Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 3245498, 8 pages

https://doi.org/10.1155/2018/3245498

## Mathematical Model for the Fluid-Gas Spontaneous Displacement in Nanoscale Porous Media considering the Slippage and Temperature

^{1}College of Geoscience and Surveying Engineering, China University of Mining & Technology, Beijing 100083, China^{2}Beijing Dadi Gaoke Coalbed Methane Engineering Technology Research Institute, Beijing 100040, China^{3}National Administration of Coal Geology in China, Beijing 100038, China

Correspondence should be addressed to Zhongyue Lin

Received 25 September 2017; Revised 10 January 2018; Accepted 23 January 2018; Published 15 February 2018

Academic Editor: Sandro Longo

Copyright © 2018 Kang Liu 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

The fracturing fluid-gas spontaneous displacement during the fracturing process is important to investigate the shale gas production and formation damage. Temperature and slippage are the major mechanisms underlying fluid transport in the micro-/nanomatrix in shale, as reported in the previous studies. We built a fracturing fluid-gas spontaneous displacement model for the porous media with micro-/nanopores, considering two major mechanisms. Then, our spontaneous displacement model was verified by the experimental result of the typical shale samples and fracturing fluids. Finally, the influences of temperature, slip length, and pore size distribution on the spontaneous imbibition process were discussed. Slippage and temperature significantly influenced the imbibition process. Lower viscosity, higher temperature, and longer slip length increased the imbibition speed. Ignoring the temperature change and slippage will lead to significant underestimation of the imbibition process.

#### 1. Introduction

During hydraulic fracturing process in the unconventional gas formation, a relatively large volume of fracturing fluid is pumped into formation, which can greatly stimulate the gas production [1, 2]. In this process, water will be imbibed into matrix in the fractured reservoir by many influences, including capillary pressure [3–5], chemical osmotic pressure [6], pore network [7–9], and clay mineral [10], which is called spontaneous imbibition. Also, leakage, lost circulation, and induced fracture in drilling process will lead to fracturing fluid being pumped into formation [11–13], which will also cause imbibition process and may change the stress field near wellbore [14] and drilling state [15, 16]. The spontaneous imbibition is the dominant mechanism of the water transport into the formation because of the high capillary pressure by nanopores [17–19]. Recently studies have shown that the spontaneous imbibition of fracturing fluid can be a driving force to enhance the gas recovery for shale gas reservoir [20, 21]. Thus, analysis of the spontaneous imbibition of shale and the potential effect on gas recovery needs urgent attention.

Shale has an ultralow permeability and porosity with abundant nanopores. Liquid flow mechanism in shale is much more complex than conventional formations. Slip flow is a major mechanism of liquid transport in nanotube [22, 23]. More studies have shown that the slip flow is significantly different from the no-slip boundary condition in the nanotubes [24, 25]. In addition, these studies reported that conventional flow equations, such as Darcy’s law, may not be valid for shale systems because of the difference in the controlling physics of liquid flow.

Fracturing fluid commonly includes hydrochloric acid, friction reducers, guar gum, and biocides. Viscosity varies with temperature significantly [26]. Reservoir temperature is also one of the key controlling factors in spontaneous imbibition. The influence of temperature on gas production is usually ignored because the temperature change is not severe. However, for spontaneous imbibition in the hydraulic fracturing process, the temperature of fracturing liquid is quite different from formation. Thus, ignoring the variation in temperature will lead to inaccurate results and errors.

#### 2. Mathematical Model

##### 2.1. Spontaneous Imbibition considering Slip Effect in a Single Capillary

Spontaneous imbibition occurs in the capillaries of shale as the wetting fluid imbibed in capillaries is driven by capillary force automatically. Liquid slip in nanoscale capillaries is especially not negligible because the slip length is the same scale with the diameter. In this section, considering liquid slip effect, we established a spontaneous imbibition model. To focus on the effect of liquid slip, we have made some simplifications as follows: the cross section of tube is circular; liquid is the wetting phase, while gas is the nonwetting phase; liquid is the Newton liquid with laminar flow, and inertial forces have been ignored; the driving force of spontaneous imbibition is the capillary force; slip occurs at tube wall; and gravity is ignored. On the basis of the Hagen–Poiseuille equation, the fluid flux considering the liquid slip can be calculated as follows:where is the fluid flux in tube, is the pressure difference on fluid, is the dynamic viscosity, refers to the length of fluid path line, is the tube’s equivalent diameter, and is the slip length. can be expressed in a dimensionless form, .

Imbibition velocity can be determined as follows:

The real capillary in shale is tortuous. Tortuous fractal dimension is introduced to express the tortuous capillaries, according to Yu and Cheng [27] and Cai et al. [28].where is the distance between meniscus and liquid intake and refers to the fractal dimension of a tortuous capillary.

If (1)–(3) are rearranged, we have the following equation:

The driving pressure of spontaneous imbibition is the capillary force. Thus, we have the following equation:where is the contact angle between liquid and tube wall and is the interfacial tension. During the imbibition process, the imbibition length is increasing with the movement of meniscus. For the tortuous tube, using the initial condition , the relationship between imbibition length and time can be derived as follows:

##### 2.2. Pore Size Distribution

Pore size distribution is also important. According to the statistical data by Diamond and Dolch [29] and Hwang and Powers [30], the pore size distribution of the porous media can be simulated by lognormal distribution function. This function is a good way to represent the pore size distribution of the porous media. Thus, the pore space is the generalized lognormal distribution, as follows [29]:where , is the equivalent diameter of pores, and are the distribution parameters characterizing the distribution properties of , is the mean or expectation of the distribution, and sigma is the standard deviation. is the percent volume of voids in diameter . The cumulative distribution function can be expressed as follows:where is the percent volume of voids in the diameters larger than . According to Diamond and Dolch [29], as decreases, pores are more concentrated on the mean or expectation of the distribution ; in addition, the peak of the curve is higher. For the capillary bundle model used in this work, the imbibition volume can be expressed as follows:where is the area of cross section.

##### 2.3. Temperature Influences

Viscosity of the fracturing fluid varies with temperature. Normally, viscosity decreases with the increase in the temperature. For simplicity, many empirical or semiempirical equations (correlations) are proposed to describe the temperature dependence of the fluid viscosity [31–33]. According to the observations on the experimental data, is a linear function of the reciprocal absolute temperature in the low temperature range. For the typical fracturing fluid, the relationship between viscosity and temperature follows the Arrhenius equation [31–33]: where is the activation energy (Arrhenius energy) of the viscous flow, is the temperature of liquid, and is the apparent viscosity, as is the preexponential factor, and is the universal gas constant. The constants can be derived by experiment. Equation (10) can be rewritten as follows:where and are the Arrhenius activation temperature [33]. Taking (3), (6), (7), and (10) into (9), the spontaneous imbibition model considering the reservoir temperature and slippage effect can be derived.

#### 3. Experiment and Validation

We used three samples from the Longmaxi Marine Shale Formation of Lower Silurian in Sichuan Basin to validate our imbibition model. The basic properties of the samples are shown in Table 1. Sample permeabilities are tested by pulse-decay method on an ultralow permeability measurement instrument. Related introductions have been attached in Appendix. Permeability results are listed in Table 1.