ARTICLE pubs.acs.org/EF
Combustion Kinetics of Heavy Oils in Porous Media Murat Cinar,* Louis M. Castanier, and Anthony R. Kovscek Energy Resources Engineering, Stanford University, Stanford, California 94305, United States ABSTRACT: In this study, the kinetics of heavy crude-oil combustion in porous media are reported. Ramped temperature oxidation (RTO) tests with effluent gas analysis are conducted to probe in situ combustion (ISC) reaction kinetics along with isothermal coke formation experiments. The role of oxygen on coke formation reactions (i.e., fuel formation for ISC) is investigated using X-ray photoelectron spectroscopy (XPS) of intermediate reaction products. The XPS data is analyzed along with companion RTO experiments to obtain a simplified multistep reaction scheme. Synthetic cases illustrate the connection between a proposed reaction scheme for oil/matrix pairs and one-dimensional combustion front propagation. Analysis of experimental results illustrate that the reaction scheme is capable of reproducing experimental results including the basic trends in oxygen consumption and carbon oxides production for RTO experiments as a function of heating rate for both good and poor ISC candidates. The combination of XPS and RTO studies indicates that the quality (or reactivity) of coke formed during the process is a function of oxygen presence/absence. Coke formed in the presence of oxygen is significantly more reactive due to additional oxygen functional groups on the coke surface in comparison to coke formed under an inert atmosphere. Additionally, this work extends relatively easy to perform RTO tests as a screening tool for ISC performance.
’ INTRODUCTION Predictive kinetic models are necessary for the estimation of in situ combustion (ISC) enhanced oil recovery processes at field conditions. ISC is the process of injecting air into a reservoir to oxidize a small portion of the hydrocarbons in situ. Viscosity is reduced significantly because of exothermic reactions and the resulting temperature increase. The oil is driven toward the production wells by a vigorous drive of combustion gases, a steam drive, and a water drive. Also oil from the combustion zone is upgraded in situ as the heaviest components burn and the lighter crude-oil components fractionate ahead of the combustion front.1 The ISC process includes aspects of almost every known enhanced oil recovery technique. One factor that has limited the application of ISC, however, is the lack of predictability at both laboratory and field scale. Chemical kinetics studies of crude oils probe the rates of chemical reactions and help to frame a constitutive model of reactions under various conditions.28 Kinetic models coupled with flow equations derived for specified reactors are used to predict performance of processes in reaction engineering. Similarly, ISC performance predictions are only reliable if proper reaction models are formulated and reliable models are only possible with improved understanding of the fundamentals of combustion in porous media. The kinetics of crude-oil combustion in porous media is complicated by two major factors: (i) the heterogeneity of reactant and (ii) the participation of rock matrix in reactions. Crude oil typically has hundreds of identified components with a large distribution of molecular weight and many different homologous series. Heteroatoms such as nitrogen, oxygen, sulfur, nickel, and vanadium reside at different locations on large polyfunctional molecules such as asphaltenes.9 These polyfunctional molecules may undergo reactions such as hydrogenation, dehydrogenation, cracking, and removal of heteroatoms. In addition, rock mineralogy, such as carbonaceous versus siliceous, r 2011 American Chemical Society
and surface area influence reactions associated with the ISC process; however, the nature of these reactions is yet to be fully explained. In previous work,10 we showed that results of crude-oil oxidation kinetics tests may be plotted and interpreted as effective activation energy versus temperature or conversion. Because, an isoconversional method11,12 is used for interpretation, we refer to such plots as an isoconversional “fingerprint” of the crude-oil oxidation kinetics. The isoconversional interpretation appears to be suitable for representing the burning quality of a particular oil/matrix pair. In this sense, isoconversional fingerprints are useful as a diagnostic tool to infer whether an oil/matrix pair is suitable for ISC processes.10 An isoconversional method enables identification of the so-called negative temperature gradient region (NTGR).13 The lack of a significant negative temperature gradient region is a key to a self-sustained burning front. The proposed method is based on kinetic cell and combustion experiments of eight different crude-oil samples. Experimental procedures and results were given in our previous study.10 The effect of the NTGR on the success of combustion is critical; however, no reaction scheme, to date, is capable of reproducing the behavior. The objective of this manuscript is to provide simplified reaction models for ISC that reproduce the general trends in kinetic experiments and the negative temperature gradient region with a minimum number of pseudo reactions. Such models must also be capable of predicting the results of onedimensional combustion tube tests where front propagation is measured. In this paper, isothermal coke formation experiments are presented along with X-ray photoelectron spectroscopy (XPS) analysis of reaction products. Additional kinetic cell Received: May 5, 2011 Revised: August 9, 2011 Published: August 15, 2011 4438
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Energy & Fuels experiments are also reported. These experiments were conducted to investigate specifically the effect of oxygen during coke formation and to infer the reaction model. Kinetics cell simulations were run using the proposed reaction scheme with synthetic parameters.
’ PREVIOUS WORK Given the complexity of ISC and the limited knowledge of the series and parallel reaction pathways involved, global reaction schemes using lumped components with little chemical insight are the main approach found in the literature. Multiple reactions both in series and parallel propagate during ISC. The first attempt to group these reactions dates to 1959. In his work on the mechanism of oil production by underground combustion, Tadema14 described two distinct reaction regimes, at about 543 and 673 K based on differential thermal analysis (DTA) of an oil sand. In addition, Tadema observed that at around 543 K a coke-like residue is formed and concluded that most of the oxygen consumed in this region is used to form water, whereas around 673 K the residue burns forming carbon oxides. Subsequently, these two regions were referred to as low temperature oxidation (LTO) and high temperature oxidation (HTO). Later, Alexander et al.15 developed a method to study the process variables of ISC and showed that LTO has an important effect on the fuel availability. In their quantitative work, Bousaid and Ramey16 estimated kinetic parameters using an effluent gas analysis (EGA) technique at isothermal conditions. They concluded that the rate for carbon burning depends on carbon concentration, oxygen partial pressure, and combustion temperature. Bae17 used thermogravimetric analysis (TGA) and DTA for ISC characterization of crude oil. By using 15 different oil samples, he identified 3 distinctive burning characteristics. No complete correlation between viscosity or composition of crude oil and burning characteristics was observed; however, low gravity, high viscosity oils were grouped into the same category. In their quantitative work, Vossoughi et. al.18 analyzed the data from TGA and differential scanning calorimetry (DSC). In addition to LTO and HTO, they identified another group of reactions between temperatures 773 and 873 K. Similarly, Fassihi2 proposed that reactions during ISC are an overlap of three consecutive set of reactions: LTO, middle temperature reactions, and HTO. Middle temperature reactions are proposed to be homogeneous gas-phase reactions in which products of pyrolysis and distillation oxidize leaving behind a heavy-oil residue on the solid matrix. There seems to be a consensus in the literature of the characterization of these reactions as it is expressed by Fassihi,2 namely, LTO, fuel formation, and fuel combustion. The literature is vast on the quantification of fuel formation and oxidation reactions. Most of these analyses are based on EGA at either isothermal16,19 or nonisothermal conditions.2,5 Most nonisothermal kinetic experiments are conducted by imposing a linear heating rate also referred to as ramped temperature oxidation (RTO). Additionally, DTA and TGA are used.18,2022 Some of the studies in the literature focused on fuel formation,4,23,24 and some researchers focused on LTO3,19,25,26 Besides EGA and DTA/TGA analyses, a different type of analysis also has its place in the literature. In their study on the thermal cracking of Athabasca Bitumen, Hayashitan et al.24 used a separation scheme based on solubility of hydrocarbons
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in different solvents. Isothermal experiments were conducted. After a predetermined time, the reactor was cooled and the sample was fractionated according to a given separation scheme. Later Adegbesan et al.3 adopted a similar method using saturates, asphaltenes, resins, and aromatics (SARA) analysis. In their extensive work on LTO, Adegbesan et al. proposed different reaction schemes and estimated reaction parameters for various sets of reactions. Milliour et al.25 elaborated by analyzing bitumen samples in the presence of rock matrix and water. Probably the most comprehensive reaction scheme is proposed by Belgrave et al.6 Fuel formation and LTO reactions are modeled with fuel combustion reactions at high temperature. Later, Freitag and Verkoczy7 and Freitag and Exelby8 explored the use of SARA fractions analysis in LTO and pyrolysis reactions. SARA analysis is based on the solubility of petroleum constituents in different solvents. Crude oil is fractionated into saturates, aromatics, resins, and asphaltenes. The kinetic analysis is applied to these fractions individually providing valuable information, but it is not clear how to superimpose the information to yield the kinetic behavior of the whole crude-oil sample. As shown later in the text, mutual solubility of different fractions affects the kinetics of the process. Recenty, Cinar et al.10,27 combined RTO experiments with isoconversional analysis.11,12 On the basis of extensive RTO and combustion tube experiments of eight different heavy and/ or viscous crude-oil samples, an excellent correlation between isoconversional fingerprint (kinetic analysis) and combustion tube results was found. Three distinct burning characteristics were identified using the isoconversional fingerprints. The good ISC candidates have lower effective activation energy (in general) and a smooth transition from NTGR to the combustion region. The poor candidates are divided into two categories. The first exhibit a high effective activation energy barrier in NTGR. The candidates in this group do not recover from NTGR. The second group of poor candidates are those that are not able to precipitate enough good quality coke. These exhibit increasing effective activation energy in the HTO region. The isoconversional method is used to infer whether an oil\matrix pair is a good or a poor candidate for combustion under the given conditions. Isoconversional analysis provides a methodology to estimate activation energy without assuming a reaction model. This technique based on a simple assumption that at a given extent of conversion, the reaction rate is only a function of conversion. Experiments at different heating rates are conducted for the analysis and at the same extent of conversion; the experiments are compared to estimate activation energy.
’ COKE FORMATION EXPERIMENTS Coke formation during thermolysis of petroleum, in the absence of porous medium, is controlled by the phase separation of asphaltenes.28 In his pioneering work, Wiehe28 used an isothermal batch reactor to study coke formation. He showed that oil, when burned in an isothermal (673 K) batch reactor (nitrogen flowing over the reactor), starts forming coke after an induction period. On the other hand, when the reactant is the asphaltene fraction of the same oil, coke is formed immediately without any induction period. During the induction period, the number of asphaltene cores that are unreactive and dissolved in maltenes increases until the solubility limit is reached. At that 4439
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Scheme 1. Wiehe’s28 Suggestions for Coke Formation (Oil in Bulk)
Figure 1. Schematic of coke formation apparatus.
Figure 4. Coke formation versus time under air flow, sample F.
Figure 2. Coke formation versus time under nitrogen flow, sample F.
Figure 3. Coke formation versus time under nitrogen flow, Hamaca.
point, asphaltene cores precipitate and coke is formed from these asphaltene cores. In our study, we investigated the applicability of the same mechanism for coke formation in porous media. Experimental procedures similar to those described by Wiehe28 were employed. A quartz tube was used as the reactor. The experiments were conducted at atmospheric pressure and at 673 K. Quartz sand mixed with oil was put in the reactor and nitrogen was injected from the bottom of the tube. A schematic is given in Figure 1. At different reaction times (15, 30, 45, 60, 75 min) the experiment was stopped and the sample cooled immediately using an ice bath. The sample was allowed to stand overnight in toluene. Then, the mixture was filtered using a coarse fritted glass filter. Coke formed is retained on the sand and the toluene soluble part filtered off. Then, the remaining sand sample waited overnight in a vacuum oven. The experiments were repeated with two different oil samples, sample F and sample Hamaca. In order to be consistent with the previous paper,10 the convention used for naming the oil samples is identical. Sample F is 21°API and Hamaca is 9.5°API. The asphaltene content of the samples was
measured with ASTM D656029 procedures. Both samples have 10% asphaltene by weight. These samples are chosen because they have the same asphaltene content with different API gravity. The solubility characteristics of asphaltenes in maltenes is assumed to be different because oil composition is different. Also in our previous study,10 we showed that sample Hamaca is a good and sample F is a poor combustion candidate at the experimental conditions. Figures 2 and 3 show pictures of the filtered material taken after the toluene soluble portion is removed. For sample F, no coke is formed for the first 60 min;, however, after 75 min a significant amount of coke is observed. Sample F indicates 6075 min of induction. The induction period for Hamaca, on the other hand, is 4560 min, Figure 3. There is a little amount of coke formation after 15 min; however, after 45 min a significant amount of coke is formed. Results indicate that under nitrogen injection, coke formation reactions in porous media may be limited by the phase separation of asphaltenes similar to oil that is reacted under identical conditions in bulk without any sand. The experimental observations suggest that independent of being bulk or in porous media, the coke formation process under nitrogen flow is similarly limited by the solubility of asphaltenes in maltenes. Thus, Wiehe’s28 suggestions for coke formation (in bulk), appear to be valid for coke formation reactions in porous media in the absence of oxygen, given by Scheme 1. The mechanism suggests the oil to be composed of three components: asphaltenes, maltenes, and volatiles. Asphaltene cores-subscript “c” and coke are two additional components for the mechanism proposed. Coke is formed from precipitated asphaltene cores. Asphaltene cores precipitate only if the amount of asphaltene cores dissolved in maltenes exceeds the solubility limit, SL. Asphaltene cores are produced from either asphaltenes or maltenes. Under nitrogen, the above mechanism is potentially responsible for coke formation in porous media. The induction period observed in coke formation bears some similarity to Freitag and Verkoczy’s7 observation of an induction period associated with LTO of saturates. Coke formation reactions do not always occur under nitrogen flow. To investigate the role played by the presence of oxygen during coke formation, the same experiments were repeated 4440
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Table 1. SARA Analysis of Sample G
wt %
saturates
aromatics
resins
asphaltenes
22.6
33.6
32.9
10.8
Table 2. Elemental Analysis of Sample G
wt %
nitrogen
carbon
hydrogen
sulfur
oxygen
0.69
83.79
10.43
4.01
0.69
under air flow. The results were quite different. Figure 4 shows the pictures taken after the toluene soluble filtrate is separated, for sample F. The results are very similar for Hamaca. Even at 8 min of reaction time, significant coke was produced in the air environment. The results indicate that in the presence of oxygen, coke is formed immediately. There is no induction period for coke formation. As a result, coke formation reactions in the presence of oxygen are not limited by the phase separation of asphaltenes. Instead coke formation reactions follow a different path. One may speculate that oxidation of maltene components (resins and so on) makes crude oil a less effective solvent for asphaltenes resulting in precipitation. At this point, more study is needed to confirm such a mechanism. Further investigation of the differences between coke formed under nitrogen and air were carried out using XPS analysis. XPS or electron spectroscopy for chemical analysis (ESCA) is widely used for surface analysis of solids. Chemical composition as well as chemical structure of the surface is determined using XPS. The technique is based on the photoelectric effect. Samples under vacuum are bombarded with monochromatic X-rays. Adsorption of X-rays by an atom on the surface leads to the release of an electron. The kinetic energy of the emitted electrons is measured and binding energies are estimated to produce a spectrum of electron intensity as a function of energy.30 Binding energies are characteristic of the element. Therefore, XPS provides elemental analysis. In addition, small shifts in binding energies allow the identification of the functional groups. In this work, XPS spectra of both coke samples, formed under nitrogen and air flow, were recorded with a PHI 5000 VersaProbe (ULVAC-PHI, Chigasaki, Japan) with Al X-ray Kα beam. Coke was formed on both quartz sand under air and nitrogen flow using 8.1°API sample G. SARA fraction and elemental analysis of sample G is given in Tables 1 and 2, respectively. The sample is a mixture of oil (1.5 g), matrix (50 g 20 mesh), and water (4 mL). After the constituents are mixed, the sample is placed in a stainless steel reactor. The pressure is fixed at 690 kPa (100 psi). A linear heating schedule is imposed (2.74 K/min). Gas is injected into the reactor at a constant rate of 2 standard liters per minute (SLPM). After the specified temperature is reached (600 K), the experiment is stopped and cooled rapidly at the fixed pressure. The pressure in the cell is fixed during cooling in order to avoid any effects of sudden pressure decrease. After the sample is cooled to room temperature, the sample is removed from the reactor. To separate the toluene soluble fraction, the samples are allowed to stand overnight in 200 mL of HPLC grade toluene. The mixture is filtered through a coarse fritted glass filter. Any remnants of oil are separated from matrix covered with coke. Finally the sample is dried in a vacuum oven at 320 K overnight. These samples were analyzed with the aforementioned XPS device. The coke formation experiments under nitrogen flow
Figure 5. High-resolution XPS C 1s spectra of coke samples formed under nitrogen flow, acquired at four different points on the sample, oil sample G, graphitic carbon (peak I, 284.6 eV), and carbon present in alcohol or ether (hydroxyl) (peak II, 286.1286.3 eV).35
were quenched at 600 K; however, the experiments conducted under nitrogen flow failed to form coke at those temperatures. This observation is consistent with our previous observation that under nitrogen flow coke formation is relatively slow. Also note that, it is not possible to recognize visually, without using toluene, whether the material formed on the matrix is coke or not. As a result, the target temperature for those experiments was chosen as 873 K, and the experiment was cooled overnight under pressure and the same procedure was repeated to separate the toluene soluble part. In the experiments with air flow, no toluene soluble fraction was observed while the samples formed under nitrogen flow indicate negligible amounts of toluene soluble hydrocarbons. For each sample, both survey scans and high-resolution scans of the C 1s region were performed. Because of the heterogeneity of the samples, these analyses were repeated at four different spots on the samples. Survey scans were collected from 0 to 1000 eV with a pass energy of 117.4 eV, while high-resolution C 1s scans were recorded from 278 to 298 eV with a pass energy of 23.5 eV. The curve fitting of the XPS spectra was carried out with a nonlinear curve fitting program (SDP v 4.3) using the Gauss/Lorentz mix with the Shirley background. For the graphitic carbon peak, the Gauss/Lorentz mix was taken as 0.9 with an asymmetry of 0.05. This asymmetric shape was necessary in order to fit the spectra in a reasonable way. This is common for graphitic carbon and well described in the literature.3133 For the other components, the mix was assumed to be 0.5.34 The carbon 1s electron binding energy corresponding to graphitic carbon peak was taken at 284.6 eV.34 Deconvolution of the C 1s spectra may include five component peaks: graphitic carbon (peak I, 284.6 eV), carbon present in alcohol or ether (hydroxyl) (peak II, 286.1286.3 eV), carbonyl groups (peak III, 287.3287.6 eV), carboxyl or ester (peak IV, 288.4288.9 eV), and carbon present in carbonate (peak V, 290.4290.8 eV).35 4441
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Figure 8. XPS survey spectra of coke samples formed under air flow, acquired at four different points on the sample, oil sample G. Figure 6. High-resolution XPS C 1s spectra of coke samples formed under air flow, acquired at four different points on the sample, oil sample G, graphitic carbon (peak I, 284.6 eV), carbon present in alcohol or ether (hydroxyl) (peak II, 286.1286.3 eV), and carboxyl or ester (peak IV, 288.4288.9 eV).35
Figure 7. XPS survey spectra of coke samples formed under nitrogen flow, acquired at four different points on the sample, oil sample G.
High-resolution XPS spectra of coke formed under both nitrogen and air flow are given in Figures 5 and 6 along with the percentages of graphitic and functional carbon atoms. The results reveal that XPS spectra of coke formed under nitrogen flow, when deconvolved in the manner described above, give two peaks, groups I and II. On the other hand, coke formed under air flow gives three peaks, groups I, II, and, IV. Apparently, the presence of oxygen during coking helps to form oxygenated functional groups (in this case carboxyl) on the coke surface whereas it decreases the graphitic content. Also an increase in hydroxyl group (I) is observed. Hydrogen and oxygen functional groups serve as active sites on the coke surface;36 thus, we may speculate that the coke
formed under air flow has more active sites than the coke formed under nitrogen flow due to the presence of oxygen during coking. Thus, when comparing the reactivity of both coke samples, we speculate that coke formed under air flow is more reactive in comparison to the coke formed under nitrogen flow. That is, reactivity appears to correlate with the concentration of active sites: hydrogen and oxygen functional groups on the coke surface. This observation helps to explain why LTO has a significant effect on the fuel availability.15 XPS surveys scans of coke precipitated under nitrogen and air flow on the quartz surface are shown in Figures 7 and 8. Major peaks results from C 1s and O 1s photoelectrons. In addition, sulfur is observed in all of the survey scans. This observation is consistent with the elemental analysis (Table 2). Silicon and nitrogen were also present. Silicon signal comes from the inner layer of quartz; thus, the percentage of silicon estimated is correlated with the thickness of the coke layer. The thickness of overlaying coke is estimated using silicon intensity. The intensity decays exponentially with depth given by37 x I ¼ I0 exp a λ sinðθÞ Here, I is the intensity, x is the total distance traveled by the electron, and λa is the attenuation length. θ is the take off angle (45° here). λa is calculated as 1.4 nm (under the assumption that elastic scattering has no influence on trajectories) using the procedure of Seah and Dench.38 For coke samples formed under nitrogen, the thickness of the coke layer is between 3.5 and 5.5 nm. Note that the thickness of the coke layer is different at different locations on the sample. For the coke sample formed under air flow, no silicon was observed in the signal. Thus, coke samples formed under air flow exhibit a thicker layer of coke in comparison to coke samples formed under nitrogen flow. This observation supports the long lasting conclusion that low temperature reactions promote the deposition of coke.15 Also, there is a difference between these coke samples in terms of the sulfur content. Coke formed under nitrogen flow has an average sulfur content of 1% while coke formed under nitrogen has an average of 1.4%. This is probably due to oxygen sulfur bonding on the surface. 4442
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Figure 10. Comparison of kinetic experiment with the preheated kinetic experiment at 2.13 K/min with sample G.
Figure 9. Kinetic experiment conducted at 2.13 K/min with sample G.
The average atomic O/C ratio for coke formed under nitrogen flow is 0.07 and 0.11 for coke formed under air flow. As expected, the O/C ratio is greater for the coke formed under air flow suggesting oxygen bonding on the surface. These ratios are only for comparison across samples. It is erroneous to accept these ratios as the elemental composition on the surface. The quantification of XPS data is complicated by the fact that oxygen and carbon from the atmosphere are adsorbed on the surface. The trends, rather than the quantity, are relevant in this context.
’ KINETICS EXPERIMENTS Kinetic experiments were conducted with sample G. The oil sample was mixed with 60 mesh sand and water (50 g of sand, 4 mL of water, and 1.5 g of oil). Kinetic experiments were conducted at 690 kPa (100 psi). Details of the experiment and the experimental apparatus are given elsewhere.27 Figure 9 gives the typical kinetic experimental results. The figure indicates that there is a lag between oxygen consumption and carbon oxides production. Oxygen consumption starts at about 450 K, whereas carbon oxides production begins at around 500 K. Also in the same region (LTO), material balance suggests that not all the oxygen consumed is produced in the form of carbon oxides. These observations are consistent with oxygen addition onto hydrocarbons; however, some portion of the oxygen reacts with hydrogen to form water. The other possibilities are nitrogen and sulfur oxides; however, our gas analyzer only measures the concentration of CO, CO2, O2, and CH4. The sample size is only 1.5 g, and the amount of methane produced was, apparently, below the detection limits of the gas analyzer. Liquids collected in the trap were mostly water with light hydrocarbons. Investigation of the temperature curve reveals deviations from the preprogrammed temperature history. The first deviation is about 375 K; here water changes phase. The other two deviations are direct results of exothermic reactions during LTO around 530 K and HTO 700 K. In order to observe the extent of coke formation, different experiments were conducted at the same conditions and quenched under pressure at different temperatures. After the sample is cooled down to room temperature, coke formed is separated. Note, coke is
observed at 500 K and this is just at about the same temperature where carbon oxides begin to form. Even if little coke is present at these temperatures, the sample is almost solid. It is hard to differentiate between coke and oil without toluene separation. Figure 9 shows the regions where oil or coke is dominant and indicates the region where these phases coexist as a hatched pattern. After 560 K, negligible amounts of oil are present in the sample. This corresponds to the start of the NTGR13 where the rate of reaction decreases with increasing temperature. Thus, the experimental data suggest that in the NTGR, coke formed in the LTO region goes through a transformation that leads to the coke that burns in the HTO region. The primary difference between LTO and HTO coke is the concentration of oxygen functional groups, in other words active sites. In order to elaborate this idea, a different set of experiments were conducted. The kinetic cell experimental procedure was modified to test the role of the preheating under nitrogen flow and the dominant phases in NTGR. In a regular RTO kinetic experiment, the sample was put in the kinetic cell and temperature is increased under air flow. In the modified version, the sample was first exposed to the same heating rate under nitrogen flow. Once the target temperature (673 K) was reached, the furnace was stopped and cooled overnight. The target temperature was based on the temperatures that HTO reactions start. Our previous observations suggest that when the kinetic cell was cooled down immediately at 673 K, it was not possible to convert all of the oil into coke. In this first stage, the sample is preheated and the entire sample was converted into a solid phase and some portion was converted to coke. The next day, the preheated sample was heated with the same heating rate under air flow (a regular kinetic experiment was conducted with preheated sample, target temperature is 923 K), burning all of the preheated producing carbon oxides. We refer to this as a preheated experiment. The results are compared with a regular kinetic experiment shown in Figure 10. Also regular and modified kinetic experiments were repeated at 5 different heating rates to give the isoconversional fingerprint.27 The comparison of fingerprints acquired with a regular kinetic experiment and the modified experiment is given in Figure 11. The experiments repeated at different heating rates have the same trends. First note, LTO and HTO for the regular and preheated experiments start at about the same temperature. Even if no liquid phase is present, oxygen addition starts before carbon oxides production suggesting that oxygen addition also occurs with the solid phase. Oxygen consumption in the LTO region is significantly lower for the preheated case; however, the carbon dioxide production is 4443
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Scheme 2. Simplified Reaction Scheme Proposed for in Situ Combustion
Figure 11. Comparison of the isoconversional fingerprint of sample G acquired from a standard RTO kinetic experiment with a fingerprint of coke formed under N2.
identical. Thus, in both cases a similar coke material burns in the LTO region. Probably, some portion of the oil is converted into coke without any need for gas-phase oxygen. The reason for the difference in oxygen consumption appears to be the difference in the nature of the oxygen addition. In both cases, oxygen addition occurs, but for the regular case, oxygen addition also occurs in the liquid phase. In the preheated case, oxygen addition only occurs at the solid surface. Also water formation is another possibility for the increase in oxygen consumption. The HTO peak is greater for the regular case. One reason is the extent of cracking reactions. In the preheated case, the coke formation reactions appear to be slower than the cracking reactions leaving less fuel for combustion. This could also be the reason for the difference in oxygen uptake in the LTO region. Further investigation is needed to confirm these observations. The isoconversional fingerprints of these tests are almost identical except at low conversion or temperature. The difference at low temperatures is as expected as more oxygen is consumed initially in the regular experiment. The isoconversional fingerprint for the preheated case exhibits a flat region in the beginning at around 45 000 J/mol. This is a good initial estimate for the activation energy for the reactions producing carbon oxides in the LTO region. This part is masked in the regular case as more oxygen is consumed. The HTO part is identical implying that in both cases the reaction process is very similar. Also both curves exhibit a NTGR, with a valley in the isoconversional plots,27 centered at a conversion of roughly 0.3 or a temperature of 600 K (see Figure 11). In the NTGR, the solid phase is dominant and plays the major role. In summary, the experimental data suggest that during the combustion process distillation is followed by oxygen addition in the liquid phase. This process paces coke formation and coke is formed in the LTO region that reacts with oxygen at low temperatures due to the greater concentration of oxygen functional groups (active sites) to produce carbon oxides. At the time carbon oxides formed, the sample is almost in the solid phase. With increasing temperature, coke goes through a transformation decreasing reactivity and thus the NTGR is observed. At the beginning of the NTGR, almost all the oil is converted into coke. The extent of the transformation and increase in temperature competes to lead HTO. One possible explanation for the transformation could be the loss of active sites due to the increase in temperature. Thus, active site concentration may be changing. Further studies were done to create a profile of active site concentration with temperature. Kinetics experiments were quenched at different temperatures
and samples were analyzed with XPS. Our efforts met with insurmountable technical difficulties. One problem was the gassing out of samples at lower temperatures that increases the pressure in the vacuum chamber making the analysis impossible. The same samples were also analyzed using Fourier transform infrared (FTIR) spectroscopy. The FTIR device employed required a pellet to be created with the material to be analyzed and a binder. In our case, the binder was KBr. The results were in conclusive as the signal from the coke was similar to background due to the small mass of coke in the sample. Further evidence, however, is found in the literature. In their study using XPS to characterize coal, Perry and Grint39 investigated the effect of heating on the functional oxygen compositions. Their data suggests that between 623 and 773 K starting from carboxyl groups, carbonyl and carboxyl groups disappear with increasing temperature. Their coal transforms to become more similar to pure carbon. This is also the main reason why the target temperature of 673 K was used for the preheated experiment. That is, to limit carbonization. On the basis of our observations, the following simplified reaction scheme is proposed to mimic all the experimental observations for oxidation of heavy oil including the isoconversional fingerprint (Scheme 2). At least two different coke formation reactions are necessary because coke formed under nitrogen and under air flow have been demonstrated to be different. In combustion tube experiments, ignition is initiated under nitrogen flow so this step is necessary. Also there should be two different coke species for LTO and HTO. There are two significant features of this reaction scheme. First, the scheme does not include any oil directly burning as experimental data strongly suggest that heavy oil does not burn directly and at different temperatures and different rates of heating different reactions are dominant. Second, it includes a reaction that represents the transformation of coke formed in the LTO region into the fuel (coke) that burns in HTO region to mimic a change in the concentration of active sites.
’ SIMULATIONS On the basis of the reactions given above, kinetic cell simulations are carried out with STARS.40 A radial model with one grid block in the z direction and four blocks in the r direction is built. The details for the grid blocks used in the r direction are given in Figure 12. The first grid block in the r direction represents the sample, the second the stainless steel tube, the third the spacing between the furnace wall and reactor, and the fourth the furnace. 4444
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Table 4. Reaction Parameters for Synthetic Model 1 A
reaction 3
2.00 1012
4
2.50 10
8
5
4.00 10
7
6
1.00 10
12
ΔH, J/mol
E, J/mol 9.30 104
1.00 105
mol/cm min
5.20 10
3.00 105
1/min
9.50 10
0.00 100
1.10 10
5.00 105
mol/cm3min 3
4 4
3
mol/cm min
5
Figure 12. Dimensions of kinetic cell and combustion tube.
Scheme 3. Reaction Model for the First Synthetic Example
Table 3. Molecular Weights of Lumped Components, Synthetic Example 1 component
MW, g/gmol
oil
515
coke1
24
coke2
12
These grid blocks were necessary to mimic the exact programmed heating schedule in the experiments. The injected fluid temperature is increased at each time step according to the heating schedule. This is similar to the experiments where tubing is tightly coiled and placed within the furnace to allow heating of the inlet gas. Thus, the injected gas reaches the furnace temperature before it reacts. The challenge in modeling the kinetic experiments was to impose a linear heating rate. There is not an option in the version of STARS that we used for the simulations to impose a linear heating rate directly. This problem is solved by changing the target temperature for the heaters at each time step. Therefore, we impose a linear heating with different rates with precalculated target temperature based on the rate. Note that this is done only for the preprogramed temperature history not for the deviations resulting from reactions or water changing phase. These are a direct result of reaction enthalpy and enthalpy of vaporization. All parameters used in the simulations are given in the Appendix. A synthetic example is generated in order to create a typical isoconversional fingerprint that includes all the features of both experiments and isoconversional fingerprints of a good and a poor ISC candidate. Here our intention is not to match the experimental data; however, realistic values are used for both reaction parameters and other properties. Our aim is to show with simulations that the isoconversional fingerprint associated with a good and a poor ISC candidate behave accordingly in combustion tube simulations. The features of the isoconversional fingerprint of a good candidate and a poor candidate differ significantly. Cinar et al.,10
Figure 13. Kinetic cell simulation results at 2.13 K/min.
Figure 14. Kinetic cell simulation results with the given reaction model at five different heating rates, synthetic example 1.
showed that good and poor ISC candidates share certain features. Thus, we show two cases: a good and a poor candidate. For the kinetic cell simulations of a good combustion candidate the reaction model given by Scheme 2 is used. For both of the cases, reactions 1 and 2 (Scheme 2) are excluded as no detailed investigation is available for reaction types. The reaction model used for synthetic example 1 is given by Scheme 3. For all of the reactions, the Arrhenius form of the rate constant is assumed. The molecular weights of the lumped components are shown in Table 3. The reaction parameters for the synthetic model 1 are given in Table 4. All reactions conserve mass. No chemical structure for the coke species is assumed; however, experimental data suggest that it is composed of mainly carbon atoms with oxygen and hydrogen functional groups. Figure 13 gives the simulation results for a heating rate of 2.13 K/min. This figure indicates that the general trends observed in the 4445
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Scheme 4. Reaction Model for the Second Synthetic Example
Figure 15. Isoconversional fingerprint, synthetic example 1.
Table 5. Molecular Weights of Lumped Components, Synthetic Example 2 component
MW, g/gmol
oil
515
coke1
24
coke2
21.6
Table 6. Reaction Parameters for Synthetic Model 2 A
reaction
Figure 16. Combustion tube results, synthetic example 1. Top: temperature history at 5.5 cm intervals along the domain. Bottom: effluent gas history.
experiments are conserved in numerical simulations. Compare Figures 13 and 9. Figure 14 shows the simulation results for all the runs at different heating rates. On the basis of these simulated data, the isoconversional fingerprint of synthetic example 1 is estimated and given by Figure 15. The fingerprint is typical of a good burning heavy crude oil/rock pair.10 In addition, combustion tube simulations were run with the same reaction model at 690 kPa (100 psi) backpressure. The 1 m long combustion tube is modeled with 912 grid blocks in the z direction and 4 in the r direction. The first block in the r direction represents the porous medium, the second block is the tube wall, the third is the insulation, and the fourth block is the relatively large grid block (at room temperature) that is used to account for the heat losses. The results are given in Figure 16. The temperature history at 5.5 cm intervals along the combustion tube is plotted as is the effluent gas rate. As it is shown by the temperature profiles, a combustion front propagates in the tube. One noticeable behavior regarding Figure 16 is the oscillations in gas rates after 2 h. Interestingly, similar oscillations are in fact
3
4.00 10
12
ΔH, J/mol
E, J/mol 3
1.00 10
1.00 105
3
4
mol/cm min
5
4 5
10
1.00 10 1.10 108
mol/cm min 1/min
7.20 10 1.00 105
3.00 105 0 100
6
2.00 1017
mol/cm3min
1.80 105
5.00 105
also present in our combustion tube experiments as well as those of different researchers.2,5,41 Most of the time, these oscillations are attributed to the local differences in oil (consequently coke) concentration due to packing41 Careful investigation of combustion tube experiments reveals that the magnitude of these oscillations correlate well with the amount of oxygen that passes through the combustion front suggesting that oxygen addition reactions are responsible for the oscillatory behavior. The amount of oxygen consumed at the combustion front is directly proportional to the amount and the reactivity of the coke deposited. The reactivity of coke deposited is determined by the conditions during coke formation. These depend mainly on the availability of oxygen. As indicated in this study, the presence of oxygen during coke formation increases the concentration of the oxygen functional groups (active sites) on the coke surface. Also, it is well-known that low temperature reactions increase the amount of coke deposited.15 Oxygen is only available for coke formation reactions if the oxygen penetrates the combustion front. That is, more oxygen is supplied than required for the combustion reaction. Note that these experiments are conducted at a constant air flow rate. In most of these experiments, the oxygen is present ahead of the combustion front where coke formation reactions take place increasing the amount and the reactivity of the coke deposited. When the combustion front reaches the region where the coke formed in the presence of oxygen, more oxygen is consumed due to the increase in the amount deposited and increased reactivity due to oxygen functional groups. More oxygen consumption decreases the amount of oxygen penetrating the combustion front. Consequently, coke deposited is less in amount and also less reactive. When this coke is burned, more oxygen penetrates and coke deposited is again more reactive completing and starting another cycle. This 4446
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Figure 17. Kinetic cell simulation results with the given reaction model at five different heating rates, synthetic example 2.
Figure 18. Isoconversional fingerprint, synthetic example 2.
Figure 19. Combustion tube results, synthetic example 2. Top: temperature history at 5.5 cm intervals along the domain. Bottom: effluent gas history.
behavior could be the reason for the oscillations in the effluent gas concentrations. In the simulations, two different kinds of coke are assumed to play a role in the combustion with different reactivity. The relative amount of these components change with the amount of oxygen available for coking reactions, as reaction 5 (Scheme 2)
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competes with reaction 3 (Scheme 2) leading to an oscillatory behavior. The second synthetic example is a case of a poor ISC candidate with a typical isoconversional fingerprint. The same reaction scheme is used with a different set of parameters and stoichiometry, Scheme 4. The molecular weights of the lumped components and reaction parameters are given in Tables 5 and 6, respectively. Simulation results for five different heating rates are given in Figure 17, and also the isoconversional fingerprint of the simulated case is shown in Figure 18. It is a typical fingerprint of a poor ISC candidate10 because of the effective activation energy barrier in the NTRG region. Poor combustion performance is confirmed by the combustion tube simulations given by Figure 19. Note that the temperature history shows decreasing temperatures characteristic of a lack of sustained combustion.
’ CONCLUSIONS This study focused on the kinetics of heavy-oil combustion in porous media. Both kinetic cell and coke formation experiments were used along with XPS analysis. These studies showed that the quality (or reactivity) of coke formed during the process is a function of the presence of oxygen that helps to create oxygenated functional groups, such as carboxyl, on the coke surface. The reactivity of the coke formed is a function of the concentration of these oxygen functional groups that are termed active sites. During the process (ISC), the concentration of active sites could change as a function of temperature to lead to the NTGR. Further investigation is needed at this point to confirm this point. Additionally, it is shown that coke formation reactions under nitrogen flow are relatively slow and may be controlled by the precipitation of asphaltenes. On the other hand, in the presence of oxygen, coke formation reactions are relatively fast. It is shown that an oil sample preheated to 673 K under nitrogen flow becomes solid, undergoes LTO, and exhibits a NTGR. Isoconversional fingerprints of preheated and regular kinetic experiments are quite similar. This leads to the conclusion that LTO is an inherent part of the combustion process. Even if the oil is in the solid phase, LTO reactions propagate creating oxygen active sites on the surface that are necessary for combustion. On the basis of the experimental data, a generalized (pseudo) reaction scheme is proposed that is capable of reproducing NTGR. With the kinetic cell and combustion tube simulations conducted, it is shown that the proposed reaction scheme is capable of producing the isoconversional fingerprints of a good and a poor combustion candidate. Finally, we confirmed with simulations that the isoconversional method fingerprints whether an oil/matrix pair is a good or a poor ISC candidate. ’ APPENDIX: INPUT DATA FOR SIMULATIONS The properties necessary to run the simulations are based on correlations and data available in the literature. In simulation studies, a heavy oil of 14°API with a molecular weight of 515 g/ gmol is assumed. Porous medium porosity and absolute permeability are assumed to be, respectively, 0.4 and 30 000 md. The thermal properties, critical properties are estimated based on data in the literature. In addition, the dead oil assumption is used; all equilibrium constants for oil are assumed to be zero. The critical properties of oil are estimated based on the correlations given by Kesler and Lee.42 Once the molecular 4447
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Table 7. Properties of the Oil Used for Synthetic Examples °API
MW g/gmol
Tb, °C
ω
Tc, °C
Pc, kPa
14
515
551.6
1.26607
697.8
837.1
Table 8. Oil Viscosity Data for Synthetic Examples (Aziz et al., 1987) T, °C
viscosity, cp
23.89 37.78
5.78 103 1.38 103
65.56
1.87 10
93.33
4.70 101
121.11
1.74 101
148.89
8.50 100
176.67
5.20 100
260.00
2.50 100
648.89 2000.00
6.95 102 1.48 103
Thermal conductivity of stainless steel 316 is tabulated with respect to temperature based on the following expression. λSS316 ðJ=cm min°CÞ ¼ 7:7661 þ 0:0095Tð°CÞ 2 106 T 2
The expression is obtained by curve fitting the data provided by the Thermophysical Properties of Matter Database.45 The thermal conductivity of coke is tabulated based on the following expression. λcoke ðJ=cm min°CÞ ¼ 1:1006 þ 0:0015Tð°CÞ 6 107 T 2
2
ðA.5Þ
The expression is obtained by curve fitting the data provided by the Thermophysical Properties of Matter Database.45 Insulation material heat capacity is 0.125 J/cm3 °C.47 Thermal conductivity of the insulation material is tabulated with the following expression. λinsulation ðJ=cm min°CÞ ¼ 0:0077 þ 1:1 105 Tð°CÞ þ 3:3 108 T 2 þ 1:9 1010 T 3
weight and the specific gravity are known, critical pressure, critical temperature, acentric factor, and boiling point are estimated. The values are given in Table 7. Thermal Conductivity for Different Components. Oil thermal conductivity is estimated based on the following correlation given by Cragoe43 and adjusted to appropriate units. λo J=mincm°CÞ ¼ 0:071 186 4 0:000 038 44Tð°CÞ ðA:1Þ The viscosity of oil is based on values provided by Aziz et al.44 The last two values are calculated by fitting an exponential function and extrapolating. The values are given by Table 8. The thermal conductivity of water is tabulated based on the following expression.
ðA.4Þ
λw ðJ=cm min°CÞ ¼ ð0:3414Þð0:0011ÞTð°CÞ 4 106 T 2 þ 109 T 3 ðA.2Þ
The expression is obtained by curve fitting the data provided by the Thermophysical Properties of Matter Database.45 Thermal conductivity of the gas phase is tabulated based on the following correlation. λg ðJ=cm min°CÞ ¼ 0:0142 þ 5 105 Tð°CÞ 2 108 T 2 þ 3 1012 T 3
ðA.6Þ The expression is obtained by curve fitting the data given by the Spaceloft subsea data sheet.48 Heat Capacity for Different Components. Specific heat capacity of the oil is estimated based on the correlations given by Watson and Nelson.49 On the basis of the assumptions (14°API and 515 g/gmol), the heat capacity values are generated with respect to temperature and a linear curve is fitted to the data. The following correlation is used for simulations. cp ðJ=mol°CÞ ¼ 524:882 þ 1:148 635 4ðTð°CÞ þ 273:15Þ
ðA:7Þ The heat capacity of sand is based on SiO2 values from NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species.50 J cp ¼ 1:6328 þ 0:001ðTð°CÞ þ 273:15Þ cm3 °C ðA:8Þ The heat capacity of stainless steel 316 is tabulated based on the following expression. J cp ¼ 3:3821 þ 0:0013ðTð°CÞ þ 273:15Þ cm3 °C
ðA.3Þ
ðA:9Þ
The gas phase is assumed to be composed of air and the expression is obtained by fitting air thermal conductivity data provided by the Thermophysical Properties of Matter Database.45 Thermal conductivity of sand is 0.1578 J/cm min °C.46 The effective thermal conductivity of the medium is estimated with volume weighting of the phase thermal conductivities by default in the simulator.40 Therefore, the thermal conductivity of rock is estimated to be 0.2528 J/cm min °C so that at 25 °C the same value (0.1578 J/cm min °C) is achieved after volume averaging of an air saturated sand.
The expression is obtained by curve fitting the data provided by the Structural Alloys Handbook.51 Heat capacity of the heater blocks is assumed to be equal to the stainless steel heat capacity, and the thermal conductivity is given by Table 9. These values are estimated based on experimental temperature profiles. The values are adjusted so that the experimental temperature profiles match the simulations ignoring reactions. Gas specific heat capacity values are estimated based on the NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species.50 The expression in the database has 4448
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Table 9. Thermal Conductivity of Heater Blocks for Kinetic Cell Simulation T, °C
λ, J/cm min °C
15
0.800
40 65
0.801 0.802
90
0.803
115
0.804
140
0.805
165
0.806
190
0.807
215
0.808
240 265
0.809 0.810
290
Table 10. Gas Specific Heat Capacity Coefficients gas
a1
a2
a3
CO
3.099 101
CO2
1.947 10
1
O2
2.860 10
1
N2
3.096 10
1
1.272 10
CH4
2.680 10
1
3
1.392 102 2
a4
3.015 105
1.415 108
5
7.565 10
6.075 10
3
3.497 10
2
5.649 10
2.011 108
5
1.493 108
5
1.107 108
5
5.471 108
2.440 10 2.549 10 9.501 10
Table 11. Parameters Used for Gas Viscosity Calculations MW, (g/gmol)
σ, (Å)
N2
28.013
0.820
O2 CO
31.999 28.010
315
0.830
CO2
340
0.840
CH4
365
0.860
390
0.880
415
0.900
440 465
0.920 0.930
490
0.940
515
0.950
540
0.960
565
0.970
590
0.980
615
0.990
where
640
1.000
kT ðA:14Þ ε Although eq A.12 is for monatomic gases, it has been found to be a good approximation for polyatomic gases.52 The values of constants in eqs A.12 and A.14 are given in Table 11 for different gases.53,54 The STARS40 simulator requires gas viscosity values to be input as
gas
ε/k (1/K)
ref
3.667
99.8
54
3.433 3.59
113.0 110.0
53 53
44.010
3.996
190.0
53
16.040
3.78
154.0
54
Table 12. Parameters for Gas Viscosity Correlation a
gas
b
N2
0.000 350
0.692 747
O2
0.000 363
0.712 099
CO CO2
0.000 332 0.000 187
0.703 732 0.775 480
CH4
0.000 161
0.745 316
T ¼
seven coefficients and has the following form: cp ðTÞ ¼ a1 T 2 þ a2 T1 þ a3 þ a4 T þ a5 T 2 þ a6 T 3 þ a7 T 4 R
ðA:10Þ 40
On the other hand STARS has the following format for the gas specific heat values. cp ðTÞ ¼ a1 þ a2 TðKÞ þ a3 T þ a4 T 2
3
μg ¼ aTðKÞb
ðA:11Þ
Therefore the data need to be transformed to the required format with the necessary unit conversions. The estimated coefficients are given by Table 10. Gas Viscosity for Different Components. The viscosity of pure monatomic gas is given in terms of Lennard-Jones parameters52 as pffiffiffiffiffiffiffiffiffiffiffiffi 5 MWT μ ¼ 2:6693 10 ðA:12Þ σ2 Ω
The calculated gas viscosity values at different temperatures are fit to eq A.15. The constants obtained for different gases are given in Table 12. Liquid compressibility is assumed to be 7.25 107 1/kPa.44 The thermal expansion coefficient is assumed to be 0.000 692 42 1/°C.44 The relative permeability expressions for the water/oil system are44 2:5 Sw Swir krw ¼ krwro ðA:16Þ 1 Sorw Swir
Here, μ = viscosity (P), σ = characteristic diameter of the molecule (Å), MW = molecular weight, Ω = collision integral for viscosity, and T = temperature (K). The collision integral for viscosity is given by eq A.1353 Ω¼
ðA:15Þ
krow ¼ kroiw
1 Sorw Sw 1 Sorw Siw
2
For the gas oil system,44
1:161 45 0:524 87 2:161 78 þ þ expð0:773 20T Þ expð2:4378T Þ T 0:14874
krog ¼ kroiw
ðA:13Þ 4449
1 Siw Sorg Sg 1 Siw Sorg
ðA:17Þ !2 ðA:18Þ
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krg ¼ krgro
Sg Sgc 1 Siw Sgc
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!1:5 ðA:19Þ
where, Swir (irreducible water saturation) = 0.45, Siw (interstitial water saturation) = 0.45, Sorw (water/oil residual oil saturation) = 0.15, Sorg (gas/oil residual oil saturation) = 0.10, Sgc (critical gas saturation) = 0.06, kroiw (oil relative permeability at interstitial water saturation) = 0.4, krwro (water relative permeability at residual oil saturation, water/oil system) = 0.1, and krgro (gas relative permeability at residual oil saturation, gas/oil system) = 0.2. No effect of temperature or hysteresis is assumed. Three phase relative permeabilities are calculated based on the modified Stone II model55 (default option in STARS40).
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT M. Cinar acknowledges the financial support of Istanbul Technical University for his doctoral studies. Additional financial support was provided by Schlumberger and the Stanford University Petroleum Research Institute Affiliates (SUPRI-A). We thank S. Taylor of DBR Technology Center for providing SARA and elemental analysis of sample 12. ’ NOMENCLATURE a = exponent in gas viscosity calculation A = pre-exponential factor b = coefficient in gas viscosity calculation cp = specific heat capacity at constant pressure, J mol1 K1 E = activation energy, J/mol I = electron intensity k = absolute permeability, md kroiw = oil relative permeability at interstitial water saturation krwro = water relative permeability at residual oil saturation, water/oil system krgro = gas relative permeability at residual oil saturation, gas/ oil system MW = molecular weight, g/gmol R = universal gas constant, 8.3145 J mol1 K1 Swir = irreducible water saturation Siw = interstitial water saturation Sorw = water/oil residual oil saturation Sorg = gas/oil residual oil saturation Sgc = critical gas saturation T = temperature, K Tb = boiling point, K Tc = critical temperature, K P = pressure, kPa Pc = critical pressure, kPa x = the total distance traveled by the electron Greek Letters
ω = accentric factor θ = take off angle, degrees k = Boltzmann’s constant, 1.380 650 3 1023 m2 kg s2 K1 ε = characteristic energy λ = thermal conductivity, J cm1 min1 °C1 λa = attenuation length, nanometer
σ = characteristic diameter of the molecule, Å μ = viscosity, cp Ω = collision integral for viscosity ΔH = reaction enthalpy, J/mol
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