A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach
Abstract
:1. Introduction
2. Mathematical Model
2.1. Model Assumptions
- The porous media is composed by a bundle of capillary bundles and a single capillary with the equivalent radius r is made up of a packing of equivalent spherical grains.
- The interspaces in porous media have fractal characteristics.
- The single phase flow is under isothermal and stress condition, which is fully developed and at steady state.
- The deformation of porous media obeys Hertzian contact theory.
- During the flow, the fluid has constant viscosity and density.
2.2. Conformable Derivative Approach to Swartzendruber Equation
2.3. Non-Linear Flow Model
3. Results and Discussion
4. Conclusions
- The proposed models indicate that average flow velocity in tight porous media is a function of microstructural parameters of the pore space, rock lithology and differential order α as well as hydraulic gradient and threshold hydraulic gradient.
- The parametric study reveals that average flow velocity increases with the rougher pore surfaces and rock elastic modulus, and decreases with increasing effective stress. “Softer” rock lithology may yield lower average flow velocity.
- This non-linear model presented here considers microstructural parameters of pore space and rock lithology; we have shown that its forecasted values are robust, at least compared to experimental data, and thus may be useful for performance predictions of non-linear flow behavior in tight porous media. Results also reveal more information about the details of specific parameters (and therefore mechanisms) that affect non-linear flow behavior in porous media. The new model presented in this work can be used to depict the non-linear flow in tight porous media, and may provide meaningful applications for design and development of tight reservoirs. In addition, as the model takes effective stress into account, it is also useful for performance predictions of the coupled flow deformation behavior (stress sensitivity) in tight porous media.
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Latin symbols | |
A | Cross sectional area of a unit cell, µm2 |
Df0 | Initial pore area fractal dimension at zero stress, dimensionless |
Df | Pore area fractal dimension, dimensionless |
e0 | Initial void ratio of porous media, dimensionless |
E | Rock elastic modulus of porous media, GPa |
f | Probability density function for pore size distribution, dimensionless |
1F1 | Kummer confluent hypergeometric function |
g | Gravitational acceleration, N/kg |
K | Absolute permeability of porous media, µm2 |
i | Hydraulic gradient, dimensionless |
I | Threshold hydraulic gradient, dimensionless |
N | Number of pores of a unit cell, dimensionless |
peff | Effective stress, MPa |
q | Flow rate in the cross section, m3/s |
Q | Total volumetric flow rate in the cross section under stress condition, m3/s |
r0 | Initial equivalent pore radius of capillary at zero stress, µm |
r | Equivalent pore radius of capillary under effective stress, µm |
u | Flow velocity in the cross section, m/s |
uav | Average flow velocity, m/s |
Greek symbols | |
α | Differential order |
β | Power law index, dimensionless |
μ | Fluid viscosity, mPa·s |
ρ | Fluid density, kg/m3 |
φ0 | Initial porosity of porous media, dimensionless |
φ | Porosity under effective stress, dimensionless |
ν | Poisson’s ratio, dimensionless |
Subscript | |
av | Average |
eff | Effective |
max | maximum values |
max0 | Initial maximum values at zero stress |
min | minimum values |
min0 | Initial minimum values at zero stress |
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Lei, G.; Cao, N.; Liu, D.; Wang, H. A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach. Energies 2018, 11, 2986. https://0-doi-org.brum.beds.ac.uk/10.3390/en11112986
Lei G, Cao N, Liu D, Wang H. A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach. Energies. 2018; 11(11):2986. https://0-doi-org.brum.beds.ac.uk/10.3390/en11112986
Chicago/Turabian StyleLei, Gang, Nai Cao, Di Liu, and Huijie Wang. 2018. "A Non-Linear Flow Model for Porous Media Based on Conformable Derivative Approach" Energies 11, no. 11: 2986. https://0-doi-org.brum.beds.ac.uk/10.3390/en11112986