as the drained fluids approach the heater, the condensed steam is revaporized and migrates upward, while the heated oil drains toward the production well. During electrical heating of oil sands, the formation around downhole heaters will be exposed to very high temperatures (above 300 °C), virtually converting the generated steam chamber to a downhole reactor. At such a high temperature the bitumen drained by gravity drainage enters the downhole reactor and will be susceptible to aquathermolysis and thermal cracking in the presence of steam. Both aquathermolysis and thermal cracking of bitumen generate undesirable gases, which may adversely impact the performance of the process. In particular, the rising level of the H2S and CO2 concentrations in the generated gas is a hazard and affects the integrity of the

Figure 8. Reaction products composition versus time for different temperatures. The mass fraction multiplied by 103 gives the grams of each species per kilogram of bitumen sample.

time for different temperatures. The mass fraction multiplied by 103 gives the grams of each species per kilogram of the reacted bitumen. The results show that temperature has a significant effect on the conversion of ASPH and HO to gaseous products and coke. Production of a significant amount of gaseous hydrocarbon (C1, C2, and C3+) was predicted. Among nonhydrocarbon gaseous species, the most notable product is H2S followed by CO2, H2, and CO. A significant amount of coke formation has been predicted for temperatures above 380 °C. The coke mass fraction reaches as high as 0.3 at 420 °C. Figure 8 7007


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Figure 9. Produced gas phase composition versus time for different temperatures. The species mole fraction multiplied by 106 gives molar based ppm.

shows only the overall composition of the products, and so to have a better understanding of the composition of the gaseous phase, which is of practical importance, the composition of the produced gas phase has been plotted in Figure 9. The mole fraction multiplied by 106 gives the molar based ppm of the produced gaseous species. The results show that the concentrations of non-hydrocarbon species generally decrease with time while those of hydrocarbon species increase in the produced gas. The major components of the produced gas stream are C1 followed by C3+, C2H6, H2S, CO2, H2, and CO. The results which show that the composition of the non-hydrocarbon gases decrease with time are consistent with the experimental observations.10,11 The effect of temperature on the composition of the produced gas is also of practical importance. Figure 10 shows the composition of the major non-hydrocarbon gases against temperature. The results show that the compositions of H2, H2S, and CO2 increase with temperature, reach a maximum, and then decline. The temperatures at which the maximum concentrations of H2, H2S, and CO2 occur are around 374, 370, and 367 °C, respectively. The main source of H2S production is gasification of asphaltene, which is dominant at the lower temperature. On the other hand, at higher temperature, thermal cracking is the leading mechanism, which results in production of significant amounts of hydrocarbon species, such as methane, as was shown earlier in Figure 9. Therefore, it is expected that the concentration of H2S in the produced gas phase decreases with increasing temperature. Hyne1 has speculated that, regardless of the temperature, an upper limit to H2S production is anticipated. However, at higher temperatures, thermal cracking prevails and the concentration of nonhydrocarbon gases decreases with time. This has been observed

Figure 10. Mole fractions of H2, H2S, and CO2 in the produced gas phase against temperature for a residence time of 1 day.

in experimental results as well being predicted by the current model. Therefore, a plateau for the concentration of H2S in the produced gas stream is not expected at higher temperatures. One of the practical challenges in high temperature stimulation of oil sands is coke formation because it may hinder oil drainage by plugging the formation. Figure 11 shows coke formation versus the operating temperature. The results show that at lower temperatures the coke formation is negligible while at higher temperatures the coke mass fraction can reach as much as 300 g per kilogram of the reacted bitumen sample for 1 day of residence time. Such a high concentration of coke imposes great challenges in field scale implementation of electrical heating. Figure 11 also shows the evolved gas mass fraction. The results show that for the bitumen sample studied for temperatures 7008


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Figure 11. Coke and gas mass fraction versus temperature for a residence time of 1 day.

Figure 13. Conversion of asphaltenes (ASPH) and heavy oil (HO) as a function of the reciprocal of temperature for a residence time of 1 day.

greater than 370 °C the coke mass fraction exceeds the gas formation. The gas evolution during thermal stimulation of bitumen depends highly on the asphaltene content. Figure 12 shows mole

ISR process and the batch experiments. Therefore, we seek a relation between the residence time of the reactants for the batch experiments and the ISR process. To do so, a global reaction rate 19 constant can be defined using kG = ∑i = 1 ki . This global rate constant represents the conversion of asphaltenes (ASPH), heavy oil (HO), and the light oil (LO) phase to gas and coke. The Arrhenius plot for the global reaction rate is shown in Figure 14, which shows a coefficient of determination (R2) close to unity.

Figure 12. Effect of asphaletene content on the evolved H2, H2S, and CO2 at different temperatures for a residence time of 1 day.

fractions of H2, H2S, and CO2 in the produced gas phase against temperature for different asphaltene contents. The results show that, irrespective of asphaltene content, there exists a temperature range (∼365−370 °C) at which the concentrations of H2, H2S, and CO2 are at a maximum. These results have been obtained based on 1 day of residence time. To gain a better understanding of the mechanisms involved, conversions of asphaltenes and heavy oil to gaseous products and coke versus the reciprocal of temperature are depicted in Figure 13. The slope of the semilog plot yields a crude estimate of the energy involved in the chemical reactions.1 The semilog plot of conversion of HO indicates a larger slope as compared with the ashpaltenes conversion, suggesting greater energy requirement. The heavy oil fraction (HO) is primarily converted to hydrocarbons as a result of thermal cracking reactions, which are more energy intensive than conversion of asphaltenes to nonhydrocarbon gases. Conversion of asphaltene to non-hydrocarbon gases involves scission of carbon−sulfur bonds and is less destructive as compared to thermal cracking, and thus, less energy is required.1 Scaling Analysis. The model described in this work simulates batch reactor experiments while the ISR process is a continuous one. To make a zeroth-order approximation of the insitu gas generation and coke formation, the residence time concept has been used to relate the evolution of products for the

Figure 14. Arrhenius plot for the global rate of reaction.

The reaction and the gravity drainage time scales can be defined as τr ∼ 1/kG and τg ∼ Hμφ/(KΔρg), respectively, where H is the formation thickness used as a characteristic length scale, μ is the oil viscosity at steam temperature, K is the effective formation permeability to oil, Δρ is the oil- and steam-phase density difference, and g is the gravitational acceleration. Therefore, the dimensionless times for the batch experiments and the ISR process are tDr = kGtr and tDg = (KΔρg/Hμφ)tg, respectively. A zeroth-order approximation suggests that, to have the same products in batch experiments and the ISR process, the dimensionless times of the two processes should be equal. While upscaling of the lab data to field scale is a great challenge, using this simple notion allows upscaling the batch reactor experiments to the ISR process. As an example, for a typical oil sand formation with an effective oil permeability of K = 1D, φ = 0.3, μ = 1 cP (at steam temperature), H = 30 m, Δρ = 790 kg/m3, and kG ≈ 1.512 × 10−5 s−1 (0.0544 h−1) at ∼360 °C the gravity drainage dimensionless time can be obtained as tDg = (8.776 × 10−7) 7009


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which can be related to the oil sand formation and fluid properties, through definition of the dimensionless time given by tD = (KΔρg/Hμφ)t. Figure 15 also shows the H2S concentration in the produced gas (molar based ppm) versus temperature for different dimensionless residence times. The results show that the concentration of H2S in the produced gas stream decreases for temperatures greater than 370 °C. This is expected due to control of gas generation by thermal cracking of HO and LO to gaseous hydrocarbons at higher temperatures. Also, the H2S concentration in the produced gas stream was decreased by increasing the retention time as a result of generation of a large volume of gaseous hydrocarbons at high residence times. These results presented here are the zeroth-order approximation of the complex processes and are based on a simple scaling analysis with simplified assumptions. This analysis does not take into account the complex fluid phase behavior and reactive multiphase flow physics of the problem; however, much insight can be gained regarding the fundamentals of the process by using the simple scaling.

tgwhereas the batch reaction dimensionless time is given by tDr = (1.512 × 10−5)tr. Comparing the two dimensionless times, one can conclude that the ISR process involves a downhole reactor with an effective reaction rate constant of kG ≈ 0.0032 h−1 (8.776 × 10−7 s−1) as compared to the batch reactor experiments with kG ≈ 0.0544 h−1. Therefore, the time required to produce the same reaction products as the batch experiment would be on the order of tg ≈ 17tr. Thus, the products generated in a 10 h batch experiment require about 7 days of ISR field scale residence time to form. Often the produced volumes of H2S and CO2 per unit volume of the extracted bitumen12,29,30 have been reported against the operation temperature, and while the concentrations of these species have generally shown the expected increasing trend with temperature, a wide scatter has been observed. For example, for a constant temperature of 255 °C, a H2S to bitumen ratio of 50− 250 L/m3 has been reported.29 The volume of gases produced during thermal stimulation depends on the residence time of the mobilized bitumen, which has been disregarded in previous studies. An analysis was conducted to examine the sensitivity of the produced gas to the residence time. Results are presented for different dimensionless times to generalize the findings. Figure 15 shows volumes of H2, H2S, CO2, CH4, and total gases per

■

SUMMARY AND CONCLUSIONS In-Situ Reflux (ISR) has been suggested recently for bitumen recovery, which relies on electrical heating of oil sand formations. The ISR process applied to oil sands involves high temperatures converting the created steam chamber to a downhole reactor. Generation of gases during thermal stimulation of oil sand formations can have a great impact on design and operation of surface facilities, environmental footprint, as well as recovery of bitumen. In addition, formation of coke at higher operation temperatures can potentially lead to formation damage. In this study a comprehensive reaction model was developed to predict the in-situ gas evolution and coke formation during thermal stimulation of a sample of an Athabasca oil sand. The kinetic model comprises 11 components, including coke, the liquid fractions (asphaltene, heavy oil, light oil), hydrocarbon gaseous species (C1, C2, C3+), and non-hydrocarbon species (H2, CO, CO2, H2S). Experimental data of thermal stimulation of an Athabasca oil sand sample in the temperature range 360−420 °C was used to estimate the reaction parameters. This study shows that significant amounts of gases can be produced as a result of conversion of ASPH and HO during the thermal stimulation of oil sands at higher temperatures. The operating temperature was found to have an important effect on conversion of ASPH and HO to gaseous products and coke. Coke formation at higher temperature was found to be significant and can reach as high as 0.3 at 420 °C. The main source of non-hydrocarbon gas generation was found to be the asphaltene. It was shown that at higher temperatures thermal cracking becomes dominant and the concentration of non-hydrocarbon gases decreases with time. This has been observed in experimental results as well as being predicted by the current model. Therefore, an important observation was that a plateau to H2S production is not expected at higher temperatures. The volume of the produced gases has been traditionally reported as a function of the operating temperature. While this type of analysis has shown an increasing trend with temperature, a significant scatter has been reported, which can be misleading, and has no association to the real physics of the in-situ gas generation. Using a simple scaling analysis, we have shown that this type of analysis appears to be irrelevant since the residence time of the mobilized bitumen has not been taken into account. Using the developed model, it was shown that the residence time

Figure 15. Volume of produced gases per metric tonne of bitumen reacted as well as concentration of H2S (molar based ppm) in the produced gas phase.

metric tonne of bitumen reacted. The results show that the retention time has a profound effect on the volume of these gases, particularly at higher temperatures. Figure 15 shows that up to 4 S m3 of CO2 may be produced per tonne of bitumen depending on the residence time. (P = 101.325 kPa (14.7 psi) and T = 15.56 °C (60 °F) were used as standard pressure and temperature, respectively.) This ratio for H2 and H2S is close to 3 and 15, respectively. The predicted volume of the produced gases shows a minimum at 370 °C and increases with temperature to about 70 S m3 per tonne of bitumen at higher residence time. The residence time of the mobilized oil as a result of thermal stimulation is a function of the formation and fluid properties, such as permeability, porosity, oil viscosity, and formation thickness, as defined earlier. In addition, the residence time is also a strong function of reservoir heterogeneity. Therefore, a wide scatter has been reported in the concentration of the produced H2S when plotted against temperature. The results shown in Figure 15 are presented based on dimensionless residence time, 7010


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reactions. Our study shows a great need for experimental data of in-situ gas generation at lower temperatures in the range 100− 350 °C. And finally, the developed model does not consider the coke induction time, which has been observed during the cracking studies.31 This means that applying the model at low temperatures, where the coke induction times can be long, may overestimate the coke formation, and this in turn may have the same effect on gas formation.

of the mobilized bitumen has a great impact on the volume of the produced gases. The developed model provides a useful tool for the estimation of the in-situ produced gas composition during thermal recovery processes at a temperature range of 360−420 °C. It has been clearly shown in the literature that the gas evolution from the oil sands depends on the type and origin of the samples. The asphaltene content and heavy fractions of bitumen play a critical role in generation of the in-situ gases. Therefore, results obtained for one bitumen sample cannot be easily extended to bitumens from different sources. Incorporation of kinetics models of in-situ gas generation into reservoir simulations of thermal processes is of great importance, and much understanding can be gained concerning the fundamentals of the process through consideration of these ⎡−kASPH ⎢ ⎢ ⎢ k1 ⎢ ⎢ ⎢ ⎢k 2 ⎢ ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ k10 ⎢ ⎢k 9 ⎢ ⎢k 7 ⎢ ⎣k8

0

■

APPENDIX

The reaction scheme in the matrix form can be written as ω̇ = [A][ω], where ω = [ASPH, HO, LO, H2, H2S, CO, CO2, CH4, C2H6, C+3 , Coke]T, ω̇ = dω/dt, and A is a sparse matrix given by

0

0

0

0

0

− ∑ ki 0

0

0

0

0

− ∑ ki 0

0

0

0

0

0 0

15 i = 11 19

k11

i = 16

0

0

k 3ϕH 2 0

0

0

0

k4ϕH 2S 0

0

0

0

0

k5ϕCO 0

0

0

0

0

0

k6ϕCO2

k12

k19

0

0

0

0

k13

k18

0

0

0

0

k15

k16

0

0

0

0

k14

k17

0

0

0

0

(A1)

ωLO(t ) = e−(λ1+ λ2)t [(β1 + β2)(λ 2 − λ3)e λ2t + β3(λ1 − λ3)e λ1t ]

where

(λ3 − λ1)(λ 2 − λ3)

kASPH = k1 + k 2 + k 7 + k 8 + k 9 + k10 + ϕH 2k 3 + ϕH 2Sk4 + ϕCOk5 + ϕCO2k6

⎛ [(β1 + β2)(λ 2 − λ3) + β3(λ1 − λ3)] ⎞ −λ t 0 ⎟e 3 + ⎜ωLO − (λ3 − λ1)(λ 2 − λ3) ⎝ ⎠

(A2)

The initial condition for the system of equations above is given by ω0 = [ASPH 0 , HO0 , LO0 , 0, 0, 0, 0, 0, 0, 0, 0]T

(A6)

(A3)

where λ3 =

This system of equations subject to the initial condition for a case with no ϕi = 1 (no deactivation) can be solved using the integrating factor method32 as given by 0 ωASPH (t ) = ωASPH e−λ1t

β2 =

10

ωHO(t ) =

β1 =

k2ω0ASPH,

and

0 k11k1ωASPH (λ 2 − λ1)

(A7)

(A8)

The concentration of non-hydrocarbon species can be obtained using

⎛ ⎞ 0 ⎟e − λ 2 t − e−λ1t + ⎜ωHO (λ 2 − λ1) (λ 2 − λ1) ⎠ ⎝ 0 k1ωASPH

19 ∑i = 16 ki ,

⎛ ⎞ k ω0 0 β3 = k11⎜ωHO − 1 ASPH ⎟ (λ 2 − λ1) ⎠ ⎝

(A4)

where λ1 = ∑i = 1 ki , 0 k1ωASPH

ωi(t ) =

(A5)

where λ 2 =

0 0 0 0⎤ ⎥ ⎥ 0 0 0 0⎥ ⎥ ⎥ ⎥ 0 0 0 0⎥ ⎥ ⎥ 0 0 0 0⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎥ ⎥ 0 0 0 0⎦

15 ∑i = 11 ki

0 kjxASPH

λ1

(1 − e−λ1t )

(A9)

where (i,j) = (H2,3), (H2S,4), (CO,5), (CO2,6). 7011


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(3) Chen, Y.; Wang, Y.; Lu, J.; Wu, C. The viscosity reduction of nanokeggin-K3PMo12O40 in catalytic aquathermolysis of heavy oil. Fuel 2009, 88 (8), 1426−1434. (4) Pitzer, K. S. Dielectric constant of water at very high temperature and pressure. Proc. Natl. Acad. Sci. U. S. A. 1983, 80, 4575−4576. (5) Katritzky, A. R.; Allin, S. M.; Siskin, M. Aquathermolysis: Reactions of organic compounds with supersaturated water. Acc. Chem. Res. 1996, 29, 399−406. (6) Clark, P. D.; Hyne, J. B.; Tyrer, J. D. Chemistry of organosulphur compound types occurring in heavy oil sands: 1. High temperature hydrolysis and thermolysis of tetrahydrothiophene in relation to steam stimulation processes. Fuel 1983, 62, 959−962. (7) Clark, P. D.; Hyne, J. B.; Tyrer, J. D. Some chemistry of organosulphur compound types occurring in heavy oil sands: 2. Influence of pH on the high temperature hydrolysis of tetrahydrothiophene and thiophene. Fuel 1984, 63, 125−128. (8) Clark, P. D.; Hyne, J. B. Chemistry of organosulphur compound types occurring in heavy oil sands: 3. Reaction of thiophene and tetrahydrothiophene with vanadyl and nickel salts. Fuel 1984, 63, 1649− 1654. (9) Monin, J. C.; Audlbert, A. Thermal cracking of heavy-oil/mineral matrix systems. SPE Reservoir Eng. 1988, 3 (4), 1243−1250. (10) Belgrave, J. D. M.; Moore, R. G.; Ursenbach, M. G. Comprehensive kinetic models for the aquathermolysis of heavy oils. J. Can. Pet. Technol. 1997, 36 (4), 38−44. (11) Belgrave, J. D. M.; Moore, R. G.; Ursenbach, M. G. Gas evolution from the aquathermolysis of heavy oils. Can. J. Chem. Eng. 1994, 72, 511−516. (12) Thimm, H. F. Prediction of hydrogen sulphide production in SAGD projects. J. Can. Pet. Technol. 2008, 47 (1), 7−9. (13) Zheng, R.; Huang, H. Review of kinetics of gas formation of steam processes in recovery of heavy oil/bitumen reservoirs. AERI/ARC Core Industry Research Program, Alberta Research Council Inc.: June 2008, Report #0809-6. pp 1−20. (14) Lamoureux, V.; Lorant, F. H2S artificial formation as a result of steam injection for EOR: a compositional kinetic approach. SPE 97810, SPE International Thermal Operations and Heavy Oil Symposium, 1−3 November, Calgary, Alberta, Canada, 2005. (15) Goicetty, J. L. B. Estimation of the H2S formation under steam injection conditions for the Orinoco oil belt. SPE 141128, SPE Annual Technical Conference and Exhibition, 19−22 September, Florence, Italy, 2010. (16) Song, G.; Zhou, T.; Cheng, L.; Wang, Y.; Tian, G.; Pi, J.; Zhang, Z. Aquathermolysis of conventional heavy oil with superheated steam. Pet. Sci. 2009, 6, 289−293. (17) Ayache, S. V.; Lamoureux-Var, V.; Michel, P.; Preux, C. Reservoir simulation of H2S production during a SAGD process using a new sulfur-based-compositional kinetic model SPE 174441, SPE Canada Heavy Oil Technical Conference, 9−11 June, Calgary, Alberta, Canada, 2015. (18) Barroux, C.; Lamoureux-Var, V.; Flauraud, E. Forecasting of H2S production due to aquathermolysis reactions SPE 164317, SPE Middle East Oil and Gas Show and Conference, 10−13 March, Manama, Bahrain, 2013. (19) Lamoureux, C., Barroux, V..Using geochemistry to address H2S production risk due to steam injection in oil sands. SPE 165437 SPE Heavy Oil Conference-Canada, 11−13 June, Calgary, Alberta, Canada, 2013. (20) Kapadia, P. R.; Kallos, M. S.; Gates, I. D. A new reaction model for aquathermolysis of Athabasca bitumen. Can. J. Chem. Eng. 2013, 91 (3), 475−482. (21) Montgomery, W.; Sephton, M. A.; Watson, J. S.; Zeng, W.; Rees, A. C. Minimising hydrogen sulphide generation during steam assisted production of heavy oil. Sci. Rep. 2015, 5, 8159. (22) Tumanyan, B. P.; Petrukhina, N. N.; Kayukova, G. P.; Nurgaliev, D. K.; Foss, L. E.; Romanov, G. V. Aquathermolysis of crude oils and natural bitumen: chemistry, catalysts and prospects for industrial implementation. Russ. Chem. Rev. 2015, 84 (11), 1145−1175.

The concentration of hydrocarbon species can be obtained as ωi(t ) =

γ1 + γ3 + γ4 λ1 +

γ6 λ3

(1 − e−λ1t ) +

γ2 + γ5 λ2

(1 − e−λ2t )

(1 − e−λ3t ) (A10)

where

γ1 =

0 knk1ωASPH (λ 2 − λ1)

(A11)

⎞ ⎛ k ω0 0 γ2 = kn⎜ωHO − 1 ASPH ⎟ (λ 2 − λ1) ⎠ ⎝

(A12)

0 γ3 = kjωASPH

(A13)

γ4 = km

γ5 = −

(β1 + β2) (λ3 − λ1)

(A14)

kmβ3 (λ 2 − λ3)

(A15)

⎛ [(β1 + β2)(λ 2 − λ3) + β3(λ1 − λ3)] ⎞ 0 ⎟ γ6 = km⎜ωLO − (λ3 − λ1)(λ 2 − λ3) ⎝ ⎠ (A16)

(C+3 ,

(i, j, m, n) = (C2H4,10,19,12), (C2H6,9,18,13), 7,16,15), (Coke,8,17,14). For a case where ϕi ≠ 1, an empirical exponential deactivation function is defined as given by ϕi(t ) = αi(e−κit − e−(κi + λ1)t )

(A17)

where i = H2, H2S, CO, or CO2, and κ and α are the temperaturedependent parameters of the deactivation rate function. Using the reaction deactivation does not permit analytical solution, and thus, the system of equations should be solved numerically.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +1 (403) 210 6645. E-mail address: [email protected] (H. Hassanzadeh). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank Nexen Energy ULC for permission to publish this paper. The authors would also like to thank the anonymous reviewers for their constructive comments. The first author would like to thank Dr. Mohsen Zirrahi for many useful discussions on parameter estimation.

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REFERENCES

(1) Hyne, J. B. Aquathermolysis − A synopsis of work on the chemical reactions between water (steam) and heavy oil sand during simulate steam stimulation, Report No. 50, 1986. (2) Wu, C.; Lei, G.-L.; Yao, C.-J.; Sun, K.-J.; Gai, P.-Y.; Cao, Y.-B. Mechanism for reducing the viscosity of extra-heavy oil by aquathermolysis with an amphiphilic catalyst. Journal of Fuel Chemistry and Technology 2010, 38 (6), 684−690. 7012


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NOTE ADDED AFTER ASAP PUBLICATION This article published August 30, 2016 with an error in the expression below equation 3, and a format change to the equation below equation 12. The corrected article reposted August 31, 2016.

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