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Fractional flow equations. A model for pressure deficit in an oil well

B. F. Martínez-Salgado, F. Alcántara-López, A. Torres-Hernandez, F. Brambila-Paz, C. Fuentes, J. López Estrada

Abstract

This article presents a novel system of flow equations that models the pressure deficit of a reservoir considered as a triple continuous medium formed by the rock matrix, vugular medium and fracture. In non-conventional reservoirs, the velocity of the fluid particles is altered due to physical and chemical phenomena caused by the interaction of the fluid with the medium, this behavior is defined as anomalous. A more exact model can be obtained with the inclusion of the memory formalism concept that can be expressed through the use of fractional derivatives. Using Laplace transform of the Caputo fractional derivative and Bessel functions, a semi-analytical solution is reached in the Laplace space.

Fractional flow equations. A model for pressure deficit in an oil well

Abstract

This article presents a novel system of flow equations that models the pressure deficit of a reservoir considered as a triple continuous medium formed by the rock matrix, vugular medium and fracture. In non-conventional reservoirs, the velocity of the fluid particles is altered due to physical and chemical phenomena caused by the interaction of the fluid with the medium, this behavior is defined as anomalous. A more exact model can be obtained with the inclusion of the memory formalism concept that can be expressed through the use of fractional derivatives. Using Laplace transform of the Caputo fractional derivative and Bessel functions, a semi-analytical solution is reached in the Laplace space.

Paper Structure

This paper contains 12 sections, 70 equations, 1 figure.

Table of Contents

  1. Previous Works
  2. Methods
  3. Fractional Calculus
  4. Bessel Functions
  5. Flow equation (fractional time derivative)
  6. Laplace Transform
  7. Bessel Functions
  8. Border conditions
  9. Flow equation with triple porosity and triple permeability with fractional time derivative
  10. Adimensionalization of the flow equations
  11. The system with fractional derivative
  12. Conclusions

Figures (1)

  • Figure 1: Behavior of pressure $p_w$ and pressure derivative $p^{(1)}_w$ for different values of $\beta_m, \ \beta_f$ and $\beta_v$. All solutions were obtained with the particular values $\kappa_f=0.75,\ \kappa_v=\omega_f=0.02,\ \omega_v=0.8,\ \lambda_{mf}=10^{-3},\ \lambda_{vf}=10^{-8}$ and $\lambda_{fv}=10^{-5}$. The solution in green in all the graphs corresponds to the classic case.