## 1. Introduction

Increasing global energy demand and recent advances in drilling techniques for highly deviated wells have accelerated the exploration and development of the carbonate gas reservoir. It has been reported that strong heterogeneity is an important characteristic of carbonate gas reservoirs [

1]. Three types of reservoir space are randomly distributed, including matrix pores, natural fractures and vugs with different degrees of development [

2]. In order to connect as many fracture-vug blocks as possible, highly deviated wells are widely used in carbonate gas reservoirs to improve the production. One of these challenges is how to analyze the pressure performance of highly deviated wells with consideration of these multiple-porosity system. In addition, unlike the conventional carbonate reservoirs, the sizes of pores and vugs are small and the connectivity of fractures is poor in the Gaoshiti-Moxi carbonate gas reservoir of the Sichuan Basin [

3,

4]. There are a series of nonlinear seepage influences such as stress sensitivity effect and threshold pressure gradient, which are caused by the low permeability of the carbonate gas reservoir [

5]. So, another prominent problem is how to deal with the nonlinearity of flow equations coupled with the above two effects.

A lot of models have been proposed to describe the heterogeneity and multiple-porosity characteristic of a carbonate gas reservoir. A dual-porosity model proposed by Warren and Root [

6] was a classical analytical model, which assumed that the reservoir was composed of matrix blocks (high-porosity but low-permeability) and fractures (high-permeability but low-porosity). Then a triple-porosity model was employed to characterize the unsteady-state interporosity flow in a carbonate reservoir [

7]. In this model, two types of matrix blocks were assumed to have different storativity and mobility. Only the single permeability of fractures was considered, that is to say, fractures were the only channels to the wellbore. Similarly, there were triple-porosity dual-permeability models [

8] and dual-fracture models [

9]. Based on these equivalent simplified models, the pressure transient analysis in a carbonate reservoir was presented with an analytical method. Multiple media models were widely used in reservoirs with fractures, like shale gas reservoir [

10]. However, for a carbonate reservoir, the distribution of the fracture and hole is strongly heterogeneous and random. This kind of simple equivalent treatment will result in a large deviation. So, some numerical models were proposed to characterize the discrete fractures and vugs. Chen Peng et al. [

11] studied the effects of location and size of vugs on pressure performance with the boundary element method. He Jie et al. [

12] presented a transient flow model coupling Darcy and Stokes flow in different area of medium. The model was solved by finite difference method and several applications were verified based on realistic geologic model. Some attempts on reservoir numerical simulation were performed for numerical well testing and secondary oil recovery [

13,

14]. These numerical models were required to depict the shape of various mediums and discretize them. It is too time-consuming and complicated to use in practice. To reduce computational complexity, some semi-analytical models were proposed to deal with multiple-porosity mediums, including linear composite models and radial composite models. Medeiros Flavio [

15] presented a linear composite model, which divided the reservoir into a series of continuously distributed rectangular blocks. The nature of each homogeneous block is different from that of the others. The pressure transient responses of the horizontal well in the heterogeneous formation were analyzed and the flux distributions along the horizontal well were compared. Olarewaju J S et al. [

16,

17] proposed a radial composite model for fractured well or high negative skin well. Based on this model, a number of investigators extended the scope of composite model. Wei M et al. [

18] presented a Blasingame production decline analysis method for multi-fractured horizontal wells with considering stimulated reservoir volume (SRV) in shale gas reservoir. Zeng et al. [

19] obtained a semi-analytical solution of multi-fractured horizontal well using a composite model. Zhang L et al. [

20] established a multi-region radially heterogeneous model for a vertical well with non-uniform thickness. Most of these studies have focused on vertical wells or multi-fractured horizontal wells, which could be simplified to two dimensions. However, for a highly deviated well, there is also the vertical flow besides 2D planar flow. It is necessary to study the pressure responses of highly deviated wells in a carbonate gas reservoir with consideration of multiple-porosity features.

The results of laboratory experiments on carbonate rocks show that porosity and permeability are sensitive to effective pressure [

21]. Wu H et al. [

22] studied the low-velocity nonlinear flow in carbonate rocks and the results showed that there was a pseudo-threshold pressure gradient during the low-velocity flow. To analyze the performance in a carbonate reservoir, Wang K et al. [

23] presented a semi-analytical model of a highly deviated well in a carbonate reservoir and obtained the solution of pressure and production based on the triple-porosity assumption. Wang Y et al. [

24] analyzed the transient pressure responses of multi-fractured horizontal wells in a triple media carbonate reservoir. These studied did not take stress sensitivity and threshold pressure gradient into account. Wei M et al. [

25] and Meng F et al. [

26] analyzed the effect of stress sensitivity on production performance, respectively, in multi-fractured horizontal wells and deviated wells. Although stress sensitivity was also considered in the model presented by Zhang L et al. [

20], the effect of threshold pressure gradient was not taken into account. Thus, they were not suitable for carbonate gas reservoirs with threshold pressure gradient.

Above all, previous studies have the following three problems: (1) They either only considered the composite model or only considered the dual-porosity media, and no combination of the two was found. (2) Most of these studies have focused on vertical wells or multi-fractured horizontal wells, which could be simplified to two dimensions. However, for highly deviated well, it involves three-dimensional flow. (3) The coupling effect of stress sensitivity and threshold pressure gradient were not taken into account. Many previous studies have considered only one non-Darcy flow effect. It is very challenging to incorporate multiple-porosity features of carbonate gas reservoirs and evaluate the pressure performance of highly deviated wells with consideration of stress sensitivity effect and threshold pressure gradient. This research mainly aims to solve the above three problems and find a way to deal with the nonlinearity of flow equation. In order to describe the pressure dynamics of highly deviated wells in carbonate reservoir more accurately and analyze the degree of stress sensitivity and threshold pressure gradient effectively, a mathematical model should be established with consideration of multiple-porosity features and the non-Darcy effect. In this paper, a dual-porosity composite model is stablished to study the pressure performance of highly deviated well in carbonate gas reservoir. Source function, Stehfest numerical inversion, Laplace transformation, Fourier transform and the perturbation method are employed to solve the mathematical model. The semi-analytical solution is obtained, and typical curves are plotted. The validity of the proposed model is verified through the comparison of transient pressure curves with results of simplified model. Moreover, the effects of relevant factors on the pressure performance are studied.