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An Application of the Results - From the Large-Scale Thermal Stimulation Method of Methane Hydrate Dissociation to the Field Tests Swanand S Tupsakhare, Samhita Kattekola, and Marco J. Castaldi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00553 • Publication Date (Web): 27 Mar 2017 Downloaded from http://pubs.acs.org on March 31, 2017
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An Application of the Results - From the Large-Scale Thermal Stimulation Method of Methane Hydrate Dissociation to the Field Tests Swanand S. Tupsakhare†, Samhita Kattekola ‡, Marco J. Castaldi*† †
Chemical Engineering Department, City College of New York, City University of New York, New York, NY, 10031, United States
‡Chemical Engineering Department, University of Delaware, Newark, DE 19716, United States *
Corresponding Author
[email protected] Abstract Methane hydrate formation and the gas recovery from the hydrates using thermal stimulation method was studied in a large scale laboratory reactor. A large-scale laboratory reactor (59 L volume) was used in this study. The efficiencies of gas recovery and energy utilization were studied over two different values of initial hydrate saturation (30% and 50%) and three different values of heating rates (20, 50 and 100 W). Results obtained from the tests demonstrate that with the initial hydrate saturation remaining constant (50% SH); total recovery of methane increases from 40% to 52% to 73% with a heating rate increase from 20W to 50W to 100W. The average thermal efficiency, however, decreases from 86% to 84% to 82% over the same heating rate range. Alternately at a constant heating rate (100W), total recovery of methane increased from 67% to 74% when an initial hydrate saturation was increased from 30% to 50% SH. We show that maintaining a constant heating rate throughout the dissociation is not the most efficient. Instead, starting with a low heating rate for 40% of the time then increasing it when methane output starts to decrease will be more effective. This suggests that accurate knowledge of the saturation percentage of a chosen hydrate reservoir will enable efficient energy use when recovering methane. We also found that the free water generated during the dissociation and the free water present initially in the reservoir plays a major role in carrying the heat front to distant locations in the reservoir hence increasing the gas recovery. We recommend that in the higher hydrate saturation reservoirs, which may lack the free water; the down-hole combustion method will benefit from an injection of hot water in the initial stages, supplying a small amount of heat for the dissociation but providing the free water for convection of heat to the outer regions. 1. Introduction Clathrate hydrates also called as gas hydrates are formed from a combination of a guest molecule such as methane, propane, cyclopentane and water molecules 1,2,3,4. These structures are stabilized under the conditions of high pressure and low temperature 4. There are a number of structures that can be formed that are determined by the water to gas ratio 1. Hydrates are formed in two stages, (1) Nucleation and (2) Growth. The rate of hydrate formation depends on the rate 1
of sub-cooling. The higher rates of hydrate formation can be achieved by using higher subcooling rates 5. The hydrate formation is thermodynamically feasible at a temperature lower than the hydrate equilibrium temperature however for significant nucleation rates; it is required to have higher sub-cooling rates conditions 5. This work reports the trends observed during the thermal dissociation of methane hydrates at a relatively higher saturation of 50% by pore volume. We use the tests at 30% hydrate saturation as a guide for comparison with the high saturation of 50% SH. At a higher hydrate saturation, the amount of free water available may be significantly reduced which changes the heat transfer characteristics of the hydrate dissociation process. We have shown previously that the understanding of heat transfer during the hydrate dissociation process is the key to achieving higher values of recovery efficiency and thermal efficiency 6,7,8. In an attempt to focus of these parameters, several lab scale tests that focus on thermal and depressurization methods have been reported 9,10. Most of these dissociation tests require a small thermal input to initiate the gas recovery. In actual field tests, steam or hot fluid is injected to increase the temperature of the sediment 7. However, the drawback of these methods is the heat lost during the transit of the fluid from ground to the location of hydrate sediment, which lowers the production efficiency. Our group has proposed in-situ combustion of the methane released from the hydrate phase, which could be used to generate energy for further dissociation 8. The current work expands on this idea by using a point electrical heater to simulate the in-situ combustion. This technique has shown to produce higher efficiencies due to the elimination of heat losses during hot fluid injection. The following gives a brief summary of the work by other researchers followed by our recent findings contributes to a developing database of hydrate-related work. Feng et al. 11 perform hydrate dissociation tests in a sandy reservoir by depressurizing the system along with the injection of warm water. They reported 38-39 °C as the optimum temperature for the hydrate dissociation based on the values of entropy, energy ratio, and thermal efficiency and stated it is uneconomical to increase the temperature further as there is a loss of energy during the transit of fluid. Nair et al. 12 used thermal stimulation in confined sediments to study the kinetics of the hydrate formation and dissociation processes and to investigate the effect of sand size. They observed that the dissociation is favored in smaller size sand and pure water as compared to the bigger size sand and sea water. Cranganu et al. 13 introduced the idea of in-situ thermal stimulation. They used a combustion chamber with a concentric pipe for gas recovery line to avoid separate gas recovery wells. This method addresses the drawbacks of depressurization combined with thermal simulation as the target area of heat delivery is directly in the gas hydrate zone (GHZ). That method eliminates heat loss to the surroundings, thereby increasing the efficiency of the process. Schicks et al. 14 designed a Large Laboratory Reservoir (LARS) modeled after the apparatus utilized in this work, as a part of the SUGAR project in Potsdam, Germany. That work uses a catalytic combustion chamber that employs an 8 Wt % Pd catalyst supported on ZrO2. They performed the thermal 2
dissociation studies reporting efficiencies of about 70 to 80 %. Schicks et al. used the LARS to study the feasibility of in situ combustion to dissociate hydrates using thermal stimulation method. They found that about 15% of the gas recovered from the hydrate phase is required to generate the heat for significant dissociation of the hydrates. Song et al. 15 studied hydrate dissociation in porous media using depressurization, two-cycle hot water injection and a combination of the two, over various initial hydrate saturations. Comparing the amount and rate of gas production and energy efficiency, they concluded that using depressurization combined with hot water injection was the most effective. Depressurization alone, while efficient during the initial stages, exhibited a decreased rate of gas production as temperature decreased in hydrate sediments. Hot water injection alone, while useful in preventing blockages and formation of ice, produced an overall lower gas production amount and rate. The combined method proved to be most efficient, irrespective of the initial hydrate saturation. Wang et al. 16 studied gas dissociation in a large-scale reactor combining depressurization with thermal stimulation. Their work focused on the influence of temperature on hydrate dissociation. They concluded that the hydrate dissociation rate is directly proportional to the difference between the hydrate dissociation temperature and injection temperature. Li et al. 17 used cyclic steam injection method to study the gas evolution rates using a 5.8 L reactor. They studied the dependence of initial hydrate saturation and the temperature of the injected fluid on the gas evolution from the system. It was found that systems with low initial hydrate saturation produced higher gas evolution rates, as there was less resistance to the path of gas flow. Pang et al. 18 studied the hydrate dissociation using the thermal stimulation, by hot water injection in a 10-liter reactor kept below 0 °C. Results indicated that hydrate dissociation process is mainly dominated by heat transfer, which is aligned with previous findings from our studies 6,7,8. Tang et al. 19 used hot water injection to study gas recovery from unconsolidated hydrate sediment. They found that the rate of water injection, the temperature of the water and the hydrate saturation, all impact the gas production thus energy efficiency. As hot water is injected, dissociation of hydrate begins, the production rate increases reaching a peak followed by a subsequent decline. Li et al. 20 conducted tests using an injection of a hot brine solution in which they evaluated the effect of geological parameters and brine injection parameters on the energy efficiency of the thermal stimulation method. They found that the energy efficiency increases with an increase in the initial temperature of the reservoir, the permeability of the sediment and the brine concentration. Nair et al. 12 and Mekala et al. 21 focused on studying the kinetics of the hydrate formation process comparing the rates of formation in pure water and seawater. They noticed that less gas is consumed during the hydrate formation using seawater as compared to pure water. Chong et al. 22 employed higher water content sediments by pressurizing with water instead of gas to simulate marine environments. They found that the methane uptake is about 81.5% at time scales of 76-408 hours. They studied different temperature driving forces during the dissociation and 3
concluded that a minimum temperature rise of 2.1 K is required to attain 90% recovery in 10 hours of the test. Another significant finding of this work is that the free water generated during the dissociation of the hydrate has an effect on the gas production rates and it is required to manage the water content to get higher efficiency values. Fitzgerald et al. 23 studied the formation and thermal dissociation of hydrates in silica beads. Numerical simulations were also performed using COMSOL software. The numerical and experimental data showed consistent trends demonstrating that higher efficiency rates could be obtained by lower heating rates. Song He et al. 24 used an unconsolidated hydrate sediment from the South China Sea for dissociation using microwaves. They report that hydrate saturation and water greatly affects the gas recovery behaviors. Several other researchers have combined the thermal stimulation method with CO2 injection 25,26,27,28. The majority of the work done to date has focused mainly on hot water or steam injection and is on a smaller scale. We make use of a relatively larger scale system that closely approximates the permafrost conditions and aspect ratios relevant to field scale to allow for the study of in situ combustion using a point electrical heating source. We show how methane recovery and thermal efficiencies are affected by the presence of free water in higher hydrate saturation tests. We make a recommendation on how the down-hole combustion method could be used in combination with a conventional hot water injection method to increase the effectiveness of the process. The apparatus utilized in this work has a well-established history of more than ten years in which we observed some unique experimental phenomena and compared our results with other researchers in this field. We have shown in our previously published research 6 that the data obtained from this setup for gas discharge during hydrate dissociation was within an order of magnitude of that was predicted by Moridis et al.29 for field simulations. The gas discharge value predicted by our experiment was 4.5 x 106 m3 whereas the value predicted by Moridis et al.29 for field simulation was 4.1 x 107 m3. The test results are comparable to the numerical results from Moridis’ model, considering the scale differences between the model and the actual experiment. Therefore, good agreement with the field simulation data by Moridis indicates that the hydrate formation and dissociation procedure followed in this work and the apparatus is capable of producing useful results that provide insights about hydrate dissociation in the field if the aspect ratios are maintained.
2. Equipment & Methods A large-scale hydrate vessel (LSHV) was used for formation and dissociation studies of methane hydrates that have been detailed in previous publications 6,7,8 and will be summarized here. 315 Stainless Steel was used for the construction of the LSHV because it does not corrode easily and can withstand high pressures associated with hydrate formation. The diameter and height of the LSHV are 0.305 m and 0.813 m respectively, providing an internal volume of 59.3 liters. It is filled with quartz sand, with a mean particle diameter of 500 μm. There is a 3 cm empty head 4
space left at the top to allow for gas accumulation and volume expansion associated with the hydrate formation process. The quartz sand is saturated with deionized water to 50 % by pore space volume. Figure 1 shows the schematic of the LSHV indicating the gas lines and instrumentation, followed by Figure 2, which shows the actual equipment used for experimentation.
Figure 1. Schematic of LSHV showing gas injection lines and instrumentation The two ends of the cylindrical body of the LSHV are closed using two end plates of 5 cm thickness each, which are secured in place by two threaded collars. The maximum value of pressure utilized in this hydrate formation study is ~ 4.8 MPa.
Figure 2. Large Scale Hydrate Vessel made with stainless steel body 5
The end plate on the top has five ports. The middle port (3/8 inch) is used to insert a custommade cartridge heater (Hotset CU375) that simulates combustion (shown in red in Figure 1). Two thermocouple sets (all the thermocouples are Omega, K-type, ungrounded, 0.0625”), T set (T1-T4) and C set (C1-C4) are inserted through two of the remaining four ports. A pressure relief valve (Swagelok, SSR3D) and a pressure transducer P1 (Trans metrics P100 series), are connected to the last two ports on the top end plate. The pressure relief valve is offset to 10.3 Mpa. The pressure transducer (P1) measures the pressure in the empty headspace of the LSHV. Another set of thermocouples, B set (B1-B4) is inserted through the wall of the cylinder body as shown in Figure 1 and Figure 2. The relative location of all the thermocouples is also highlighted in Figure 1. The placement of the thermocouples forms symmetry vertically, horizontally and azimuthally. Two additional pressure transducers, P2 and P3, are located in the gas sampling port and the CO2 injection line, respectively. P2, located in the gas sampling line, is used for measuring the hydrostatic pressure in the sand matrix. The CO2 injection line and the sample ports are not used in this tests. These two are used when the apparatus is used for the tests with a combination of thermal stimulation with CO2 exchange. The bottom end plate of the LSHV is equipped with four gas injection ports used for the injection of CH4 into the system. The CH4 gas utilized in this work is Ultra High Purity (UHP 5.0, Praxair) supplied from T size bottles. The entire setup is kept in a walk-in refrigerator from KOLPAC (Model PR195MPD208) to maintain the LSHV at desired temperatures and also to keep the constant far-field temperature. The setup in this work is kept at a temperature of 2.5oC. The rate of gas injection and the gas released during the dissociation is measured using flow meters. For the injection of methane into the reactor, Omega FMA-873A-V flow meter (0-50 SLPM), calibrated for CH4 is used. An Aalborg 17 series flow meter (0-5 SLPM) is used to measure the release of gas during dissociation. All the thermocouples, pressures transducer, flow meters and flow controllers are connected to the Omega USB data acquisition system, which uses Personal DaqViewTM software to record the data. 3. Results & Discussion 3.1 Methane Hydrate Formation A desired methane hydrate saturation (SH) is obtained by pressurizing the reactor above the equilibrium pressure at a temperature of 2.5 °C. The equilibrium pressure for pure methane hydrate at 2.5 °C is 3.33 MPa. The reactor filled with quartz sand is pre-wetted with deionized water and saturated to 50 % by volume of the pores. After the whole system reaches the equilibrium temperature of 2.5 °C, the methane gas is injected into the reactor to establish pressures of 4.8 MPa which is well above the equilibrium pressure of the gas-water-hydrate system at 2.5 °C. The gas fills in the pore space as well as the empty head space at the top of the reactor. The onset of hydrate formation is observed from a temperature increase at various
6
locations in the reactor. The temperature rise is between 1 to 6.5 °C depending on the hydrate saturation and the location of the thermocouple.
Figure 3.Pressure-Temperature plot during hydrate formation (data points) with predictions from equilibrium calculations (solid line) Figure 3 shows the temperature-pressure profile at selected locations (data points) plotted with the equilibrium predictions (solid line). The pressure in MPa is plotted on the Y-axis and Temperature in °C is plotted on the X-axis. Figure 3 shows that the LSHV follows the gashydrate-water phase equilibrium closely, indicating the presence of methane hydrates in the reactor. Figure 4 shows temperature in °C on the Y-axis and time in hours on the X-axis for hydrate formation test to achieve 50 % hydrate saturation. This data is for the C set of thermocouples, the relative location of the C set thermocouple is shown in the inset. The 50 % hydrate saturation is achieved in 5 steps, each giving nearly 10 % hydrate saturation by pore volume. The five peaks in Figure 4 correspond to the five cycles of hydrate formation after each methane gas injection. In one cycle, the reactor is pressurized 4.8 MPa, and the number of moles of hydrates formed is calculated by using the pressure drop in the gas phase as the gas is incorporated into hydrate. This process is repeated several times to achieve the desired initial hydrate saturation in the reactor. For example, in Figure 4, 50 % hydrate saturation is obtained by pressurizing the system with methane for five times. As soon as the system exceeds the equilibrium pressure of 3.33 MPa, the system shows a rise in temperature at various locations. This system forms a combination of pore filling and grain cementing type of hydrates as reported previously7. The following equations are used to calculate the number of moles of hydrate formed (Equation 1) and the total hydrate saturation (Equation 2) in the reactor. 𝜂ℎ𝑦𝑑 =
𝑉𝑝𝑜𝑟𝑒 (𝑃𝑖 𝑍𝑓 −𝑃𝑓 𝑍𝑖 ) 𝑍𝑓 (𝑅𝑇−
𝑃𝑓 ) 𝜌ℎ𝑦𝑑
(1)
7
𝑆𝐻 =
𝜂ℎ𝑦𝑑
1 𝜌ℎ𝑦𝑑
𝑉𝑝𝑜𝑟𝑒
(2)
In the above equations, Vpore is the pore volume available for hydrate formation, Pi and Pf are the pressure values at the beginning and the end of each pressurization cycle respectively. Zi and Zf are the compressibility values of methane at a pressure of Pi and Pf respectively. R and T are universal gas constant and temperature during injection respectively. ρhyd is the density of hydrates formed. Equation 1 also takes into consideration the change in the density during the hydrate formation process. Equation 2 is used to calculate the hydrate saturation in the reactor. The total number of moles of hydrates formed were 53 and 83 in the case of 30% and 50% SH tests respectively.
Figure 4. Temperature (°C) vs. Time (hours) for 50 % hydrate saturation test for C set thermocouple The hydrate formation process is exothermic releasing 54 KJ of energy per mole of hydrate formed. Therefore as the hydrates form there is an observed rise in the temperature at various thermocouple locations in the reactor. These measurements in temperature are shown in Figure 4. From Figure 4 it can be observed that the temperature rises above the equilibrium value by 1 ° C (C1) to 6.5 °C (C4) (depending upon location), indicating the onset of hydrate formation. To ensure hydrates are fully formed we wait until the temperature returns to 2.5 °C after approximately 24 hours. At this point, the reactor is again pressurized with methane to get a second cycle of hydrate formation and so on. From Figure 4, the temperature at locations C2, C3, and C4 track together in almost all five cycles. The peak temperature recorded by C2, C3 and C4 decreases from the first cycle to the fifth cycle. As can be seen in Figure 4, the temperature at location C1, which is located just near the empty head space of the reactor, does not significantly rise as compared to other locations. This is likely due to heat removal occurring to a larger extent from the C1 location as compared to other locations. To explain this further, C2, C3 and C4 lie deep inside the sand matrix of the reactor 8
and they have a higher associated thermal mass. Although, the set point of the refrigerator is 2.5 ° C the temperature at the walls oscillates +/- 0.75 °C due to the limits on the accuracy of the cooling room. This oscillation affects the temperature of the region near the walls of the reactor yet has no effect on the solid matrix of the reactor. As a result, C1 has a faster rate of heat removal (being at the interface of the solid matrix and gas headspace) than C2, C3, and C4; First order calculations using capillary rise phenomena are done to investigate this further and determine if the location C1 is deficient in water. The following equations are used to calculate the capillary rise. 𝑁𝑝 =
6(1−𝜀) 𝜋𝑑𝑝 3
𝑆𝑣 = 𝑁𝑝 (𝜋𝑑𝑝 2 ) 𝑑𝑝𝑜𝑟𝑒 = ℎ=
4𝑉𝑝𝑜𝑟𝑒𝑠 𝑆𝑣
2𝛾𝑐𝑜𝑠𝜃 𝜌𝑔𝑟
(3) (4) (5) (6)
Where Np, Ɛ, and dp are number of particles, porosity, and particle diameter respectively. Sv, dpore, Vpores, are a surface area, pore diameter, and volume of the pores, respectively. Equation 6 is the Young-Laplace equation used for calculating capillary rise. ϒ, θ, ρ, g and r are surface tension, contact angle, density, gravity and radius of pore respectively. The contact angle used was 2 degrees taken from Weisbrod et al 30. and is considered consistent with our understanding of the matrix system. The pore diameter value calculated using above equations is 5.56x10 -6 m, and the capillary rise height is 2.66 meters. The total height of the reactor is 0.914 meters. Therefore, 2.66 m is almost three times the reactor height. This gives an indication that the location C1 is not deficient in water. This leaves us to conclude that there may be sufficient hydrate formation taking place near C1. However, the measurement is impacted due to heat being removed quickly through the gas head space.
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Figure 5. Temperature (°C) vs. Time (hours) for 50 % hydrate saturation test for B set thermocouple Figure 5 shows the temperature response of the B set thermocouple during the 50 % hydrate saturation test plotted as temperature (°C) on the Y-axis and time (hours) on the X-axis. Unlike the C set thermocouples (which are located in the top half of the reactor) the B set thermocouples have a different response. The peak temperature recorded by each of the B set thermocouples increase from the first cycle to the fifth cycle. The peak temperatures registered by the B set is nearly 4 °C in first three cycles and almost 6 °C for last two cycles. This response was observed in a repeatable manner in this system. Combining the information from the C set temperature response (Figure 4) and B set temperature response (Figure 5) it can be concluded that the hydrate formation begins in the top half of the reactor and then progresses to the bottom half of the reactor in subsequent pressurization cycles. It is likely that a small amount of hydrate formation does occur in the lower region of the reactor in the first pressurization, but the extent is lower compared to the top half of the reactor. It is hypothesized that the onset of hydrate formation in the top half of the reactor is due to the presence of the empty head-space that carries methane gas. As soon as the gas molecule is consumed during hydrate formation anywhere in the reactor, the gas molecules from the empty headspace can take its place, and this is more favorable for the top region of the reactor than the bottom region. In short, the driving force or the concentration gradient for hydrate formation is higher for the area near the empty headspace than away from the head space and this favors the onset of hydrate formation in the top half region.
Figure 6. Temperature profile for B set thermocouple during second and third formation cycle indicating secondary hydrate formation Another marked feature in Figure 5 is the secondary hydrate formation observed during the second and third formation cycles. To explain this further, Figure 6 shows a magnification of the timeframe between 20 to 70 hours of Figure 5. In Figure 6, the temperature at location B3 and 10
B4 (solid and open triangle respectively) shows a primary peak near 28 hours and then appears to have settled at about 31 hours. However, a rise is observed once again at 31 hours giving a secondary peak at about 34 hours. Similarly, a secondary peak is observed for B3 and B4 in third formation cycle at approximately 58 hours. This delayed temperature rise is very likely the secondary hydrate formation. This phenomenon of secondary hydrate formation in which a slight temperature rise is observed after the primary hydrates have ceased to grow. This phenomenon can be explained based on the system being mass transfer limited. It is possible that the free water gets entrapped in the shell of the hydrates and limits the exposure to the free gas. In the later stages, due to the expansion associated with hydrate formation and water movement due to capillary forces this water may become available later for hydrate formation. The temperature profile of the T set thermocouples (which is again located in the top region of the reactor) shows similar trends as the C set, with higher peak temperatures recorded in the first cycle and this peak temperature decreases in the subsequent pressurizations. 3.2 Molar recovery of methane in heating tests at constant initial saturation & different heating rates The experiments were designed to control hydrate saturation and heating rate in the LSHV. Table 1 gives the test parameters used in this work.
Table 1. Summary of test parameters for the thermal stimulation tests showing % hydrate saturation and heating rate (W) Test# 1 2 3 4 5 6
% Hydrate Saturation (SH) 30% 30% 30% 50% 50% 50%
Heating Rate (W) 100 W 50 W 20 W 100 W 50 W 20 W
Following the formation of the hydrates is the recovery of methane gas, measured in terms of moles of methane recovered, by performing dissociation tests. All dissociation tests are initiated when the system is at 2.5 °C with the desired initial hydrate saturation of 30% or 50%. The pressure in the reactor is maintained at 3.5 MPa (slightly above the equilibrium of 3.33 MPa) for the entire duration of the test using a back pressure regulator. Changes in pressure cause a change in the hydrate dissociation temperature and thus, maintaining the pressure at a constant 11
value allows variability in only one parameter, i.e., temperature. Due to the sensitivity of the back-pressure regulator, the pressure in the reactor oscillates by 10 psi (0.07 MPa); however, calculations show that this affects the hydrate equilibrium temperature only by 0.35-0.4 °C and can be neglected. The duration of each test depends on the initial hydrate saturation and the heating rate used. Nevertheless, the test is terminated once the flow rate of methane exiting the reactor approaches the lower detection limit of the flow meter. 3.2.1 Cumulative molar recovery at 30 % SH & different heating rates Figure 7 correlates the cumulative moles of methane recovered from the reactor with time at heating rates of 20, 50 and 100 W for 30% SH. Increasing the heating rate has directly increased the recovery efficiency. As shown in Figure 7, nearly 43, 35 and 24 moles of methane were recovered at the end of each heating test at 100 W, 50 W, and 20 W respectively. The total number of moles of methane in the hydrate phase for 30 % saturation was roughly 52.5. This gives the recovery efficiency of 82 %, 67 % and 46 % for 100 W, 50 W, and 20 W heating respectively.
Figure 7. Cumulative moles of methane recovered from the system at 30 % saturation and three different heating rates (100 W, 50 W, 20 W) It is evident from Figure 7 that cumulative moles of CH4 for 100 W and 50 W heating rates overlap closely for first 7 hours. This trend strongly suggests that maintaining a constant heating rate throughout the test may not be the most efficient way for dissociation. In the first 7 hours, approximately the same number of moles of methane as 100 W are recovered with half the heating rate of 50 W. Therefore; it would be more efficient to start at a lower heating rate, which dissociates the hydrates in the proximity of the heater. Eventually, increasing the heating rate would allow for the heat front to reach farther and dissociate the hydrates located far away from the heater. This is justified by performing the following calculations. It is observed that the cumulative moles recovered in first 7 hours of Test 1 (30% SH, 100 W) are the same as that of Test 2 (30% SH, 50 W). Starting the heating test at 50 W and increasing the heating rate to 100 12
W after 7 hours would save 1260 KJ of energy in the first 7 hours, which would significantly decrease the extraction cost in a large-scale scenario. Importantly there is a limit to the heating rate that can be applied at the lower end. Figure 7 shows that the 20 W heating rate yields less methane at all times. This is related to a balance between the heat required for dissociation and the heat removal rate compared to the heat delivery at 20 W. The goal is to supply a sufficient amount of heat to maintain the temperature above the dissociation value while compensating for heat loss throughout the system. However, the dissociation of the hydrate has the largest impact and can be seen in Figure 8. 3.2.2 Cumulative molar recovery at 50 % SH & different heating rates Figure 8, shows the cumulative moles of methane recovered from the reactor with time at heating rates of 20, 50 and 100 W for 50% SH. In Tests 4, 5 and 6 which were performed at 50 % initial saturation and three different heating rates of 100, 50 and 20 W respectively; nearly 61, 43 and 33 moles of methane at the end of each test respectively were obtained. There was a total of 83 moles of methane in the hydrate phase initially, in each of the three tests. That gives recovery of 73 %, 52 % and 40 % in each of the Tests 4, 5 and 6 respectively.
Figure 8. Cumulative moles of methane recovered from the system at 50 % saturation and three different heating rates (100 W, 50 W, and 20 W) Unlike Figure 7, Figure 8 does not indicate much of an overlap between any of the curves. It can be inferred that the optimum heating rate in the initial stages is primarily a function of hydrate saturation. At 50% SH, there was a significant increase in recovery rate as you increase the heating rate, thereby reducing the duration of the test. Test 4 (50% SH, 100 W) was able to recover ~61 moles in about 42 hours whereas Test 5 (50% SH, 50 W) was able to recover ~43 moles in almost the same time. Although more energy is supplied for 100 W test as compared to 50 W test, less energy is dissipated in the unproductive heating of the sand matrix and water, due to higher initial saturation of 50% as compared to 30 % 13
SH test. In the case of 30% SH, more energy is taken up by the unproductive heating of sand matrix, which decreases the performance. It is important to note that, in Figure 8, at a heating rate of 20 W, the cumulative recovery of methane stabilizes after ~60 hours of the test, giving diminishing returns. This is mainly because, after 60 hours, the un-dissociated hydrates are now located too far away from the heat source where the heat front cannot reach. Therefore, only 31 of 83 moles of methane are recovered at the end of 60 hours. Additionally, only 2 moles of methane are recovered over the next 37 hours. The remaining methane is still trapped in the hydrate phase, as the regions in which hydrates exist are still well within the hydrate stability zones. The heat front is too weak to reach those locations even after allowing for an extended time. This also suggests that increasing the heating rate in the later stages will improve the recovery. An ideal situation would be to start with lower heating rate and eventually increase the heating rate when there is a decrease in the methane that is released until the majority of the methane is recovered. Importantly this presents a configuration that should be used in the field. If the aspect ratios are maintained, the radius of influence can be calculated for a given heat rate and hydrate saturation. Figure 9a and 9b shows the temperature profile of selected thermocouples at the heating rates of 20 W and 100 W respectively. The temperature profiles as shown in Figure 9a and 9b, support the claim that, during Test 6 (at 20 W), the heat front did not reach farther locations as opposed to the case of 100 W heating test. For a better understanding Table, 2 shows the physical distance of the thermocouples from the top plate of the reactor.
Figure 9a. Temperature profile of selected thermocouple during dissociation in Test 6 at 50 % SH and 20 W heating
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Figure 9b. Temperature profile of selected thermocouple during dissociation in Test 6 at 50 % SH and 100 W heating, T2 is plotted on the right Y-axis in °C Table 2- Physical distances of the thermocouples (the vertical distances shown are measured as depth from the top plate and the radial distances from geometric center) TC Label C1 C2 C3 C4
Radial Distanc e [r] 2" 2" 2" 2"
Vertical Distanc e [Z] 3 3/8" 12 3/4" 14 3/4" 18"
TC Label B1 B2 B3 B4
Radial Distanc e [r] 0 0 0 2"
Vertical Distanc e [Z] 34" 29" 24" 24"
TC Label T1 T2 T3 T4
Radial Distanc e [r] 4" 4" 4" 4"
Vertical Distanc e [Z] 6 1/4" 12 3/4" 18 1/2" 22 1/4"
The dotted line at 3 °C shows the methane hydrate equilibrium temperature at the system pressure of 3.5 MPa. Methane hydrates are stable below this line and dissociate at temperatures above 3 oC. The data points are shown for locations C1, T1, B1, B2, and T2. It should be noted that to get a realistic flow rate of methane and assist the dissociation; the entire hydrate matrix must be heated to approximately 8 to 10 °C higher than the equilibrium temperature 6. This is because the energy supplied is not only used for the dissociation of hydrates, but some of it is also taken during the unproductive heating of the sand matrix. Figure 9a reveals that there are several locations in the reactor, which are either below the hydrate stability temperature or only slightly above the hydrate stability temperature when the heating rate is 20 W. Increasing the heating rate allows the heat front to reach farther locations and increase the recovery efficiency. As seen in Figure 9b, with a 100W heating rate, thermocouple locations B2 and T2 are at a significantly higher temperature after 10 hours of the 15
test, and the rest of the thermocouples also reach higher temperatures after 30 hours of the test, which was not the case with a 20W heating rate. With a heating rate of 100W, dissociation can prevail faster, at a higher rate and even a farther location. In addition to this, it is important to note that the low methane recovery in the case of 20 W heating test does not imply non uniform distribution of hydrates in the reactor. We expect the hydrate saturation to be uniform in the case of 30% as well as 50% SH and there are no discontinuities present in the hydrate matrix. This was confirmed using the Young-Laplace equation to quantitatively show the capillary forces overcome gravity, thus ensuring uniform saturation of water. Thermocouples within the matrix exhibit similar temperature increases suggesting similar hydrate formation amounts. Finally, the rates of methane recovery (see Figure 7) exhibit continuous functions. If the system was not uniform we would expect to observe plateaus in the production rates. 3.3 Molar recovery at constant heating rates and different initial saturations This section compares the molar recovery of the methane with constant heating rate and different initial saturation. Figure 9, Figure 10 and Figure 12 plot cumulative moles of methane on the Yaxis and time on the X-axis, for heating rates of 100, 50 and 20 W respectively. The solid circles indicate 50% SH and hollow circles indicate 30% SH. 3.3.1 Cumulative molar recovery at 100 W and different initial saturations Figure 9 compares the number of moles of CH4 recovered from the reactor with different initial hydrate saturation when subjected to the heat of 100 W. At the end of Test 1 and 4, 35 and 61 moles of methane were recovered. That gives the percent recovery of 67 % and 74 % respectively in 30 % and 50 % initial saturation case. At any time, the number of moles recovered for 50 % SH case was more than 30 % SH case. This will have a marked effect on the thermal efficiency since the heat input was the same in both cases. The effect on thermal efficiency is discussed in the next section.
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Figure 9. Cumulative moles of methane recovered from the system at 100 W for two different initial saturations (30 % SH and 50 % SH) 3.3.2 Cumulative molar recovery at 50 W and different initial saturations Figure 10 shows the molar recovery for two tests, both at 50 W heating but different initial saturations of 30 % and 50 % respectively. Nearly 35 and 46 moles were recovered at the end of Test 2 and Test 5 at 30 % and 50 % initial saturation respectively. That gives the percentage recovery of about 67 % and 55 % respectively. This contradicts what was measured and predicted in the case with 100 W heating rate in Figure 9. In Figure 9 a higher initial hydrate saturation resulted in a greater percentage recovery. However, Figure 10 indicates that a higher initial saturation yields lower percentage recovery. This is because there is a deficiency of heat in the case with 50% SH. Although there is sufficiently high saturation of the hydrate, the heat supplied is not enough to reach all the locations where the hydrates are present, which results in a lower percentage recovery. In Figure 10, it is noticed that there is a faster recovery of methane in the first 10 hours in the case of 30% SH. Hydrate saturation away from the heater starts playing a role after these 10 hours and is more efficient in case of 50% SH.
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Figure 10. Cumulative moles of methane recovered from the system at 50 W for two different initial saturations (30 % SH and 50 % SH) It is important to note that in Figure 10, for the first 8 hours the output from Test 2 (30 % SH) was higher than the output from Test 5 (50 % SH). Figure 11 shows a magnification of the data during first 8 hours.
Figure 11. Focusing first 8 hours of Figure 10 for methane recovered from the system at 50 W for two different initial saturations (30 % and 50 %) Figure 11 shows that output for Test 2 (at 30 % SH, 50 W) was higher than Test 5 (at 50 % SH, 50 W) for ~ 8 hours. This can be explained on the basis that there is an availability of free water for the convection of heat and thus progression of a heat front. Calculations show that the free water available for Test 2 is 5.7 liters whereas it is only 2.6 liters during Test 5. The availability of a higher amount of free water in the 30 % SH case allows the heat front to move away from the heater at a faster rate than in the 50% SH case, thus affecting the cumulative molar recovery of methane in the initial stages. However, during the later stages, there is more free water available for the convection of heat in the 50% SH case, due to dissociation of more hydrates, increasing 18
the methane production. In other words, all tests had the same amount of water to sand ratio. The amount of hydrate formation dictates the free water that remains. For example, a 30% hydrate saturation will have, initially, more free water than a 50% hydrate saturation. Once the amount of dissociated hydrates reaches a point where both have the same amount of hydrates, then the free water will be the same. Another interesting trait observed while comparing Figure 9 and Figure 10, is the molar recovery. The molar recovery of methane in Figure 9 is affected solely by the initial saturation (heating rate remaining constant), whereas, it is influenced by the initial saturation as well as the availability of free water in Figure 10, indicating the need for optimizing the heat rate and saturation. 3.3.3 Cumulative molar recovery at 20 W and different initial saturations Figure 12 shows the cumulative moles of methane recovered at the 20 W heating rate and different initial saturations of 30 % and 50 %.
Figure 12. Cumulative moles of methane retrieved from the system at 20 W for two different initial saturations (30 % SH and 50 % SH) In Test 3 and 6 which were performed at 20 W heating and 30 % and 50 % initial saturation, 24 and 33 moles were recovered at the end of the tests respectively. That gives a percentage recovery of 46 % and 40 % in Test 3 and 6 respectively. Figure 12, shows trends, which are similar to the case of 50 W heating rate, where higher initial saturation resulted in lower percentage recovery. 20 W heating rate was only effective in the early stages for 50% SH, but the yields are diminished during later stages. 3.4 Thermal efficiency of the heating tests at constant initial saturation & different heating rates
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This section reports and compares the thermal efficiency of the tests at constant initial saturation and different heating rates. The number of moles of methane recovered from the system and the heat supplied via the heating element is used in the calculation of the efficiency. Equation 7 is used for the calculation of the thermal efficiency of the heating tests. 𝜂=
𝑑𝑛𝑔𝑎𝑠 𝐻ℎℎ𝑣 −𝑄ℎ𝑒𝑎𝑡 𝑑𝑡 𝑑𝑛𝑔𝑎𝑠 𝐻ℎℎ𝑣 𝑑𝑡
(7)
Where, ngas is the number of moles of methane released; Hhhv is the higher heating value of methane and Qheat is the heat supplied through the heater in joules per second. Individual data points are obtained by finding ngas at the end of each hour. Practically, the numerator of Equation 7 represents the difference between the energy content of the methane released and the heat supplied, and the denominator is the total energy content of methane gas released during a test. 3.4.1 Thermal efficiency at constant initial saturation (30 %) and different heating rates (100W, 50W, 20W) Figure 13 and Figure 14 presents the relation between thermal efficiency, plotted on Y-axis, and time, plotted on X-axis, for different heating rates at constant initial saturation of 30% and 50% respectively.
Figure 13. Thermal efficiency at constant initial saturation (30 % SH) and different heating rates (100W, 50W, 20W) In Figure 13, which shows the thermal efficiency data for 30% SH, it is observed that the lower the heating rate, the higher the efficiency. The peak efficiency is highest for the test with 20 W, 20
compared to 50 W and then 100 W. In the case of a low heating rate, the unproductive heating of the sand pack and the free water is reduced; hence most of the energy supplied is used for hydrate dissociation, that is a minimal over-temperature is achieved. The peak efficiencies of Tests 1, 2, and 3 at 100, 50 and 20 W with 30 % initial saturation were~86 %, 91% and 96 % respectively. The efficiency decreases in the later stages as the heat is continuously being supplied while the recovery of methane is slowing. 3.4.2 Thermal efficiency at constant initial saturation (50 %) and different heating rates (100W, 50W, 20W) Figure 14 presents the thermal efficiency for the tests at 50 % initial saturation and three different heating rates of 100W, 50W, and 20 W respectively. This directly connects to Figure 8, which was the cumulative moles of methane recovery. Recall, the 20 W heating test showed a point where methane production ceased at approximately 50 hours. Continued heat input is wasted. This is clearly seen in Figure 14 at the 50-hour mark. Since the lower heating rate was able to yield the methane up to that time, it is more efficient. However, continued heating shows a continued reduction in efficiency. Eventually, the 20 W efficiency falls below the 50 W efficiency and if the test were to continue it would fall below the 100 W efficiency. Figure 14 shows that at the 50-hour mark, the efficiency of 50 W test is about 78% whereas it is nearly 85% for the 20 W condition. However, at 100 hours, the 20 W efficiency is down to 75%, which is lower than the 50 W test. This scenario sets a boundary on when to stop heat injection into a reservoir.
Figure 14. Thermal efficiency at constant initial saturation (50 % SH) and different heating rates (100W, 50W, 20W) Trends similar to Figure 13 are observed here as well, with the peak values of efficiencies of 90%, 89% and 92% in Tests 4, 5 and 6 respectively. The lowest heating rate (20 W) gave the highest peak value and, also remained at a higher efficiency for the majority of the duration of
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the test. Nevertheless, an interesting trend is observed in the first 15 hours of the test in Figure 14. Figure 15, is a magnification of the first 15 hours of Figure 14.
Figure 15. Thermal efficiency at constant initial saturation (50 % SH) and different heating rates (100W, 50W, 20W). Magnification of the data from Figure 14 for first 15 hours Figure 15 reveals that Test 6 always produced the highest efficiency. However, Test 4 initially yields higher efficiency than Test 5 and this efficiency decreases once the hydrate saturation near the heater decreases. This is due to a condition we consider an over-temperature, where the heat rate into the reservoir is too high compared to the heat front movement into undissociated hydrates. Therefore the continued heat input is going into heating the unproductive sediment. This demonstrates that the optimal heating rate is one that provides sufficient energy and matches the rate of dissociation, heat loss, and heat front velocity. Although Test 4 (50 % SH & 100 W heating) gives lower thermal efficiency than Test 5 (50 % SH & 50 W heating), the recovery efficiency of Test 4 is higher than that of Test 5. One has to make a choice between higher recovery efficiency and higher thermal efficiency. If higher recovery efficiency is desired, it is optimal to start at a lower heating rate and eventually increase the heating rate during the later stages. Otherwise, if higher thermal efficiency is desired, it is optimal to start at a higher heating rate and eventually decrease the heating rate during the later stages. 3.6 Summary of test results Figure 16 shows the results of dissociation tests as a function of the heating rate. Black data points show the methane recovery values at the end of the test whereas red data points show the peak value of thermal efficiency in each test. Note that the lines are drawn only to represent the trend clearly.
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Figure 16. Percent methane recovery (black data points) and percent thermal efficiency (red data points) with heating rate (W) In Figure 16, to the left of the graph (at lower heating rates) although we are able to achieve higher peak thermal efficiency values; the percent recovery is low, for both 30 % as well as 50 % SH. As we increase the heating rate to 50 W the peak thermal efficiency value decreases slightly for both 30 % as well as 50 % SH tests. At the same time, we see an increase in the methane recovery value. The slope of the segment ‘AB’ is greater than the slope of the segment ‘DE'. There is a 32% recovery increase for a 3% decrease in thermal efficiency for saturation at 30%. The 50% saturation shows a 25% increase in recovery for nearly the same thermal efficiency decline. This is because of the availability of free water. Line segment ‘AB’ corresponds to 30 % SH which means the amount of free water present is greater in quantity than in the case of line segment ‘DE’. This additional amount of free water in the case of 30 % hydrate saturation helps dissociate hydrates away from the heater due to convection currents. When the heating rate is further increased from 50 to 100 W, segment ‘EF’ maintains the same slope as segment ‘DE’, however, the slope of segment ‘BC’ drops to a lower value than the slope of segment “AB’. In other words, 50 % hydrate saturation test continues to achieve higher values of recovery efficiency at the previously achieved rate; however, 30 % hydrate saturation test fails to maintain the same rate of increase in the recovery efficiency values. This indicates that hydrate saturation in the reactor plays a major role at higher values of heating rates. Although there is enough energy and free water, the amount of hydrates becomes the controlling factor in the case of 30 % hydrate saturation. In the case of 50 % hydrate saturation, there is sufficient energy for dissociation, free water generation as well as sufficient hydrates in the reactor. Once again from Figure 16, at lower hydrate saturation; thermal efficiency decreased with increase in the heating rate. At a relatively higher saturation, thermal efficiency first decreased as the heating rate was increased from 20 to 50 W. However, the increase of heating rate from 50 to 100 W caused an increase in the thermal efficiency. Points C and C’ correspond to the value of recovery efficiency and peak thermal efficiency in the case of 30 % hydrate saturation test. Point 23
C has already achieved a value of greater than 80% in such a case, increasing the heating rate beyond 100 W may not cause a significant increase in the recovery efficiency as the majority of the hydrates are now present further away from the heater; however, it will cause a steep drop in the value of peak efficiency. On the other hand, in the case of 50 % hydrate saturation point F and F’ denote the values of recovery efficiency and peak thermal efficiency. Only about 70 % of the hydrates are recovered at point F, and there is still potential for further recovery. Location of points F and F’ (as opposed to C and C’) suggests that increasing the heating rate beyond 100 W might cause a significant increase in the recovery efficiency and this increase in the methane recovery might offset the anticipated drop in the thermal efficiency. 4. Conclusion The fundamentals of the hydrate formation and dissociation in a large-scale laboratory reactor were studied using a thermal stimulation method. Pure methane hydrate formations of 30 % and 50 % initial saturations were achieved. An attempt has been made to connect the laboratory scale results to a large scale field tests. The results show that at higher values of hydrate saturation; the recovery efficiency is a linear function of the heating rate and increases with the heating rate. At relatively lower values of hydrate saturation; the recovery efficiency increases with the heating rate at first. However, it starts to drop as the certain value of heating rate is reached. Maintaining a constant heating rate throughout the test may not be the most efficient way to extract the methane gas from the reservoir. Because in the initial stages the saturation of hydrates near the heating source will be higher, the lower heating rate proves to be effective as shown in the tests at 30% SH with 50 and 100 W heating. In these two tests, the output of methane gas was almost equal for the first 7 hours. This will have a huge impact on the thermal efficiency on a large scale. In the actual field test, this will resemble either increasing the amount of fuel supplied for combustion or relocating the heating source to a different zone. On the other hand, increasing the heating rate in the later stages after the hydrate saturation depletes, may help increase the recovery efficiency. As seen in the test at 50 % SH and 20 W heating, only a small portion of the initially present methane was recovered due to lower heating rates, increasing the heating rate in the later stages would help increase the recovery efficiency. Also, the highest value of heating rate should be decided based on the concentration of hydrates in the reservoir, increase beyond a certain point may not be effective. An accurate knowledge of the hydrate saturation in the reservoir is required to help increase the efficiency of the process. A 50% SH condition yielded a higher value of recovery efficiency (and hence thermal efficiency) at the heating rate of 100 W, as compared to the test at 30% SH and 100 W heating. The free water in the hydrate reservoir plays a major role in heat transfer and ultimately in the recovery efficiency, as evidenced by tests at 30 and 50 % SH, both at the heating rate of 50 W. In 24
these two tests, 30% SH test there is more free water present as compared to 50% SH. As a result, 30 % SH test gave higher recovery efficiency in the initial stages than 50% SH. Alternatively, a conventional hot water injection method could be modified to combine with downhole combustion. As a result, the majority of the heat required for the dissociation could be provided via downhole combustion whereas, a small amount of hot water injection is possible which will partially provide heat as well as create free water for convection of heat in the farther regions of hydrate from the combustion source. The critical role of free water in gas recovery has also been reported by other researchers 22 5. References (1) Ahuja, A.; Zylyftari, G.; Morris, J.F. Yield stress measurements of cyclopentane hydrate slurry. J. Non-Newtonian Fluid Mech. 2015, 220,116-125. (2) Ahuja, A.; Zylyftari, G.; Morris, J.F. Calorimetric and rheological studies on cyclopentane hydrate-forming water-in-kerosene emulsions. J. Chem. Eng. Data 2014, 60(2), 362-368. (3) Karanjkar, P.U.; Ahuja, A.; Zylyftari, G.; Lee, J.W.; Morris, J.F. Rheology of cyclopentane hydrate slurry in a model oil-continuous emulsion. Rheol. Acta 2016, 55(3), 235-243. (4) Zylyftari, G.; Ahuja, A.; Morris, J.F. Modeling oilfield emulsions: comparison of cyclopentane hydrate and ice. Energy Fuels 2015, 29(10), 6286-6295. (5) Zylyftari, G.; Ahuja, A.; Morris, J.F. Nucleation of cyclopentane hydrate by ice studied by morphology and rheology. Chem. Eng. Sci. 2014, 116, 497-507. (6) Zhou, Y.; Castaldi, M.J.; Yegulalp, T.M. Experimental investigation of methane gas production from methane hydrate. Ind. Eng. Chem. Res. 2009, 48(6), 3142-3149. (7) Tupsakhare, S.S.; Fitzgerald, G.C.; Castaldi, M.J. Thermally Assisted Dissociation of Methane Hydrates and the Impact of CO2 Injection. Ind. Eng. Chem. Res. 2016, 55(39), 10465-10476. (8) Castaldi, M.J.; Zhou, Y.; Yegulalp, T.M. Down-hole combustion method for gas production from methane hydrates. J. Pet. Sci. Eng. 2007, 56(1), 176-185. (9) Lee, J.; Park, S.; Sung, W. An experimental study on the productivity of dissociated gas from gas hydrate by depressurization scheme. Energy Convers. Manage. 2010, 51(12), 2510-2515. (10) Rutqvist, J.; Moridis, G.J.; Grover, T.; Collett, T. Geomechanical response of permafrostassociated hydrate deposits to depressurization-induced gas production. J. Pet. Sci. Eng. 2009, 67(1), 1-12. 25
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(23) Fitzgerald, G.C.; Castaldi, M.J.; Schicks, J.M. Methane hydrate formation and thermal based dissociation behavior in silica glass bead porous media. Ind. Eng. Chem. Res. 2014, 53(16), 6840-6854. (24) He, S.; Liang, D.; Li, D.; Ma, L. Experimental investigation on the dissociation behavior of methane gas hydrate in an unconsolidated sediment by microwave stimulation. Energy Fuels 2010, 25(1), 33-41. (25) Castellani, B.; Rossetti, G.; Tupsakhare, S.; Rossi, F.; Nicolini, A.; Castaldi, M.J. Simulation of CO2 storage and methane gas production from gas hydrates in a large scale laboratory reactor. J. Pet. Sci. Eng. 2016, 147, 515-527. (26) Ota, M.; Abe, Y.; Watanabe, M.; Smith, R.L.; Inomata, H. Methane recovery from methane hydrate using pressurized CO2. Fluid Phase Equilib. 2005, 228, 553-559. (27) Liu, Y.; Strumendo, M.; Arastoopour, H. Simulation of methane production from hydrates by depressurization and thermal stimulation. Ind. Eng. Chem. Res. 2008, 48(5), 2451-2464. (28) Zhao, J.; Xu, K.; Song, Y.; Liu, W.; Lam, W.; Liu, Y.; Xue, K.; Zhu, Y.; Yu, X.; Li, Q. A review on research on replacement of CH4 in natural gas hydrates by use of CO2. Energies, 2012, 5(2), 399-419. (29) Moridis, G.J. January. Numerical studies of gas production from methane hydrates. In SPE Gas Technology Symposium. Society of Petroleum Engineers, 2002 (30) Weisbrod, N.; McGinnis, T.; Rockhold, M.L.; Niemet, M.R.; Selker, J.S. Effective Darcy scale contact angles in porous media imbibing solutions of various surface tensions. Water Resour. Res. 2009, 45(4). 6. Acknowledgements The authors acknowledge the insightful feedback and assistance from Dr. Jeffrey LeBlanc from the City College of New York and Dr. Garrett Fitzgerald from Rocky Mountain Institute.
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