Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs
Abstract
:1. Introduction
2. Problem Statement
3. Analytical Solution of the Problem
4. Results
5. Discussion
6. Conclusions
- The new fractal analytical solution for a constant rate describes the pressure-transient behavior for partially penetrating wells in a single-porosity naturally fractured reservoir and includes the traditional Euclidean solution as a special case.
- The proposed fractal solution generates a power-law response at late times during the transient period after the wellbore storage, mechanical skin, and partial penetration effects have ended. This behavior occurs when the radial fractal parameters are different from the Euclidean values, i.e., dfr < 2 and θr > 0.
- A different behavior to the power-law response occurs when dfz < 1 and θz > 0. The effect of these parameters is shown only during the partial penetration period, and after this period, the traditional radial behavior (if dfr = 2 and θr = 0) or a power-law behavior (when dfr < 2 and/or θr > 0) can be present.
- The typical spherical flow regime due to partial penetration is only present when the fractal parameters in the radial direction have the Euclidean values, i.e., dfr = 2 and θr = 0.
- An expression is provided to evaluate the pseudo-skin due to the partial penetration effects that consider fractal behavior in both the radial and vertical directions.
- To determine the pseudo-damage due to restricted penetration, horizontal permeability, vertical to horizontal permeability ratio, mechanical skin, and the four fractal parameters, it is necessary to resort to a type–curve matching process of the pressure data and its semi-logarithmic derivative using a robust optimizer that minimizes the difference between the real data and the analytical solution.
Author Contributions
Acknowledgments
Conflicts of Interest
Nomenclature
ct | Compressibility [psi−1] |
C | Wellbore storage constant [bbl/psi] |
CD | Dimensionless wellbore storage constant |
dfr | Fractal dimension in the radial direction (1 ≤ dfr ≤ 2) |
dfz | Fractal dimension in the vertical direction (0 ≤ dfz ≤ 1) |
h | Formation thickness [ft] |
hp | Producing interval [ft] |
hpD | Dimensionless producing interval |
hwD | Dimensionless depth at bottom of producing interval |
kr | Permeability in the radial direction, [mD] |
kz | Permeability in the vertical direction, [mD] |
krw | Reference radial permeability at the wellbore, [mD] |
kzw | Reference vertical permeability at the top of the anticline, [mD] |
Kν | Modified Bessel function of order ν |
Jν | Bessel function of order ν |
p | Pressure, [psi] |
pi | Initial pressure, [psi] |
pwD | Dimensionless wellbore pressure |
q | Production rate, [bpd] |
rw | Wellbore radius, [ft] |
rD | Dimensionless radius |
s | Laplace domain parameter |
S | Mechanical skin factor |
t | Time, [hours] |
tD | Dimensionless time |
z | Vertical depth from the top of the dome, [ft] |
zD | Dimensionless vertical depth |
zw | Top position of open interval, [ft] |
zwD | Dimensionless depth at top of open interval |
zwmD | Dimensionless depth at medium of open interval |
μ | Fluid viscosity, [cp] |
ø | Porosity |
θr | Connectivity index in the radial direction (0 ≤ θr ≤ 1) |
θz | Connectivity index in the vertical direction (0 ≤ θz ≤ 1) |
ξ(s) | Pseudo-skin due to partial penetration considering fractal behavior |
Appendix A. Solution in the Radial Direction
Appendix B. Solution in the Vertical Direction
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Posadas-Mondragón, R.; Camacho-Velázquez, R.G. Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs. Fractal Fract. 2019, 3, 23. https://doi.org/10.3390/fractalfract3020023
Posadas-Mondragón R, Camacho-Velázquez RG. Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs. Fractal and Fractional. 2019; 3(2):23. https://doi.org/10.3390/fractalfract3020023
Chicago/Turabian StylePosadas-Mondragón, Ricardo, and Rodolfo G. Camacho-Velázquez. 2019. "Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs" Fractal and Fractional 3, no. 2: 23. https://doi.org/10.3390/fractalfract3020023
APA StylePosadas-Mondragón, R., & Camacho-Velázquez, R. G. (2019). Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs. Fractal and Fractional, 3(2), 23. https://doi.org/10.3390/fractalfract3020023