- Research
- Open Access
Theory and practice of micro water-oil displacement efficiency based on mathematical models
- Haohan Liu^{1, 2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13362-019-0061-z
© The Author(s) 2019
- Received: 3 September 2018
- Accepted: 10 April 2019
- Published: 29 April 2019
Abstract
Theoretical study about quantitative influence of the micro parameters (interfacial tension, wetting angle, distance, unit length, bottom radius, boundary distance, etc.) on the macro parameters (oil displacement efficiency, pressure gradient, etc.) in water-oil displacement process is very important. This paper focuses on the micro oil displacement efficiency calculation method in oil reservoir, building a new bridge between these parameters. Combining fluid motion equation, continuity equation to pressure gradient equation, the occurrence radius equation of remaining oil in different capillary is established; Taking mercury intrusion porosimetry experiment and constructing the mathematical model, new normal probability distribution of pore radius is achieved, and then a new probability function for injected water entering pore throat is established based on the occurrence radius equation and pore radius distribution equation; By using the Darcy’s Law and definite integral theory, new micro oil displacement efficiency equation is obtained. The Nuclear Magnetic Resonance experiments show: new method gets high calculating accuracy, and can be used to guide oil reservoir development.
Keywords
- Occurrence radius
- Probability function
- Definite integral theory
- Oil displacement efficiency
- Nuclear Magnetic Resonance experiment
1 Introduction
Experts and scholars focus on water-oil displacement efficiency for years. Richardson [1] shows the oil displacement efficiency in weak hydrophilic rock samples is higher than that in strong hydrophilic rock samples. And then, Morrow [2] find there is a negative correlation between the water-oil displacement efficiency and rock sample hydrophilic characteristics. Li [3] and He [4] study the Permeability influence on water-oil displacement efficiency in two different stages. Wang [5] gives the Positive correlation of feature value function and water-oil displacement efficiency. Cai [6] and Shen [7] and Chen [8] show the negative correlation of inhomogeneity of pore structure and water-oil displacement efficiency. Yu [9] studied the Oil and water viscosity ratio influence on water-oil displacement efficiency. Lu [10] studies the indication phenomenon is a key factor influence the oil displacement efficiency. Yan [11] gives that the water-oil displacement efficiency increased by Injection multiples. Jiao’s research [12] shows that the EOR is logarithmic to the injected pore volume. Fu [13] shows the replacement pressure is positive correlative to the oil displacement efficiency in the early development stage. Fan Li [14], Rogério Soares Silva [15], Zheng [16] and Dong [17] also work on how to improve the EOR. However, there is seldom theoretical research on micro factors influences (such as interfacial tension) on macro factors (oil displacement efficiency), and the explanation of oil field development phenomena is usually acted by our experiences.
In previous time, my team studied oil displacement efficiency from different aspects: based on the water flooding location, the water drive characteristic curve, the statistic methods ANN and \(GM(1,n)\) and the time-varying system [18–24]. After years’ study, we find there is a necessary quantitative connection between water-oil displacement efficiency and these micro factors. This manuscript aims to establish new relationships between macro parameter water-oil displacement efficiency, pressure gradient with the micro parameters interfacial tension, wetting angle, distance, unit length, bottom radius, boundary distance, etc. Then, we will get the interaction mechanism between the macro parameters and the micro parameters.
2 Remaining oil occurrence radius in capillary
For the incompressible single-phase liquid, the seepage process in a homogeneous reservoir is isothermal steady.
For some particular reservoir with bottom water or lumpy oil layer, etc, the liquid flow in the bottom well is spherical directed flow.
Hence, the maximum occurrence radius changed by pressure gradient \(p_{e} - p_{w}\), wetting angle θ, the unit distance, the unit length \(\delta_{R}^{ - 1}\), the bottom radius \(r_{w}\), the boundary distance \(r_{e}\) and the interfacial tension σ.
For polymer flooding, the polymerized substance can change the wetting angle and interfacial tension, and then the occurrence radius changed. By increasing displacement PV, the \(r_{D}\) will also be changed, we can also quantitatively compare the difference of water drive and polymer flooding efficiency with parameter \(r_{D}\).
For heterogeneous oil reservoirs, the fluid motion equation and continuity equation will be changed, we can also use the above-mentioned steps to obtain \(r_{D}\).
3 New oil displacement efficiency calculation method
Here, \(E_{D}(r_{D})\) is an error function, \(E_{D}(r_{D}) = \operatorname{erf}(r_{D})\), math software-Matlab can be used to calculate the numerical solution.
By introducing related reservoir engineering measurement to change related factors can lead to the change of \(E_{D}\).
4 Results and discussion
Parameter values
\(r_{\min}\), μm | \(r_{\max}\), μm | \(\delta_{R}\), m | \(r_{e}\), m | \(r_{w}\), m | θ | σ | \(s_{\mathrm{wc}}\) |
---|---|---|---|---|---|---|---|
70 | 2 | 0.1 | 300 | 0.2 | \(\frac{3}{4}\pi\) | 0.025 \(\mathrm{N} \cdot \mathrm{m}^{ - 1}\) | 0.15 |
Parameter values in a constant displacement pressure gradient
R, m | \(r_{D}\), μm | \(J_{k}\), % | \(E_{D}\) |
---|---|---|---|
5 | 6.465 | 94.18 | 0.937788 |
10 | 12.93 | 83.47 | 0.89356 |
15 | 19.395 | 60.36 | 0.836094 |
20 | 25.86 | 45.89 | 0.806094 |
25 | 32.325 | 36.21 | 0.782941 |
30 | 38.79 | 29.05 | 0.683835 |
35 | 45.255 | 24.33 | 0.653835 |
40 | 51.72 | 20.1 | 0.573835 |
45 | 58.185 | 18.63 | 0.423835 |
Figure 5 shows that the Percentage of water entries into rock pore decreased by occurrence radius, when the occurrence radius is small enough, the water can 100% enter the pore capillary.
Figure 6 shows: the oil displacement efficiency decreased by occurrence radius. When the occurrence radius is smaller than 25 μm, the oil displacement efficiency is over 80%, and when \(R=5\) m, \(r_{D} = 6.465\) μm, the oil displacement efficiency is near to 0.937788 in the GD7-42-J195TG block. Hence, the oil displacement efficiency decreased by occurrence radius.
When the displacing time and pressure gradient is big enough, the remaining oil saturation equals to the residual oil saturation, it is \(s_{\mathrm{or}} = s_{o}\), the oil displacement efficiency maximized.
Using Eq. (26) and data of Table 2, water saturation is achieved, \(s_{o} ( \Delta p = 0.2~\mathrm{Mpa} ) = 0.22744\).
Theoretical calculating results of oil saturation
Sample | Remaining oil saturation, % | ||
---|---|---|---|
0.05 mL/min | 0.5 mL/min | 2 mL/min | |
A7 | 34.34 | 27.27 | 19.94 |
B9 | 34.28 | 26.32 | 20.19 |
C112 | 36.16 | 27.28 | 21.37 |
Remaining oil saturation with NMR experiment
Samples | Remaining oil saturation, % | ||
---|---|---|---|
0.05 mL/min | 0.5 mL/min | 2 mL/min | |
A7 | 33.13 | 26.34 | 21.45 |
B9 | 35.7 | 25.93 | 20.65 |
C12 | 35.21 | 25.13 | 20.27 |
In Fig. 10, Fig. 11 and Fig. 12 the oil is red and the water is blue, separately. Figure 10, Fig. 11 and Fig. 12 show: the remaining oil saturation can be achieved and the residual mechanism can be concluded.
Figure 13 show that the theoretical calculating results of oil saturation is close to the experimental results. The oil saturation variance of sample-A7 is 2.116367, sample-B9 is 3.3992, and sample-C112 is 2.978333. Hence, new method can be used to guide the oilfield development.
5 Conclusions
(1) Occurrence radius of remaining oil is decided by pressure gradient, wetting angle, well distance, the unit length, the bottom radius, the boundary distance and the interfacial tension, influencing the oil displacement efficiency, it collects micro factors and macro factor in a special way.
(2) MIP experiments data show that the distribution of pore radius is normal, and the probability expression can be worked out by numerical fitting method, and it can be used to calculate pore radius distributions of new blocks in the same oil reservoir.
(3) Not all of the injected water enters the capillary pores, the residual part, the discontinuous part and the possible formation pathway should be taken into consideration.
(4) The micro oil displacement efficiency decreased by occurrence radius.
(5) NMR experiments can be used to test the accuracy of the new method, and the experiments data of rock samples A7, B9, C12 show: the NMR experiments values (oil saturations) are in accordance with the values calculated by new method.
Declarations
Acknowledgements
I always acknowledge reviewers and the associate editor and the support of Scientific Project of the Scientific Project of the vocational colleges teaching steering committee in ministry of education of China, No. 2018GGJCKT200 and the scientific project of Sichuan science and technology department, No. 2018CC0109.
Availability of data and materials
Fig. 1 in Sect. 3 is based on the MIP experiment data of SL oilfield in China (the initial data figure). Figure 6, Fig. 7 and Fig. 8 are MRI of samples of A7, B9 and C112 in different flow speed in SL oilfield (the initial data figure). Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Authors’ information
Haohan Liu (1985-08), Ph.D. of oil and gas development engineering, vice professor, post-doctor, engaged in oil and gas development for 8 years, interested in optimization and statistics theory and application.
Funding
Here, I offer my regards and blessings to the Scientific Project of the vocational colleges teaching steering committee in ministry of education of China and the and the scientific project of education department of Sichuan Province, helping me taking the MIP experiment and collecting the initial data here.
Authors’ contributions
The main idea of this paper was proposed by HL, and the author prepared the manuscript initially and performed all the steps of the proofs in this research. The author read and approved the final manuscript.
Competing interests
I declare that there is no competing interest.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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