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To improve the carbonate gas reservoir development and production, highly deviated wells (HDW) are widely used in the field. Production decline analysis of HDW is crucial for long-term gas reservoir development. However, it is a new challenge to incorporate the complex pore structure of naturally fractured-vuggy carbonate gas reservoirs and evaluate the production performance of HDW. This paper presents a semianalytical model to analyze the pressure and production behavior of HDW in naturally fractured-vuggy carbonate gas reservoirs, which consist of fractures, vugs, and matrix. The primary flow occurs only through the fracture and the outer boundary is closed. Introducing pseudopressure and pseudotime, the Laplace transformation, Fourier transformation, and its inverse and Stehfest numerical inversion were employed to establish a point source and line source solutions. Furthermore, the validity of the proposed model was verified by comparing a field data from the Arum River Basin in Turkmenistan. Finally, the effects of major parameters on the production decline curves were analyzed by using the proposed model and it was found that they had influences at different stages of gas production history and the sensitivity intensity of each parameter was different. With its high efficiency and simplicity, this semianalytical model will serve as a useful tool to evaluate the well production behavior for the naturally fractured-vuggy carbonate gas reservoirs.

Carbonate gas reservoirs are widely distributed throughout the world. In these carbonate reservoirs, the original gas in place (OGIP) and gas production account for more than half of gas reserves and production in the world [

In the past decades, a lot of researchers reported some relevant works. The theory and applications on dual-porosity flow model for fractured reservoirs have been well researched [

Reservoirs composed of matrix, fracture, and vugs are called triple-porosity systems. A lot of triple-porosity models have been proposed through analytical, semianalytical, and numerical methods for analyzing gas flow in fractured-vuggy carbonate reservoirs. With regard to analytical methods, a triple-porosity and single-permeability model was first proposed by Abdassah and Ershaghi [

A substantial amount of research has focused on the pressure behavior and production performance of deviated wells. Abbaszadeh and Hegeman first proposed analytical solution for the pressure drawdown behavior of a deviated limited-entry well with different combinations of boundaries. The authors used the method of source and Green’s functions to derive these solutions [

Currently, only few suitable analytical models exist to evaluate the production behavior of HDW in fractured-vuggy carbonate gas reservoirs. In this study, a semianalytical flow model was established by employing Laplace transformation, Fourier transformation and inversion, Stehfest numerical inversion algorithm, and point source function. Based on the proposed model, the effects of major parameters on production performance were studied.

Figure

HDW are located in a laterally infinite gas reservoir.

The gas reservoir is assumed to be horizontal with equal thickness and have two impermeable boundaries at the top and bottom.

During the well production, gas flows only through the fracture system into the wellbore; interporosity flow occurs from the vugs and the matrix systems to the fracture system.

Flow is single phase, isothermal and follows Darcy's law. Gravitational and capillary effects are negligible.

The pressure is uniform initially throughout the gas reservoir and is equal to

The HDW produces at a constant rate

The permeability

Schematic diagram of HDW in a fractured-vuggy carbonate gas reservoir.

Flow sketch map for a cell [

In this work, the interporosity flow from vugs and matrix to fractures is assumed to be pseudosteady state. Combining the continuity equation with transport equation as well as the equation of state and the introduction of pseudopressure and pseudotime (the detailed procedure for pseudopressure and pseudotime is documented in Appendix

For the matrix system the equation is

For the vugs system the equation is

where

The initial pseudopressure

The outer boundary is closed:

At the inner boundary, there is a continuous point source (

where

At the top and bottom, the boundaries are impermeable:

where

Based on the definitions of dimensionless variables in Table

Definitions of dimensionless variables.

Dimensionless variables | Definitions |
---|---|

Dimensionless pseudopressure | |

Dimensionless pseudotime | |

Dimensionless radius | |

Dimensionless formation radius | |

Dimensionless distance of | |

Dimensionless distance of | |

Dimensionless distance of mid-perforation in | |

Dimensionless distance of mid-perforation in | |

Dimensionless formation thickness | |

Dimensionless vertical distance | |

Dimensionless vertical distance of mid-perforation | |

Dimensionless infinitesimal variable | |

Dimensionless deviated well length | |

Dimensionless storativity ratio | |

Dimensionless interporosity flow coefficient | |

Dimensionless production rate | |

The dimensionless governing equation of fracture, matrix, and vugs systems are

Transformed initial condition:

Transformed outer boundary condition:

Transformed inner boundary condition:

Transformed top and bottom boundaries:

Laplace transformation, Fourier transformation, and inversion were employed to solve the dimensionless model (Eq. (

where

where

To obtain the pressure solution of HDWs, the wellbore was treated as a uniform flux line source. It was assumed that there was an infinitesimal point on the HDW. Based on the principle of superposition for a point source, integration was carried out along the deviated line for the line source solution. Cinco (1975) successfully got the

where

Based on the principle of superposition, the pressure solution with wellbore storage and skin effect in Laplace domain can be obtained as

where

According to Van Everdingen and Hurst [

Equation (

Field data from the Arum River Basin in Turkmenistan was used to validate the proposed model and to demonstrate its practical application. Arum River Basin is a large scale sedimentary basin, which is located between the border of Turkmenistan and Uzbekistan. The reservoir bed possesses carbonate rock that is formed by postdepositional diagenesis, including dissolution and dolomitization. Vugs, fractures, and dissolution pores are highly dispersed in the reservoir. The details of relevant parameters needed for the model evaluation are listed in Table

The values of relevant parameters for verification and sensitivity analysis.

Parameters | Value |
---|---|

Wellbore radius, | 0.0889 |

Formation radius, | 400 |

Formation thickness, | 30 |

Fracture permeability, | 1 |

Interporosity flow coefficient between fractures and vugs, | |

Interporosity flow coefficient between fractures and matrix, | |

Fractures porosity, | 0.002 |

Matrix porosity, | 0.04 |

Vugs porosity, | 0.017 |

Inclination angle, | 75 |

Length of the highly deviated well, | 300 |

Initial reservoir pressure, | 24 |

Bottom-hole pressure, | 20 |

Reservoir temperature, | 381.15 |

Gas production rate, ^{3}/d) | 600,000 |

Gas viscosity, | 0.02 |

Gas compressibility factor, ^{3}/m^{3}) | 0.97 |

Figure

Comparing simulated results and production rates of Arum River Basin HDW.

where

Moreover, it could be seen that the trend of simulated production rate in the following 1000 days and even longer predicted the gas production trend for the future with a constant bottom-hole pressure. Based on this application, it is confirmed that the proposed model can be used to describe the gas production behavior of HDW in naturally fractured-vuggy carbonate gas reservoir. Therefore, the estimated input parameters of the model will be used as the basis for the sensitivity analysis.

In order to understand the production behavior with triple-porosity flows, different flow regime analysis was conducted to study the effects of various flow mechanisms on the overall production behavior. Figure

Typical pressure curves for HDW in fractured-vuggy carbonate gas reservoir under closed circle boundary.

Based on the field data, the production sensitivity analyses for different key parameters are discussed. In this section, the HDW produces with a constant bottom-hole pressure. The key parameters include formation thickness, fracture permeability, inclination angle of the HDW, formation radius, interporosity flow coefficient between fractures and vugs and vugs storativity ratio. The influences of these parameters are apparent and their estimates are essential for future production analysis and forecasting.

Figure

Effect of formation thickness on production decline curves.

From Figure

Effect of fracture permeability on production decline curves.

Figure

Effect of gas reservoir radius on production decline curves.

Figure

Effect of interporosity flow coefficient between fractures and vugs on production decline curves.

Effect of vugs storativity ratio on production decline curves.

Figure

Effect of inclination angle on production decline curves.

According to Figures

Effect of parameters at

parameters | 10 days | 100 days | 1000 days | |
---|---|---|---|---|

| | | ||

| 1/0.5 | 22.1 | 18.4 | 18.2 |

| 800/400 | 3.8 | 12.8 | 20.9 |

| 0.0002/0.00002 | 4.6 | 1.1 | 1.2 |

| 0.1/0.01 | 3.7 | 7.5 | 5.9 |

| 80/40 | 23.1 | 21.8 | 18.7 |

Based on Table

Trends of percentage difference of parameters by extracting production rate at

This research investigated the production decline behavior of highly deviated well (HDW) in naturally fractured-vuggy carbonate gas reservoir. The validity of the proposed model was verified by matching it with the production history of a field case from the Arum River carbonate gas reservoir in Turkmenistan. Furthermore, the effects of relevant parameters on production performance were studied. The main conclusions drawn are as follows:

The proposed production semianalytical model could be used to predict the gas production trend with a constant bottom-hole pressure.

Pressure type curves for HDW in naturally fractured-vuggy carbonate gas reservoir were divided into five flow stages. Among them, the second stage was dominated by the inclination angle (

Using the proposed model, sensitivity analyses were conducted and it was found that formation thickness (

In (

Equations (

The dimensionless governing equations of fractures, matrix, and vugs systems in the Laplace domain are as follows:

The outer boundary condition is transformed as

The inner boundary condition is transformed as

The top and bottom boundaries are transformed as

where

To eliminate the variable

where

By employing the Fourier cosine transform (Eq. (

where

The outer boundary condition is transformed as

The inner boundary condition is transformed as

Then (

The general solution of (

Based on the outer and inner boundary conditions (Eq. (

Wellbore storage coefficient,

Total compressibility,

Average relative error, %

Formation thickness,

Permeability of vugs system,

High deviated well length,

Pseudopressure,

Well bottom-hole pseudopressure,

Pressure,

Well bottom-hole pressure,

Pressure at standard condition,

Production rate from point source,

Gas production rate,

Simulated production rate from the proposed model,

Field production rate,

Radial distance,

Wellbore radius,

Formation radius,

Skin factor

Laplace transform variable

Pseudotime, day

Pseudotime, day

Reservoir temperature,

Temperature at standard condition,

Directional coordinates

Distance of mid-perforation in x, y, and z coordinates,

Z-factor of gas, dimensionless

Shape factors of vugs and matrix,

Interporosity flow coefficient, dimensionless

Storativity ratio, dimensionless

Inclination angle, degree

Porosity, fraction

Gas viscosity,

Constant,

Vugs system

Fractures system

Vugs system

Horizontal direction

Initial condition

Initial condition

Vertical direction

Dimensionless.

Laplace domain

Fourier domain

Field production data.

The field data from the Arum River Basin in Turkmenistan used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors would like to acknowledge the support provided by Xin Zhao and Jun Shi in the initial stages of this study. This article is funded by National Science and Technology Major Project of China (