A Pressure Transient Model for Power-Law Fluids in Porous Media Embedded with a Tree-Shaped Fractal Network

This work studies the pressure transient of power-law fluids in porous media embedded with a tree-shaped fractal network. A pressure transient model was created based on the fractal properties of tree-shaped capillaries, generalized Darcy’s law and constitutive equation for power-law fluids. The dimensionless pressure model was developed using the Laplace transform and Stehfest numerical inversion method. According to the model’s solution, the bi-logarithmic type curves of power-law fluids in porous media embedded with a tree-shaped fractal network are illustrated. The influences of different fractal factors and Powerlaw fluids parameters on pressure transient responses are discussed.


Introduction
Power-law fluids flow in porous media has always been a subject of great interest owing to its fundamental and pragmatic significance.Consequently, over the years, a voluminous amount of research has been conducted to gain insight into phenomena that is related to fluids flow.Traditionally, numerical methods have played a significant role in the analysis of power-law fluids flow in porous media.Lopez et al. [1] determined the flow of power-law fluids in porous media using network models.Through the use of numerical methods, a network model is capable of predicting each of the parameters associated with a pore, such as pressure drop, velocity, and superficial viscosity.Network model predictions are superior to those that are based on simplified capillary bundle models.Kumari and Nath [2] used numerical methods to study non-Darcy mixed convection of power-law fluids in porous media.Orgéas et al. [3] created a flow model for power-law fluids in anisotropic porous media at the low-pore Reynolds number.Pascal [4] researched the transient flow of power-law fluids in a porous incline using the Kármán momentum integral method.Non-Newtonian constitutive equations, used as one of the numerical methods, have been incorporated into the conventional lattice Boltzmann method (LBM) to simulate flow of power-law fluids [5][6][7][8][9].However, the LBM demands tedious calculations and necessitates accuracy of model.These constraints limit the LBM's ability to describe flow of power-law fluids in porous media.
Yun et al. [10] have proposed the starting pressure gradient model for the flow of Bingham fluids in fractal porous media, based on the fractal nature of pore size distribution in porous media.They also analyzed the effect of capillary pressure on the starting pressure gradient.Results showed that the effect of capillary pressure on the starting pressure gradient at a low porosity is significant and cannot be neglected.However, at high porosity, the effect of capillary pressure on the starting pressure gradient is found to be negligible.Li and Yu [11]  Yun et al. [10] and Li and Yu [11] was further developed by Wang and Yu [12].They derived the starting pressure gradient model for Bingham fluids flow in tree-shaped fractal porous media by embedding the tree-shaped fractal network into porous media.Based on the tree-shaped fractal model, Wang et al. [13] proposed a starting pressure gradient model for Bingham fluids flow in porous media that is embedded with randomly distributed tree-shaped fractal networks.

Physical Modeling
Figure 1 shows a coordinate system for radial flow toward the center point, , from the external boundary.The center point, , has an internal boundary,  in .ℎ is the system thickness.The coordinate system consists of two parts: the tree-shaped fractal network and the porous media.During a tree-shaped fractal network generation, the branches on each level must end up on the same circle with all circles having the same center of origin, . tubes, which start at , make up the tree-shaped fractal network.The tube's initial length and diameter are  0 and  0 , respectively.The double branches ( = 2), whose angles are  ( < /2) and total network branch levels are , are applied in this network.Furthermore, two scale factors are used in this fractal network, length ratio, , and diameter ratio, .
The physic model assumptions are as follows.
(1) The branch tube in tree-shaped fractal network is assumed to be smooth and tube wall thickness is ignored.
(2) The porous media is divided into  annular sections by a tree-shaped fractal network.(3) Each of the tree-shaped fractal network section's properties is different, but each of the porous media section's properties is identical.(4) Porous media permeability is much lower than treeshaped fractal network permeability, and fluids only flow to the center point, , through the tree-shaped fractal network.
(5) Rock and liquid are considered slightly compressible, with each having a constant and small compressibility.(6) Isothermal and single direction flow is considered.(7) Capillary pressure and gravity effects are neglected.(8) Non-Newtonian fluids obey the power-law principle and are considered as pseudoplastic fluids (0 <  < 1).
(9) The internal boundary is considered to have constant flow rate, while the external boundary is considered to be closed or at constant pressure.(10) At time  = 0, pressure is uniformly distributed in all system and is equal to initial pressure (  ).

Mathematical Modeling
3.1.Flow Parameters in the Tree-Shaped Fractal Network.The flow velocity through a single tube for power-law fluids is given by [14] where  is shear stress,  is the consistency index, γ is shear rate, and  is the power index.
Ikoku and Ramey Jr [15] defined the flow velocity for power-law fluids as where  is permeability and  eff is effective viscosity.Substituting (1) into (2), permeability for power-law fluids in a single tube can be expressed by effective viscosity: When  = 1 and  =  eff =  in (3), the permeability expression for power-law fluids reduces to that for Newtonian fluids which agrees with the research result of Xu et al. [16].
To characterize the branching structures, let the length and diameter of a typical branch at some intermediate level  ( = 0, 1, . . ., ) be   and   , respectively.Two scale factors length ratio, , and diameter ratio, , are defined as For the th level tube, length and diameter are given by The interface radius is defined as the distance from the internal boundary to each section's boundary [16]: The external flow radius is defined as the distance from the internal boundary to the external boundary: The single tube permeability in th section is expressed by Considering th section in tree-shaped fractal network tortuosity   , the single tube permeability in th section can be expressed as where Substituting ( 7) into (11), the permeability in th section of tree-shaped fractal network is The total system volume in the th section of tree-shaped fractal network can be calculated by The volume in the th section of tree-shaped fractal network can be calculated by The tree-shaped fractal network porosity in the th section of tree-shaped fractal network can be obtained by dividing (15) by (14): Under the condition of double branches ( = 2), the permeability,   , and porosity,   , in the th section of treeshaped fractal network are equal; that is, Substituting ( 13) and ( 16) into (17), we can obtain  = 2,  = 1, and  = 0.707.

Mathematical Model.
According to the physical model, the mathematical pressure transient model for power-law fluids in porous media embedded with a tree-shaped fractal network can be described as follows.
The dimensionless interface radius expression can be derived by substituting ( 8) into (A.3): The expression of the function   () is as follows: where The fluid capacitance coefficient,   , can be obtained by substituting ( 16) into (A.5), and the interporosity flow coefficient,   , can be obtained by substituting ( 13) into (A.6).

Flow Behavior Characteristics
4.1.Flow Regime Recognition.  ,   and   ( = 0, 1, . . ., ) can be obtained by solving the general solution equations ( 40) and (42) and definite condition equations (32)-(35).We can obtain dimensionless internal boundary pressure in real space,   in , by the Stehfest numerical inversion method.The bilogarithmic type curves for powerlaw fluids in porous media embedded with a tree-shaped fractal network can be illustrated.In a condition of closed external boundary, the transient flow process, which has four flow regimes, can be clearly shown (Figure 2).The full and dashed lines represent pressure and pressure derivative curves, respectively.Regime 1 is interporosity flow in tree-shaped fractal network regime.Power-law fluids start to flow in tree-shaped fractal network.Dimensionless pressure increases rapidly before slowing slightly, because it is hard to flow for power-law fluids (0 <  < 1) at the beginning.
Regime 2 is radial flow in tree-shaped fractal network regime.All fluids have flowed in tree-shaped fractal network, and pressure waves spread all over the tree-shaped fractal network.The pressure derivative curve converges to a horizontal line, which depicts the response of the pressure dynamic in tree-shaped fractal network.
Regime 3 is the interporosity flow regime of porous media to tree-shaped fractal network.Power-law fluids in porous media start to flow into tree-shaped fractal network.The pressure derivative curve is Vshaped, which depicts the response of interporosity flow between the tree-shaped fractal network and porous media.It is influenced by spread of the pressure wave through the porous media.
Regime 4 is the total system radial flow regime in porous media embedded with a tree-shaped fractal network.The pressure derivative curve converges to a horizontal line, which depicts the response of the pressure dynamic balance state in the whole system.

Parameter Influence.
Figure 3 shows the type curve characteristics affected by the power index of power-law fluids, .As  decreases, it is harder to flow for power-law fluids (0 <  < 1) in tree-shaped fractal network and porous media.
A larger  leads to a lower location of the dimensionless Figure 4 exhibits the type curve characteristics affected by the length ratio, .The porosity of tree-shaped fractal network,   , decreases with an increase in .A smaller   leads to a lower supplying capacity in tree-shaped fractal network, which would result in a lower location of the dimensionless pressure curve in regimes 2, 3, and 4 and an earlier transition to a longer lasting regime 3. When  increases, regime 3 occurs earlier, which manifests in a deeper and wider V-shaped type curve.
Figure 5 exhibits the type curve characteristics affected by diameter ratio, .The permeability of tree-shaped fractal network,   , and the porosity of tree-shaped fractal network,   , increase with an increase in .A larger   leads to larger flow capacity in tree-shaped fractal network and a later transition to regime 2. A larger   leads to greater supplying capacity in tree-shaped fractal network with a later and shorter regime 3. A larger  leads to a lower location of the dimensionless pressure curve and regime 3 occurs later, resulting in a shallower and narrower V-shaped type curve.When  equals the critical value of 0.707 (see in (17)), the horizontal line representing regime 4 equals 0.5.When  is smaller than 0.707,   increases with an increase in radius, , and the horizontal line is higher than 0.5.When  is greater than 0.707,   decreases with radius, , and the horizontal line is higher than 0.5.
Figure 6 shows the type curve characteristics affected by branch angle, .Permeability of tree-shaped fractal network,   , decreases with an increase in .A smaller   leads to lower flow capacity in tree-shaped fractal network, with an earlier transition to regime 3. Porosity of tree-shaped fractal  network,   , increases with an increase in .A larger   leads to greater supplying capacity in the tree-shaped fractal network, and regime 3 occurs later with a shorter duration.

Conclusions
A pressure transient model for power-law fluids in porous media embedded with a tree-shaped fractal network has been developed and is expressed as a function of the power index of power-law fluids, tree-shaped branches number, diameter ratio, length ratio, and other parameters of porous media.This model is established and solved, type curves are illustrated, and dual fractal flow behavior characteristics are analyzed.Four flow regimes for pressure type curves can be established.Type curves are dominated by parameters of tree-shaped fractal network and porous media.These various parameters affect type curves differently.Power-law fluids flow in a porous media embedded with a tree-shaped fractal network is an interesting and challenging topic, and this work is in processing.
Solution to Mathematical Model.The dimensionless mathematical model for power-law fluids in porous media embedded with a tree-shaped fractal network is shown in the Appendix B. Introduce the Laplace transform based on   .The dimensionless mathematical model in Laplace space is as follows.