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Transport zones of oil confined in lipophilic nanopores: a technical note

  • Fengrui SunEmail author
  • Yuedong YaoEmail author
  • Guozhen LiEmail author
  • Xiangfang Li
ICCESEN 2017
Part of the following topical collections:
  1. Geo-Resources-Earth-Environmental Sciences

Abstract

Shale resource is becoming more important in the energy supply market. While the shale oil resource is widely distributed throughout the world, the study on the oil transport mechanisms through nanopores of shale reservoir is still at the starting stage. In this paper, a novel idea is proposed by dividing the oil in the nanopore into three sub-zones: the absorbed zone, transition zone, and bulk zone. In each zone, the oil viscosity is calculated independently. The governing equations for each zone are presented. Then, coupling with the boundary equations obtained from the continuum law, three analytic equations are developed. Finally, the equation for total mass flow rate of oil transport through nanopores is obtained linearly. The model presented in this paper is the eve before the birth of a continuous model, and the new model can also shed light on the simulation of oil transport in nanoscale pores, which is the basis of multi-scale shale oil simulation.

Keywords

Shale oil Lipophilic nanopore Transport modeling A multi-zone model Analytic equation Continuous theory 

Introduction

Unconventional resources, such as heavy oil, tight oil, and shale oil, have become the hotspots in the present petroleum industry (Sun et al. 2017a, b, c, d, e, f, g). In recent years, the exploration and development of unconventional oil and gas has gradually increased (Sun et al. 2018a, b c, d, e, f, g, h, i, j, k, l, m). Different from conventional resources, which can be developed by water flooding, a series of unconventional methods, such as thermal recovery, polymer flooding, hydraulic fracturing, and surfactant flooding, are adopted for certain resources (Sun et al. 2018n, o, 2019a; Yu et al. 2019; Sun et al. 2018p, 2019b, 2018q, 2017h, 2018r, s, t, 2018, u, v).

In this paper, a simple but useful model is proposed for shale oil transport in nanopores of shale reservoirs. One of the salient features of shale reservoir is that the formation is rich in nanoscale pores (Fathi and Akkutlu 2013). It has been revealed that the classic Darcy’ law, which has been successfully applied to conventional reservoir simulation, is no longer useful in modeling oil and gas transport performances in shale reservoirs (Monteiro and Rycroft 2012; Nie et al. 2012). By means of modern physics (Deng et al 2019; Duan et al. 2018; Wen et al. 2018), the study on shale structures showed that the shale matrix is mainly composed of organic and inorganic matter (Wang et al. 2009). The modeling of fluid transport through nanoscale pores has become a hotspot (Wu et al. 2015a, b, 2016, Feng et al. 2018; Li et al. 2018a, b; Zeng et al. 2017). However, it is never easy to do so due to the complexity of multiple mechanisms of fluid transport through nanopores (Yao et al. 2017; Feng et al. 2017; Ariketi et al. 2017; Shar et al. 2018; Koteeswaran et al. 2018; Kudapa et al. 2017).

Based on experimental methods, Majumder et al. (Majumder et al. 2005) studied the transport performances of some typical fluids through aligned carbon nanotubes, and found that the mass flux of these fluids are 4 to 5 orders of magnitude greater than the predicted results from the non-slip Poiseuille equation. In the petroleum industry, Cui et al. (Cui et al. 2017) presented a model which was adapted from Majumder et al.’s work. Then, Sun et al. (Sun et al. 2019a) studied the effect of critical thickness of the absorbed layer on the transport behaviors of oil through nanopores of organic shale matrix. This model enlightens the discovery of transition zone of oil confined in a nanopore. In fact, there is a transition zone between the absorbed oil and bulk oil in the nanopores.

In this paper, a simple analytic equation is developed for oil transport through nanopores considering the oil state confined in a nanopore of shale. The oil confined is divided into three zones: bulk oil zone, transition zone, and absorbed zone. The equation for each zone is built and then a coupled model is developed based on the three equations.

Physical description

When oil flows through lipophilic nanopores, the oil near the wall is absorbed on the surface. The viscosity of absorbed oil is significantly higher than that of bulk oil. However, the changing of oil viscosity in the nanopore is a continuous process. Therefore, there exists a transition zone between the absorbed zone and bulk zone, as shown in Figs. 1 and 2.
Fig. 1

Cross section of the nanopore

Fig. 2

The distribution of absorbed oil zone (confined oil zone), transition oil zone, and bulk oil zone in the nanopore

Analytic model

The governing equation of bulk oil transport in nanopores can be expressed as (Myers 2011):
$$ \frac{\mu_{\mathrm{b}}}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {v}_{\mathrm{b}}}{\partial r}\right)=\frac{\partial p}{\partial z},r\in \left[0,{r}_{\mathrm{b}}\right] $$
(1)
where μb denotes the bulk oil viscosity; r denotes the radii in the radial direction; vb denotes the transport velocity of bulk oil; p denotes the bulk oil pressure; z denotes the distance in the flow direction.
The governing equation of oil transport in transition zone through nanopores can be expressed as:
$$ \frac{\mu_1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {v}_1}{\partial r}\right)=\frac{\partial p}{\partial z},r\in \left[{r}_{\mathrm{b}},{r}_1\right] $$
(2)
where μ1 denotes the viscosity of oil in transition zone; v1 denotes the transport velocity of oil in transition zone.
The governing equation of oil transport in absorbed zone through nanopores can be expressed as:
$$ \frac{\mu_0}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {v}_0}{\partial r}\right)=\frac{\partial p}{\partial z},r\in \left[{r}_1,{r}_0\right] $$
(3)
where μ0 denotes the viscosity of oil in absorbed zone; v0 denotes the transport velocity of oil in absorbed zone.
As mentioned above, the changing of oil viscosity in the nanopore is a continuous process. The boundary conditions of these three governing equations are listed below:
$$ {\left.\frac{\partial {v}_{\mathrm{b}}}{\partial r}\right|}_{r=0}=0 $$
(4)
$$ {v}_{\mathrm{b}}{\left|{}_{r={r}_{\mathrm{b}}}={v}_1\right|}_{r={r}_{\mathrm{b}}} $$
(5)
$$ {v}_1{\left|{}_{r={r}_1}={v}_0\right|}_{r={r}_1} $$
(6)
$$ {\mu}_{\mathrm{b}}\frac{\partial {v}_{\mathrm{b}}}{\partial r}{\left|{}_{r={r}_{\mathrm{b}}}={\mu}_1\frac{\partial {v}_1}{\partial r}\right|}_{r={r}_{\mathrm{b}}} $$
(7)
$$ {\mu}_1\frac{\partial {v}_1}{\partial r}{\left|{}_{r={r}_1}={\mu}_0\frac{\partial {v}_0}{\partial r}\right|}_{r={r}_1} $$
(8)
$$ -{l}_{\mathrm{st}}\frac{\partial {v}_1}{\partial r}{\left|{}_{r={r}_{\mathrm{b}}}={v}_1\right|}_{r={r}_{\mathrm{b}}} $$
(9)
$$ -{l}_{\mathrm{st}}\frac{\partial {v}_0}{\partial r}{\left|{}_{r={r}_1}={v}_0\right|}_{r={r}_1} $$
(10)
where lst denotes the total slip length.
Coupling with the boundary equations, the distribution of oil transport velocity can be developed as:
$$ {v}_{\mathrm{b}}=\left[\frac{1}{4{\mu}_{\mathrm{b}}}\left({r_{\mathrm{b}}}^2-{r}^2\right)+\frac{1}{4{\mu}_1}\left({r_1}^2-{r_{\mathrm{b}}}^2\right)+\frac{r_1}{2{\mu}_1}{l}_{\mathrm{st}}\right]\frac{\nabla p}{L},r\in \left[0,{r}_{\mathrm{b}}\right] $$
(11)
$$ {v}_1=\left[\frac{1}{4{\mu}_1}\left({r_1}^2-{r}^2\right)+\frac{r_1}{2{\mu}_1}{l}_{\mathrm{st}}\right]\frac{\nabla p}{L},r\in \left[{r}_{\mathrm{b}},{r}_1\right] $$
(12)
$$ {v}_0=\left[A\left({r_1}^2-{r}^2\right)+\frac{r_1}{2{\mu}_1}{l}_{\mathrm{st}}\right]\frac{\nabla p}{L},r\in \left[{r}_1,{r}_0\right] $$
(13)
By integrating in the radial direction, we can obtain:
$$ {q}_{\mathrm{b}}={\int}_0^{r_{\mathrm{b}}}{v}_{\mathrm{b}} dA=\frac{\pi }{8{\mu}_{\mathrm{b}}}\left[{r_{\mathrm{b}}}^2\left({r_{\mathrm{b}}}^2+2\frac{\mu_{\mathrm{b}}}{\mu_1}\left({r_1}^2-{r_{\mathrm{b}}}^2+2{r}_1{l}_{\mathrm{st}}\right)\right)\right]\frac{\nabla p}{L} $$
(14)
$$ {q}_1={\int}_{r_{\mathrm{b}}}^{r_1}{v}_1 dA=\frac{\pi }{8{\mu}_1}\left[\left({r_1}^2-{r_{\mathrm{b}}}^2\right)\left({r_1}^2-{r_{\mathrm{b}}}^2+4{r}_1{l}_{\mathrm{st}}\right)\right]\frac{\nabla p}{L} $$
(15)
$$ {q}_0={\int}_{r_1}^{r_0}{v}_0 dA=\frac{\pi }{8{\mu}_1}\left[\left({r_0}^2-{r_1}^2\right)\left(4{\mu}_1A{r_1}^2-4{\mu}_1A{r_0}^2+4{r}_1{l}_{\mathrm{st}}\right)\right]\frac{\nabla p}{L} $$
(16)
Equations (14) to (16) are the analytic equations for the above mentioned three zones. Then, the total mass flow rate of oil in the nanopores can be obtained linearly as:
$$ q={\int}_0^{r_{\mathrm{b}}}{v}_{\mathrm{b}} dA+{\int}_{r_{\mathrm{b}}}^{r_1}{v}_1 dA+{\int}_{r_1}^{r_0}{v}_0 dA $$
(17)

Conclusions

At present, the study on oil transport through nanopores of shale reservoir is at the starting stage. There are many questions waited to be explored. In this paper, a novel idea is proposed by dividing the oil in the nanopore into three sub-zones: the absorbed zone, transition zone, and bulk zone. In each zone, the oil viscosity is independent, which is quite different from previous works. This is because this method can be the basis for an integral model, which is quite from the discontinuous model. This paper can also shed light on the simulation of oil transport in nanoscale pores, which is the basis of multi-scale shale oil simulation.

Notes

Acknowledgements

The research was supported by the National Science and Technology Major Projects of China (no. 2016ZX05042, no. 2017ZX05039, and no. 2016ZX05039) and the National Natural Science Foundation Projects of China (no. 51504269, no. 51490654, and no. 40974055). The authors also acknowledge Science Foundation of China University of Petroleum, Beijing (no. C201605), the National Basic Research Program of China (2015CB250900), and the Program for New Century Excellent Talents in University (grant no. NCET-13-1030) in supporting part of this work.

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Copyright information

© Saudi Society for Geosciences 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Petroleum Resources and ProspectingChina University of Petroleum – BeijingBeijingPeople’s Republic of China
  2. 2.College of Petroleum EngineeringChina University of Petroleum – BeijingBeijingPeople’s Republic of China
  3. 3.China University of Petroleum – BeijingBeijingPeople’s Republic of China

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