## Abstract

To know the dynamic characteristics of shale gas reservoirs’ production and to improve shale gas well production, it is very important to research on shale gas seepage mechanism and production evaluation. According to the shale gas seepage mechanism, adsorption and desorption characteristics, the diffusion mechanism and mass conservation theory in shale gas development, the dual pore medium shale gas reservoir mathematical model is set up.

#### Authors

##### Yin Daiyin, Wang Dongqi, Zhang Chengli and Duan Yingjiao

*Northeast Petroleum University, Key Laboratory of Enhanced Oil and Gas Recovery of Ministry of Education, Daqing, China.*

Received: April 21, 2015 Revised: May 17, 2015 Accepted: June 01, 2015.

© Daiyin *et al.*

The mathematical model is built by the finite difference method based on starting pressure gradient, slippage effect and the isothermal adsorption principle, then program to solve it. Analyze the Langmuir volume, Langmuir pressure, starting pressure gradient and slippage coefficient and other factors on the impact of shale gas wells production.

## INTRODUCTION

The exploration and development of the shale gas, a kind of unconventional oil and gas resources [1-3], is getting more and more attention in the oil industry. Global shale gas resource is about 456.24 10^{12}m^{3}. Shale gas resources in our country is very abundant, which is estimated about 26x10^{12}m^{3} by Zhang Jinchuan. Shale gas is unconventional gas generated by mudstone and shale under various geological conditions, and mainly reserved in free and absorbed states, which is the rise of high quality clean energy in recent years. The occurrence forms includes adsorbed, free and dissolved states, and the adsorption state account for 20% ~ 80% of shale gas [4], so it is very significant for effective exploitation of shale gas to research seepage mechanism and migration characteristics. Shale gas production research can predict shale gas well production, determine production dynamic characteristics, optimize well pattern arrangement and completion program to lay the foundation on gas well proration and gas reservoir engineering study. So considering starting pressure gradient, slippage effect, adsorption and desorption characteristics, seepage mechanism and the law of conservation of mass, use finite difference method to build numerical model and program the corresponding program to solve it, then analyze the various factors on the impact of shale gas production [5].

## SEEPAGE MECHANISM

Shale gas reservoir is a typical self-generated and self-stored reservoir forming patterns, and main occurrence forms are adsorbed and free states [6, 7]. Adsorbed states mainly reserves in the surface of clay and organic matter particles, and free states reserve in shale matrix pore and fracture. However the dissolved gas content is less, dissolved in kerogen and asphaltene.

Shale reservoir, a porous medium, mainly is divides into four parts, respectively no-organic substrate, organic kerogen, natural fracture and hydraulic fracturing cracks.

Shale gas flow mechanism mainly includes the following three process [8-10]: (1) when the formation pressure drop to desorption pressure in mining development, gas desorbs step by step from shale substrate surface becoming free gas;(2) as the shale gas desorbs, internal and surface concentration difference is generated between interior and surface of matrix system, and desorption gas enters into natural or hydraulic fracturing cracks by diffusion;

(3) gas seeps in the natural and artificial fracturing cracks, eventually into the wellbore.

### Mathematical Model Establishment

When shale gas seepage mathematical model is set up, the following basic assumptions are needed: (1) the shale gas reservoir is uniform distribution of dual porosity reservoir; (2) rock, gas can be compressed; (3) each point' temperature is constant in the shale gas reservoir, the isothermal seepage process; (4) single-phase gas flow ignoring gravity, and capillary force.

(1) The equations of motion considered starting pressure gradient and slippage effect

The gas seepage law is not in conformity with the darcy law in low permeability and porosity shale gas reservoir. Besides viscous resistance, the absorbed layer resistance is overcome. When plus pressure gradient is higher than the starting pressure gradient, gas begins to flow; otherwise the flow does not occur; when pressure gradient is lower, it will deviation from linear. At present, use a simplified model of starting pressure gradient [11, 12]:

Seepage velocity influenced by slipping effect:

Combining type (1) with type (2) can get seepage velocity:

(2) Shale gas absorption equation [13]

Isothermal adsorption equation:

The diffusion equation:

Diffusion amount equation:

(3) Shale gas state equation [14, 15]

Gas compressibility:

Pore compressibility:

Comprehensive compressibility:

(4) The basic seepage equation

The matrix system:

The fracture system:

According to the law of conservation of mass, integrating all kinds of equation can get shale gas seepage equation:

Initial conduction:

The boundary conditions:

Outer boundary conditions:

(5) Finite difference on seepage equation

By not equidistant subdivision for gas reservoir seepage areas, use block center grid system:

if

the outer radius of the *i *piece

the inside radius

, outer boundary

Both the matrix and fracture equations for each grid do finite difference, and tri-diagonal equations about fracture system pressure can be got through after taking finite difference for matrix system pressure into fracture system pressure. Then based on solution condition, calculate production after getting the formation pressure.

Differential equation for matrix system:

Differential equation for fracture system:

Where,

(6) Production calculation for some time

By the seepage formulas:

Integrate type (19):

For stable seepage, viscosity μ, Z is μ, *Z *corresponding to average pressure. Get the result by integrating type (20):

Basing on state equation can get:

Combining (1) with (2) can get gas volume flow formula corresponding to standard state:

Do dimensionless for production and time into:

Where *v*: seepage velocity, m/s; *k*: permeability, μm^{2}; *μ*: ∂p/∂r pressure gradient, MPa/m; η- starting pressure gradient, MPa/m; *V**E*: isothermal adsorption quantity, m^{3}/m^{3}; *V**L*: Langmuir volume; *P**L*: pressure corresponding to 50% of maximum absorption gas content, MPa; *P**g*: gas pressure, MPa; *V**m*: average absorption gas content, m^{3}/m^{3}; *D**m*: diffusion coefficient; *F**s*: shape factor; π : absorption time constant; *q _{m}*: diffusion amount, m

^{3}/(m

^{3}·d);

*Fg*geometrical factor; b: slipping coefficient; h: thickness, m; M: shale gas molar mass; p

_{m}: matrix pressure, MPa; p

_{f}fracture pressure , MPa; Q: production, MPa; p

_{w}: bore pressure, MPa;

*p*

_{e}: boundary pressure, MPa;

*R*: general gas constant, getting 0.008314MPa·m

^{3}/(kmol·k);

*r*: supply radius, m;

_{e }*r*: bore radius, m;

_{w }*r*: the distance between the center of wellbore and some point of formation;

*S*: skin factor;

*T*: absolute temperature, K;

*t*: production time,

*d*;

*Z*: compressibility factor; φ: porosity;

*N*: total grid number;

*n*: time step; Subscript

*sc*, D: respectively representing standard conditions and dimensionless quantity.

### Verification of the Model and Error Analysis

After the prediction model is established, program to calculate it. The model is verified for shale gas wells in some field. Comparing predicting curve and practical curve can we see two curves match well, average production relative error is 3.82%.

The main cause of error has the following three aspects: do mean processing for reservoir parameters controlled by shale gas wells, such as thickness, porosity and so on, leading to errors in calculating production; cannot make a clear distinction between the artificial and natural fracture permeability, leading to errors; The each grid con ductivity for artificial fracture is different, so fracture spread conductivity should be improved. During calculating the result is nearly consistent with production practice, so as to satisfying the engineering calculating need for some important parameters fitting the measured data (Fig. **1**).

**Fig. (1****)****. **The result of predicting production.

## INFLUENCE FACTOR

### Slipping Effect Influence

Draw IPR curves corresponding to different slip coefficient (Fig. **2**), it is reflects that under the same pressure difference, as the slip coefficient increase the production increase. Gas well production capacity is increased. In low flow pressure phase, production is greatly influenced by slippage, and at a high flow pressure it is not obvious.

**Fig. (2****)****. **IPR curve for fracturing wells in different slip coefficients.

### Starting Pressure Gradient

As the starting pressure gradient increases, the gas well production decreases. gas wells produce gas only when production pressure difference must be greater than start the pressure difference, which better illustrates gas well resuming production after shutting needs starting pressure difference as Fig. (**3**).

**Fig. (3****)****. **The starting pressure gradient influence.

**The Langmuir Pressure Influence**

It is reflects that under the same Langmuir volume, as the Langmuir pressure increase the pressure spread is slower, and production decrease is less as Fig. (**4**).

**Fig. (4****)****. **The Langmuir pressure influence.

### The Langmuir Volume Influence

It is reflects that under the same Langmuir pressure, as the Langmuir volume increase the pressure spread is slower. Further contrast can be seen that the Langmuir volume influence is linear, and the Langmuir pressure influence is non-linear as Fig. (**5**).

**Fig. (5****)****. **The langmuir volume influence.

## CONCLUSION

Shale gas reservoir permeability and low porosity are very low. In gas seeping, considering start-up pressure gradient is more close to the actual stratum, and considering the adsorption and desorption effect makes the gas well production decline slow and the production time longer.

Establishing the double medium model of considering slippage effect, can accurately describe desorption process, crossflow between matrix and fracture systems, and low speed non-darcy seepage in the fracture system.

And the model is applied to the production practice verified that it can meet the needs of engineering calculation, the relative error is less than 5%.

By analysis of geological parameters and laboratory test can conclude that Langmuir volume influence is linear, and the Langmuir pressure influence is nonlinear.

#### CONFLICT OF INTEREST

The authors confirm that this article content has no conflict of interest.

#### ACKNOWLEDGEMENTS

This work was financially supported by the Natural Science Foundation of China under Grant (No. 51474071).

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