Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells
Abstract
:1. Introduction
2. Methods
- The reservoir was layered, the values of permeability remained constant across the thickness for each layer in the j-th layer;
- The logarithm of the permeability of each layer was represented as a surface spline. A spline surface is a model of a thin plate bent under the action of external forces applied at some points (interpolation nodes). To construct a spline surface, it was necessary to solve the variational problem of finding the minimum free energy of the plate, which led to the following formula [34]:
- The value of permeability in the i-th well in the j-th layer was calculated by the formula , where is the unknown coefficient, and is the a priori value of .
3. Results
4. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
boundary of the reservoir | |
part of the boundary on which the boundary condition of the first kind is specified | |
part of the boundary on which the boundary condition of the second kind is specified | |
inflow boundary () | |
sensitivity matrix | |
formation volume factor of phase α | |
g | gradient of the residual function |
approximate matrix of second derivatives | |
I | unit matrix |
residual function | |
absolute permeability | |
relative permeability of phase α | |
gas phase relative permeability of oil–gas system | |
oil phase relative permeability of oil–gas system | |
oil phase relative permeability of oil–water system | |
water phase relative permeability of oil–water system | |
length of the perforated zone of the i-th well | |
total number of measurements for all wells | |
number of measurements on i-th well | |
number of wells | |
outward unit normal to the boundary | |
pressure | |
bottom hole pressure of the i-th well | |
time | |
external sources and sinks of component α | |
calculated values of the total liquid production rate | |
measured values of the total liquid production rate | |
radius of the i-th well | |
drainage radius of the i-th well | |
gas solubility | |
saturation of phase α | |
critical water saturation in the oil–water system | |
volumetric velocity of phase α | |
vector of identification parameters | |
Dirac delta function at | |
Marquardt parameter | |
viscosity of phase α | |
porosity | |
Subscripts and Superscripts | |
a—priori values; c—calculated values; G—gas component; g—gas phase; O—-oil component; o—oil phase; tr—true values; W—water component; w—water phase |
Appendix A. The Method of the Simultaneous Solution for Black Oil Model
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p, MPa | |||
---|---|---|---|
5 | 1.025 | 0.02 | 300 |
10 | 1.045 | 0.01 | 600 |
15 | 1.07 | 0.006 | 900 |
20 | 1.09 | 0.0045 | 1200 |
25 | 1.11 | 0.0035 | 1400 |
0.2 | 0.00 | 1.00 | 0.0 | 0.00 | 1.00 |
0.3 | 0.05 | 0.40 | 0.05 | 0.00 | 0.60 |
0.4 | 0.10 | 0.10 | 0.1 | 0.01 | 0.30 |
0.6 | 0.30 | 0.005 | 0.2 | 0.05 | 0.10 |
0.8 | 0.60 | 0.00 | 0.3 | 0.10 | 0.01 |
1.00 | 1.00 | 0.00 | 0.4 | 0.20 | 0.00 |
0.6 | 0.50 | 0.00 | |||
0.7 | 0.70 | 0.00 | |||
0.8 | 1.00 | 0.00 |
2.08 × 10−4 | 5.42 × 10−5 | 0.014 | 6 | |
0.00450 | 0.00428 | 1.593 | 11 | |
0.00454 | 0.00435 | 1.708 | 11 | |
0.00437 | 0.00388 | 1.355 | 11 | |
0.02179 | 0.02055 | 7.389 | 10 | |
0.02823 | 0.02273 | 4.095 | 13 | |
0.02144 | 0.01535 | 4.298 | 10 | |
0.04355 | 0.03958 | 15.127 | 11 | |
0.08357 | 0.04937 | 23.377 | 9 | |
0.04497 | 0.03683 | 14.607 | 10 |
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Elesin, A.V.; Kadyrova, A.S.; Nikiforov, A.I.; Tsepaev, A.V. Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells. Mathematics 2022, 10, 4558. https://doi.org/10.3390/math10234558
Elesin AV, Kadyrova AS, Nikiforov AI, Tsepaev AV. Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells. Mathematics. 2022; 10(23):4558. https://doi.org/10.3390/math10234558
Chicago/Turabian StyleElesin, Andrey V., Alfiya Sh. Kadyrova, Anatoliy I. Nikiforov, and Aleksey V. Tsepaev. 2022. "Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells" Mathematics 10, no. 23: 4558. https://doi.org/10.3390/math10234558
APA StyleElesin, A. V., Kadyrova, A. S., Nikiforov, A. I., & Tsepaev, A. V. (2022). Reservoir Permeability Identification under Three-Phase Filtration Using a Priori Information on Wells. Mathematics, 10(23), 4558. https://doi.org/10.3390/math10234558