Abstract
We have studied simplified, pulsating, one-dimensional, in situ combustion processes. For two cases, with different reaction stoichiometry, oscillations in temperature, flue gas rate, and flue gas composition are demonstrated and the parameter space resulting in oscillatory behavior is identified. To understand the role of different parameters, linear stability of the problem is studied. Because linear stability analysis requires the solution of uniform front propagation, we investigated an asymptotic analytical solution of the problem. We found an original formula for the front propagation velocity. The analytical solution enabled us to define four dimensionless parameters including Zeldovich (Ze) number, Damkohler (Da) number, a specialized air-fuel ratio (B), and a ratio incorporating air and rock heat capacities (Δ1). Using linear stability analysis, we show that the stability of the problem is also governed by these four parameters. Because Δ1 ≈ 1 for typical laboratory conditions, the set of (Ze, Da, B) is used to construct the stability plane; consequently, several important design considerations are suggested. Both larger air injection rate and air enriched in oxygen increase the front propagation speed but push the system toward oscillatory behavior. Conversely, the introduction of catalysts and metal additives, that decrease the activation energy of reactions, increases the front speed and stability. Similarly, increasing the amount of fuel available for the combustion makes the design more stable and drives the combustion front to propagate more quickly.
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Acknowledgements
The authors acknowledge Ecopetrol for their help in funding this project. We thank Dr. Franck Monmont for introducing us to the topic of oscillatory ISC. We also thank the members of the SUPRI-A (Stanford University Petroleum Research Institute) Industrial Affiliates.
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Appendix
Appendix
1.1 7.1 Front Velocity Calculation
We define the non dimensional parameter
By plotting Ψ vs. T (Fig. 14), we observe empirically that at T = TR, Ψ ≈ 1.
So we have the following
From Eq. 8 (with \(\frac {\partial }{\partial \theta }= 0\)), we have
By assuming that \(\frac {\partial T}{\partial \xi } \) is constant between TR and TF and integration we write:
Additionally, from Eq. 13 and knowing that at ξ = 0, C = C0,and T = TR, we have:
Hence, we reach (22).
1.2 7.2 Non-dimensional Form of the Energy Balance Equation
Starting from the energy balance equation
and using Eq. ?? and dimensionless temperature, we have:
Dividing both sides by TF − Ti and introducing Δ1 we have:
in which we have added and subtracted the \(\overset {\frown }{m} (\frac {A\ Vc_{s}}{L} )\frac {\partial \overset {\frown }{T}} {\partial \overset {\frown }{\xi } } \) term. By using Eq. 22, \(\frac {\lambda \ A}{L^{2}}\) in the first term of the LHS, is written as:
By inserting L and 𝜃 in the RHS of the equation, we see that all the terms contain \(\frac {A\ c_{s} \ddot {R}}{C_{0}}\). By dividing the equation by \(\frac {A\ c_{s} \ddot {R}}{C_{0}}\) and rearranging we reach Eq. 29.
1.3 7.3 Convergence and Stability
We show the convergence of our simulation by temporal and spatial refinement of the simulation. We also show the stability of our numerical simulation to both infinitesimal and finite perturbations.
In general, for σ < 1, we found that the fully resolved system is achieved by having Δx = 0.3 mm and Δt = 0.01 min. Figures 15 and 16 show the flue gas rates and oxygen composition for the case of Δx = 0.3 mm and Δx = 0.15 mm. We see that convergence is achieved.
For σ > 1, we found that the convergence is slower both in temporal and spatial domain. Figure 17 shows the flue gas rate for a time step size of 0.01 min having Δx = 0.3, 0.15,and 0.075 mm. Figure 18 shows the flue gas rates for Δx = 0.3 mm having time step sizes of 0.001, 0.0005, and 0.0001 min.
We also have tested the stability of our numerical simulation against infinitesimal and finite perturbations. The results show that that an infinitesimal change in the parameters does not cause deviation from the original results. Also a finite change in the parameters, causes the results to be in finite difference with the original results. Figure 19 shows the stability of the original numerical simulation (with with Ea= 120000 J/mol) toward toward infinitesimal change (Ea= 120000.001 J/mol) and finite change (Ea= 120100 J/mole). As we can see from Fig. 19, the infinitesimal change of activation energy does not change the final result. Also, a finite change in activation energy makes the result to be in finite difference with the original solution.
1.4 7.4 Results
It is interesting to see how the oscillation develops with time for different scenarios. Here, we show the results of flue gas compositions, temperature profile, and flue gases rate for cases 1.2, 1.3, 1.4, 1.5, 2.2, 2.3, 2.4, and 2.5 in Figs. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, and 34.
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Bazargan, M., Kovscek, A.R. Pulsating linear in situ combustion: why do we often observe oscillatory behavior?. Comput Geosci 22, 1115–1134 (2018). https://doi.org/10.1007/s10596-018-9741-9
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DOI: https://doi.org/10.1007/s10596-018-9741-9