Analytical Solution of Unsteady-state Forchheimer Flow Problem in an Infinite Reservoir: The Boltzmann Transform Approach

For several decades, attempts had been made by several authors to develop models suitable for predicting the effects of Forchheimer flow on pressure transient in porous media. However, due to the complexity of the problem, they employed numerical and/or semi-analytical approach, which greatly affected the accuracy and range of applicability of their results. Therefore, in order to increase accuracy and range of applicability, a purely analytical approach to solving this problem is introduced and applied. Therefore, the objective of this paper is to develop a mathematical model suitable for quantifying the effects of turbulence on pressure transient in porous media by employing a purely analytical approach. The partial differential equation (PDE) that governs the unsteady-state flow in porous media under turbulent condition is obtained by combining the Forchheimer equation with the continuity equation and equations of state. The obtained partial differential equation (PDE) is then presented in dimensionless form (by defining appropriate dimensionless variables) in order to enhance more generalization in application and the method of Boltzmann Transform is employed to obtain an exact analytical solution of the dimensionless equation. Finally, the logarithms approximation (for larger times) of the analytical solution is derived. Moreover, after a rigorous mathematical modeling and analysis, a novel mathematical relationship between dimensionless time, dimensionless pressure, and dimensionless radius was obtained for an infinite reservoir dominated by turbulent flow. It was observed that this mathematical relationship bears some similarities with that of unsteady-state flow under laminar conditions. Their logarithm approximations also share some similarities. In addition, the results obtained show the efficiency and accuracy of the Boltzmann Transform approach in solving this kind of complex problem.


Introduction
Darcy's equation (which assumes a linear relationship between pressure gradient and velocity) is one of the fundamental equations of porous media flow. However, the Darcy's equation is only valid for viscous (laminar) flow. This is due to the fact that in turbulent flow, the relationship between pressure gradient and velocity is not linear. The Forchheimer equation emerged as an attempt to model the departure from Darcy's law at high velocity flow. The Forchheimer equation is given as; − dp dr = K v + βρv 2 The above equation governs non-Darcy flow in porous media. Attempts had been made by several authors to establish the validity of the Forchheimer equation and also demonstrate its applicability to porous media flow. Geertsma (1974), developed a relationship between Forchheimer coefficient and rock properties and also validated the equation by experimental data [1]. Hassanizadeh and Gray (1987) provided physical basis for the Forchheimer equation and identified the source of the non-linearity [2]. Giorgi (1997) derived the Forchheimer equation using matched asymptotic expansion [3]. Zimmerman et al. (2004), confirmed the existence of a weak inertia regime and showed that the Forchheimer equation can probably be used over the entire range of Reynolds numbers [4]. Fourar et al. (2004), analyzed the effect of space dimensions on development of flow regimes [5]. Lucas et al. (2007), presented that quadratic deviation in Forchheimer law appear as a result of non-periodicity in porous media and fractures [6]. Several other authors have modelled the Forchheimer flow in porous media by employing the numerical method [7][8][9][10][11][12][13], and semi-analytical approach [14][15][16][17][18][19][20][21][22][23][24][25].
The objective of this paper is to develop a mathematical model suitable for quantifying the effects of turbulence on pressure transient in porous media by employing a purely analytical approach. The partial differential equation (PDE) that governs the unsteady-state flow in porous media under turbulent condition is obtained by combining the Forchheimer equation with the continuity equation and equations of state. The obtained partial differential equation (PDE) is then presented in dimensionless form (by defining appropriate dimensionless variables) in order to enhance more generalization in application and the method of Boltzmann Transform is employed to obtain an exact analytical solution of the dimensionless equation. Finally, the logarithms approximation (for larger times) of the analytical solution is derived. Moreover, after a rigorous mathematical modeling and analysis, a novel mathematical relationship between dimensionless time, dimensionless pressure, and dimensionless radius was obtained for an infinite reservoir dominated by turbulent flow.

Research Methodology
The flow chart below shows the processes involved in proffering solution to the unsteady-state forchheimer flow problem in an infinite reservoir by employing the Boltzmann Transform Approach.

Mathematical Development
In this section, the equation that governs Forchheimer flow in porous media under unsteady-state condition is derived. This equation is also presented in dimensionless form by defining the appropriate dimensionless variables in order to ensure a more generalized application.

Mathematical Modeling of Forchheimer Flow
The mathematical modeling of fluids transport in porous media involves the combination of three categories of equations, namely; the continuity equation, transport equation and the equations of state.
The continuity equation for radial flow in porous media is given as: Finally, the set of equations of state for fluid and formation compressibility as given as follow: Equations 3(a) and 3(b) describe fluids compressibility while Equation 3(c) is the equations for rock formation (porous media) compressibility. In order to obtain the mathematical expression for Forchheimer flow under unsteadystate condition, Equations 1 to 3 must be combined. Expanding the right-hand-side (RHS) of the Equation 1, we have: Substituting Equations 3(b) and 3(c) into the above equation yields: Simplifying further, we have: for v, we have: where F is the parameter that accounts for the departure from Darcy's viscous flow.
Substituting Equation 6 into 5 we have: Expanding the above equations and simplifying further, we obtain the following equation The above equation governs Non-Darcy flow in porous media under unsteady-state condition.

Dimensionless Transformation
Presenting equations in dimensionless form has always been of great advantage in Science and Engineering as it makes results to be applicable on a more global scale. By defining the following dimensionless variables;

Analytical Solution
In the previous section, the following mathematical problem was developed: In this section, the method of Boltzmann transformation is employed to obtain the exact analytical solution of the above problem.

Boltzmann's Transformation
In this section, the mathematical problem above is transformed into an ordinary differential equation that is easier to solve via Boltzmann transform. The Boltzmann transform variable is defined as; It can be seen that Equations 16(a) and 16(b) above are the same, this confirms the validity of the Boltzmann transformation for this problem as the "collapsing" of the initial and outer boundary conditions must occur for the Boltzmann transform to be technically valid. Also, transforming the inner boundary condition, we have:

Solution of the Transformed Problem
In the previous section, the following transformed form of the mathematical problem developed in section (2) was obtained;

Logarithm Approximation
Although Equation 18 obtained in the previous section looks very simple, the Efunction associated with it is computationally rigorous and requires numerical scheme. Therefore, an approximated form of Equation 18 that is valid for large times and does not involve the Efunction is presented in this section.
The Equation 19 is another form of Equation 18 that is valid for large times.

Results and Discussion
After a rigorous mathematical analysis and modeling of Forchheimer flow in an infinite reservoir, the results obtained are summarized in mathematical form as follows; ( , ) = Equation 19 above is a mathematical relationship between dimensionless pressure ( ), dimensionless radius (r D ), dimensionless time (t ), and the forchheimer factor (F). the equation shows that an inverse relationship exist between the dimensionless pressure and the Forchheimer factor. The implication of this is that the dimensionless pressure increases as the Forchheimer factor decreases. The graph below shows the effect of the Forchheimer factor on the dimensionless pressure.

Conclusion
A rigorous mathematical analysis of Forchheimer flow in an infinite reservoir has been carried out and simplified mathematical relationships between the parameters involved have been obtained. The significance of this study cannot be overemphasized as the Forchheimer factor has a significant effect on the instantaneous production rate, reservoir pressure, reservoir production history, and the economic analysis of hydrocarbon reservoirs. From the results above, it is evident that an inverse relationship exist between pressure drop and the Forchheimer factor. This is a result that can be applied to maximize production from a reservoir. Since pressure is the primary energy of a reservoir, minimizing the rate of pressure decline will definitely extend the production life of a reservoir while also deferring the implementation of the secondary and tertiary recovery techniques which are costly. The results above suggest that the rate of pressure decline can be minimized by keeping the Forchheimer factor as high as possible. This can be done by deploying some engineering techniques that can help to modify the magnitudes of the reservoir and fluid properties that control the magnitude of the Forchheimer factor. Finally, it is worthy to note that the mathematical results obtained in this paper are only valid for the case of homogeneous and isotropic porous media. Therefore a more rigorous mathematical adventure must be embarked on in order to obtain similar results for heterogeneous and anisotropic porous media. Furthermore, Equations 19 and 21 must be applied with caution as we must ensure that the condition for their validity is fully satisfied.

Data Availability Statement
The data presented in this study are available in article.

Funding
The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Competing Interest
The author declares that there is no conflict of interests regarding the publication of this manuscript. In addition, the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancies have been completely observed by the author.