A Transient Productivity Prediction Model for Horizontal Wells Coupled with Oil and Gas Two-Phase Seepage and Wellbore Flow

Abstract

Capacity prediction is the basis for the optimization of oil and gas well production work systems and parameter optimization design. Horizontal wells are becoming increasingly popular for oil and gas extraction. However, the seepage law of reservoirs produced with horizontal wells is more complicated than that of reservoirs produced with vertical wells, especially when the bottom hole flowing pressure or formation pressure is less than the saturation pressure of crude oil in the reservoir. Oil and gas two-phase seepage can occur in a part or all areas of the wellbore and reservoir. Because the oil and gas two-phase seepage characteristics of reservoir oil well production will be reduced—possibly greatly reduced—the formation seepage law is complex. Thus, it is very important to better predict the horizontal well capacity. To address this, a method and process of establishing a transient calculation model of two-phase flow in horizontal wells are introduced in detail from three aspects: fluid physical properties, reservoir oil and gas two-phase seepage, and the coupling model of the inflow performance and flow in the wellbore. The model is found to be reliable through verification with production data from five wells in two oilfields. The established model simplifies the reservoir model, does not involve very complex meshing, and only simulates one well. Therefore, the calculation speed will be faster than that of other reservoir numerical simulation methods under the same conditions.

1. Introduction

Productivity prediction is an important task in oil and gas development with horizontal wells. When the bottom hole flowing pressure or formation pressure is less than the saturation pressure of the crude oil in the formation, oil and gas two-phase seepage will occur in the reservoir and wellbore, and the corresponding prediction model will be more complicated.

In 1958, Merkulov [1,2] published an article proposing a formula for predicting horizontal well production. He assumed that the reservoir shape was box-shaped and that the horizontal well was in the center of the reservoir. He then used the seepage mechanics method to derive the production formula of the horizontal well under steady seepage conditions. In 1964, Borisov [3] systematically summarized the development process of horizontal wells and inclined wells, introduced the production principle of horizontal wells, and comprehensively applied the seepage mechanics principle and mathematical derivation methods to obtain the yield analysis formula of horizontal wells under steady seepage conditions. In 1984, Giger [4,5,6] used basically the same assumptions as those adopted with Borisov’s formula to derive the oil recovery index equation for a horizontal well in the center of a reservoir. In the same year, he proposed a formula for calculating the horizontal well capacity of heterogeneous reservoirs, which is obtained by replacing the original permeability with the equivalent permeability on the basis of the homogeneous reservoir capacity formula. The Giger equation, similar to Borisov’s equation, does not account for the limitations of the horizontal length of horizontal wells and ignores the effect of the wellbore pressure drop.

In 1986, Joshi [7,8,9] established a single-phase seepage model of horizontal wells based on the principle of electric field flow. This model simplifies the three-dimensional ellipsoid seepage problem of horizontal wells into two two-dimensional seepage problems in the horizontal plane and vertical plane and uses potential energy theory to derive the steady-state productivity equation of horizontal wells in homogeneous isotropic reservoirs. The assumptions are as follows: (1) the flow is single-phase, steady-state flow; (2) the fluid is weakly compressible; (3) the oil reservoir is homogeneous, and the skin effect is not considered; (4) the outer boundary is the constant pressure boundary; and (5) the horizontal well is in the middle of the reservoir. In 1988, Babu [10] proposed the productivity equation for horizontal wells in the box-type closed reservoir he studied. This productivity equation is different from the productivity equation of other scholars and considers quasisteady flow. In 1990, Renard and Dupuy [11] studied the impact of formation damage on horizontal wells based on the summary of the productivity equation of Joshi and Giger horizontal wells and obtained the productivity equation of horizontal wells considering the skin effect. The equation is suitable for circular oil drainage areas, elliptical oil drainage areas, and rectangular oil drainage areas.

In 1996, Dou Hongen [12] regarded a horizontal well as a vertical well across an infinite formation and derived a horizontal well capacity formula according to the potential superposition principle. In 1996, Shedid explored the difference in the seepage mechanism between the heel and toe of a horizontal section of a horizontal well and described the shape of the oil drainage area of horizontal wells through two rectangles and a semicircle. After a series of studies, he proposed a horizontal well oil production index formula applicable to gas cap and bottom water reservoirs. In 2008, based on the research of Joshi and Giger on the production formula of horizontal wells, Chen Yuanqian [13,14] used the area equivalence method to equivalently convert the elliptical oil drainage area to a quasicircular oil drainage area. At the same time, the length of the horizontal section was changed into a quasicircular production tunnel according to the principle of production equivalence, and a new horizontal well production calculation formula was finally obtained by using the equivalent flowing resistance method. In 2010, Liu Wenchao [15] divided the elliptical oil drainage area of a horizontal well into an inner zone, middle zone, and outer zone, derived a steady seepage capacity calculation formula of each area according to the seepage characteristics of each zone, and finally obtained the production formula for a horizontal well in a heavy oil reservoir by assuming that the fluid flow through the boundaries of the zones was consistent.

In 2017, Soleimani M. [16] combined fracture modeling and elastic gridding, which improves flexibility, especially in complex reservoirs. The proposed model revises conventional modeling fractures by hard rigid planes that do not change during production. This is a dubious assumption, especially in reservoirs with a high early production rate. The proposed elastic fracture modeling considers changes in the fracture properties, shapes, and apertures through the simulation. This strategy is reliable for only naturally fractured reservoirs with high fracture permeabilities and less permeable matrices and parallel fractures with fewer cross-connections. In the same year, he proposed an integrated optimization procedure [17] for well performance analysis in a heterogeneous oilfield from southwest Iran. The proposed integrated strategy optimizes well performance in an iterative manner, while fluid properties and geological and petrophysical data are analyzed separately by new advanced methods.

In 2019, based on the analysis of typical seepage characteristics of horizontal wells, Jia Xiaofei et al. [18] considered planar elliptical flow and deduced a new comprehensive productivity formula of horizontal wells by using the water and electricity similitude principle and the equivalent flowing resistance method. This formula is more adaptable than the commonly used formula and can be used to calculate the horizontal well capacity under different drainage shapes, penetration ratios, etc.

In the same year, Li Yongming et al. [19] determined that the stress-sensitive effects of natural fractures and the hydraulic fracture morphology and seepage characteristics should be accounted for when considering productivity. Therefore, based on dual-medium seepage theory, a shale gas reservoir seepage model was established by considering the adsorption, desorption, and diffusion mechanisms (pseudosteady-state and transient diffusion) of shale gas in reservoirs and the stress-sensitive effects of natural fractures. The limited conductivity of hydraulic fractures and hydraulic fracture azimuth were considered in the hydraulic fracture model. The source function method was used to discretize cracks, and the hydraulic fracture model was then superimposed. Finally, the two models were coupled to obtain the unstable seepage model and productivity model of a horizontal well completed with multistage fracturing in a shale gas reservoir.

In 2021, based on the potential superposition principle and the mirror reflection principle, Gao Yihua et al. [20] established a calculation model of the radial flow distribution along the wellbore of horizontal wells across plugging faults under two modes, obtained the productivity prediction method of horizontal well cross faults in complex fault-block oilfields, and thus formed a reservoir engineering method to quickly optimize the sectional length of horizontal wells across plugging faults in each fault block.

In 2023, Shuwei Ma et al. [21] established a physical model that divides the fluid seepage area into the following three sections and five small zones for horizontal well and volume fracturing for shale oil production: the horizontal wellbore zone, highly transformed zone, weakly transformed zone, matrix drainage zone, and matrix nondrainage zone. The Joukowski transformation was then introduced before a mathematical solution was deduced, and the equivalent seepage resistance law and material balance method were applied.

In the same year, Yunhao Zhang et al. [22] developed and validated a robust and pragmatic method to describe the two-phase flow behavior of a multifractured horizontal well (MFHW) in a shale gas formation. Regarding a fracture subsystem, the permeability modulus, non-Darcy flow coefficient, and slippage factor were defined and embedded into the governing equation, while an iterative method was applied to update the gas/water saturation in each fracture segment within discrete fracture networks. Regarding a matrix subsystem, a skin factor on a fracture face is defined and introduced to represent the change in the relative permeability in the matrix domain at each timestep, while the adsorption/desorption term is incorporated into the diffusivity equation to accurately calculate the shale gas production by taking the adsorbed gas in nanoscale porous media into account. Then, the theoretical model can be applied to accurately capture the two-phase flow behavior in different subdomains.

However, most of these earlier research results are based on steady-state and single-phase or water and gas two-phase systems, which can no longer meet the needs of the current development of smart oilfields. Therefore, in this paper, a transient model of horizontal well capacity prediction coupled with oil and gas two-phase seepage and wellbore flow is established, and the model is verified by comparing its results with those of an existing model.

2. Establishment of Oil and Gas Two-Phase Seepage and Its Coupling Model with the Wellbore

According to the basic principle of reservoir oil and gas seepage, the basic equation of oil and gas two-phase seepage can be obtained. Regarding the oil phase, the following equation is used:

$$\nabla \cdot \left[\frac{K_{ro}\left(S_o\right)}{{\mu }_o\left(p\right)\cdot B_o\left(p\right)}\nabla p\right]=\frac{\varphi }{K}\frac{\partial }{\partial t}\left[\frac{S_o}{B_o\left(p\right)}\right]$$

(1)

Regarding the gas phase, the following equation is used:

$$\nabla \cdot \left[\frac{K_{rg}\left(S_o\right)}{{\mu }_g\left(p\right)\cdot B_g\left(p\right)}\nabla p\right]+\nabla \cdot \left[\frac{R_s\left(p\right)K_{ro}\left(S_o\right)}{{\mu }_o\left(p\right)\cdot B_o\left(p\right)}\nabla p\right]=\frac{\varphi }{K}\frac{\partial }{\partial t}\left[\left(1-S_o-S_{wc}\right)\frac{1}{B_g\left(p\right)}+\frac{R_s\left(p\right)}{B_o\left(p\right)}{S}_o\right]$$

(2)

where $K_{ro}$ and $K_{rg}$ are the relative permeabilities of oil and gas, respectively, which are functions of the oil saturation $S_o$. $B_o$ and ${\mu }_o$ are the volume factor and viscosity of the oil phase, respectively, which are functions of the pressure. $B_g$, ${\mu }_g$, and $R_s$ are the volume factor, viscosity, and dissolved gas–oil ratio, respectively. These parameters are also functions of the pressure.

Equations (1) and (2) show that the functions to be solved are the pressure $p$ and saturation $S_o$. However, in the two equations, the coefficient terms are often functions of these two functions. Thus, such an equation has strong nonlinearity, and at the same time, the obtained solutions (such as the pressure and saturation) have a strong dependence on these state parameters (such as the relative permeability, volume factor, dissolved gas–oil ratio, and viscosity).

The establishment of the above equation implies that the pressure of the oil layer is lower than the saturation pressure of crude oil. Therefore, oil and gas two-phase seepage will occur in the formation, so the above two equations are valid. Since the oil layer pressure is lower than the original saturation pressure, with the continuous degassing of crude oil, the properties of crude oil, such as the crude oil volume coefficient and viscosity and dissolved gas–oil ratio, are changed.

2.1. Simplified Model Calculation of Fluid Physical Properties under Average Pressure Conditions in a Certain Region

The basic equation of two-phase seepage in oil and gas is a nonlinear partial differential equation, which is commonly solved by approximate methods. The assumptions of the material balance method are as follows:

(1)

At any moment, the porosity, fluid saturation, and relative permeability of the reservoir are uniform.

(2)

Regardless of the gas zone and oil zone in the reservoir, the formation pressure is the same, and the volume coefficient, viscosity, and gas dissolution amount of gas and oil are the same.

(3)

The influence of gravity is not considered.

(4)

At any moment, the oil phase and the gas phase are balanced.

(5)

No water intrusion, other than the water output, occurs.

Rs, Bo, and Bg are all functions of the average formation pressure P, determined by high-pressure physical property experiments, as shown in Figure 1.

Production gas–oil ratio:

$$R=\frac{\frac{\mathrm{d}{V}_g}{\mathrm{d}P}}{\frac{\mathrm{d}{V}_o}{\mathrm{d}P}}=\frac{\frac{R_s}{B_o}\frac{\mathrm{d}{S}_o}{\mathrm{d}P}+\frac{S_o}{B_o}\frac{\mathrm{d}{R}_s}{\mathrm{d}P}-\frac{R_s{S}_o}{B_o^2}\frac{\mathrm{d}{B}_o}{\mathrm{d}P}+\left(1-S_o-S_{wr}\right)\frac{\mathrm{d}{B}_g^{\prime }}{\mathrm{d}P}-B_g^{\prime }\frac{\mathrm{d}{S}_o}{\mathrm{d}P}}{\frac{1}{B_o}\frac{\mathrm{d}{S}_o}{\mathrm{d}P}-\frac{S_o}{B_o^2}\frac{\mathrm{d}{B}_o}{\mathrm{d}P}}$$

(3)

Since the production gas–oil ratio is the ratio of the gas flow (including dissolved gas and free gas) converted to standard conditions to the oil flow rate converted to standard atmospheric conditions, the production gas–oil ratio can also be written as follows:

$$R=\frac{Q_g{B}_g^{\prime }+\frac{Q_o}{B_o}{R}_s}{\frac{Q_o}{B_o}}=B_o{B}_g^{\prime }\frac{K_g}{K_o}\frac{{\mu }_o}{{\mu }_g}+R_s$$

(4)

where $Q_g$ is the gas flow rate under oil layer conditions, $Q_g=\frac{K_g}{{\mu }_g}2\pi rh\frac{\mathrm{d}p}{\mathrm{d}r}$, cm³/s; $Q_o$ is the oil flow rate under oil layer conditions, $Q_g=\frac{K_o}{{\mu }_o}2\pi rh\frac{\mathrm{d}p}{\mathrm{d}r}$, cm³/s; and ${\mu }_g$ and ${\mu }_o$ are the viscosities of gas and oil, respectively, and are functions of the average formation pressure $\overline{p}$, mPa·s.

The relationship between the average formation pressure and oil saturation can be obtained by equating the above two equations:

$$\frac{\mathrm{d}{S}_o}{\mathrm{d}p}=\frac{\frac{S_o}{B_o{B}_g^{\prime }}\frac{\mathrm{d}{R}_s}{\mathrm{d}p}+\frac{S_o}{B_o}\frac{K_g}{K_o}\frac{{\mu }_o}{{\mu }_g}\frac{\mathrm{d}{B}_o}{\mathrm{d}p}+\left(1-S_o-S_{wr}\right)\frac{1}{B_g^{\prime }}\frac{\mathrm{d}{B}_g^{\prime }}{\mathrm{d}p}}{1+\frac{K_g}{K_o}\frac{{\mu }_o}{{\mu }_g}}$$

(5)

In the above formula, the viscosities of crude oil and natural gas are also functions of pressure, as shown in Figure 2.

According to the above formula, the saturation of crude oil in the formation under different average formation pressure conditions can be obtained, and the relationship between the recovery degree and the average formation pressure can be further obtained, as shown in Figure 3.

2.2. Simplified Model Calculation of Average Pressure in a Certain Region

The steady-state sequential replacement method can be used to obtain the following conclusions:

Each instantaneous moment of the whole process of unsteady oil and gas two-phase seepage can be approximated as a steady state, and the unsteady state of the whole process can be regarded as a superposition of many steady states.

From the relationship curve between the average formation pressure and formation of crude oil saturation, the pressure can be divided into several intervals, and the pressure value and saturation value in each interval are taken as the median value:

$$\overline{p}=\frac{p_i+p_{i+1}}{2}$$

(6)

$$S_o=\frac{S_{oi}+S_{oi+1}}{2}$$

(7)

In each small pressure interval, it is considered that the oil and gas seepage is steady, and the physical characteristics of the internal fluid are consistent.

2.3. Calculation of the Three-Dimensional H (Potential) of a Horizontal Well under the Condition of Oil and Gas Two-Phase Flow

When the gas-mixed oil seeps into the well due to the dissolved gas drive, its flow state is unsteady, and the well production (or bottom hole pressures) changes with time. However, although the process of gas-mixed oil seepage reflects an unsteady state, at every moment in the total process, it can be approximately regarded as a steady-state process.

That is, in a certain short period, the formation pressure and oil saturation change little. If this time interval is sufficiently small, the pressure and saturation can be considered independent of time. That is, steady seepage occurs. At this time, the oil well production formula obtained according to the steady state will basically conform to the actual situation.

Corresponding to the oil-gas-water two-phase flow, although the oil has been produced, the oil-phase flow always exists throughout the production process. Therefore, it is more appropriate to consider the flow process in the oil-gas-water three-phase system with the phase seepage law of the oil phase. The seepage law of the oil phase resembles the single-phase oil seepage law in the literature (Liu P, et al.) [23]. The instantaneous (one sink point in space) production is as follows:

$$q_o=\frac{K_o}{{\mu }_o{B}_o}A\frac{\mathrm{d}p}{\mathrm{d}r}$$

(8)

Importing $K_{ro}=\frac{K_o}{K}$, we can obtain

$$q_o=\frac{K}{{\mu }_o{B}_o}{K}_{ro}A\frac{\mathrm{d}p}{\mathrm{d}r}=4K\pi r^2\frac{K_{ro}}{{\mu }_o{B}_o}\frac{\mathrm{d}p}{\mathrm{d}r}$$

(9)

$$q_o\frac{1}{4K\pi r^2}\mathrm{d}r=\frac{K_{ro}}{{\mu }_o{B}_o}\mathrm{d}p$$

(10)

$$-q_o\frac{1}{4K\pi r}+C={\displaystyle \int \frac{K_{ro}}{{\mu }_o{B}_o}\mathrm{d}p}$$

(11)

If $\mathrm{d}H=\frac{K_{ro}}{{\mu }_o{B}_o}\mathrm{d}p$, then

$$H=-\frac{q_o}{4K\pi r}+C$$

(12)

$$-q_o\frac{1}{4K\pi r}+C={\displaystyle \int \frac{K_{ro}}{{\mu }_o{B}_o}\mathrm{d}p}=H$$

(13)

$$-q_o\frac{1}{4\pi r}+C=K{\displaystyle \int \frac{K_{ro}}{{\mu }_o{B}_o}\mathrm{d}p}=KH$$

(14)

We compare the single-phase steady-state point sink function $\varphi =-\frac{q}{4\pi r}+C$ and the single-phase instantaneous point source function as follows:

$$\Delta p(r,t)=\frac{q\mu }{4\pi \mathrm{K}r}erfc(\frac{r}{2\sqrt{{\eta }_{r}t}})$$

(15)

$${\eta }_r=\frac{\mathrm{K}}{\mu \varphi c_t}$$

(16)

Then, the instantaneous point sink function of the oil phase in the oil and gas two-phase system is as follows:

$$\Delta H(r,t)=-\frac{q}{4\pi r}erfc(\frac{r}{2\sqrt{{\eta }_{r}t}})$$

(17)

Similarly, single-phase seepage can be used to obtain the instantaneous line sink function of oil and gas two-phase flow and the calculation function of horizontal well H (potential) in different types of reservoirs, resulting in a coupling model and solution method.

That is, the H (potential) generated by an entire horizontal segment in space ($X$,$Y$,$Z$) is as follows:

$$H={\displaystyle \sum _{i=1}^m{H}_i=-\frac{q}{4\pi KL}{\displaystyle \sum _{i=1}^m({\displaystyle {\int }_{x_{si}}^{x_{ei}}f(x,y_{si},z_{si},t)dx+{\displaystyle {\int }_{y_{si}}^{y_{ei}}g(x,y,z_{si},t)dy}+{\displaystyle {\int }_{z_{si}}^{z_{ei}}h(x,y,z,t)dz}})}}$$

(18)

The H (potential) on the right side of the equation is obtained by calculating and accumulating several intervals separately.

Oil well productivity prediction is also closely related to the type of reservoir. Different types of reservoirs have different potentials in the formation for each segment of the horizontal well. The bottom water reservoir is taken as an example for illustration, and other reservoir calculations can be deduced by referring to Liu Xiangping [24] et al. If the two-phase seepage of oil and gas occurs in a reservoir with closed upper and lower boundaries, the calculation method of the three-dimensional horizontal well H (potential) in the reservoir is as follows.

2.4. Calculation of the Potential of Uniform Inflow into a Horizontal Section in a Closed Reservoir

For the reservoir with closed upper and lower boundaries, as shown in Figure 4, a horizontal well with length $L$ is divided into $N$ segments.

According to the mirror reflection principle, as shown in Figure 5, we can obtain the following:

$${\varphi }_j(X,Y,Z,t)=-\frac{q_j}{4\pi }\{{\displaystyle \sum _{n=-\infty }^{\infty }[\xi (x,y,2nh+z,X,Y,Z,t)+\xi (x,y,2nh-z,X,Y,Z,t)}]\}+C_j$$

(19)

where ${\varphi }_j$ is the potential generated by the j-th line sink at any point in the oil layer; $q_j$ is the flow rate of the j-th line sink; $h$ is the oil thickness; $z$ is the distance between each part of the well and the bottom of the oil layer; $C_j$ is a constant; and $\xi$ is the function defined by the following formula:

$$\begin{array}{ll}{\xi }_j(x,y,2nh+z,X,Y,Z,t)=& \frac{1}{L_j}{\displaystyle \sum _{i=1}^m({\displaystyle {\int }_{x_{si}}^{x_{ei}}f(x,y_{si},z_{si},t)dx+{\displaystyle {\int }_{y_{si}}^{y_{ei}}g(x,y,z_{si},t)dy}}}\\ & +{\displaystyle {\int }_{4nh+z_{si}}^{4nh+z_{ei}}h(x,y,z,t)dz)}\end{array}$$

(20)

where $L_j$ is the length of the j-th line sink; $x_{s1}$ and $x_{em}$ are the starting and ending x values of the j-th line sink in the x-axis direction; and the other parameters are the $y$ and $z$ direction coordinates.

That is, the H (potential) generated by a certain horizontal segment in space ($X$, $Y$, $Z$) is as follows:

$$H_j=\frac{{\varphi }_j(X,Y,Z,t)}{K}$$

(21)

2.5. Horizontal Well Flow

According to the potential superposition principle, the potential generated by the whole horizontal well in the oil layer is as follows:

$$\varphi (X,Y,Z,t)={\displaystyle \sum _{j=1}^N{\varphi }_j(X,Y,Z,t)}+C=-{\displaystyle \sum _{j=1}^N\frac{q_j}{4\pi }}{\phi }_j+C$$

(22)

For different types of reservoirs, the formula ${\phi }_j$ is equal to the expression in brackets in Equation (19).

The following equation can be obtained from Equation (22):

$${\varphi }_e={\displaystyle \sum _{j=1}^N{\varphi }_{je}}+C$$

(23)

The formula ${\varphi }_e$ is the potential function at the constant pressure boundary or oil drain boundary, and ${\varphi }_{je}$ is the potential generated by the j-th line sink at the constant pressure boundary or the oil drain boundary.

The following equation is obtained by Equations (22) and (23):

$$\varphi (X,Y,Z,t)={\varphi }_e+{\displaystyle \sum _{j=1}^N[{\varphi }_j(X,Y,Z,t)-{\varphi }_{je}]}$$

(24)

According to the potential function Equation (13), we can obtain the following:

$$KH(X,Y,Z,t)=K{\mathrm{H}}_e+{\displaystyle \sum _{j=1}^N[{\varphi }_j(X,Y,Z,t)-{\varphi }_{je}]}$$

(25)

The H (potential) on the left side of the equation can be obtained by regional integration according to Equation (13) as follows:

$${\displaystyle \int \frac{K_{ro}}{{\mu }_o{B}_o}\mathrm{d}p}=H$$

(26)

where $p$ is the pressure at any point in the oil layer (or the comprehensive pressure after considering the potential energy difference); $K$ is the permeability of the oil layer; $K_o$ is the permeability of the oil phase; $K_{ro}$ is the relative permeability of the oil phase; ${\mu }_o$ is the viscosity of the crude oil; and $B_o$ is the volume factor of the crude oil.

2.6. Coupling Model of the Inflow Performance and Flow in a Wellbore and Its Solution

The three-dimensional unsteady-state seepage of fluid in the oil layer and the flow in the wellbore are interrelated. The formation seepage accounts for the oil and gas two-phase flow, and the wellbore seepage is also calculated by using the multiphase flow method considering the conservation of mass. The assumptions are as follows:

(1)

The circular drainage area is controlled by each well.

(2)

The parameters of Kx, Ky, Kz, thickness H, and porosity ϕ are constant throughout the reservoir.

The pressure at the midpoint of the $j$-th line sink on the horizontal well is $p_{w,j}$, and the potential generated by the $i$-th line sink at the midpoint of the $j$-th line sink is ${\Phi }_{ij}$. The linear equation system reflecting the production of each segment is obtained according to Equations (22), (25), and (26), and the production of each segment is obtained by solving these equations.

According to the variable mass calculation method and multiphase flow calculation method, the pressure drop in the wellbore is calculated, and the pressure at a point of segment $j$ in the wellbore is as follows:

$$p_{w,j}=p_{1,j}-0.5d{p}_{w,j} (j=1, 2, \dots , \mathrm{N})$$

(27)

where $p_{2,N}=p_{wf}$ and $p_{wf}$ is the flowing pressure at the heel end of the wellbore.

$$p_{1,j+1}=p_{2,j}=p_{1,j}-\Delta p_{w,j} (j=1, 2, \dots , \mathrm{N})$$

(28)

The total production of the whole well is

$$Q_o=\frac{(q_{s,1}+q_{s,2}+q_{s,3}+\cdot \cdot \cdot +q_{s,N})}{B_o}$$

(29)

In the abovementioned coupled model, both $q$ and $p_w$ are unknowns, which can be solved using the iterative method. First, by assuming a set of $p_w$ values, Equations (22), (25) and (26) can be used to solve the $q$ array. Then, the $q$ array is substituted into the pressure drop formula and Equation (27) to update the $p_w$ array from heel to toe. Then, the $q$ array is updated from the linear equation system of formula production until both the $p_w$ and $q$ arrays reach a certain computational accuracy. Finally, the total well production can be obtained from Equation (29).

3. Example of the Transient Productivity Prediction Calculation of Oil and Gas Two-Phase Seepage in Horizontal Wells

3.1. Sample Calculation

Known conditions:

(1)

The basic parameters are shown in

Table 1. Parameters of the oil reservoir and horizontal well.

Name	Value	Unit
Horizontal permeability	10	mD
Vertical permeability	10	mD
Crude oil volume factor	1.281
porosity	0.3
Total compressibility coefficient of formation	0.0002	1/MPa
Crude oil viscosity	0.6365	mPa.s
Reservoir thickness	30	m
Horizontal well length	400	m
Original formation pressure	30	MPa
Bottom hole flowing pressure	28	MPa
Completion method	Open hole completion
Oil reservoir type	Reservoir with closed upper and lower boundaries
Saturation pressure	15	MPa
Reservoir temperature	80	℃
Production time	1	year

.

(2)

The phase penetration data are shown in Figure 6.

(3)

The software interface is shown in Figure 7. The required input parameters and the physical property calculation methods used are shown.

(4)

The result is shown below.

Figure 8, Figure 9, Figure 10 and Figure 11 show that under the same production pressure difference conditions, the production is lower for oil and gas two-phase seepage than for single-phase flow. The more significant the degassing is, the lower the production. The calculation result is as follows:

3.2. Verification of the Productivity Prediction of Two Oilfields

To test the transient coupling model of the established horizontal well productivity prediction, inflow performance verification was carried out by taking the test production horizontal wells in the Iran Masjed-I-Suleyman (MIS) oilfield and the test production horizontal wells in the Hafaya oilfield as examples.

(1)

MIS oilfield, Iran

Wells MIS 320 C/N-H7 and MIS 322 C/N-H2 are located on the northeastern flank of the MIS anticline, as shown in Figure 12 and Figure 13. They are the first and the third new wells of the twelve (12) horizontal wells drilled by CNPCI/NESCO for the redevelopment project.

The basic parameters and test production data of these two wells are shown in

Table 2. Reservoir parameters of the two horizontal wells.

Well No.	Oil Layer Thickness m	Porosity Decimal	Comprehensive Compression Factor 1/MPa	Absolute Permeability (in the X and Y Directions) mD	Absolute Permeability (in the Z Direction) mD
MIS 320 CN-H7	157.79	0.116	0.001276264	43	43
MIS 322 C/N-H2	157.79	0.116	0.001276264	578	578

,

Table 3. Characteristics of single-phase crude oil seepage in two horizontal wells.

Well No.	Crude Oil Volume Factor	Crude Oil Viscosity	Relative Density of Crude Oil
MIS 320 CN-H7	1.09	1.8	0.832
MIS 322 C/N-H2	1.09	1.8	0.832

,

Table 4. Reservoir types and completion parameters of two horizontal wells.

Well No.	Well Type	Reservoir Type	Oil Drainage Area Length (x) M	Oil Drainage Area Length (y) m	Wellbore Length m	Wellbore Diameter in	Skin Coefficient in Well Completion	Well Reservoir Factor (m3/MPa)	Formation Pressure MPa
MIS 320 CN-H7	Horizontal wells	Infinitely large homogeneous reservoirs			394.1	6 1/8”	10	7.093	2.91
MIS 322 C/N-H2	157.79	Infinitely large homogeneous reservoirs	1524	1524	422.34	6 1/8”	−3	7.093	3.059

,

Table 5. Experimental production data of the MIS 320 CN-H7 well.

No.	Duration	ESP (Hz)	Pwf (psi)	Δp (psi)	Rate (bbl/d)	PI (bbl/d/psi)
1	22:00–2:00	35	351	72	255	3.86
2	2:04–6:15	40	346	77	330
3	6:19–10:15	45	333	90	398
4	10:21–16:00	50	252	171	596

and

Table 6. Experimental production data of the MIS 322 CN-H2 well.

No.	Oil Rate (bbl/d)	ΔP (psi)	Pwf (psi)	Watercut (%)	Choke Size (1/64”)	ESP (Hz)	PI (bbl/d/psi)
1	3392	25	419	0.2	42	45	136.57
2	3820	28	416	0.2	52	47
3	4260	31	413	0.1	50	50

.

The prediction results of different productivity prediction methods of the MIS 320 CN-H7 well and the error analysis with the test production data are shown in Figure 14 and

Table 7. Calculation results of the different productivity prediction methods and error analysis for the MIS 320 CN-H7.

Method	Merkulovb	Giger	Joshi	Borisov	Babu & Odech	Renard	Elgaghad	Dou Hongen	Multiphase Flow Transient Model
Absolute average relative error (decimal)	0.369	0.442	0.378	0.422	2.859	0.422	0.413	0.120	0.096

below, respectively.

The prediction results of different productivity prediction methods of the MIS 322 CN-H2 well and the error analysis with the test production data are shown in Figure 15 and

Table 8. Calculation results of the different productivity prediction methods and error analysis for the MIS 322 CN-H2.

Method	Merkulovb	Giger	Joshi	Borisov	Babu & Odech	Renard	Elgaghad	Dou Hongen	Multiphase Flow Transient Model
Absolute average relative error (decimal)	0.519	0.808	0.617	0.775	0.806	0.775	0.787	0.097	0.103

below, respectively.

(2)

Hafaya oilfield

Located in southeastern Iraq, the Halfaya oilfield is one of the country’s seven major oilfields and contains an estimated 4.1 billion barrels (650,000,000 m³) of oil. China National Petroleum Corporation (CNPC) has already started operating in this field.

The basic parameters and test production data of the three wells in Hafaya are shown in

Table 9. Reservoir parameters of the three horizontal wells.

Well No.	Oil layer Thickness m	Porosity Decimal	Comprehensive Compression Factor 1/MPa	Absolute Permeability (in the X and Y Directions) mD	Absolute Permeability (in the Z Direction) mD	Oil Layer Temperature ℃
HF003-S001H	70	0.177	0.0013154	0.076	0.02736	81.2
HF002-M001H	30	0.115	0.001285	13.4		93.78
HF001-N002H	8	0.197	0.00174	861		114.27

,

Table 10. Characteristics of the single-phase crude oil seepage in three horizontal wells.

Well No.	Crude Oil Volume Factor	Crude Oil Viscosity	Relative Density of Crude Oil
HF003-S001H	1.237	1.33	0.904
HF002-M001H	1.55	1.643	0.794
HF001-N002H	1.444	0.64	0.872

,

Table 11. Reservoir types and completion parameters of three horizontal wells.

Well no.	Well Type	Reservoir Type	Wellbore Length m	Wellbore Diameter in	Skin Coefficient in Well Completion	Well Reservoir Factor (m3/MPa)	Formation Pressure MPa
HF003-S001H	Horizontal wells	Boundary homogeneous reservoirs	532.1	0.15	−5.69	2.49	30.38
HF002-M001H	Horizontal wells	Infinitely large homogeneous reservoirs	579	0.15	−5.47	5.99	30.659
HF001-N002H	Horizontal wells		273	0.15	−3.51		40.22

,

Table 12. Experimental production data of the HF003-S001H well.

Date	Time	Choke Size (in)	WHP (psi)	Flowing-Pressure (psi)	Differential Pressure (psi)	Production Rate
						Oil (bbl/d)	Gas (Mscf/d)
29.8.2011	14:53	16/64 “	/	2522.46	1760.4	241.6	161.2
30.8.2011	2:23	20/64 “	/	2141.11	2141.75	280.6	167.1
30.8.2011	10:42	24/64 “	/	1990.89	2291.97	255.6	221.8
30.8.2011	16:23	16/64 “	/	2301.37	1981.49	213.5	109.7

,

Table 13. Experimental production data of the HF002-M001H well.

Choke Size (in)	WHP (psia)	Flowing Pressure (psia)	Differential Pressure (psi)	Production Rate
				Fluid (bbl/d)	Oil (bbl/d)	Gas–Oil Ratio (scf/bbl)
48/64	450	2870.405	1454.19	1992	1992	976

and

Table 14. Experimental production data of the HF001-N002H well.

Oil Production Rate (bbl/d)	Flowing Pressure (psi)
3263.6	5693.99

.

The prediction results of the different productivity prediction methods of the HF003-S001H well and the error analysis with the experimental production data are shown in Figure 16 and

Table 15. Calculation results of the different productivity prediction methods and error analysis for the HF001-N002H.

Method	Merkulovb	Giger	Joshi	Borisov	Babu & Odech	Renard	Elgaghad	Dou Hongen	Multiphase Flow Transient Model
Absolute average relative error (decimal)	0.196	0.731	0.092	0.104	0.318	0.102	0.309	0.356	0.098

below, respectively.

The prediction results of different productivity prediction methods of the HF002-M001H well and the error analysis with the experimental production data are shown in Figure 17 and

Table 16. Calculation results of the different productivity prediction methods and error analysis for theHF002-M001H.

Method	Merkulovb	Giger	Joshi	Borisov	Babu & Odech	Renard	Elgaghad	Dou Hongen	Multiphase Flow Transient Model
Absolute average relative error (decimal)	1.195	1.849	1.436	1.552	2.308	1.542	1.332	0.297	0.257

below, respectively.

The prediction results of the different productivity prediction methods of the HF001-N002H wells and the error analysis with experimental production data are shown in Figure 18 and

Table 17. Calculation results of the different productivity prediction methods and error analysis for the HF001-N002H.

Method	Merkulovb	Giger	Joshi	Borisov	Babu & Odech	Renard	Elgaghad	Dou Hongen	Multiphase Flow Transient Model
Absolute average relative error (decimal)	1.829	1.987	1.901	1.940	1.580	1.940	2.230	0.194	0.116

below, respectively.

The absolute average relative error statistics and average error calculation of the five wells are shown in

Table 18. Total average error statistics of the five wells.

Forecasting Methodology	Merkulovb	Giger	Joshi	Borisov	Babu & Odech	Renard	Elgaghad	Dou Hongen	Multiphase Flow Transient Model
MIS 320 CN-H7	0.369	0.442	0.378	0.422	2.859	0.422	0.413	0.12	0.096
MIS 322 CN-H2	0.519	0.808	0.617	0.775	0.806	0.775	0.787	0.097	0.103
HF003-S001H	0.196	0.731	0.092	0.104	0.318	0.102	0.309	0.356	0.098
HF002-M001H	1.195	1.849	1.436	1.552	2.308	1.542	1.332	0.297	0.257
HF001-N002H	1.829	1.987	1.901	1.94	1.58	1.94	2.23	0.194	0.116
Average error	0.8216	1.1634	0.8848	0.9586	1.5742	0.9562	1.0142	0.2128	0.134

.

Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 and Table 18 show that through the comparison and error analysis of the prediction results of various horizontal well production capacity prediction methods with the actual test well data, the established multiphase flow transient model has a smaller error, and the average absolute average relative error of the five wells is 13.4%.

4. Conclusions

By fully considering oil and gas two-phase seepage below the saturation pressure and the seepage law of horizontal wells, the productivity prediction model of horizontal wells with oil and gas two-phase transient seepage is established, the solution method is provided, and the simulation calculation verification is carried out. The following conclusions are drawn:

(1)

If the pressure in the reservoir is lower than the saturation pressure, it is difficult to predict the two-phase transient seepage capacity of horizontal wells. Moreover, the reservoir phase permeability changes with pressure, and the variation in the crude oil volume coefficient also changes with pressure. These parameters are a function of pressure. They potentially make the calculation of the horizontal well extremely complicated, and the parameters that vary with pressure need to be considered together. Nevertheless, the relationship between the horizontal well potential (pressure) and production can still be established by using the superposition principle. Then, a capacity prediction model that couples two-phase seepage and wellbore pipe flow in the formation is established. The solution method of the model is given.

(2)

The simulation calculation of the established model shows that when the pressure is higher than the saturation pressure, the producing pressure drops, and the oil production rate are the same. When the bottom hole flowing pressure of the well is just lower than the saturation pressure, oil and gas two-phase seepage occurs in the near-well area, while the surrounding area (with a pressure higher than the saturation pressure) undergoes single-phase seepage, and the production decreases. When the formation pressure and bottom hole flowing pressure further decrease, the oil and gas two-phase seepage area in the reservoir will expand, and production will be greatly reduced. Therefore, production should be maintained as high as possible under high formation pressure conditions.

(3)

Through the verification of the productivity prediction of the five wells in two oilfields, the error of the new model is small, and the prediction result is more reliable, which shows that the new model is feasible. Therefore, the new model can predict the oil and gas two-phase productivity of horizontal wells with a simplified reservoir model, no very complex meshing, and only one simulated well. Therefore, the calculation speed will be faster than that of other reservoir numerical simulation methods under the same conditions.