# Non-Darcian Displacement of Oil by a Micellar Solution in Fractal Porous Media

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## Abstract

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## 1. Introduction

## 2. Rheological Equation of State

#### 2.1. Calculation of Mobility

#### 2.1.1. The Buckley–Leverett Equation

#### 2.1.2. Capillary Pressure

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Afolabi, F.; Mahmood, S.M.; Yekeen, N.; Akbari, S.; Sharifigaliuk, H. Polymeric surfactants for enhanced oil recovery: A review of recent progress. J. Pet. Sci. Eng.
**2022**, 208, 109358. [Google Scholar] [CrossRef] - Sheng, J. Status of Surfactant EOR Technology. Petroleum
**2015**, 79, 97–105. [Google Scholar] [CrossRef] [Green Version] - Manrique, E.; Thomas, C.; Ravikiram, R.; Izadi, M.; Lantz, M.; Romero, J.; Alvarado, V. EOS: Current Status and Opportunities, 2010. In Proceedings of the SPE IOR Symposium, Tulsa, OK, USA, 24–28 April 2010. [Google Scholar]
- Raffa, P.; Broekhuis, A.A.; Picchioni, F. Polymeric surfactants for enhanced oil recovery: A review. J. Pet. Sci. Eng.
**2016**, 145, 723–733. [Google Scholar] [CrossRef] [Green Version] - Ghannam, M.T.; Selim, M.Y.E.; Zekri, A.Y.; Esmail, N. Thixotropic Assessment of Some Enhanced Oil Recovery used Polymers. Int. J. Eng. Res. Technol.
**2020**, 9, 1683–1693. [Google Scholar] [CrossRef] - Turcio, M.; Reyes, J.; Camacho, R.; Lira-Galeana, C.; Vargas, R.; Manero, O. Calculation of effective permeability for the BMP model in fractal porous media. J. Pet. Sci. Eng.
**2013**, 103, 51–60. [Google Scholar] [CrossRef] - Othman, A.; AlSulaimani, M.; Aljawad, M.S.; Sangaru, S.S.; Kamal, M.S.; Mahmoud, M. The Synergetic Impact of Anionic, Cationic, and Neutral Polymers on VES Rheology at High-Temperature Environment. Polymers
**2022**, 14, 1145. [Google Scholar] [CrossRef] [PubMed] - Holmberg, K. (Ed.) Novel Surfactants: Preparation, Applications, and Biodegradability; Dekker: New York, NY, USA, 1998. [Google Scholar]
- Lagerman, R.; Clancy, S.; Tanner, D.; Johnston, N.; Callian, B.; Friedli, F. Synthesis and performance of ester quaternary biodegradable softeners. J. Am. Oil Chem. Soc.
**1994**, 71, 97–100. [Google Scholar] [CrossRef] - Wilk, K.A.; Bieniecki, A.; Burczyk, B.; Sokolowski, A. Synthesis and hydrolysis of chemodegradable cationic surfactants containing the 1,3-dioxolane moiety. J. Am. Oil Chem. Soc.
**1994**, 71, 81–85. [Google Scholar] [CrossRef] - Wang, G.W.; Liu, Y.C.; Yuan, X.Y.; Lei, X.G.; Guo, Q.X. Preparation, Properties, and Applications of Vesicle-Forming Cleavable Surfactants with a 1,3-Dioxane Ring. J. Colloid Interface Sci.
**1995**, 173, 49–54. [Google Scholar] [CrossRef] - West, C.A.; Sanchez, A.M.; Hanon-Aragon, K.A.; Salazar, I.C.; Menger, F.M. Preparation and characterization of a simple destructible surfactant. Tetrahedron Lett.
**1996**, 37, 9135–9138. [Google Scholar] [CrossRef] - Manero, O.; Bautista, F.; Soltero, J.; Puig, J. Dynamics of worm-like micelles: The Cox-Merz rule. J. Non-Newton. Fluid Mech.
**2002**, 106, 1–15. [Google Scholar] [CrossRef] - Spayd, K.; Shearer, M. The Buckley–Leverett Equation with Dynamic Capillary Pressure. SIAM J. Appl. Math.
**2011**, 71, 1088–1108. [Google Scholar] [CrossRef] - Olajire, A.A. Review of ASP EOR (alkaline surfactant polymer enhanced oil recovery) technology in the petroleum industry: Prospects and challenges. Energy
**2014**, 77, 963–982. [Google Scholar] [CrossRef] - Mandelbrot, B. The Fractal Geometry of Nature; Freeman: New York, NY, USA, 1982. [Google Scholar]
- Xu, P.; Mujumdar, A.S.; Sasmito, A.P.; Yu, B.M. Multiscale modeling of porous media. In Heat and Mass Transfer in Drying of Porous Media; Taylor and Francis: Oxfordshire, UK, 2019. [Google Scholar]
- Katz, A.J.; Thompson, A.H. Fractal Sandstone Pores: Implications for Conductivity and Pore Formation. Phys. Rev. Lett.
**1985**, 54, 1325–1328. [Google Scholar] [CrossRef] [PubMed] - Wu, J.; Yu, B. A fractal resistance model for flow through porous media. Int. J. Heat Mass Transf.
**2007**, 50, 3925–3932. [Google Scholar] [CrossRef] - Brambila, F. Fractal Analysis; IntechOpen: Rijeka, Croatia, 2017. [Google Scholar] [CrossRef]
- Bautista, F.; de Santos, J.; Puig, J.; Manero, O. Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model. J. Non-Newton. Fluid Mech.
**1999**, 80, 93–113. [Google Scholar] [CrossRef] - Cavatorta, N.; Tonini, R.N. Dimensionless velocity profiles and parameter maps for non-Newtonian fluids. Int. Commun. Heat Mass Transf.
**1987**, 14, 359–369. [Google Scholar] [CrossRef] - Li, Y.; Yu, B.; Chen, J.; Wang, C. Analysis of Permeability for Ellis Fluid Flow in Fractal Porous Media. Chem. Eng. Commun.
**2008**, 195, 1240–1256. [Google Scholar] [CrossRef] - McWhorter, D.; Sunada, D. Exact Integral Solutions for Two-Phase Flow. Water Resour. Res.
**1990**, 26, 399–413. [Google Scholar] [CrossRef] - Andersen, P.Ø; Nesvik, E.K.; Standnes, D.C. Analytical solutions for forced and spontaneous imbibition accounting for viscous coupling. J. Pet. Sci. Eng.
**2020**, 186, 106717. [Google Scholar] [CrossRef]

**Figure 1.**Normalized fluidity as a function of the stress normalized by the plateau stress. When the normalized stress is equal to one, the fluidity is ${\left({\phi}_{\infty}/{\phi}_{0}\right)}^{1/2}=10$, which coincides with ${\phi}_{N}/{\phi}_{0}=10$.

**Figure 2.**Mobility ratio as a function of applied stress. For small stresses, the relative mobility is $M\left(\sigma \to 0\right)=\frac{{\phi}_{N}}{{\phi}_{0}}=10$. For high stresses, the relative mobility is $M\left(\sigma \to \infty \right)=\frac{{K}_{2}}{{K}_{3}}\frac{{\phi}_{N}}{{\phi}_{\infty}}=0.2$. At the plateau stress, $M=\frac{{K}_{2}}{{K}_{3}}\frac{{\phi}_{N}}{\sqrt{{\phi}_{0}{\phi}_{\infty}}}=2$. $\frac{{K}_{2}}{{K}_{3}}$ and the ratio is variable.

**Figure 5.**The derivative of $F\left(S\right)$ as a function of saturation for the three mobility ratios.

**Figure 6.**The function $F\left(S\right)$ and its derivative as functions of the saturation S for three mobility ratios (

**a**) M = 0.3, (

**b**) M = 2.5, and (

**c**) M = 10. The tangent and point of intersection with abscise $\alpha $ are also shown.

**Figure 8.**The derivative $\frac{\partial f}{\partial S}$ as a function of the saturation for three different mobility ratios (

**a**) M = 0.3, (

**b**) M = 2.5, and (

**c**) M = 10, illustrating the equal areas criterion. Here, ${f}^{\prime}\left({S}^{*}\right)$ denotes the ordinate at which the equal area criterion holds.

**Figure 9.**Saturation as a function of propagation speed for three mobility ratios (

**a**) M = 0.3, (

**b**) M = 2.5, and (

**c**) M = 10, illustrating the location of the shock.

**Figure 11.**(

**a**) Rarefaction wave solution for ${S}_{l}=0.2,$${S}_{r}=0.4,\tau =1$, and $\epsilon =0.01$. The corresponding flux functions for three different mobility ratios (

**b**) M = 0.3, (

**c**) M = 2.5, and (

**d**) M = 10.

**Figure 12.**(

**a**) Rarefaction wave which evolves into to shocks as the mobility ratio diminishes for ${S}_{l}=0.6,$${S}_{r}=0.4$, and $\epsilon =0.05$. (

**b**) The rarefaction wave location in the upper region of the flux curve. (

**c**,

**d**) The shocks are represented by straight lines within the saturation interval $\left(0.6,0.4\right)$.

**Figure 13.**(

**a**) The rarefaction wave evolves into a shock trailing the under-compressible shock for $M=2.5$ and alternatively evolving into an under-compressive shock for $M=10$. (

**b**–

**d**) The shock behaviour changes from a rarefaction wave at low mobility ratios into a small shock trailing an under-compressive shock at high mobility ratios.

**Figure 14.**(

**a**) The undercompressive shock for ${S}_{l}=0.8,$${S}_{r}=0.2$, and $\epsilon =0.05$. The corresponding flux functions for three different mobility ratios (

**b**) M = 0.3, (

**c**) M = 2.5, and (

**d**) M = 10.

**Figure 15.**Saturation as a function of velocity for various values of the characteristic times $\tau $ (the same example as that described in Figure 13a for $M=2.5$).

**Table 1.**Mobility ratios with $\alpha $ values and the corresponding average saturations and tangents.

M | $\mathit{\alpha}$ | ${\mathit{S}}_{\mathit{ave}}$ | m |
---|---|---|---|

0.3 | 0.476 | 0.645 | 1.55 |

2.5 | 0.845 | 0.916 | 1.09 |

10 | 0.953 | 0.976 | 1.02 |

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**MDPI and ACS Style**

Ramírez-Casco, R.; Vargas, R.O.; Lira-Galeana, C.; Ramírez-Jaramillo, E.; Turcio, M.; Manero, O.
Non-Darcian Displacement of Oil by a Micellar Solution in Fractal Porous Media. *Fluids* **2022**, *7*, 377.
https://doi.org/10.3390/fluids7120377

**AMA Style**

Ramírez-Casco R, Vargas RO, Lira-Galeana C, Ramírez-Jaramillo E, Turcio M, Manero O.
Non-Darcian Displacement of Oil by a Micellar Solution in Fractal Porous Media. *Fluids*. 2022; 7(12):377.
https://doi.org/10.3390/fluids7120377

**Chicago/Turabian Style**

Ramírez-Casco, Rafael, René O. Vargas, Carlos Lira-Galeana, Edgar Ramírez-Jaramillo, Marcos Turcio, and Octavio Manero.
2022. "Non-Darcian Displacement of Oil by a Micellar Solution in Fractal Porous Media" *Fluids* 7, no. 12: 377.
https://doi.org/10.3390/fluids7120377