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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 Power-law fluids parameters on pressure transient responses are discussed.

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. [

Yun et al. [

Figure

Porous media embedded with a tree-shaped fractal network.

The physic model assumptions are as follows.

The branch tube in tree-shaped fractal network is assumed to be smooth and tube wall thickness is ignored.

The porous media is divided into

Each of the tree-shaped fractal network section’s properties is different, but each of the porous media section’s properties is identical.

Porous media permeability is much lower than tree-shaped fractal network permeability, and fluids only flow to the center point,

Rock and liquid are considered slightly compressible, with each having a constant and small compressibility.

Isothermal and single direction flow is considered.

Capillary pressure and gravity effects are neglected.

Non-Newtonian fluids obey the power-law principle and are considered as pseudoplastic fluids

The internal boundary is considered to have constant flow rate, while the external boundary is considered to be closed or at constant pressure.

At time

The flow velocity through a single tube for power-law fluids is given by [

Ikoku and Ramey Jr [

Substituting (

When

To characterize the branching structures, let the length and diameter of a typical branch at some intermediate level

For the

The interface radius is defined as the distance from the internal boundary to each section’s boundary [

The external flow radius is defined as the distance from the internal boundary to the external boundary:

The single tube permeability in

Considering

Substituting (

The total system volume in the

The volume in the

The tree-shaped fractal network porosity in the

Under the condition of double branches

Substituting (

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.

Governing differential equations for tree-shaped fractal network [

Governing differential equations for porous media [

Initial condition:

Interface connecting conditions of each section, pressure continuity [

Interface connecting conditions of each section, rate continuity [

Internal boundary condition [

External boundary condition (infinite):

External boundary condition (constant pressure):

External boundary condition (closed):

The dimensionless mathematical model for power-law fluids in porous media embedded with a tree-shaped fractal network is shown in the Appendix

Governing differential equations for tree-shaped fractal network:

Governing differential equations for porous media:

Initial condition:

Interface connecting conditions of each zone, pressure continuity:

Interface connecting conditions of each zone, rate continuity:

Internal boundary condition:

External boundary condition (infinite):

External boundary condition (constant pressure):

External boundary condition (closed):

Substituting (

In (

The dimensionless interface radius expression can be derived by substituting (

The expression of the function

The fluid capacitance coefficient,

The general solution of

Then, the derivative of

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 (

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 V-shaped, 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.

Pressure type curves for power-law fluids in porous media embedded with a tree-shaped fractal network (

Figure

Effect of power index (

Figure

Effect of length ratio (

Figure

Effect of diameter ratio (

Figure

Effect of branch angle (

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.

To simplify the mathematical model and its solution, dimensionless parameters are defined as follows [

The dimensionless pressure in the

The dimensionless pressure in the

The dimensionless interface radius:

The dimensionless time:

The fluid capacitance coefficient in the

The interporosity flow coefficient in the

All kinds of dimensionless definitions are shown in the Appendix

Governing differential equations for tree-shaped fractal network:

Governing differential equations for porous media:

Initial condition:

Interface connecting conditions of each section, pressure continuity:

Interface connecting conditions of each section, rate continuity:

Internal boundary condition:

External boundary condition (infinite):

External boundary condition (constant pressure):

External boundary condition (closed):

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful for financial support from the National Science Fund for Distinguished Young Scholars of China (Grant no. 51125019) and the Science and Technology Innovation Fund of Southwest Petroleum University (Grant no. GIFSB0701).