A mathematical model for the gas-water two-phase flow in tight gas reservoirs is elaborated. The model can account for the gas slip effect, stress sensitivity, and high-speed non-Darcy factors. The related equations are solved in the framework of a finite element method. The results are validated against those obtained by using the commercial software CMG (Computer Modeling Group software for advanced recovery process simulation). It is shown that the proposed method is reliable. It can capture the fracture rejection characteristics of tight gas reservoirs better than the CMG. A sensitivity analysis of various control factors (initial water saturation, reservoir parameters, and fracturing parameters) affecting the production in tight gas wells is conducted accordingly. Finally, a series of theoretical arguments are provided for a rational and effective development/exploitation of tight sandstone gas reservoirs.

The recoverable reserves of conventional oil and natural gas continue to decrease. Unconventional oil and gas resources, especially shale gas, have gradually become a hot topic for researchers [

Fractured horizontal wells can make up for the deficiencies of vertical wells in some aspects. Especially in developing tight gas reservoirs, fractured horizontal wells have shown their unique advantages [

As mentioned above, analytical and numerical models are the two most widely used models for fractured horizontal well capacity prediction [

A great deal of previous work has been done to improve mathematical models for multistage fractured horizontal wells in unconventional gas reservoirs (e.g., tight gas reservoirs). They have been developed mainly for the single-phase flow of natural gas [

Based on the flow characteristics of different media systems in low permeability tight reservoirs modified by fractured horizontal wells [

In addition, many literatures have performed parametric sensitivity analyses. The influencing factors have not been comprehensively considered [

Next, this paper is based on the characteristics of dense sandstone gas reservoirs. Based on the discrete fracture model. A mathematical model of gas-water two-phase nonlinear seepage is established. The model integrates the matrix system slip effect, stress sensitivity, and high-speed non-Darcy flow in the artificial fracture system. The model was solved using the finite element method. In

In this paper, we assume that the original tight gas reservoir has no microfracture development. It is a homogeneous reservoir. A multi-stage fractured horizontal well modifies the reservoir. The fluid seeps from the reservoir matrix into the artificial fractures. Finally, it flows to the wellbore through the hydraulic fractures. The model sets the external boundary of the tight gas reservoir as a closed boundary. The well type is set as a multi-stage fractured horizontal well, as shown in

Multi-scale flow mechanisms are considered. Slip effects are considered for gas-phase flows in matrix systems. Stress-sensitive and high-velocity non-Darcy effects are considered for the artificial fracture system. The following assumptions are also given for the above model for calculation purposes:

There are only two types of fluids in the gas reservoir. There is a gas phase and a water phase. And the gas phase is insoluble in the water phase.

The rock and water are slightly compressible. And the compression coefficient is constant. Gas-phase is compressible.

The effect of the capillary force of gas and water phases is neglected.

The effect of gravity is neglected.

The flow process is isothermal percolation.

The source-sink term is considered. Based on the principle of conservation of matter, the continuity equation of the gas-water two-phase is obtained by using the infinitesimal unit analysis from reference [

Gas-phase

Water phase

The gas and water saturation satisfy the following equation:

The capillary pressure satisfies the following equation:

Here, the subscript

For the two-dimensional matrix system, the gas slip effect [

To simplify the model, we assume that the capillary force profile of the matrix system is a function depending on the saturation [_{c}

For the fracture system, the description is proposed to use the

where

In addition, the effective stress on the fracture increases as the formation pressure decreases, and the flow conductivity of the fracture decreases a lot, so it is necessary to consider the presence of stress sensitivity in the fracture system. The apparent permeability of the fracture system considering the stress-sensitive effect is given by^{−1};

The water saturation in the fracture can be expressed by the water saturation in the matrix, then the differential equation for gas-water two-phase flow in a discrete fracture model for a multi-stage fractured horizontal well in a tight gas reservoir can be abbreviated as

The gas-phase pressure equation for the matrix is

The equation for the water phase saturation of the matrix system is

The pressure equation in the fracture system is

In the above equation

The three flow equations above are solved by the finite volume method, and the specific steps to derive the equivalent ∭ integral weak form are as follows, taking

Applying Gauss’s theorem to the Darcy formula term on the left side of

By approximating the pressure gradient as the quotient of the pressure difference

In the above equation,

The cumulative term on the right-hand side of

Solving them sequentially yields

To verify the correctness of the model proposed in this paper and the reliability of the calculation results. In this paper, the daily production data of a fractured gas well in a tight gas reservoir are fitted to the history. And the calculation results of this simulator are compared with those of the commercial software CMG. The burial depth of the model is about 3280 m∼3740 m. Other essential parameters are shown in

Parameter | Value | Parameter | Value |
---|---|---|---|

Gas reservoir temperature | 70°C | Number of artificial fracture stages | 4 |

Permeability of matrix system | 0.01 mD | Original water saturation | 0.45 |

Relative density of gas-phase | 0.65 | Artificial fracture half-length | 50 m |

Porosity of matrix system | 10% | Well radius | 1 m |

Artificial fracture width | 0.003 m | Artificial fracture inflow capacity | 1 D⋅cm |

Water viscosity | 0.3457 mPa⋅s | Epithermal factor | 1 |

Rock compression factor | 1.5e−4 MPa^{−1} |
Formation water compression | 5.8e−4 MPa^{−1} |

The original water saturation (Sw) of a reservoir is an important parameter in the development of oil and gas fields. During the development process, the value of water saturation is constantly changing. For tight gas reservoirs containing lateral and bottom water. It is particularly important to consider the effect of water saturation on production. To investigate the effect of raw water saturation on the production of tight gas reservoirs. In this paper, the basic parameters of the conceptual model are kept constant. Only the pristine water of the reservoir is changed. The production dynamics of a multi-stage fractured horizontal well in a tight gas reservoir are simulated for 300 days under different pristine water saturation conditions. The specific scenarios are initial water saturation of 0.55, 0.65, and 0.75.

As shown in ^{3}/day, respectively. When Sw is 0.65, the value is 50,000 m^{3}/day. When Sw is 0.75, the value is 205,200 m^{3}/day. This indicates that increasing reservoir pristine water saturation increases the resistance to gas flow. This resulted in a decrease in gas production from the wells. Specifically from the daily water production curve, when Sw is 0.75, the daily water production after 100 days of production is about 3 m^{3}/day.

The effect of matrix permeability on the production of fractured horizontal wells in tight gas reservoirs is investigated. In this paper, while keeping the basic parameters of the conceptual model unchanged, only the matrix permeability parameters were changed. The matrix permeability was set to 0.0001, 0.001, 0.01 and 0.1 mD to simulate the production dynamics of multi-stage fractured horizontal wells in tight gas reservoirs for 300 days.

Fracture conductivity (Fcd) is an important parameter in fracturing construction. It is defined as the product of artificial fracture closure and artificial fracture permeability. It is mainly influenced by proppant type, fracture closure pressure, and fluid properties. To investigate the effect of artificial fracture inflow on the production of fractured horizontal wells in tight gas reservoirs. In this paper, while keeping the basic parameters of the conceptual model unchanged, only the parameter of artificial fracture inflow is changed. In this paper, the production dynamics of a multi-stage fractured horizontal well in a tight gas reservoir are simulated for 300 days under different artificial fracture inflow capacities. The specific scenario is to set the inflow capacity to 1, 5, 20, and 30 D⋅cm.

From

Through the above research, the paper mainly draws the following conclusions:

The mathematical model established in this paper considers the gas-slip effect of the matrix system, the stress sensitivity of the artificial fracture system, and the high-speed non-Darcy phenomenon. It can more accurately describe the dense gas flow characteristics in the production process. Compared with the commercial numerical simulation software CMG, the proposed method is more reliable and effective.

In this paper, the gas-water flow law is analyzed under different initial water saturation. The initial water saturation was set to 0.55, 0.65 and 0.75. The initial daily production rate decreased by 10.7% and 29%, respectively. It was found that the decrease in matrix permeability led to a reduction in both daily gas and daily water production. This paper also simulates the effect of different fracture permeability on its production. The production was found to decrease gradually with the increase of fracture permeability.

Although the seepage law of tight gas reservoirs is systematically considered in this paper. However, capillary force and gravity are not considered. It still has some limitations.