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 Fig. 1.

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:(1)

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 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 [14].

Gas-phase(1) ∇[kkrgμgBg∇pg]+qgBg=∂∂t(ϕSgBg) where k is the permeability, mD; krg is the relative permeability of the gas phase, mD; μg is the viscosity of the gas, mPa⋅s; pg is the gas pressure, MPa; Bg is gas compressible coefficient; ϕ is the porosity; Sg is the gas saturation.

Water phase(2) ∇[kkrwμwBw∇pw]+qwBw=∂∂t(ϕSwBw) where krw is the relative permeability of the gas phase, mD; μw is the viscosity of the gas, mPa⋅s; pw is the gas pressure, MPa; Bw is the gas compressible coefficient; Sw is the gas saturation.

The gas and water saturation satisfy the following equation:(3) Sw+Sg=1

The capillary pressure satisfies the following equation:(4) Pc(Sw)=Pg−Pw where Pc is a capillary force between the gas and water phase.

Here, the subscript m represents the bedrock system, f represents the fracture system, w represents the water phase, and g represents the gas phase.

For the two-dimensional matrix system, the gas slip effect [28] is considered, and the apparent permeability is introduced as(5) ke=k(1+bp¯) where b parameters are determined mainly by experiments; km is the reservoir permeability without correction

To simplify the model, we assume that the capillary force profile of the matrix system is a function depending on the saturation [29], which is neglected in the fractured system.(6) pc=−Bcln⁡(Se)Se=Sw−Srw1−Srg−Srw where B_c is capillary force parameters; Srw is residual water saturation; Srg is residual gas saturation.

For the fracture system, the description is proposed to use the Forchheimer formula considering the high-speed non-Darcy phenomenon generated by the high-speed motion of the fluid. Neglecting the non-Darcy percolation of the water phase then, the non-Darcy correction factor of the gas phase under multiphase flow conditions is assumed to be ξn [30].

(7) vn=ξnkkrgμg∇p(n=x,y,z)

where vn the gas flow velocity.

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(8) Ke=Ki(1+bp¯)e−αf(pi−p) where Ki is absolute reservoir permeability, mD; α is sensitivity index, MPa⁻¹; Pi is original formation pressure, MPa.

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(9) M∇(kmekrgμgBg∇Pgm)+∇[kkrwμwBw∇pgm]−∇[kkrwμwBw∇pcm]−δqmfT∗=Mp∂pgm∂t

The equation for the water phase saturation of the matrix system is(10) ∇[kmekrgμgBg∇pgm]−δqmfg∗=−Cg1∂Swm∂t+Cg2∂pwm∂t

The pressure equation in the fracture system is

(11) M∂∂l(kkrgfμgf∂pgf∂l)+∂∂l(kkrwfμwf∂pgf∂l)+Qwf+Qgf+δqmfT∗=Mp∂pgf∂t

In the above equation M=(Cw1−Cw2dpcdSw)Cg1 , Cg1=ϕBg , Cg2=ϕSgBg(Cg+Cϕ) , Cw1=ϕBg , Cw2=ϕSwBw(Cw+Cϕ) , qmfT∗=qmfw∗+qmfg∗ are total tamper flow. δ={10Source and sinkNo

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 Eq. (9) as an example.(12) ∫∫∫Vi⁡M∇(KmeKrgμgBg∇Pgm)dVi+∫∫∫Vi⁡∇[KKrwμwBw∇pgm]dVi−∫∫∫Vi⁡∇[KKrwμwBw∇pcm]dVi=∫∫∫Vi⁡(Mp∂pgm∂t+δqmfT∗)dVi

Applying Gauss’s theorem to the Darcy formula term on the left side of Eq. (12).(13) ∫∫∫Vi⁡M∇(KμB∇P)dVi=∫∫◯s⁡KμB∇P⋅nds

By approximating the pressure gradient as the quotient of the pressure difference ∇P between grids i and j and the node distance, Eq. (13) can be rewritten as(14) ∫∫∫Vi⁡M∇(KμB∇P)dVi=∑jTij(Pj−Pi)

In the above equation, Tij is the flow coefficient, defined as Tij=(KμB)Aijlij .

The cumulative term on the right-hand side of Eq. (12) can be rewritten as(15) ∫∫∫Vi⁡(Mp∂pgm∂t+δqmfT∗)dVi=ViΔt[(Mppgm)n+1−(δqmfT∗)n+1+(δqmfT∗)n] where the superscript n represents the current time step, and n + 1 represents the next time step.

Solving them sequentially yields pgm , Swm and pgf .

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 Figs. 6 and 7, the decreasing rates of daily gas production and daily water production still exhibit a fast and slowly decreasing rate. In addition, a single well’s daily gas production decreases as the basal raw water saturation increases. In contrast, the daily water production of single well increases. When Sw is 0.55, the daily gas production at the beginning of production is 280,000 m³/day, respectively. When Sw is 0.65, the value is 50,000 m³/day. When Sw is 0.75, the value is 205,200 m³/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 m³/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.

Figs. 8 and 9 show the production dynamics of fractured horizontal wells in tight gas reservoirs under different matrix permeability conditions. Figs. 8 and 9 show the effect of reservoir matrix permeability on production. With the increase of reservoir matrix permeability, daily gas and water production will increase significantly. The daily gas and water production curves corresponding to different matrix permeabilities are compared. It can also be found that the smaller the matrix permeability is, the less significant its effect size on the fractured horizontal wells in the matrix pore space of tight gas reservoirs. This is mainly because when the reservoir matrix permeability is too small, the flow of tight gas in the matrix pore space is complicated. This makes the matrix contribution to production smaller.

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 Figs. 10 and 11, the daily gas production and daily water production of fractured horizontal wells in tight gas reservoirs increase with the increase of artificial fracture inflow. Specifically, observe the cumulative production curve. It can be found that: the cumulative production increases less and less with the increase of fracturing inflow. When Fcd is 10 D⋅cm and Fcd is 30 D⋅cm, the cumulative production curves overlap. In addition, the decreasing rate of daily gas and water production still shows the characteristics of fast and then slow.