Geofluids

Volume 2018, Article ID 8302782, 14 pages

https://doi.org/10.1155/2018/8302782

## The Seepage Model Considering Liquid/Solid Interaction in Confined Nanoscale Pores

^{1}Department of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, China^{2}Harold Vance Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843, USA^{3}Daqing Oilfield Limited Company, Daqing 163712, China

Correspondence should be addressed to Erlong Yang; nc.ude.upen@gnolregnay and Kaoping Song; nc.ude.upen@pks

Received 20 October 2017; Revised 7 January 2018; Accepted 3 June 2018; Published 19 August 2018

Academic Editor: Zhenhua Rui

Copyright © 2018 Xiaona Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Different from conventional reservoirs, nanoscale pores and fractures are dominant in tight or shale reservoirs. The flow behaviors of hydrocarbons in nanopores (called “confined space”) are more complex than that of bulk spaces. The interaction between liquid hydrocarbons and solid pore wall cannot be neglected. The viscosity formula which is varied with the pore diameter and interaction coefficient of liquids and solids in confined nanopores has been introduced in this paper to describe the interaction effects of hydrocarbons and pore walls. Based on the Navier-Stokes equation, the governing equation considered liquid/solid effect in two dimensions has been established, and approximate theoretical solutions to the governing equations have been achieved after mathematic simplification. By introducing the vortex equation, the complex numerical seepage model has been discretized and solved. Numerical results show that the radial velocity distribution near the solid wall has an obvious change when considering the liquid/solid interaction. The results consist well with that approximate mathematical solution. And when the capillary radius is smaller, the liquid and solid interaction coefficient is greater. The liquid and solid interaction obviously cannot be neglected in the seepage model if the capillary radius is small than 50 nm when . The numerical model has also been further validated by two types of nanopore flow tests: from pore to throat and inversely from throat to pore. There is no big difference in flow regularity of throat to pore model considering when liquid/solid interaction or not, whereas the liquid/solid interaction of pore to throat model totally cannot be overlooked.

#### 1. Introduction

Unconventional oil and gas are playing an increasingly important role in global energy supply. Unlike conventional reservoirs, tight or shale reservoirs possess unique characteristics, such as ultralow permeability and strong heterogeneity [1]. The length scales associated with transport phenomena in tight and shale formations are rich, from nanoscale phenomena to field-scale applications [2–4]. Nanoscale pores and nanoscale fractures are still dominant in tight or shale reservoirs. The extremely small size of tight or shale pores, leads to strong intermolecular forces between fluid particles and solid surface [5–7]. The flow behaviors of hydrocarbons in nanopores (called “confined space”) are more complex than those in bulk spaces [8]. The deviations in the prediction of the phase behavior and production in confined nanoscale tight or shale reservoirs from that of conventional oilfields are needed to be calibrated [9–11].

In macroflow, fluid can be assumed as continuous medium due to its characteristic size of the liquid molecules is bigger than the mean free path. In microscale flow [12, 13], when the characteristic size and the mean free path are in the same magnitude, some macroscopic and laws based on the continuous medium would not work anymore [14]. The viscosity coefficient is also needed to be discussed again. Newtonian fluid presents the properties of non-Newtonian fluid in nanochannels [15], such as the boundary layer effect. The fluid in porous media can be classified into body fluid and boundary layer fluid. Body fluid which is close to the axial of porous media channel is not affected by the boundary effect. The fluid which is affected by boundary effect is easy to form a boundary layer on the pore wall [16].

Many researches on the formation of boundary layer and its influence on porous media seepage have been carried out. Engelhardt [17] explains the reason the oil seepage deviating from Darcy’s law is mainly due to the active component adsorption on the surface of rock wall. Karimov [18] confirmed that the active surface substances in fluid could be adsorbed on rock particle surface by experiments. Secondly, tight rock such as shale and mudstone can adsorb the salt components in water. Bhasin [19] found that oil viscosity of boundary layer is 10–15 times higher than the body oil. The distribution of crude oil composition is orderly in the boundary layer, and the viscosity increased. The water presents non-Newton characteristics in the tiny pore under the effect of liquid-solid interface [20–22]. Lilatov pointed out that the boundary layer thickness of polymer solution is 160 nanons [23]. Zhang and Xiangan described the rheological properties of the oil film by transforming the constitutive Bingham equation, in view of the increasing viscosity and yield stress of oil film along the radial direction [24]. Hara and Tsuchiya examined water-(pyroclastic) rock interactions using flow-through experiments to deduce the effect of mass transport phenomena on the reaction process [25]. Bea et al. presented a reactive transport formulation coupled to thermohydrodynamic and simplified mechanical processes, taking the heat and mass transport and water-rock interactions into account [26].

For the Newtonian fluid, exact solutions and transformations of equations as well as different problems on the hydrodynamic boundary layer have been addressed by many researchers. Pavlovskii began to obtain invariant solutions to the steady-state boundary layer equations in 1961 [27]. And invariant-irreducible, partially invariant solutions of steady-state boundary layer equations have been worked out [28]. Vereshchagina fibered the spatial unsteady boundary layer equations [29]. Then, many similarity solutions of the two-dimensional unsteady and steady-state boundary-layer equations came out [30, 31]. The reduction method of partial differential equations has been proposed to describe a laminar stationary flat boundary layer [32, 33]. Explicit solutions of the boundary-layer equations have been found [34–39]. Prandtl’s boundary layer equation for radial flow is derived [40]. General solution and quasiexact solution of equation is found, and norm of the residual function is presented. Equations of unsteady axisymmetric boundary layer are studied [41]. The solutions are obtained with a new method (direct method of functional separation of variables) based on using particular solutions to an auxiliary ODE [42]. Su et al. put forward a method based on a mixed control volume-discontinuous finite element formulation to accurately simulate multiphase flow in fractured shale reservoir. It has solved the problem of discontinuous or near discontinuous behavior of saturation in real oilfield [43].

Previous researches have studied flow behavior in big channel considering the liquid and solid interaction, and many numerical reservoir simulation models in terms of shale or tight oil and gas have been investigated. Monteiro et al. completed mathematical modeling of the flow in nanoporous rocks based on the hypothesis that the permeability of the inclusions substantially depends on the pressure gradient; boundary layers are considered in this model [44]. Li applied engineering density functional theory (DFT) combined with the Peng-Robinson equation of state (EOS) to investigate the adsorption and phase behavior of pure substances and mixtures in nanopores [45]. Wang incorporated the capillary pressure effect and pore space compaction in a compositional reservoir simulator to evaluate their effects on production [46]. Mi et al. has established the DFN model with fracture network to study the effect of the stress-dependent fracture conductivity on the shale gas well performance [47, 48]. Wang et al. developed a new semianalytical model for multiple-fractured horizontal wells (MFHWs) with stimulated reservoir volume (SRV) in tight oil reservoirs by combining source function theory with boundary element idea [49]. However, there are no researches about the seepage description in confined nanopores considering the liquid and solid interaction. This paper establishes a two-dimensional numerical model of seepage flow in confined nanopores. Vortex equation has been applied to discretize the complex model. The numerical results are compared with the approximate mathematical results. The numerical model has also been validated by two types of nanopore flow: from pore to throat and inversely from throat to pore.

#### 2. Liquid/Solid Interaction in Confined Pores

Molecular interactions include three parts, orientating effect, inducing effect, and dispersing effect. The force of water molecules in boundary layer which is affected by inner interaction action of water molecules and solid pore wall would influence the viscosity. Since viscosity can represent molecular attraction, when the molecular attraction is greater, the viscosity will be greater. According to this, we can assume viscosity is proportional to intermolecular attraction. The viscosity value in boundary layer is composed of two parts: bulk fluid viscosity and additional viscosity imposed by solid surface. In formula, the viscosity coefficient of fluid [50] in nanoflow boundary layer can be arranged as

From (1), on the solid surface where , the viscosity coefficient of water molecule is infinity, and the water molecules would stand still, which is consistent with the classical boundary layer theory without slip conditions. While on the solid surface where , the water viscosity is just normal.

#### 3. Mathematical Modeling Description

##### 3.1. Governing Equation

Rheological equation of nanoscale porous media fluid in boundary layer has been revised based on the Navier-Stokes equation considering the viscosity coefficient formula of boundary layer fluid. Then, the governing equation of boundary layer fluid in the nanochannel has established.

Velocity component is equal to zero in two-dimensional flow; therefore all velocity variables has nothing to do with , that is, its partial derivative is equal to zero. The continuity equation is as follows:

Momentum equations:

For steady incompressible fluid, (3), (4a), and (4b) are as follows: where , (5a), (5b), and (5c) are the fluid flow control equations considering solid-liquid interaction.

##### 3.2. Approximate Solution

If the effects of viscosity and heat diffusion in boundary layer simply react in thin layer close to the surface, the change of velocity and temperature mainly exists in the thin layer of this object plane. These two kinds of thin layer thickness can be expressed as velocity boundary layer thickness and temperature boundary layer thickness . Compared with , they are so small that the order quantity is , so quantity order within the boundary layer is . Velocity component changes from the speed 0 in the boundary layer to the mainstream speed beyond boundary layer. Taking and as characteristic values, the order quantity is one. At the same time, the continuity equation can be approximatively written as

Of the viscous force and inertial force item in (4a), truncating the order items with small quantity, respectively, it can be obtained:

According to (7), if the viscous force and inertial force items are in the same amount of order, the order quantity of should be , so it is still the highest item in (4b):

For two-dimension nanoflow which is steady and incompressible in boundary layer, when substituting (1) () into (7), the simplified continuity equation and momentum equation can be obtained as follows:

Boundary conditions are , (9) and (10) are approximation equations of two-dimension nanoflow which is steady and incompressible.

Taking the flow of Newtonian fluid in circular cross capillary tube as an example, the velocity distribution considering liquid-solid effect and not have been obtained by solving the approximation equations.

As shown in Figure 1, take the tube axes as axis, radial coordinates as axis, both the circumferential and radial velocity is zero, horizontal velocity component is which is the only function of , and the pressure on each cross section is a constant value. When nanoflow boundary layer equations are expressed in cylindrical coordinates, (9) and (10) can be simplified to: