## Abstract

This paper aims to study the dynamics of the unsteady pressure flow of coal gases caused by the temperature conditions and compressibility in the presence of a nano-porous medium using soft computing technique. To immaculately understand the mechanism, a novelty in the partial differential equation is augmented by considering the fractional-order Caputo derivative, which produces theoretically significant and accurate approximation. Subsequently, the constructed model is experimentally simulated by means of artificial neural network (ANN) and a stochastic process based on a firefly algorithm (FFA). ANN has the ability to approximate and transform the differential equation into an error minimization problem, while FFA efficiently minimizes the error function and optimizes the unknown weights of the constructed network. Furthermore, two error measuring tools; mean absolute error and root mean square error, is also formulated to evaluate the performance index of the designed scheme. Accordingly, the designed scheme is systematically elaborated to assess the pressure sorption of coal gases such as nitrogen (N_{2}) and carbon dioxide (CO_{2}). The accuracy of the obtained approximation shows the competitiveness of the considered scheme. Notably, the deliberation provides substantial indications about the dynamical behaviour of coal gases, which can be implemented significantly on various dynamical problems.

This is a preview of subscription content, access via your institution.

## References

Abbasbandy S (2012) Numerical study on gas flow through a micro-nano porous media. Acta Phys Pol Ser A 121(3):581

Agarwal RP, O’Regan D (2002) Infinite interval problems modeling the flow of a gas through a semi-infinite porous medium. Stud Appl Math 108(3):245–257

Agatonovic-Kustrin S, Beresford R (2000) Basic concepts of artificial neural network (ANN) modeling and its application in pharmaceutical research. J Pharm Biomed Anal 22(5):717–727. https://doi.org/10.1016/S0731-7085(99)00272-1

Ahmadi MA, Ahmadi A (2016) Applying a sophisticated approach to predict CO

_{2}solubility in brines: application to CO_{2}sequestration. Int J Low-Carbon Technol 11(3):325–332. https://doi.org/10.1093/ijlct/ctu034Ahmadi MA, Golshadi M (2012) Neural network based swarm concept for prediction asphaltene precipitation due to natural depletion. J Petrol Sci Eng 98–99:40–49. https://doi.org/10.1016/j.petrol.2012.08.011

Ahmadi MA, Ahmadi MR, Hosseini SM, Ebadi M (2014a) Connectionist model predicts the porosity and permeability of petroleum reservoirs by means of petro-physical logs: application of artificial intelligence. J Petrol Sci Eng 123:183–200

Ahmadi MA, Ebadi M, Yazdanpanah A (2014b) Robust intelligent tool for estimating dew point pressure in retrograded condensate gas reservoirs: application of particle swarm optimization. J Petrol Sci Eng 123:7–19

Ara A, Khan NA, Razzaq OA, Hameed T, Raja MAZ (2018) Wavelets optimization method for evaluation of fractional partial differential equations: an application to financial modelling. Adv Differ Equa 2018(1):8. https://doi.org/10.1186/s13662-017-1461-2

Ayesh AI (2016) Metal/metal-oxide nanoclusters for gas sensor applications. J Nanomater 2016:17. https://doi.org/10.1155/2016/2359019

Chan KY, Liu Z (2018) A learning strategy for developing neural networks using repetitive observations. Soft Comput 23(13):4853–4869

Cybenko G (1989) Approximation by superpositions of a sigmoidal function. Math Control Signals Syst 2:303–314

Hassanizadeh SM, Das DB (2005) Upscaling multiphase flow in porous media: from pore to core and beyond. Springer, Berlin

Ho CK, Webb SW (2006) Gas Transport in Porous Media. In: Clifford SWW, Ho K (series ed) Theory and applications of transport in porous media, vol 20. Springer, Dordrecht

Kazem S, Rad JA, Parand K, Shaban M, Saberi H (2012) The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method. Int J Comput Math 89(16):2240–2258. https://doi.org/10.1080/00207160.2012.704995

Khan NA, Hameed T, Razzaq OA, Ayaz M (2019) Intelligent computing for Duffing-harmonic oscillator equation via the bio-evolutionary optimization algorithm. J Low Freq Noise Vib Act Control. https://doi.org/10.1177/1461348418819408

Kidder R (1957) Unsteady flow of gas through a semi-infinite porous medium. J Appl Mech 27:329–332

Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987–1000. https://doi.org/10.1109/72.712178

Li Y, Peng C, Wang Y, Jiao L (2019) Optimization based on nonlinear transformation in decision space. Soft Comput 23(11):3571–3590

Mai-Duy N, Tran-Cong T (2001) Numerical solution of differential equations using multiquadric radial basis function networks. Neural Netw 14(2):185–199. https://doi.org/10.1016/S0893-6080(00)00095-2

Malek A, Beidokhti RS (2006) Numerical solution for high order differential equations using a hybrid neural network—Optimization method. Appl Math Comput 183(1):260–271. https://doi.org/10.1016/j.amc.2006.05.068

Mall S, Chakraverty S (2014) Chebyshev Neural Network based model for solving Lane-Emden type equations. Appl Math Comput 247:100–114. https://doi.org/10.1016/j.amc.2014.08.085

Masood Z, Majeed K, Samar R, Raja MAZ (2017) Design of Mexican Hat Wavelet neural networks for solving Bratu type nonlinear systems. Neurocomputing 221:1–14. https://doi.org/10.1016/j.neucom.2016.08.079

Meade AJ, Fernandez A (1994) The numerical solution of linear ordinary differential equations by feedforward neural networks. Math Comput Model 19(12):1–25. https://doi.org/10.1016/0895-7177(94)90095-7

Monteiro PJ, Rycroft CH, Barenblatt GI (2012) A mathematical model of fluid and gas flow in nanoporous media. Proc Natl Acad Sci 109(50):20309–20313

Muskat M, Wyckoff RD (1946) The flow of homogeneous fluids through porous media. Ann Arbor, Michigan: Edwards

Noor MA, Mohyud-Din ST (2009) Variational iteration method for unsteady flow of gas through a porous medium using He’s polynomials and Pade approximants. Comput Math Appl 58(11–12):2182–2189

Pakdaman M, Ahmadian A, Effati S, Salahshour S, Baleanu D (2017) Solving differential equations of fractional order using an optimization technique based on training artificial neural network. Appl Math Comput 293:81–95. https://doi.org/10.1016/j.amc.2016.07.021

Parand K, Hemami M (2017) Application of Meshfree method based on compactly supported radial basis function for solving unsteady isothermal gas through a micro-nano porous medium. Iran J Sci Technol Trans A Sci 41(3):677–684. https://doi.org/10.1007/s40995-017-0293-y

Rad JA, Parand K (2010) Analytical solution of gas flow through a micro-nano porous media by homotopy perturbation method. World Acad Sci Eng Technol 61:546–550

Raja MAZ (2014) Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP. Connect Sci 26(3):195–214

Raja MAZ, Khan JA, Zameer A, Khan NA, Manzar MA (2017) Numerical treatment of nonlinear singular Flierl–Petviashivili systems using neural networks models. Neural Comput Appl. https://doi.org/10.1007/s00521-017-3193-3

Raja MAZ, Umar M, Sabir Z, Khan JA, Baleanu D (2018a) A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head. Eur Phys J Plus 133(9):364

Raja MAZ, Zameer A, Kiani AK, Shehzad A, Khan MAR (2018b) Nature-inspired computational intelligence integration with Nelder–Mead method to solve nonlinear benchmark models. Neural Comput Appl 29(4):1169–1193

Rezaei A, Parand K, Pirkhedri A (2011) Numerical Study on Gas Flow Through a Micro-Nano Porous Media Based on Special Functions. J Comput Theor Nanosci 8(2):282–288. https://doi.org/10.1166/jctn.2011.1690

Rodrigues CF, Lemos de Sousa MJ (2002) The measurement of coal porosity with different gases. Int J Coal Geol 48(3):245–251. https://doi.org/10.1016/S0166-5162(01)00061-1

Shekofteh Y, Jafari S, Rajagopal K (2018) Cost function based on hidden Markov models for parameter estimation of chaotic systems. Soft Comput 23:4765–4776

Wazwaz A-M (2001) The modified decomposition method applied to unsteady flow of gas through a porous medium. Appl Math Comput 118(2–3):123–132

Weniger P, Kalkreuth W, Busch A, Krooss BM (2010) High-pressure methane and carbon dioxide sorption on coal and shale samples from the Paraná Basin, Brazil. Int J Coal Geol 84(3):190–205. https://doi.org/10.1016/j.coal.2010.08.003

Yang X-S (2010) Firefly algorithm, stochastic test functions and design optimisation. arXiv preprint arXiv:1003.1409

Yang X-S (2014a) Chapter 8—firefly algorithms. In: Yang X-S (ed) Nature-inspired optimization algorithms. Elsevier, Oxford, pp 111–127

Yang X-S (2014b) Nature-inspired optimization algorithms. Luniver Press: Elsevier, London

Zhang Z, Lin B, Li G, Ye Q (2013) Explosion pressure characteristics of coal gas. Combust Sci Technol 185(3):514–531. https://doi.org/10.1080/00102202.2012.729112

Zhong Y-B, Xiao G, Yang X-P (2019) Fuzzy relation lexicographic programming for modelling P2P file sharing system. Soft Comput 23(11):3605–3614

## Author information

### Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interests.

### Ethical approval

This article does not contain any studies with animals performed by any of the authors.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by V. Loia.

## Appendix: Derivation of fractional-order unsteady flow of gases

### Appendix: Derivation of fractional-order unsteady flow of gases

To obtain a similarity solution, taking the fractional-order derivative (\( D_{\tau }^{\beta } = \frac{{\partial^{\beta } }}{{\partial \tau^{\beta } }} \)) of Eq. (6) in the Caputo derivative sense, we get,

The gradient of Eq. (6) is,

The pressure gradient of Eq. (7) with respect to *z* and \( \eta \) will be,

Multiplying Eq. (A.3) by the pressure \( P \) and taking its gradient with respect to \( z \) we attain,

Now, taking the fractional-order derivative of Eq. (6) with respect to \( \tau \) and using the property of the fractional derivative \( \partial^{\beta } {\text{P = }}\varGamma (\beta + 1)\partial {\text{P}} \),

Substitute Eqs. (A.1) and (A.4) in Eq. (A.6)

By replacing the values of Eqs. (A.5) and (A.7) in Eq. (3), the outcomes of the equation suggest that we must take \( c = - \frac{\beta }{2} \), thus the system generates Eqs. (8)–(9).

## Rights and permissions

## About this article

### Cite this article

Khan, N.A., Hameed, T. & Razzaq, O.A. Modelling and simulation of coal gases in a nano-porous medium: a biologically inspired stochastic simulation.
*Soft Comput* **24, **5133–5150 (2020). https://doi.org/10.1007/s00500-019-04267-x

Published:

Issue Date:

### Keywords

- Coal gases
- Unsteady gas equation
- Artificial neural network
- Firefly algorithm