## 1. Introduction

Shale gas reservoirs have received increasing attention due to the depletion of conventional oil and gas resources [

1]. According to the roadmap on natural gas development in China, the total production of natural gas will reach 4.5 - 5 × 10

^{12} m

^{3} in 2020 and the proportion of unconventional gas will then exceed 30% of the total gas production [

2]. Due to the low permeability, high water saturation, and complex pore structure of shale gas reservoirs, the gas flow in a low-pressure gradient zone is always slow and non-Darcy [

3,

4,

5,

6]. This low-velocity non-Darcy flow has been emphasized in low-permeability reservoirs for the following two issues [

3,

7,

8], which are still unsolved:

(A) Does a threshold pressure gradient (TPG) exist in low-permeability reservoirs?

(B) How is the non-Darcy flow manifested in low-permeability reservoirs?

For the first issue, many experimental results [

7,

9,

10] have shown that there is indeed a resisting force to prevent the fluid from the flow initiation until the pressure difference across the core reaches a critical value. For the second issue, some typical low-velocity non-Darcy flow models have been proposed and evaluated [

3,

5,

7,

11]. For example, Cai [

3] developed a fractal approach for the low-velocity non-Darcy flow in a low-permeability porous medium. He incorporated the fractal characteristics of complex pore structure into the analysis of the TPG. However, his model did not consider the effect of residual water saturation. Li et al. [

5] numerically investigated the effect of the TPG on transient pressure and still missed the mechanism for the determination of the TPG. Zhu et al. [

7] conducted an experiment to measure the TPG in water-bearing tight gas reservoirs and their results observed that the TPG increased with water saturation, but their experiment did not consider the effect of pore structure on the TPG. Zhang et al. [

11] stated that gas flow was subjected to more non-linear multiphase flow due to water saturation. Their model used a non-Darcy coefficient under constant water saturation that did not vary with the complex pore structure. Among these models, the threshold pressure gradient (TPG) has been formulated in a simple manner and is feasible for characterizing the non-Darcy flow in low-permeability reservoirs. However, the combined effects of pore pressure, water saturation, and complex pore structure on the TPG in low-permeability porous media have not been fully investigated.

Figure 1 shows a typical schematic diagram for the low-velocity non-Darcy flow with TPG, which has been modified after the literature [

8,

12,

13,

14]. This curve can be divided into three zones according to its flow mechanism. In zone 1, the viscous force between the fluid and the solid is dominant. The pressure difference applied across the core is not sufficient to drive fluid flow. As the pressure difference continues to increase until the critical point C (defined as zone 2), a highly nonlinear relationship is observed between the fluid flow rate and the pressure difference. In addition, gas slip occurs in zone 2, thus resulting in a decrease of gas effective permeability. This may also be an important contributor to the nonlinear flow. After zone 2, a pseudo-linear flow follows in zone 3. If the straight line on the linear flow is extended to the abscissa, the pseudo threshold pressure gradient is obtained. This value is widely used in the non-Darcy flow model and is called the TPG [

15].

Miller and Low [

15] discovered the existence of the TPG in the water flow through the soils with rich clay content. Based on the flow experiments in sandstones, Prada and Civan [

16] developed the following empirical expression for TPG:

where

$k$ is the permeability of porous medium and

$\mu $ is the fluid viscosity. This expression indicates that the TPG is related to the ratio of permeability of porous media to fluid viscosity or mobility. Wang et al. [

17] proposed another empirical formula for heavy oil flow based on the Kozeny–Carman equation:

where

${\tau}_{0}$ represents the yield stress and

$\varphi $ is the porosity of rock.

There is an approximately linear relationship between the TPG and yield stress, if the ratio of permeability to porosity of porous medium is constant. More TPG correlations are summarized in

Table 1. Moreover, Liu et al. [

18] derived an analytical solution for one-dimensional flow with the TPG in a semi-infinite porous medium. The results showed that the effect of the TPG on the distribution of the pore pressure was more significant when the boundary condition was a constant pressure instead of a constant flow rate. Zhu et al. [

7] experimentally investigated the existence of TPG using cores with different water saturations from the Sichuan gas field in China. Song et al. [

4] developed a low-velocity non-Darcy model with the TPG. The results showed that the theoretical model considering the TPG was more accurate in predicting gas production and had better agreement with the experimental data. Ding et al. [

9] suggested that permeability, water saturation, and pore pressure were the controlling parameters for the TPG in tight gas reservoirs. Their results showed that the increase in pore pressure caused a significant decrease in the TPG. Tian et al. [

6] experimentally revealed that the TPG increased exponentially with a higher water saturation or lower permeability. The above-mentioned laboratory investigations verified the existence of the TPG and some theoretical models with controlling parameters have been proposed. However, the effects of complex pore structure in low-permeability porous media on the TPG were not investigated, although the TPG was very sensitive to these tiny pore-throats. Therefore, the TPG should be carefully studied through the analysis of complex pore structures.

On the other hand, fractal theory is an effective tool to characterize the complex pore geometry and structure [

22,

23]. Yu and Cheng [

24] developed a fractal model to evaluate the permeability for both porous particles and porous fabrics media. However, this model was only applicable to Newtonian fluids and cannot take the effect of the TPG into account. Wang and Yu [

25] established a fractal model for Bingham fluids in porous media embedded with a fractal-like tree network. The effect of diameter ratio at different branching lengths on the TPG was investigated, however, this fractal model could not describe the flow non-linearity induced by the existence of water saturation. Cai [

3] pointed out that the scaling behavior between the TPG and the permeability of porous medium could be further expressed by the tortuosity fractal dimension D

_{T},

$TPG~{k}^{-{D}_{T}/(1+{D}_{T})}$. Ye et al. [

26] proposed a fractal model for the TPG in tight oil reservoirs. In this model, the residual water saturation was treated as connate water by forming a boundary layer fluid, which could hinder the gas flow. Their results indicated that a higher residual water saturation would greatly increase the TPG, especially when the permeability was less than 0.01 mD. Although the above literature presented different TPG models, the coupling of multi-scale geometry and physical mechanisms in the TPG is still unclear. Therefore, it is necessary to investigate the flow mechanism of the TPG with complex pore structure and to explore its application in numerical simulations.

This study aimed to develop a reliable semi-empirical formula for the TPG in a numerical model of non-Darcy flow that can also be used to provide a better understanding of flow mechanism in porous media with complex pore structures. The rest of this paper is organized as follows. First, a semi-empirical formula is proposed considering permeability, water saturation, and pore pressure. Then, this semi-empirical formula is verified using three sets of experimental data. Third, a fractal model of capillary tubes is developed to link the TPG formula with the micro-structural parameters of porous media, residual water saturation, and capillary pressure. Finally, conclusions are drawn in

Section 4.