# Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Assumptions and Models

- The total length of the capillary tube is the same as that of the Y-shaped branching model;
- The specific surface area of the capillary bundle model is equal to that of the Y-shaped branching model, and the radius of the capillary bundle model is determined by this feature;
- The porosity of the capillary bundle model is equal to that of the Y-shaped branching model. According to the same porosity, the number of capillaries is determined;
- The permeability of the capillary bundle model is equal to that of the Y-shaped branching model. By keeping the specific surface area and porosity of the two models equal, the permeability of the two models is approximately equal.

**j**+ 1)th branch and

**j**th branch is defined as Equation (2).

**j**th and the first branching levels, respectively. $\alpha $ is generally assumed to be smaller than 1 as in natural network systems.

**j**+ 1)th branching level to that at the

**j**th branching level is defined as Equation (4).

**j**th and first branching levels, respectively.

## 3. Derivation and Calculation

#### 3.1. Model Features

**j**th level, which is expressed as ${k}_{j}={m}^{j-1}$.

**S**as the ratio of the $\mathrm{cos}\theta $ between the (

**j**+ 1)th branch and the

**j**th branch,

#### 3.2. Derivation of Flow Process

**k**th branch, the capillary pressure is

**j**th level with the length ${l}_{j}$ is obtained as Equation (18).

## 4. Results and Discussions

#### 4.1. Effect of Proportional Variation in Wetting Strength on the Flow

_{d}. The dimensionless distance l

_{d}is the total length of the Y-shaped branching model divided by the length of the first branch, which is normalized to avoid the value of the model length and to facilitate the comparison under different conditions. The relationship between dimensionless time T

_{d}and dimensionless length l

_{d}is shown in Figure 6. The result confirms that the imbibition process in the capillary bundle model obeys the rule described by the Washburn equation. However, the imbibition in the tree-like network model did not obey the Washburn equation. With the same l

_{d}, a shorter imbibition time was observed in the tree-like network model. The minimum dimensionless time T

_{d}= 0.5154 was calculated precisely under the conditions of s = 1, α = 0.5143, and β = 0.6557. Hence, the imbibition process can be accelerated in the tree-like network models.

#### 4.2. Effect of Random Variation in Wetting Strength on Flow

## 5. Conclusions

_{d}and the time of fluid flow to the end of the model with different wettabilities were studied, and the sensitivity of dimensionless time to wettability changes was investigated. A three-stage branching model was selected to investigate the dynamics of capillary flow with random wettability changes. This paper shows the law of spontaneous imbibition of Newtonian fluids in Y-shape branching models with different wettabilities, which is of significance for the study of fluid flow in the pore space of tight sandstone reservoirs.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

_{c,2}is equal to:

## Appendix B

_{d}from Equation (A12).

_{d}and α. The results were obtained by solving Equation (23) (our model) and Equation (A12) when β = 1.6 and n = 4. The results of our model match those observed in the earlier study, in which T

_{d}decreases in the early stages and increases in the later stages as α increases. All the T

_{d}values calculated in our model are larger than those predicted in Shou et al.’s model. This is mainly because the specific surface area and permeability were considered in our model, which added more resistance to the fluid flow; thus, a longer imbibition time was predicted in our model. In addition, according the results reported in Shou et al.’s research, the shortest T

_{d}was obtained at an α of around 0.5. However, as discussed in the previous section, the shortest T

_{d}was predicted at an α of around 0.6 in our model. The difference made the red curves shown in Figure A1 move to the right compared to the curve predicted by Shou et al. The movement of the red curve made the difference in the predicted results on the left of Figure A1 larger than that on right.

_{d}and l

_{d}. Compared to Shou et al.’s model, the dimensionless time expected in our model was greater when the imbibition reached the end of the capillary tube, because the specific surface and permeability factors were considered in our model. Due to the shorter first stage of our model, a shorter imbibition time at the first stage of our model was predicted; hence, the imbibition process would enter the second stage earlier in our model. In addition, the radius ratio between different stages of Shou et al.’s model was larger than ours, causing the radius variation between different stages of their model to be more significant than ours, making the imbibition in their model faster and the dimensionless time smaller than ours in the late imbibition stages. The time difference between early and late imbibition stages resulted in the intersection of the two curves.

## References

- Berkowitz, B.; Ewing, R.P. Percolation Theory and Network Modeling Applications in Soil Physics. Surv. Geophys.
**1998**, 19, 23–72. [Google Scholar] [CrossRef] - Zhang, L.; Xu, C.; Guo, Y.; Zhu, G.; Cai, S.; Wang, X.; Jing, W.; Sun, H.; Yang, Y.; Yao, J. The Effect of Surface Roughness on Immiscible Displacement Using Pore Scale Simulation. Transp. Porous Media
**2021**, 1–13. [Google Scholar] [CrossRef] - Yang, Y.F.; Liu, J.; Yao, J.; Kou, J.L.; Li, Z.; Wu, T.H.; Zhang, K.; Zhang, L.; Sun, H. Adsorption behaviors of shale oil in kerogen slit by molecular simulation. Chem. Eng. J.
**2020**, 387, 124054. [Google Scholar] [CrossRef] - Yang, Y.F.; Yang, H.Y.; Tao, L.; Yao, J. Microscopic Determination of Remaining Oil Distribution in Sandstones With Different Permeability Scales Using Computed Tomography Scanning. J. Energy Resour. Technol.-Trans. ASME
**2019**, 141, 092903. [Google Scholar] [CrossRef] - Yang, Y.F.; Yao, J.; Wang, C.C.; Gao, Y.; Zhang, Q.; An, S.Y.; Song, W.H. New pore space characterization method of shale matrix formation by considering organic and inorganic pores. J. Nat. Gas Sci. Eng.
**2015**, 27, 496–503. [Google Scholar] [CrossRef] - Balberg, I.; Berkowitz, B.; Drachsler, G.E. Application of a percolation model to flow in fractured hard rocks. J. Geophys. Res. Solid Earth
**1991**, 96, 10015–10021. [Google Scholar] [CrossRef] - Liu, J.; Regenauer-Lieb, K. Application of percolation theory to microtomography of rocks. Earth-Sci. Rev.
**2021**, 214, 103519. [Google Scholar] [CrossRef] - Cai, J.; Jin, T.; Kou, J.; Zou, S.; Xiao, J.; Meng, Q. Lucas-Washburn Equation-Based Modeling of Capillary-Driven Flow in Porous Systems. Langmuir
**2021**, 37, 1623–1636. [Google Scholar] [CrossRef] [PubMed] - Meng, Q.; Cai, J. Recent advances in spontaneous imbibition with different boundary conditions. Capillarity
**2018**, 1, 19–26. [Google Scholar] [CrossRef][Green Version] - Zhang, L.; Jing, W.L.; Yang, Y.F.; Yang, H.N.; Guo, Y.H.; Sun, H.; Zhao, J.L.; Yao, J. The Investigation of Permeability Calculation Using Digital Core Simulation Technology. Energies
**2019**, 12, 3273. [Google Scholar] [CrossRef][Green Version] - Parteli, E.J.R.; da Silva, L.R.; Andrade, J.S., Jr. Self-organized percolation in multi-layered structures. J. Stat. Mech. Theory Exp.
**2010**, 2010, P03026. [Google Scholar] [CrossRef][Green Version] - Kuuskraa, V.; Stevens, S.H.; Moodhe, K.D. Technically Recoverable Shale Oil and Shale Gas Resources: An Assessment of 137 Shale Formations in 41 Countries outside the United States; Adiministation, U.S. Energy Information: Washington, DC, USA, 2013.
- Li, N.; Ran, Q.; Li, J.; Yuan, J.; Wang, C.; Wu, Y.-S. A Multiple-Continuum Model for Simulation of Gas Production from Shale Gas Reservoirs. In Proceedings of the All Days, Abu Dhabi, United Arab Emirates, 16 September 2013; p. 13. [Google Scholar]
- Yang, Y.F.; Wang, K.; Zhang, L.; Sun, H.; Zhang, K.; Ma, J.S. Pore-scale simulation of shale oil flow based on pore network model. Fuel
**2019**, 251, 683–692. [Google Scholar] [CrossRef] - Cao, Y.; Tang, M.; Zhang, Q.; Tang, J.; Lu, S. Dynamic capillary pressure analysis of tight sandstone based on digital rock model. Capillarity
**2020**, 3, 28–35. [Google Scholar] [CrossRef] - Gao, L.; Yang, Z.; Shi, Y. Experimental study on spontaneous imbibition characteristics of tight rocks. Adv. Geo-Energy Res.
**2018**, 2, 292–304. [Google Scholar] [CrossRef] - Cai, J.; Yu, B. Advances in studies of spontaneous imbibition in porous media. Adv. Mech.
**2012**, 42, 735–754. [Google Scholar] [CrossRef] - Meng, Q.; Liu, H.; Wang, J. A critical review on fundamental mechanisms of spontaneous imbibition and the impact of boundary condition, fluid viscosity and wettability. Adv. Geo-Energy Res.
**2017**, 1, 1–17. [Google Scholar] [CrossRef] - Lucas, R. Rate of capillary ascension of liquids. Kolloid Z.
**1918**, 23, 15–22. [Google Scholar] [CrossRef] - Washburn, E.W. The Dynamics of Capillary Flow. Phys. Rev.
**1921**, 17, 273–283. [Google Scholar] [CrossRef] - Terzaghi, K. Theoretical Soil Mechanics; Wiley: New York, NY, USA, 1943. [Google Scholar]
- Handy, L.L. Determination of Effective Capillary Pressures for Porous Media from Imbibition Data. Trans. AIME
**1960**, 219, 75–80. [Google Scholar] [CrossRef] - Cuiec, L.E.; Bourbiaux, B.; Kalaydjian, F. Oil recovery by imbibition in low-permeability chalk. SPE Form. Eval.
**1994**, 9, 200–208. [Google Scholar] [CrossRef] - Kazemi, H.; Gilman, J.R.; Eisharkawy, A.M. Analytical and numerical solution of oil recovery from fractured reservoirs with empirical transfer functions. SPE Reserv. Eng.
**1992**, 7, 219–227. [Google Scholar] [CrossRef] - Li, K.W.; Horne, R.N. An analytical scaling method for spontaneous imbibition in gas/water/rock systems. SPE J.
**2004**, 9, 322–329. [Google Scholar] [CrossRef] - Li, K.W.; Horne, R.N. Generalized scaling approach for spontaneous imbibition: An analytical model. SPE Reserv. Eval. Eng.
**2006**, 9, 251–258. [Google Scholar] [CrossRef] - Mattax, C.C.; Kyte, J.R. Imbibition Oil Recovery from Fractured, Water-Drive Reservoir. Soc. Pet. Eng. J.
**1962**, 2, 177–184. [Google Scholar] [CrossRef] - Shouxiang, M.; Morrow, N.R.; Zhang, X. Generalized scaling of spontaneous imbibition data for strongly water-wet systems. J. Pet. Sci. Eng.
**1997**, 18, 165–178. [Google Scholar] [CrossRef] - Benavente, D.; Lock, P.; Del Cura, M.A.G.; Ordonez, S. Predicting the capillary imbibition of porous rocks from microstructure. Transp. Porous Media
**2002**, 49, 59–76. [Google Scholar] [CrossRef] - Leventis, A.; Verganelakis, D.A.; Halse, M.R.; Webber, J.B.; Strange, J.H. Capillary Imbibition and Pore Characterisation in Cement Pastes. Transp. Porous Media
**2000**, 39, 143–157. [Google Scholar] [CrossRef] - Li, Y.; Yu, D.; Niu, B. Prediction of spontaneous imbibition in fractal porous media based on modified porosity correlation. Capillarity
**2021**, 4, 13–22. [Google Scholar] [CrossRef] - Mandelbrot, B.B. The Fractal Geometry of Nature; Times Books: New York, NY, USA, 1982. [Google Scholar]
- Wheatcraft, S.W.; Tyler, S.W. An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour. Res.
**1988**, 24, 566–578. [Google Scholar] [CrossRef] - Cai, J.C.; Yu, B.M.; Zou, M.Q.; Luo, L. Fractal Characterization of Spontaneous Co-current Imbibition in Porous Media. Energy Fuels
**2010**, 24, 1860–1867. [Google Scholar] [CrossRef] - Wang, W.D.; Su, Y.L.; Zhang, X.; Sheng, G.L.; Ren, L. Analysis of the Complex Fracture Flow in Multiple Fractured Horizontal Wells with the Fractal Tree-Like Network Models. Fractals-Complex Geom. Patterns Scaling Nat. Soc.
**2015**, 23, 1550014. [Google Scholar] [CrossRef] - Li, C.; Shen, Y.; Ge, H.; Su, S.; Yang, Z. Analysis of Spontaneous Imbibition in Fractal Tree-Like Network System. Fractals
**2016**, 24, 1650035. [Google Scholar] [CrossRef][Green Version] - Shou, D.; Ye, L.; Fan, J. Treelike networks accelerating capillary flow. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2014**, 89, 053007. [Google Scholar] [CrossRef] [PubMed] - Lin, D.; Wang, J.; Yuan, B.; Shen, Y. Review on gas flow and recovery in unconventional porous rocks. Adv. Geo-Energy Res.
**2017**, 1, 39–53. [Google Scholar] [CrossRef] - Buckley, J.S.; Liu, Y. Some mechanisms of crude oil/brine/solid interactions. J. Pet. Sci. Eng.
**1998**, 20, 155–160. [Google Scholar] [CrossRef] - Zhu, G.P.; Kou, J.S.; Yao, B.W.; Wu, Y.S.; Yao, J.; Sun, S.Y. Thermodynamically consistent modelling of two-phase flows with moving contact line and soluble surfactants. J. Fluid Mech.
**2019**, 879, 327–359. [Google Scholar] [CrossRef][Green Version] - Zhu, G.P.; Kou, J.S.; Yao, J.; Li, A.F.; Sun, S.Y. A phase-field moving contact line model with soluble surfactants. J. Comput. Phys.
**2020**, 405, 109170. [Google Scholar] [CrossRef] - Yang, Y.; Cai, S.; Yao, J.; Zhong, J.; Zhang, K.; Song, W.; Zhang, L.; Sun, H.; Lisitsa, V. Pore-scale simulation of remaining oil distribution in 3D porous media affected by wettability and capillarity based on volume of fluid method. Int. J. Multiph. Flow
**2021**, 143, 103746. [Google Scholar] [CrossRef]

**Figure 3.**Dimensionless time curves of capillary flow at n = 2. (

**a**) The effect of radius ratio α at different s values, (

**b**) the effect of length ratio β at different s values, and (

**c**) the effect of radius s values at different radius ratios α and length ratios β.

**Figure 4.**Dimensionless time curves of capillary flow at n = 3. (

**a**) The curve of change with radius ratio $\alpha $ at different s values, (

**b**) curve of change with length ratio $\beta $ at different s values, (

**c**) curve of change with radius s values at different radius ratios $\alpha $ and length ratios $\beta $.

**Figure 5.**Dimensionless time curves of capillary flow at n = 4. (

**a**) The curve of change with radius ratio $\alpha $ at different s values, (

**b**) curve of change with length ratio $\beta $ at different s values, (

**c**) curve of change with radius s values at different radius ratios $\alpha $ and length ratios $\beta $.

**Figure 6.**Comparison of T

_{d}from capillary bundle model and tree-like network model with changing l

_{d.}

**Figure 7.**Dimensionless time varies with the distance of liquid flow in the first group of assumptions.

**Figure 8.**Dimensionless time varies with the distance of fluid flow in the second group of realizations.

Case No. | cosθ_{1} | cosθ_{2} | cosθ_{3} | |
---|---|---|---|---|

First Group | #1 | 0.8 | 0.2 | 0.2 |

#2 | 0.2 | 0.8 | 0.2 | |

#3 | 0.2 | 0.2 | 0.8 | |

Second Group | #4 | 0.2 | 0.8 | 0.8 |

#5 | 0.8 | 0.2 | 0.8 | |

#6 | 0.8 | 0.8 | 0.2 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cai, S.; Zhang, L.; Kang, L.; Yang, Y.; Jing, W.; Zhang, L.; Xu, C.; Sun, H.; Sajjadi, M. Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities. *Water* **2021**, *13*, 2370.
https://doi.org/10.3390/w13172370

**AMA Style**

Cai S, Zhang L, Kang L, Yang Y, Jing W, Zhang L, Xu C, Sun H, Sajjadi M. Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities. *Water*. 2021; 13(17):2370.
https://doi.org/10.3390/w13172370

**Chicago/Turabian Style**

Cai, Shaobin, Li Zhang, Lixin Kang, Yongfei Yang, Wenlong Jing, Lei Zhang, Chao Xu, Hai Sun, and Mozhdeh Sajjadi. 2021. "Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities" *Water* 13, no. 17: 2370.
https://doi.org/10.3390/w13172370