Introduction

As global efforts intensify to balance energy supply and demand, unconventional energy sources have gained increasing research interest. This is due to their substantial geological reserves and significant potential for development (Islam et al. 2014; Zhang H et al. 2017; Caineng et al. 2018; Jiao et al. 2019). Tight sandstone gas reservoirs, in particular, have shown promise. In their initial development stages, these reservoirs maintain significant pressure and energy. This ensures stable fluid extraction from the well bottom. However, problems arise as the reservoir develops. Its pressure decreases, leading to a reduction in gas production and a decline in the recovery rate. Most existing research in gas–water two-phase flow theory has focused on specific areas. These include gas–water two-phase flow equations, steady-state inflow dynamics in production wells, and the derivation of mathematical models for well tests and production analysis (Xiaoping and Birong 1999; Sun et al. 2018a; Wang W et al. 2019; Wang Q et al. 2020; Shi H et al. 2021; Mo et al. 2022; Yong et al. 2022). To accurately predict production behavior in tight sandstone gas reservoirs, it is ideal to integrate a geological model and develop a fully-calibrated, full-physics reservoir simulation model (Cheng et al. 2013; Jianlong et al. 2015; Guo et al. 2017; Sun et al. 2018b; Huang X et al. 2019; Luo H et al. 2019). However, numerical simulations carry limitations, including significant computational costs and time requirements (Zhang X et al. 2009; Song et al. 2017; Mutailipu et al. 2017, 2018). In contrast, the productivity equation, an analytical tool, can predict production and provide direct insights into the development of tight gas wells, making its development of great practical significance. Therefore, numerous scholars have made various improvements to the production model. For example, they have introduced new fluid flow models to explain the behavior of fluid transport at macro, micro, and meso scales (Aldhuhoori et al. 2021a, 2021b; Alkuwaiti et al. 2021; Belhaj et al. 2019). These improvements also include optimization of the traditional Darcy model to more accurately predict Darcy and non-Darcy flow characteristics in porous media (Belhaj et al. 2003a, 2003b, 2003c, 2003d; Al Hameli et al. 2022b). Some research has also introduced additional parameters into the existing theoretical models to more precisely simulate the complex fluid flow behavior in reservoirs (Nouri et al. 2003; Haroun et al. 2009; Belhaj et al. 2013; Shi S et al. 2018; Adegbite et al. 2021; Al Hameli et al. 2022a; Prempeh et al. 2022). However, these models generally face a challenge: they require a large amount of experimental data for validation and support in practical applications, which increases the difficulty of their implementation (Akilu et al. 2021).

This paper aims to provide a more scientific and rational theoretical foundation for the development planning of tight sandstone gas reservoirs. We propose a new two-phase flow production equation under pseudo-steady-state conditions. Additionally, we introduce a novel non-Darcy gas–water two-phase production prediction model that considers stress-sensitive conditions. Utilizing existing gas–water two-phase flow theory, the model can be solved using optimal fitting methods. By using only production data, it is possible to determine a single well's control reserves, relative permeability profiles, and formation pressure distribution, as well as to make predictions about the well's production. It also helps in understanding the variation of formation water saturation in gas wells. The findings from this study promise to be invaluable for predicting production and understanding the development of tight gas wells.

Gas–water two-phase flow model for a single well

Joshi suggested that during gas reservoir development, a single well can be approximated as a pseudo-Darcy flow (Joshi 1988). Based on this concept, the following assumptions have been made for the gas–water two-phase flow prediction model:

  1. (1)

    The reservoir fluid is assumed to be isothermal.

  2. (2)

    Only gas and water phases are present in the gas reservoir.

  3. (3)

    The impact of gravity and capillary forces has been neglected.

  4. (4)

    The slip effect and start-up pressure have not been considered in the gas–water two-phase flow model. These assumptions have been made to simplify the model and make it more practical to use for prediction and analysis purposes.

Motion equation for the gas–water two-phase flow

According to Darcy's law of percolation, the equation of motion of the gas and water phases can be expressed as follows:

Gas phase:

$$v_{g} = - \frac{{kk_{rg} }}{{\mu_{g} }}\nabla p$$
(1)

Water phase:

$$v_{w} = - \frac{{kk_{rw} }}{{\mu_{w} }}\nabla p,$$
(2)

where vg, vw are gas-phase and water-phase percolation velocities, 10−3 m/s; μg, μw are gas-phase and water-phase viscosities, mPa·s; krg, krw are gas-phase and water-phase relative permeabilities; k is the absolute permeability, D; \(\nabla p\) is the pressure gradient, MPa/m.

Continuity equation for the gas–water two-phase flow

During the flow process, conservation of matter within the formation unit leads to the establishment of the continuity equation, which connects the inflow and outflow masses within a dt time frame. For the gas and water phases, the continuity equations can be represented as:

$$- \left( {\frac{{\partial \rho_{i} v_{x} }}{\partial x} + \frac{{\partial \rho_{i} v_{y} }}{\partial y} + \frac{{\partial \rho_{i} v_{z} }}{\partial z}} \right) = \frac{{\partial \varphi S_{i} \rho_{i} }}{\partial t},\;\;i = g,w.$$
(3)

By dividing this equation by ρi and converting it to sub-surface volume, the gas–water two-phase continuity equation can be expressed using the Hamiltonian operator. This results in the following equation:

$$- \nabla \left( {\frac{{v_{i} }}{{B_{i} }}} \right) = \frac{\partial }{\partial t}\left( {\frac{{\varphi S_{i} }}{{B_{i} }}} \right),\;\;i = g,w,$$
(4)

where ρi is the density of gas or water phase, kg/m3; Bi is the volume factor of gas or water phase, m3/m3; Si is the saturation of gas or water phase, decimal; φ is the porosity of rock, %.

State equation for the gas–water two-phase flow

The state equation for real gas can be formulated as:

$$pV_{g} = nZRT,$$
(5)

where p is pressure, MPa; Vg is the molar volume of gas, cm3; n is Mole; Z is the gas compressibility factor; and R is the gas constant.

This state equation can be converted into a representation of subsurface volume as follows:

$$\rho_{g} = {{\gamma_{g} M_{air} p} \mathord{\left/ {\vphantom {{\gamma_{g} M_{air} p} {ZRT}}} \right. \kern-0pt} {ZRT}},$$
(6)

where γg is the relative density of the gas; and Mair is the relative molecular mass of air.

The saturation constraint equation can be expressed as:

$$S_{g} + S_{w} = 1,$$
(7)

where Sg and Sw are the gas and water saturation, respectively.

Production model for the gas–water two-phase Darcy flow

The gas–water two-phase Darcy flow model is used to calculate gas and water in tight sandstone reservoirs, excluding the influences of the oil phase, water-soluble gas, and condensate. The model assumes linear Darcy flow of the gas and water in the formation and considers the formation fluid to comprise two components: free gas and formation water.

Based on Eq. (1) and Eq. (2), the differential form of the subsurface production formula for radial flow in the gas and water planes can be expressed as follows:

$$q_{g} = - 2\pi rh\frac{{kk_{rg} }}{{B_{g} \mu_{g} }}\frac{dp}{{dr}},$$
(8)
$$q_{w} = - 2\pi rh\frac{{kk_{rw} }}{{B_{w} \mu_{w} }}\frac{dp}{{dr}},$$
(9)

where r is the plane radial flow radius, m; h is the reservoir thickness, m.

By integrating Eq. (8) and Eq. (9), assuming pseudo-steady flow in both the gas and water phases and taking into account the skin factor, we can obtain:

$$q_{g} = \frac{0.00708kh}{{\left( {\ln \frac{{r_{e} }}{{r_{w} }} - 0.75 + S_{a} } \right)}}\int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp,$$
(10)
$$q_{w} = \frac{0.00708kh}{{\left( {\ln \frac{{r_{e} }}{{r_{w} }} - 0.75 + S_{a} } \right)}}\int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp,$$
(11)

where Sa is the skin factor; rw is the radius of the bottom of the well, m; re is the flow radius, m; pr is the average reservoir pressure, MPa.

Defining the gas phase pseudo-pressure and water phase pseudo-pressure as follows:

Gas phase pseudo-pressure:

$$\Delta m(p)_{g} = \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp.$$
(12)

Water phase pseudo-pressure:

$$\Delta m(p)_{w} = \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp.$$
(13)

Concisely, Eq. (12) and Eq. (13) can be expressed as:

$$q_{g} = C\Delta m(p)_{g} ,$$
(14)
$$q_{w} = C\Delta m(p)_{w} ,$$
(15)

where C is the production coefficient, m3/d.

The ratio from Eq. (14) and Eq. (15) can be expressed as the production gas–water ratio:

$$R_{pgw} = \frac{{q_{g} }}{{q_{w} }} = \frac{{C\left[ {\left( {\frac{{kk_{rg} }}{{B_{g} \mu_{g} }}} \right)} \right]}}{{C\left[ {\left( {\frac{{kk_{rw} }}{{B_{w} \mu_{w} }}} \right)} \right]}},$$
(16)

where Rpgw is the production gas–water ratio, m3/m3.

By simplifying Eq. (16), it turns into:

$$R_{pgw} = \frac{{q_{g} }}{{q_{w} }} = \left( {\frac{{kk_{rg} }}{{kk_{rw} }}} \right)\left( {\frac{{B_{w} \mu_{w} }}{{B_{g} \mu_{g} }}} \right).$$
(17)

Using Eq. (17), the effective permeability of the water and gas phases can be expressed as follows:

$$k_{g} = kk_{rg} = R_{pgw} (kk_{rw} )\left( {\frac{{B_{g} \mu_{g} }}{{B_{w} \mu_{w} }}} \right),$$
(18)
$$k_{w} = kk_{rw} = \frac{{kk_{rg} }}{{R_{pgw} }}\left( {\frac{{B_{w} \mu_{w} }}{{B_{g} \mu_{g} }}} \right).$$
(19)

The preceding equations can be combined to yield the following system of equations:

$$\begin{array}{*{20}c} {q_{g} = C\Delta m{(}p{)}_{g} {, }q_{w} = C\Delta m{(}p{)}_{w} } \\ {\Delta m(p)_{g} = \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp} \\ {\Delta m(p)_{w} = \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp} \\ \end{array} ,$$
(20)

The set of Eq. (20) represents a novel model for gas–water two-phase production under pseudo-steady-state conditions. This set of equations can be solved using the relative permeability of one phase and the production coefficient C to determine the relative permeability of the other phase.

Formation pressure distribution

By integrating the equations of motion, state, and continuity, we can construct the mathematical model for a gas–water two-phase stable flow as follows:

$$\nabla \left[ {\frac{{k_{rg} }}{{B_{g} \mu_{g} }}\nabla p} \right] = 0,$$
(21)
$$\nabla \left[ {\frac{{k_{rw} }}{{B_{w} \mu_{w} }}\nabla p} \right] = 0.$$
(22)
$${\text{At the well wall}}:\;p(r_{w} ) = p_{wf} .$$
(23)
$${\text{At the outer boundary}}:\;p(r_{e} ) = p_{e} .$$
(24)

Substituting Eq. (12) into Eq. (21) yields:

$$\nabla^{2} \left( {\Delta m(p)_{g} } \right){ = }\nabla \left[ {\left( {\frac{{k_{rg} }}{{B_{g} \mu_{g} }}} \right)\nabla p} \right] = 0.$$
(25)

Then the mathematical model Eqs. (23)–(25) can be simplified to the following form:

$$\left\{ \begin{gathered} \nabla^{2} \Delta m(p)_{g} = 0 \hfill \\ \Delta m(r_{w} )_{g} = \Delta m_{wf} \hfill \\ \Delta m(r_{e} )_{g} = \Delta m_{e} \hfill \\ \end{gathered} \right..$$
(26)

The differential equation for planar radial steady flow of gas–water two-phase can be expressed as follows:

$$\frac{{\partial^{2} \Delta m_{g} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial \Delta m_{g} }}{\partial r} = 0.$$
(27)

The general solution of the differential equation is expressed as follows:

$$\Delta m_{g} = C_{1} \ln r + C_{2} .$$
(28)

Substituting the boundary conditions yields:

$$C_{1} = \frac{{\Delta m_{e} - \Delta m_{wf} }}{{\ln {{r_{e} } \mathord{\left/ {\vphantom {{r_{e} } {r_{w} }}} \right. \kern-0pt} {r_{w} }}}},\;\;C_{2} = \Delta m_{e} - \frac{{\Delta m_{e} - \Delta m_{wf} }}{{\ln {{r_{e} } \mathord{\left/ {\vphantom {{r_{e} } {r_{w} }}} \right. \kern-0pt} {r_{w} }}}}\ln r_{e} .$$
(29)

After incorporating values of C1 and C2 into Eq. (28), we derive the following expression for the pressure function:

$$\Delta m_{g} = \Delta m_{e} - \frac{{\Delta m_{e} - \Delta m_{wf} }}{{\ln {{r_{e} } \mathord{\left/ {\vphantom {{r_{e} } {r_{w} }}} \right. \kern-0pt} {r_{w} }}}}\ln \frac{{r_{e} }}{r}.$$
(30)

Phase relative permeability curve and water saturation

The flows of gas-phase and water-phase at r can be deduced from the gas–water two-phase flow production equations, as per Eq. (31) and Eq. (32):

$$q_{g} = C\int\limits_{{p_{wf} }}^{p} {\left( {\frac{{kk_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp,$$
(31)
$$q_{w} = C\int\limits_{{p_{wf} }}^{p} {\left( {\frac{{kk_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp.$$
(32)

The empirical equation for the phase relative permeability data is expressed as follows (Huang BG, 2004):

$$k_{rg} = \left( {1 - \overline{S}_{w} } \right)^{2} \left( {1 - \overline{S}_{w}^{{\frac{5 - D}{{3 - D}}}} } \right),$$
(33)
$$k_{rw} = \overline{S}_{w}^{{\frac{11 - 3D}{{3 - D}}}} ,$$
(34)

where \(\overline{S}_{w}\) is the average water saturation of the formation, decimal; D is the phase relative permeability coefficient.

Applying Eq. (33) and Eq. (34), krg/krw can be formulated as a function of water saturation Sw:

$$\frac{{k_{rg} }}{{k_{rw} }} = \frac{{(1 - \overline{S}_{w} )^{2} \left( {1 - \overline{S}_{w}^{{\frac{5 - D}{{3 - D}}}} } \right)}}{{\overline{S}_{w}^{{\frac{11 - 3D}{{3 - D}}}} }}$$
(35)

Jokhio (Jokhio and Tiab 2002) suggested a method to calculate each relative permeability ratio using the gas–water production ratio Rpgw. This approach allows the depiction of krg/krw as a function of Rpgw under gas–water two-phase flow conditions:

$$\frac{{k_{rg} }}{{k_{rw} }} = R_{pgw} \left( {\frac{{B_{g} \mu_{g} }}{{B_{w} \mu_{w} }}} \right)$$
(36)

In Eq. (36), the ratio of gas-phase relative permeability to water-phase relative permeability, krg/krw, is represented as a function of average water saturation and pressure, inclusive of gas and water viscosities (μg and μw) and gas and water formation volume factors (Bg and Bw). Employing the formation pressure p and the production gas–water ratio Rpgw, the values of krg/krw, the average water saturation, and subsequently, the relative permeability of gas and water phases, krg and krw, can be determined. Merging Eq. (35) with Eq. (36) derives Eq. (37).

$$\frac{{(1 - \overline{S}_{w} )^{2} (1 - \overline{S}_{w}^{{\frac{5 - D}{{3 - D}}}} )}}{{\overline{S}_{w}^{{\frac{11 - 3D}{{3 - D}}}} }} = R_{pgw} \frac{{B_{g} (p)\mu_{g} (p)}}{{B_{w} (p)\mu_{w} (p)}}$$
(37)

By uniting Eq. (37) and the pressure distribution function from Eq. (28), the water saturation function appears as follows:

$$\left\{ \begin{gathered} \Delta m_{g} = \Delta m_{e} - \frac{{\Delta m_{e} - \Delta m_{wf} }}{{\ln {{r_{e} } \mathord{\left/ {\vphantom {{r_{e} } {r_{w} }}} \right. \kern-0pt} {r_{w} }}}}\ln \frac{{r_{e} }}{r} \hfill \\ \frac{{(1 - \overline{S}_{w} )^{2} (1 - \overline{S}_{w}^{{\frac{5 - D}{{3 - D}}}} )}}{{\overline{S}_{w}^{{\frac{11 - 3D}{{3 - D}}}} }} = R_{pgw} \frac{{B_{g} (p)\mu_{g} (p)}}{{B_{w} (p)\mu_{w} (p)}} \hfill \\ \end{gathered} \right.$$
(38)

Bottom-hole pressure in gas wells

The bottom-hole pressure in a gas well (Kidnay et al. 2019) can be estimated using Eq. (39):

$$p_{wf} = p_{ts} e^{{\frac{{0.03415\gamma_{g} H}}{{T_{av} Z_{av} }}}} ,$$
(39)

where pwf is the bottom-hole pressure, MPa; pts is the tubing pressure, MPa; H is the depth in the middle of the reservoir, m; Tav is the average temperature in the middle of the reservoir, K; Zav is the gas compressibility factor at average reservoir pressure.

Average water saturation of the formation

In gas reservoirs where both gas and water occupy the pore spaces, the capacity of the reservoir can be described as follows:

$$G = \frac{{\left( {1 - S_{wi} } \right)V}}{{B_{gi} }},$$
(40)

where V is the total pore volume of gas reservoir, m3; G is the reservoir volume, m3; Swi is the original water saturation, decimal; Bgi is the original volume factor, m3/m3.

The volume of water bodies in the gas reservoir is expressed as:

$$W = S_{wi} V$$
(41)

Subsequently, the average water saturation in the gas reservoir at any time during production is given by:

$$\overline{S}_{w} = \frac{{W - W_{p} }}{V} = \frac{{\frac{{S_{wi} }}{{1 - S_{wi} }}GB_{gi} - W_{p} }}{{\frac{1}{{1 - S_{wi} }}GB_{gi} }},$$
(42)

where Wp is the total water production of gas reservoir, m3.

Single well control reserves

The material balance equation for a gas reservoir as follows:

$$\frac{p}{Z}\left[ {1 - C_{c} (p_{i} - p) - \frac{{W_{e} - W_{p} B_{w} }}{{GB_{gi} }}} \right] = \frac{{p_{i} }}{{Z_{i} }}\left( {1 - \frac{{G_{p} }}{G}} \right),$$
(43)

where Cc is the expansion coefficient of the reservoir, MPa−1; We is the volume of water intrusion in the gas reservoir, m3; pi is the initial formation pressure, MPa; Zi is initial gas compressibility factor.

In gas reservoirs, the expansion of gas is considerably more prominent compared to the contributions from formation water and rock, allowing these latter factors to be neglected. Furthermore, the simplified equations do not account for the influx of water at the bottom edge, as the predominant source of water production in dense sandstone gas reservoirs is intra-formation water. Consequently, the equations can be simplified as follows:

$$\frac{p}{Z} = \frac{{p_{i} }}{{Z_{i} }}\left[ {\frac{{1 - \frac{{G_{p} }}{G}}}{{1 + \frac{{W_{p} B_{w} }}{{GB_{gi} }}}}} \right].$$
(44)

The physical parameters of gas and water in the gas–water two-phase Darcy flow model include:

  1. (1)

    The volume factor of natural gas,

  2. (2)

    The viscosity of natural gas,

  3. (3)

    the compressibility factor of natural gas,

  4. (4)

    The volume factor of formation water.

The calculations primarily refer to the PR (Peng-Robinson) equation (Peng and Robinson 1976).

Non-Darcy flow model considering stress sensitivity for gas–water two-phase

Numerous laboratory experiments have unequivocally demonstrated that stress sensitivity has a significant impact on gas–water two-phase flow. Therefore, it becomes paramount to take into account stress sensitivity when formulating a gas–water two-phase flow production prediction model. An exponential relationship between the permeability of the porous medium and effective stress has been identified and can be articulated as follows (Xu et al. 2018; Huang X et al. 2021; Luo X et al. 2022):

$$K(p) = K_{0} e^{{ - \beta p_{eff} }} = K_{0} e^{{ - \beta (p_{i} - p)}} ,$$
(45)

where K0 is the absolute permeability of the reservoir rock at the initial state, D; β is the stress sensitivity coefficient of rock, MPa−1; peff is the effective stress of the formation, MPa.

Expanding the exponential term e−β(pi−p) to the first two terms of the Maclaurin series, we obtain:

$$K(p) = K_{0} e^{{ - \beta (p_{i} - p)}} \approx K_{0} \left[ {1 + \beta (p - p_{i} )} \right].$$
(46)

Therefore, the equations of motion for the gas and water phases considering stress sensitivity are as follows:

$$v_{i} = - \frac{{K_{0} \left[ {1 + \beta_{i} (p - p_{i} )} \right]k_{ri} }}{{\mu_{i} }}\nabla p,\;\;i = g,w$$
(47)

The differential form of the radial flow in the gas phase plane for Eq. (47):

$$q_{g} = - 2\pi rh\frac{{K_{0} \left[ {1 + \beta_{g} (p - p_{i} )} \right]k_{rg} }}{{B_{g} \mu_{g} }}\frac{dp}{{d{\text{r}}}}.$$
(48)

This calculation enables us to establish the gas production equation, taking into account stress sensitivity:

$$q_{g} = C\left( {\int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp - \beta_{g} \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{(p_{i} - p)k_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp} \right).$$
(49)

Defining the gas-phase proposed pressure function as follows:

$$\Delta m(p)^{\prime}_{g} = \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{(p_{i} - p)k_{rg} }}{{B_{g} \mu_{g} }}} \right)} dp.$$
(50)

Combining Eq. (12), Eq. (49), and Eq. (50) yields:

$$q_{g} = C\left[ {\Delta m(p)_{g} - \beta_{g} \Delta m(p)^{\prime}_{g} } \right].$$
(51)

The water production equation considering stress sensitivity can be expressed as follows:

$$q_{w} = C\left( {\int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp - \beta_{w} \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{(p_{i} - p)k_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp} \right).$$
(52)

Similarly, the proposed pressure function for the water phase can be defined as:

$$\Delta m(p)^{\prime}_{w} = \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{(p_{i} - p)k_{rw} }}{{B_{w} \mu_{w} }}} \right)} dp.$$
(53)

Combining Eq. (13) and Eq. (53) yields the water production equation:

$$q_{w} = C\left[ {\Delta m(p)_{w} - \beta_{w} \Delta m(p)^{\prime}_{w} } \right].$$
(54)

The equations for pressure, water saturation, production coefficient, and water and gas production for the gas and water phases are combined to derive the production equation that takes stress sensitivity into account.

$$\begin{aligned} q_{i} &= C\left[ {\Delta m(p)_{i} - \beta_{i} \Delta m(p)^{\prime}_{i} } \right] \\ \Delta m(p)_{i} &= \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{k_{ri} }}{{B_{i} \mu_{i} }}} \right)} dp \\ \Delta m(p)^{\prime}_{i} &= \int\limits_{{p_{wf} }}^{{p_{r} }} {\left( {\frac{{(p_{i} - p)k_{ri} }}{{B_{i} \mu_{i} }}} \right)} dp \\ \end{aligned} ,\;\;i = g,w$$
(55)

Equation (55) signifies that the individual production equation for each phase in the gas–water system can be derived solely through the use of the gas phase relative permeability krg, water phase relative permeability krw, production coefficient C, and stress sensitivity coefficient β.

Solution of the model

Utilizing production data from a sample well located in the western region of the Sichuan Basin, China, the gas–water two-phase flow production model for a single well can be partitioned into two key sections: data fitting and production prediction.

Production data fit

The goal of the model is to fit the production coefficient C, phase relative permeability coefficient D, stress sensitivity coefficient β, and single well dynamic gas reserves G with the measured values. The optimal values of these parameters are obtained by seeking the best fit between theoretical and measured values, which can be expressed as follows:

$$\sum\limits_{{i^{\prime} = 1}}^{N} E = \sum\limits_{{i^{\prime} = 1}}^{N} {\left( {q_{{gsci^{\prime}}} \left( {C,D,\beta_{g} ,G} \right) - q_{{gsci^{\prime}}} } \right)^{2} } + \sum\limits_{{i^{\prime} = 1}}^{N} {\left( {q_{{wsci^{\prime}}} \left( {C,D,\beta_{w} ,G} \right) - q_{{wsci^{\prime}}} } \right)^{2} } ,$$
(56)

where qgsci’ (C, D, βg, G), qwsci’ (C, D, βw, G) are the fitted gas production and water production, m3/d, respectively; qgsci’, qwsci’ are the actual gas production and water production, m3/d, respectively; E is the fitted objective function.

The above equation is a non-linear least squares problem, demanding an automatic fitting method to locate a set of optimal parameters that minimize the objective function. Figure 1 presents a block diagram of the gas–water two-phase non-Darcy flow model software.

Fig. 1
figure 1

Block diagram for solving the gas–water two-phase non-Darcy flow model

Full size image

The gas–water two-phase non-Darcy flow model can be calculated using the following steps:

  1. (1)

    Input key parameters such as original formation pressure, formation temperature, well depth, initial water saturation, oil pressure, actual gas production, actual water production, and gas components of the gas well.

  2. (2)

    Determine bottom-hole pressure pwf from the wellhead oil pressure.

  3. (3)

    Calculate total gas and water production from the actual production data, establish current formation pressure pr using Eq. (44), and compute formation fluid physical parameters.

  4. (4)

    Set initial values for production coefficient C, phase relative permeability coefficient D, stress sensitivity coefficient β, and dynamic gas reserves G, and compute the average water saturation of the formation based on the water saturation expression.

  5. (5)

    Determine the daily gas and water production of the gas well using Eq. (55).

  6. (6)

    Compare the model-predicted daily gas and water production with the actual production data, and calculate the deviation \(E_{{i^{\prime}}}\). Ascertain whether the total number of calculated deviations matches the production duration of the gas well. If not, return to step 5.

  7. (7)

    Change the current deviation \(\sum\nolimits_{{i^{\prime} = 1}}^{N} {E_{{i^{\prime}}}^{j - 1} }\) with the production coefficient C, stress sensitivity coefficient β, phase relative permeability coefficient D, and dynamic gas reserves G. If the current primary deviation \(\sum\nolimits_{{i^{\prime} = 1}}^{N} {E_{{i^{\prime}}}^{j} }\) is less than a specific deviation \(\left| {\sum\nolimits_{{i^{\prime} = 1}}^{N} {E_{{i^{\prime}}}^{j - 1} } - \sum\nolimits_{{i^{\prime} = 1}}^{N} {E_{{i^{\prime}}}^{j} } } \right| < error\), exit the cycle. Otherwise, adjust the value of C, D, G, β, and repeat step 4.

Production data forecasting

The steps for implementing the non-Darcy flow model are as follows:

  1. (1)

    Input the basic parameters. These should include the original formation pressure, formation temperature, well depth, original water saturation, oil pressure, actual water production, and gas components of gas wells.

  2. (2)

    Input the cumulative gas and water production. Subsequently, calculate the current formation pressure pr based on Eq. (44), and solve for the gas–water physical parameters.

  3. (3)

    Utilize historical fitting data to ascertain key coefficients and reserves. These include the production coefficient C, phase relative permeability coefficient D, stress sensitivity coefficient β, and dynamic gas reserves G for the gas well. Also, compute the average water saturation in the gas well based on the relevant water saturation expression.

  4. (4)

    Calculate the bottom-hole pressure pwf from the wellhead oil pressure using Eq. (39) for constant oil pressure production.

  5. (5)

    Evaluate the relative permeability for the gas and water phase using the relative permeability calculation expression, and calculate the gas and water production from the gas well for constant oil pressure production.

  6. (6)

    Export the final results.

Example calculation and analysis

The basic parameters of the sample wells are shown in Table 1.

Table 1 Parameters of the sample wells
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The natural gas components are shown in Table 2.

Table 2 Parameters of gas components of the sample
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The phase diagram of the sample can be obtained by solving the PR equation for the gas component parameters in Table 2, as shown in Fig. 2. The two-phase critical temperature of this sample is 200.003 K, the two-phase critical pressure is 5.351 MPa, the critical condensation pressure is 5.501 MPa, and the temperature at the critical condensation pressure is 203.442 K. These values classify this sample as a dry gas reservoir.

Fig. 2
figure 2

Phase diagram of the sample

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Based on the known formation parameters listed in Table 1, the pressure–volume-temperature (PVT) properties of the gas were calculated using the PR equation of state. The resulting p/Z, viscosity, compressibility factor, and volume factor curves of the sample gas, as a function of pressure, are depicted in Fig. 3.

Fig. 3
figure 3

PVT properties diagram of sample gas

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The p/Z, viscosity, compressibility factor, and volume factor of the sample gas varied with pressure, as shown in Fig. 3. Production data from the sample wells were used to fit the stress-sensitive model. This fitting process determined the production coefficient C, the phase relative permeability coefficient D, and the dynamic gas reserves G for the sample wells. The results of the fitting, are shown in Table 3.

Table 3 Table of production coefficients, phase relative permeability coefficients and dynamic gas reserves of sample wells
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The Darcy flow model and non-Darcy flow model were used to fit and predict the daily and total gas and water production data of the sample wells, generating Fig. 4. The total gas and water production obtained with the Darcy flow model were 14.01844 million m3 and 3394.613 m3, respectively, with deviations of 0.0062% and 0.328%. The Darcy model assumes laminar flow within the reservoir, which may not always represent the complex flow dynamics, especially when considering high flow rates or low permeability regions. However, the Darcy flow model only provides an overall average view of well development, resulting in significant deviations in the daily production data. In contrast, the non-Darcy flow model incorporates stress-sensitive coefficients βg and βw. These coefficients consider the impact of turbulence and inertial effects, which become significant in high rate flows and near-wellbore regions, leading to non-linear flow behavior. By integrating these coefficients, the model can adapt to dynamic reservoir conditions and account for the deviations from the Darcy flow. This allows real-time revisions to be made during development. This model not only satisfies the total production (with deviations of 0.2396% and 0.2114% for total gas and water production, respectively) but also provides high accuracy for the daily production data (with an average absolute deviation of 22.8%, an average relative deviation of 0.1815%, and an average absolute deviation of daily production of 41.45% and average relative deviation of − 0.2677%). The current recovery level of the sample wells is 78.78%, and both models predict a higher recovery level after 1000 days of development, with the gas reservoir predicted to be 82.43% (Darcy flow model) and 88.40% (non-Darcy flow model). It's worth noting that these predictions, especially from the non-Darcy model, reflect a more realistic assessment of the reservoir's response over time by accounting for intricate flow mechanics. The average water saturation of the gas reservoir is currently 39.798%, and the two models predict a slightly lower saturation after 1,000 days of development, with average water saturation of 39.453% (Darcy flow model) and 39.281% (non-Darcy flow model). The simulation and analysis of the sample well's water intrusion distribution curve is shown in Fig. 4, based on the relevant results of the sample parameters.

Fig. 4
figure 4

Graph of model fitting and prediction results

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Figure 5 shows the relationship between the lifting radius and the permeability recovery, indicating that as the lifting radius increases, the permeability recovery shows rapid growth initially, then levels off and stabilizes at around 2 m. Thus, it is recommended to remove the liquid lock damage at 2 m in the near-well zone, and this results in a permeability recovery of 0.143 mD, with an agent dosage of 0.911 m3 per unit layer thickness.

Fig. 5
figure 5

Schematic diagram of the amount of agent used to remove different radii of liquid lock damage and the recovery of permeability

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Based on the relevant sample parameters, daily gas production after removing the water intrusion damage in the near-well zone is simulated and analyzed, and Fig. 6 shows the results.

Fig. 6
figure 6

Schematic diagram of the distribution of water saturation and recovery of daily gas production after solving the liquid lock damage at different radii

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In Fig. 6, it is evident that the daily gas production can be restored to 4,484 m3/D after the removal of the liquid lock damage of 2 m in the near-well zone.

Conclusions

  1. (1)

    Based on the principles of the equation of motion and the continuity equation, we established two production models: the gas–water two-phase Darcy flow model and the gas–water two-phase non-Darcy flow model, which incorporates stress sensitivity. While the Darcy flow model, considering the production process as a pseudo-steady state flow, effectively fit the total production data with a deviation of less than 3%, it struggles with accurately fitting the daily production data.

  2. (2)

    Demonstrative calculations revealed that the non-Darcy flow model, taking into accounting stress sensitivity, maintains high precision in the total production data, with deviation of 0.2396% for total gas production and 0.2114% for total water production. Moreover, this model shows a high level of accuracy in predicting daily production data: the average absolute deviation of daily gas production is 22.8%, the average relative deviation is 0.1815%, and for daily water production, the average absolute and relative deviations are is 41.45% and − 0.2677%, respectively).

  3. (3)

    Utilizing the non-Darcy flow model, we simulated both the current water intrusion distribution curve and daily gas production recovery of the sample well. The results demonstrate that permeability can be restored from 0.045 to 0.143 mD, and daily gas production can increase from 1,458 to 4,484 m3/D following mitigation of liquid lock damage at 2 m distance from the well. In conclusion, our study highlights the effectiveness of the non-Darcy flow model with stress sensitivity in accurately predicting and analyzing production data in gas–water two-phase flow systems.