Determine the Inflow Performance Relationship of Water Producing Gas Well Using Multiobjective Optimization Method

Abstract

During the development of water drive gas reservoirs, the phenomena of gas escaping from water and water separating out from gas will change the seepage characteristics of formation fluid. Therefore, the traditional gas-water two-phase inflow performance relationship (IPR) models are not suitable for calculating the water producing gas well inflow performance relationship in water drive gas reservoirs. Based on the basic theory of fluid mechanics in porous medium, using the principle of mass conservation, and considering the process of dissolution and volatilization of gas and water formation, this paper establishes a new mathematical model of gas-water two-phase flow. Multiobjective optimization method is used to automatically match the sample well production data in water drive gas reservoirs and then we can achieve the sample well’s productivity equation, relative permeability curve, water influx intensity, and single well controlled reserves. In addition, the influence of different production gas water ratios (GWR) and gas-soluble water coefficients on absolute open flow rate (𝑞_AOF) is discussed. This method remedied the limitation of well testing on site and was considered to be a new way to analyze the production behaviors in water producing gas well.

1. Introduction

Well productivity is one of primary concerns in field development and provides the basis for field development strategy [1]. Xiaoping and Birong [2] put forward a method to deduce the binomial productivity equation which could calculate the inflow performance relationship (IPR) curve of water producing gas well and presented the application of the IPR curve in determining gas and water production rate from water producing gas well. Zhu et al. [3] proposed three new formation evaluation parameters for low permeability gas reservoir, on the basis of the rate controlled mercury injection, nuclear magnetic resonance, and physical simulation technologies. Wang et al. [4] analyzed gas and water phase relative permeability through cores with three different permeability leaves by the establishment of physical simulation experiment system and experimental process of water-gas mutual flooding.

Park et al. [5] proposed a fuzzy nonlinear programming approach to design production systems of gas fields. The synthetic optimization method could find a globally compromised solution and offer a new alternative with significant improvement over the existing conventional techniques. Cardoso [6] found that reduced-order model is well suited for reservoir simulation. Han et al. [7] presented a multiobjective evolutionary algorithm applied to history matching of water flooding projects, that is, to search a feasible set of geological properties showing the reliable future performance. Cancelliere et al. [8] discussed benefits, limitations, and drawbacks of assisted history matching, based on multiobjective optimization and heuristic strategies. Attention was focused on the possibility offered by these methodologies of obtaining a number of calibrated reservoir models. Shelkov et al. [9] described a comparison of single- and multiobjective history matching of a medium-sized field in Western Siberia with nearly 100 wells and over 10 years of history. And they compared the performance of both single- and multiobjective versions of particle swarm optimization. Tan et al. studied the transient flow and two-phase flow behaviors in porous media [10–12]. Some of their research results can be used to solute the problem of inflow performance relationship of water producing gas well.

2. Gas-Water Two-Phase IPR Equation

We assume that the reservoir is homogeneous with uniform thickness and total compressibility of rock and fluid is low and constant. The water phase flow is isothermal and Darcy flow, and the gas phase flow is isothermal and non-Darcy flow at high velocity, ignoring the impact of gravity and capillary forces. No chemical reaction exists between gas and water phase. Fundamental filtration equations for the gas and water phase are defined as [13] 𝑑𝑝=𝜇𝑑𝑟𝑘𝑘^𝑔_𝑟𝑔𝑣_𝑔+𝛽_𝑔𝜌_𝑔𝑣²_𝑔,𝑑𝑝=𝜇𝑑𝑟𝑘𝑘^𝑤_𝑟𝑤𝑣_𝑤, (1) where 𝑝 is pressure, 𝑟 is radial distance, 𝜇_𝑔, 𝜇_𝑤are gas and water viscosity, respectively, 𝑘 is absolute permeability, 𝑘_𝑟𝑔, 𝑘_𝑟𝑤are gas and water phase relative permeability, respectively, 𝑣_𝑔, 𝑣_𝑤are gas and water phase flow velocity, 𝛽_𝑔 is turbulence velocity coefficient, and 𝜌_𝑔 is gas density.

Under the boundary condition of steady radial state flow, the integral of (1) can be written as follows [14]: ^𝑝_{𝑅𝑝𝑤𝑓}𝑘𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑑𝑝=^𝑟𝑒_𝑟𝑤𝑞_𝑔𝑞2𝜋𝑟ℎ𝑑𝑟+^2𝑔4𝜋²ℎ^{2𝑟𝑒𝑟}_𝑤𝛽_𝑔𝜌_𝑔𝑘𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑟²𝑑𝑟,_{𝑝𝑅𝑝𝑤𝑓}𝑘𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤𝑑𝑝=^𝑟𝑒_𝑟𝑤𝑞_𝑤2𝜋𝑟ℎ𝑑𝑟, (2) where 𝑝_𝑅 is reservoir pressure, 𝑝_𝑤𝑓 is bottom hole flowing pressure, 𝑟_𝑒, 𝑟_𝑤are external boundary and wellbore radius, respectively, 𝐵_𝑔𝐵_𝑤are gas and water volume factor, respectively, 𝑞_𝑔𝑞_𝑤are gas and water production rate, respectively, and ℎ is reservoir thickness.

Fevang and Whitson [15] defined the gas and water phase pseudopressure in two-phase filtration. The equations are as follows: Δ𝑚𝑝_𝑔=^𝑝𝑅_𝑝𝑤𝑓𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤𝑅_𝑠𝑔𝑤+𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑝𝑑𝑝,Δ𝑚_𝑤=^𝑝𝑅_𝑝𝑤𝑓𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤+𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑅_𝑠𝑤𝑔𝑑𝑝, (3) where 𝑅_𝑠𝑔𝑤 is solution gas-water ratio and 𝑅_𝑠𝑤𝑔 is solution water-gas ratio.

By combing (2) and (3), the gas and water phase pseudopressure can be expressed as Δ𝑚𝑝_𝑔=𝑞_𝑔1𝑅_𝑝𝑔𝑤^𝑟𝑒_𝑟𝑤𝑅_𝑠𝑔𝑤2𝜋𝑟ℎ𝑘𝑑𝑟+^𝑟𝑒_𝑟𝑤1+𝑞2𝜋𝑟ℎ𝑘𝑑𝑟_2𝑔4𝜋²ℎ^{2𝑟𝑒𝑟}_𝑤𝛽_𝑔𝜌_𝑔𝑘𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑟²𝑝𝑑𝑟,Δ𝑚_𝑤=𝑞_𝑤^𝑟𝑒_𝑟𝑤12𝜋𝑟ℎ𝑘𝑑𝑟+𝑅_𝑝𝑔𝑤^𝑟𝑒_𝑟𝑤𝑅_𝑠𝑔𝑤+𝑞2𝜋𝑟ℎ𝑘𝑑𝑟_2𝑤𝑅²_𝑝𝑔𝑤4𝜋²ℎ^{2𝑟𝑒𝑟}_𝑤𝛽_𝑔𝜌_𝑔𝑘𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑟²𝑑𝑟, (4) where 𝑅_𝑝𝑔𝑤 is production gas-water ratio.

In order to simplify the expressions of gas and water phase pseudopressure, we define four parameters: 𝐴=^𝑟𝑒_𝑟𝑤112𝜋𝑟ℎ𝑘𝑑𝑟,𝐵=4𝜋²ℎ^{2𝑟𝑒𝑟}_𝑤𝛽_𝑔𝜌_𝑔𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑟²𝐶𝑑𝑟,_𝑔=^𝑟𝑒_𝑟𝑤𝑅_𝑠𝑔𝑤𝐶2𝜋𝑟ℎ𝑘𝑑𝑟,_𝑤=^𝑟𝑒_𝑟𝑤𝑅_𝑠𝑤𝑔2𝜋𝑟ℎ𝑘𝑑𝑟. (5)

By combing (4), we obtain the gas-water two-phase IPR equation 𝑞_𝑔=−1𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑔+1𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑔²+4𝐵⋅Δ𝑚𝑝_𝑔(2𝐵)⁻¹,𝑞_𝑤=−𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑤+𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑤²+4𝐵𝑅²_𝑝𝑔𝑤⋅Δ𝑚𝑝_𝑤×2𝐵𝑅²_𝑝𝑔𝑤⁻¹,Δ𝑚𝑝_𝑔=^𝑝𝑟_𝑝𝑤𝑓𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤𝑅_𝑠𝑔𝑤+𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑝𝑑𝑝,Δ𝑚_𝑤=^𝑝𝑟_𝑝𝑤𝑓𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤+𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑅_𝑠𝑤𝑔𝑑𝑝. (6)

This equation is determined by the four parameters 𝐴,𝐵,𝐶_𝑔, and 𝐶_𝑤, where 𝐴 is laminar coefficient, 𝐵 is turbulence coefficient, 𝐶_𝑔 is gas-soluble coefficient, representing dissolved gas within gas well control range, and 𝐶_𝑤 is water-soluble coefficient, representing dissolved formation water in gas within the well control range.

3. Gas-Water Two-Phase Comprehensive Model and the Solution

3.1. Gas and Water Relative Permeability

Gas and water phase relative permeabilities 𝑘_𝑟𝑔, 𝑘_𝑟𝑤 are needed in order to calculate the gas-water two-phase IPR equation. 𝑘_𝑟𝑔, 𝑘_𝑟𝑤 are a function of water saturation 𝑆_𝑤, the empirical equations are represented as follows [15]: 𝑘_𝑟𝑔=1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)},𝑘_𝑟𝑤=𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)}, (7) where 𝑆_𝑤 is water saturation in reservoir, 𝑆_𝑤𝑖 is initial water saturation, and 𝐷 is relative permeability index.

From (7), we obtain 𝑘_𝑟𝑔𝑘_𝑟𝑤=1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)}𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)}. (8)

Using the ratio of gas and water production, a method that aimed to obtain the ration of gas and water phase relative permeability is proposed by Jokhio and Tiab [16].

Consider 𝑘_𝑟𝑔𝑘_𝑟𝑤=𝐵_𝑔𝜇_𝑔𝐵_𝑤𝜇_𝑤𝑅_𝑝𝑔𝑤−𝑅_𝑠𝑔𝑤1−𝑅_𝑝𝑔𝑤𝑅_𝑠𝑤𝑔. (9)

The parameters 𝜇_𝑔,𝜇_𝑤,𝐵_𝑔,𝐵_𝑤,𝑅_𝑠𝑤𝑔, and 𝑅_𝑠𝑔𝑤 in (9) are functions of pressure 𝑝. Therefore, the ratio of gas and water phase relative permeability 𝑘_𝑟𝑔/𝑘_𝑟𝑤 can be obtained by means of 𝑝 and 𝑅_𝑝𝑔𝑤. Then 𝑆_𝑤 can be calculated. At last, 𝑘_𝑟𝑔,𝑘_𝑟𝑤 can be obtained.

By combining (8) and (9), we obtain 1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)}𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)}=𝐵_𝑔𝜇_𝑔𝐵_𝑤𝜇_𝑤𝑅_𝑝𝑔𝑤−𝑅_𝑠𝑔𝑤1−𝑅_𝑝𝑔𝑤𝑅_𝑠𝑤𝑔. (10)

From (7) and (10), the equations for calculating 𝑘_𝑟𝑔, 𝑘_𝑟𝑤 are defined as follows: 𝑘_𝑟𝑔=1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)},𝑘_𝑟𝑤=𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)},1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)}𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)}=𝐵_𝑔𝜇_𝑔𝐵_𝑤𝜇_𝑤𝑅_𝑝𝑔𝑤−𝑅_𝑠𝑔𝑤1−𝑅_𝑝𝑔𝑤𝑅_𝑠𝑤𝑔. (11)

Based on the material balance equation in the water drive gas reservoir, the relationships between average formation pressure and geologic reserve, cumulative gas production, and water invasion intensity can be obtained as follows [17]: 𝑝𝑧=𝑝_𝑖𝑧_𝑖1−𝐺_𝑝/𝐺1−𝐺_𝑝/𝐺^𝑅, (12) where 𝑝, 𝑝_𝑖are current and initial reservoir pressure, respectively, 𝑧, 𝑧_𝑖are gas deviation factor under current and initial reservoir pressure, respectively, 𝐺_𝑝, 𝐺are cumulative gas production and dynamic reserves, respectively, and 𝑅 is water invasion coefficient.

3.2. Gas-Water Two-Phase Comprehensive Model

By combining (6), (10), and (12), the gas-water two-phase comprehensive model can be expressed as 𝑞_𝑔=−1𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑔+1𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑔²+4𝐵⋅Δ𝑚𝑝_𝑔(2𝐵)⁻¹,𝑞_𝑤=−𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑤+𝐴+𝑅_𝑝𝑔𝑤𝐶_𝑤²+4𝐵𝑅²_𝑝𝑔𝑤⋅Δ𝑚𝑝_𝑤×2𝐵𝑅²_𝑝𝑔𝑤⁻¹,Δ𝑚𝑝_𝑔=^𝑝𝑟_𝑝𝑤𝑓𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤𝑅_𝑠𝑔𝑤+𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑝𝑑𝑝,Δ𝑚_𝑤=^𝑝𝑟_𝑝𝑤𝑓𝑘_𝑟𝑤𝐵_𝑤𝜇_𝑤+𝑘_𝑟𝑔𝐵_𝑔𝜇_𝑔𝑅_𝑠𝑤𝑔𝑘𝑑𝑝,_𝑟𝑔=1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)},𝑘_𝑟𝑤=𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)},1−𝑆_𝑤−𝑆_𝑤𝑖²1−𝑆_𝑤−𝑆_𝑤𝑖^{(5−𝐷)/(3−𝐷)}𝑆_𝑤−𝑆_𝑤𝑖^{(11−3𝐷)/(3−𝐷)}=𝐵_𝑔𝜇_𝑔𝐵_𝑤𝜇_𝑤𝑅_𝑝𝑔𝑤−𝑅_𝑠𝑔𝑤1−𝑅_𝑝𝑔𝑤𝑅_𝑠𝑤𝑔,𝑃𝑍=𝑃_𝑖𝑍_𝑖1−𝐺_𝑝/𝐺1−𝐺_𝑝/𝐺^𝑅. (13)

3.3. The Solution of Gas-Water Two-Phase Comprehensive Model

As (13) shows above, laminar coefficient 𝐴, turbulence coefficient 𝐵, gas-soluble coefficient 𝐶_𝑔, water-soluble coefficient 𝐶_𝑤, relative permeability index 𝐷, water invasion coefficient 𝑅, and single well controlled reserves 𝐺 are needed to solve. An automatic fitting multiobjective optimization method is given in this paper to solve the problem in the complicated percolation model mentioned above. The essence of this method is seeking the best fitting between theoretical value and measured value. The solution is defined as follows: 𝐸=^𝑛_𝑖=1𝑞_𝑔𝐴,𝐵,𝐶_𝑔,𝐶_𝑤,𝐷,𝑅,𝐺−𝑞_𝑔²+^𝑛_𝑖=1𝑞_𝑤𝐴,𝐵,𝐶_𝑔,𝐶_𝑤,𝐷,𝑅,𝐺−𝑞_𝑤², (14) where 𝑞_𝑔(𝐴,𝐵,𝐶_𝑔,𝐶_𝑤,𝐷,𝑅,𝐺) and 𝑞_𝑤(𝐴,𝐵,𝐶_𝑔,𝐶_𝑤,𝐷,𝑅,𝐺) are theoretical gas and water production rate, respectively, 𝑞_𝑔 and 𝑞_𝑤 are actual gas and water production rate, respectively, and𝐸 is expressed as the target function to be fitted. The proper parameters can be obtained to minimise the target function by means of the automatic fitting method. The flow chart for plotting type curves is shown in Figure 1.

Figure 1 

4. Case Analysis

4.1. Calculating the Target Parameter

During the process of acquiring the parameters like laminar coefficient 𝐴, turbulence coefficient 𝐵, gas-soluble coefficient 𝐶_𝑔, water-soluble coefficient 𝐶_𝑤, relative permeability index 𝐷, water invasion intensity 𝑅, and single well controlled reserve 𝐺, the theoretical gas and water production can be obtained based on (13). By fitting the practical gas and water production on the basis of the automatically fitting method in (14), the result shown in Figure 2 can be acquired. It is clear that the theoretical results were verified with the practical ones which indicates the reliability of the results. The basic parameters of an actual well are shown in Table 1.

Table 1 

The basic parameters of an actual well.

(a)

(b)

Figure 2 

The parameters like laminar coefficient 𝐴, turbulence coefficient 𝐵, gas-soluble coefficient 𝐶_𝑔, water-soluble coefficient 𝐶_𝑤, relative permeability index 𝐷, water invasion intensity 𝑅, and single well controlled reserve 𝐺 are shown in Table 2.

Table 2 

The target parameter in an actual well.

The value of water-soluble coefficient in this table is very small which suggests that the content of the formation water dissolving into the natural gas is little. And the energy of the formation water in the well controlled range is weak when the water invasion intensity is greater than 4. The gas and water phase relative permeability curve in different water saturations is obtained. The results are shown in Figure 3.

Figure 3 

4.2. The IPR Curves of Water Producing Gas Well

The IPR curves in different production gas-water ratios (GWR) are expressed in Figure 4. It is noted that the IPR curves are expressed with a left offset when production gas-water ratio decreases. Because when the GWR decreases, the water saturation in formation increases, and the flow resistance increases too. It becomes harder to flow in formation when the GWR decreases. The absolute open flow rates (𝑞_AOF) of water producing gas well in different production gas-water ratios can be shown in Table 3. From the curves, when the production gas-water ratios are 1, 0.5, 0.2, and 0.1, respectively, the absolute open flow rates are reduced to 32.76, 31.71, 28.79, and 24.61 × 10⁴ m³/d accordingly. The absolute open flow rates are reduced by 3.21%, 12.12%, and 24.88%, when compared to the one whose production gas-water ratio is 0.1. It is shown that water invasion will greatly reduce the gas production capacity of water producing gas well.

Table 3 

The absolute open flow rates in different production gas water ratios.

Figure 4 

The IPR curves in different gas-soluble coefficients are shown in Figure 5. It can be concluded that the IPR curves are expressed with a left offset when the gas-soluble coefficient increases. Gas-soluble coefficient represents the solubility of gas in water, which leads to a bigger flow resistance. The absolute open flow rates (𝑞_AOF) of water producing gas well in different gas-soluble coefficients can be shown in Table 4. From the curves, it is clear that when gas-soluble coefficients are 0.1, 0.5, 1, and 2, respectively, absolute open flow rates are 32.94, 29.57, 25.91, and 20.22 × 10⁴ m³/d, respectively. The absolute open flow rates are reduced by 10.23%, 21.34%, and 38.62%, when compared to the one whose gas-soluble coefficient is 0.1. It is shown that the more gas dissolved in the formation water, the more liquid phase will exist in the formation fluid. It will increase the gas flowing resistance and result in the greater productivity impairment.

Table 4 

The absolute open flow rate in different gas-soluble coefficients.

Figure 5 

5. Conclusions

Based on the basic theory of fluid mechanics in porous medium, taking the solution and volatilization of gas and water into consideration, a gas-water two-phase IPR equation was established. Combining with gas-water two-phase IPR equation, relative permeability equation, and material balance equation in the water drive gas reservoir, we deduced the gas-water two-phase comprehensive model, which is influenced by laminar coefficient, turbulence coefficient, gas-soluble coefficient, water soluble coefficient, relative permeability index, water invasion intensity, and single well controlled reserve. The influences of different production gas-water ratios and gas-soluble coefficients on the absolute open flow rate were discussed. The method proposed in this paper provided a new theoretical method for single well analysis of productivity and inflow performance and got rid of the limitation that the well productivity can only be determined according to field well test.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful for financial support from the National Science Fund for Distinguished Young Scholars of China (Grant no. 51125019).