# Analytical modeling of gas production rate in tight channel sand formation and optimization of artificial fracture

- Ruifei Wang
^{1, 2}, - Hongqing Song
^{2, 4}Email author, - Hewei Tang
^{2}, - Yuhe Wang
^{3}, - John Killough
^{2}and - Gang Huang
^{4}

**Received: **7 October 2015

**Accepted: **18 April 2016

**Published: **27 April 2016

## Abstract

Permeability variation in tight channel sand formation makes an important role in gas production. Based on the features of channel sand formation, a mathematical model has been established considering anisotropy of permeability. The analytical solutions were derived for productivity of both vertical wells and vertically fractured wells. Simulation results show that, gas production rate of anisotropic channel sand formation is less than that of isotropic formation. For vertically fractured well, artificial fracture direction, drainage radius, permeability ratio and fracture half-length have considerable influence on production rate. The optimum fracture direction should be deviated less than π/8 from the maximum permeability direction (or the channel direction). In addition, the analytical model was verified by in situ measured data. The research provides theoretical basis for the development of tight channel sand gas reservoirs.

### Keywords

Tight sandstone gas Channel sand formation Permeability anisotropy Analytical solution## Background

With development of fracturing technology, unconventional oil and gas production are growing fast (Clarkson et al. 2012; Ahn et al. 2014; Shanley et al. 2004; Dai et al. 2012; Rivard et al. 2014). Tight gas reservoirs are widespread in many major basins in China such as the Ordos Basin, the Sichuan Basin, the Bohai Bay Basin, the Songliao Basin, and the Junggar Basin (Zhu et al. 2012). Tight sandstone gas reservoirs are characterized by low porosity, high heterogeneity, extensive hydrocarbon generation, short-distance petroleum migration and complex formation pressure system (Wu et al. 2012; Zou et al. 2014; Deveugle et al. 2014; Wei et al. 2014).

The typical tight sandstone gas reservoir, represented by braided river sedimentary sediments, has various sedimentary micro-facies including braided bar, channel filling, flood basin and crevasse splay. Effective reservoir is mainly coarse sand deposited in braided bar or channel filling. According to the core analysis data, those coarse sand in braided bar and channel filling has the excellent permeability (0.8–2.0 × 10^{−3} μm^{2}) with 35–40 % thickness of the total sand. The sediment with braided bar micro-facies contributes about 75 % of the coarse sand and the rest are mostly from channel filling (Xie et al. 2015; Wei et al. 2014; Zhao et al. 2010). The sands deposited along the channel direction, which is the paleo-current direction, and have relatively high permeability (0.5–2.0 × 10^{−3} μm^{2}). Sands deposited perpendicular to the direction of the channel has low permeability (0.06–1.0 × 10^{−3} μm^{2}). This phenomenon demonstrates high permeability anisotropy in sand reservoir with fluvial deposition.

The permeability anisotropy is the basic property of channel sand formation, especially for those with coarse fluvial sand. It leads to certain negative influence on the reservoir development index prediction. Conventionally numerical simulation methods were implemented to model the impact of permeability anisotropy (Bai et al. 1993; Liu et al. 2013; Aghighi and Rahman 2010). The numerical simulation methods fall into two categories: (1) Numerical values of the three-dimensional three-phase anisotropy permeability are applied in simulation model. The finite difference method and complete implicit iterated method are used. (2) Stochastic simulation method based on geological statistics, which describes the anisotropy of permeability and obtains the effective permeability (Wang et al. 2011; Khan and Teufel 2000; Farrell et al. 2014; Fauzi 2011; Hajizadeh et al. 2011). There are numerous challenges to apply the numerical simulation in real cases. The simulation results are highly dependent on the mesh generation, while a coarse grid model brings uncertainty to detect and describe the geological heterogeneity (Song et al. 2015). In addition, in order to obtain an accurate prediction, it requires to adjust various factors for historical fitting. The high demand for calculation and adjusting parameters limits the further development of the numerical simulation method.

Analytical method has the advantages over numerical simulation method by directly deriving the relationship among major parameters. It simplifies the calculation procedure, with advanced methodology to describe the percolation theory and exploitation law (Zhang et al. 2008; Song et al. 2014). Currently, most analytical methods are derived in ideal conditions, which cannot reflect the influence of anisotropy of permeability.

What this paper intends to do are as follows: (1) based on the features of channel sand formation, establish a mathematical model considering artificial fracture; (2) derive analytical solutions for productivity of both vertical wells and vertical fractured wells; (3) analyze the influence of production parameters on production rate.

## Geological description and physical model

If the vertical well was used to exploit gas from the sandstone gas reservoir, it is easy to use Darcy’s Law to calculate the production rate. However, gas flow in fracture may not satisfy the Darcy’s Law. As shown in Fig. 1, the elliptical flow could play an important role in the performance of fractured tight gas well. We use elliptical flow to reflect the non-Darcy effect in artificial fracture. If fracturing is applied to enhance the gas recovery, the final production rate will be influenced by the direction of fracture (α) due to the anisotropy of the permeability. Fracture direction α is the angle between fracture and the direction of largest permeability in the x–y coordinate system.

In order to get the analytical solutions of differential equations modeling flow in vertical fractured well, two coordinate systems were set up. The x–y coordinate system is the absolute coordinate system, and the x-axis is paralleling to the direction of largest permeability. The other system x′–y′ is the ellipse coordinate, and the x-axis is paralleling to the long axis of ellipse. In addition, \(\varphi\) is the angle between fracture and x-axis in the x–y coordinate system, and θ is the angel fracture and y′-axis in the x′–y′ coordinate system. Obviously, \(\varphi\) is the summation of θ and α.

- 1.
The reservoir is anisotropic and horizontally infinite with uniform thickness of

*h*. - 2.
The gas flows in terms of planar flow in steady state with constant viscosity.

- 3.
The fracture has negligible width and uniform height the same to reservoir thickness.

## Mathematical model and analytical solutions

### Production rate of a vertical well

For linearization of Eq. (12), assume that \(\mu \phi\) equals approximately to \(\overline{\mu \phi }\) which is value under average formation pressure \(\bar{p}\).

*r*

_{e}is the drainage radius of the elliptical flow,

*r*

_{w}is the radius of the well. And

*m*

_{e}is the pseudo-pressure on the boundary of elliptical flow,

*m*

_{w}is the pseudo-pressure of the well.

*k*can be described by \(k_{{\rm max} }\), \(k_{{\rm min} }\) under the polar coordinates as follows:

*x*and

*y*directions, as shown in Fig. 1.

### Production rate of a fractured vertical well

*x*axis direction, as shown in Fig. 1, Cartesian coordinates, (

*x*′,

*y*′) can be transformed into elliptical coordinates, using the following relationship:

*L*(length of fracture) as focal length (Lou et al. 2013). Assume that the production rate in the elliptical area of the fractured vertical well will follow Darcy’s law:

*θ*and eccentric angle \(\eta\) of an ellipse:

*θ*

*a*and

*b*in the equation are to be determined from boundary conditions

*p*is only a function of \(\xi\), we introduce a modified variable \(\bar{r}\) in Darcy’s law, as shown in Eq. (20), where \(\bar{r}\) is defined as:

### Calculation processes

Based on the known parameters, permeability distribution should be first derived by Eq. (22), different formation permeability ratio and fracture direction will directly influence the distribution of permeability. Pressure distribution can be obtained by Eqs. (23) and (24), the values of drainage area and fracture half-length will influence the parameter of Eq. (23), and thus influence the initial pressure distribution. With permeability and pressure distribution being settled, a loop computing of production rate, following Eq. (29), is introduced to obtain the production rate at each pressure drawdown. Finally, the data are output and drawn as diagrams.

## Results and discussion

Basic parameters for numerical simulation of a channel sand reservoir

Parameters | Value (unit) | Parameters | Value (unit) |
---|---|---|---|

Reservoir thickness | 10 (m) | Pressure at boundary | 20 (MPa) |

Gas viscosity | 0.01 (mPa s) | Bottom hole pressure | 12 (MPa) |

Reservoir temperature | 383 (K) | Drainage radius | 1000 (m) |

Temperature at std. state | 293 (K) | Maximum permeability | 5 × 10 |

Pressure at std. state | 0.1 (MPa) | Minimum permeability | 0.1 × 10 |

Gas Z factor | 0.89 | Fracture half-length | 100 (m) |

Gas Z factor std. state | 1 |

### Comparison between vertical well and fractured vertical well

### Comparison between isotropic and anisotropic formation

### The influence of fracture direction

### The influence of drainage radius

As drainage radius increases, the production rate at first declines rapidly, then declines more smoothly. Besides, the pressure drawdown has no big influence on the decline trend. It is because of the fact that with pressure drawdown settled, the energy for fluid flow is as well steady. As drainage radius increase, the energy gradient via space will decrease, so will reduce the production rate.

### The influence of permeability ratio

As permeability ratio increases (keeping the average permeability is the constant), the production rate will decrease. At first, the production rate declines with permeability ratio rapidly, then the production rate becomes almost constant. In other words, the production rate has a big relationship with maximum and minimum permeability ratio when keeping the average permeability is the constant. As we known, the permeability ratio means the homogeneousness of the tight gas reservoirs. Therefore, the permeability is more homogeneous when the ratio is smaller, whereas it is more heterogeneous. It can be seen that the production rate of gas in the homogeneous reservoir is bigger than in the heterogeneous one. Plus, as pressure drawdown increases, the production rate of gas will also increase.

### The influence of fracture half-length

### In-situ measurement of production rate

Comparison of measured data and calculated results

Production well no. | Pressure drawdown/MPa | Tested Max. permeability/m | Permeability ratio | Measured production rate/10 | Calculated production rate/10 | |||
---|---|---|---|---|---|---|---|---|

Anisotropic | Difference rate/% | Isotropic | Difference rate/% | |||||

Shan 319 | 7.59 | 0.663 × 10 | 9 | 33.08 | 37.45 | 13.2 | 39.65 | 19.9 |

Shan 356 | 12.01 | 0.789 × 10 | 2 | 50.39 | 54.07 | 7.3 | 59.63 | 18.3 |

Shan 361 | 10.19 | 0.054 × 10 | 7 | 10.24 | 11.65 | 13.8 | 13.02 | 27.1 |

Su 124 | 7.99 | 0.42 × 10 | 6 | 36.95 | 40.85 | 10.6 | 43.36 | 17.3 |

Su 192 | 7.94 | 0.06 × 10 | 8 | 11.44 | 12.39 | 8.3 | 13.62 | 19.1 |

## Conclusions

- 1.
The permeability anisotropy of channel sand formation, mathematical models have been established for both vertical wells and vertical fractured wells. Analytical solutions for production rate considering elliptical flow were derived. For vertical fractured wells, numerical methods were developed to analyze the effect of fracture direction, drainage radius, permeability ratio and fracture half-length on production rate.

- 2.
For both vertical wells and vertical fractured wells, the production rate of anisotropic formation will be less than formation with isotropic permeability.

- 3.
Both the formation properties such as drainage radius and permeability ratio and the fracture properties such as its direction and half-length will influence the production rate considerably. The maximum production rate can be gained when the fracture direction is along or less than π/8 from the maximum permeability.

- 4.
Some actual field measured data were listed and compared with the calculated results. It is shown that the difference rate between measured data and calculated results is acceptable in engineering. Therefore, the analytical model in sandstone gas reservoir can be used in gas recovery.

The combined method of analytical and numerical used by this paper provided a good example of studying this kind of anisotropic tight gas channel sand reservoir.

## Declarations

### Authors’ contributions

RW proposed this topic and provided the actual data of Sulige Gas field; HS and JK supervised the work; HT and YW was in charge of model establishment and analytical solution; GH organized the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

We gratefully acknowledge the PetroChina Innovation Foundation under Grant No. 2015D-5006-0106, the National Nature Science Foundations of China under Grant 51404024 and 51104119 for financial support. We also express gratitude to researchers of the Sulige Gas field for their cooperation.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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