Xiangji Dou^{*}, Xinwei Liao, Xiaoliang Zhao, Huan Wang, Huawei Zhao and Dongfeng Zhao  
Department of Petroleum Engineering, China university of Petroleum, Beijing, China  
Corresponding Author :  Xiangji Dou Department of Petroleum Engineering China university of Petroleum, Beijing, China Tel: +861089733256 Email: [email protected] 
Received September 24, 2014; Accepted February 28, 2015; Published March 07, 2015  
Citation: Dou X, Liao X, Zhao X, Wang H, Zhao H, et al. (2015) Determination of Dynamic Deliverability Equation for Fractured Horizontal Well in Tight Gas Reservoir. J Pet Environ Biotechnol 6:210. doi:10.4172/21577463.1000210  
Copyright: © 2015 Dou X, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
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The deliverability equation of gas well is important for productivity evaluation and performance forecasting. For fractured horizontal well in stresssensitive tight gas reservoir, due to the pressuredependencies of gas properties, reservoir/fracture permeability and highvelocity nonDarcy flow, the deliverability equation would not be constant but is also dynamic. Traditionally, the relationship between reservoir/fracture permeability is commonly be obtained by laboratory experiments, which is expensive and timecost. Besides, the predecessor’s studies which neglect the pressuredependence of high velocity nonDarcy flow would also lead inaccuracy. Therefore, this paper presented a new method to quantify pressuredependence of reservoir/fracture permeability as well as obtain dynamic deliverability equation for multifractured horizontal well in tight gas reservoir considering nonconstant nonDarcy flow. This method is validated by an actual well in Sulige tight gas field in Ordos Basin, China. The result shows that this method is accurate and can contribute to the effective development of stresssensitive tight gas reservoir
Keywords 
Tight gas reservoir; Fractured horizontal well; Deliverability equation; Nondarcy flow; Stresssensitivity 
Nomenclature 
Field variables 
D_{res} nonDarcy flow coefficient in reservoir, (m^{3}/d)^{1} 
D_{f} nonDarcy flow coefficient in fracture, (m^{3}/d)^{1} 
F_{C} fracture conductivity, md•m 
F_{CDIM} the ratio of F_{C} (p_{r}) to F_{C} (p_{i}) 
q_{g} gas rate, m^{3}/d 
h formation thickness, mm 
k formation permeability, md 
k_{f} hydraulic fracture permeability, md 
P pressure, MPa 
P_{i} initial pressure, MPa 
P_{r} average formation pressure, MPa 
P_{wf} bottom hole flowing pressure 
x_{f} fracture halflength, m 
ω fracture width, m 
Introduction 
It is widely accepted that for some tight gas reservoirs, reservoir permeability are sensitive to pressure change. To investigate this pressuredependence, a large amount of experiments were conducted. Gray et al, Brace, Jones and Owens [13] proved that an exponential relationship between permeability and pressure are found in several typical rocks. 
(1) 
Besides, with widely application of fractured well, the issue that how fracture parameters would change with pressure has attracted plenty of attention. Furthermore, for fractured horizontal well, highvelocity nonDarcy is very important because of relatively high productivity. Due the existence of fracture, nonDarcy flow is influenced by fracture parameters as well as reservoir properties. The significance of non Darcy flow has been emphasized in some papers. However, the stresssensitive nonDarcy flow coefficients in both reservoir and fracture still remain to be investigated. For field application, determining this relationship with core experiment is not economical, so an easier and accuracy method is needed to obtain the pressuredependencies of these parameters. Deliverability equation is essential to determine productivity of gas well and thus forecast well performance. There are three formats of deliverability equations. They are characterized by pressure, square of pressure and pseudopressure, respectively. Among these, the equation characterized by pseudopressure (Equatation. 4) is the most accurate but may lead to complexities while the one characterized by pressure (Equatation. 2) may result in significant inaccuracy. In practice, deliverability equation characterized by square of pressure (Equatation. 3) is common used in field case. 
(2) 
(3) 
(4) 
Where A represents degree of percolation resistance force in reservoir and viscous force in fracture, B represents degree of inertial resistance caused by turbulence flow. However, for tight gas reservoir, it is always very costly and moneyconsuming to conduct actual productivity well test to determine deliverability equation and openhole capacity due to ultralow permeability. Besides, the pressuredependencies of parameters mentioned above can lead the deliverability to be dynamic. It is also impractical to conduct productivity well test for several times during well life to obtain the dynamic deliverability equation for different production phase. Therefore, an effective and fast simulation method is required to obtain the dynamic deliverability equation and investigate the influence of pressuredependent parameters on this equation. The objective of the paper is, therefore, to provide a new method to determine the pressuredependencies of several parameters and obtain dynamic deliverability equations for fractured horizontal well in tight gas reservoir. 
Determination of Dynamic Deliverability Equation 
For a fractured horizontal well with several fractures, due to the complexities of reservoir and fractures, it is difficult to determine deliverability equation exactly with analytical method. On the other hand, field test methods are always costly and timeconsuming, which cannot meet the demand of tight gas reservoir development. Modified isochronal well test simulation, a new method to determine deliverability is adopted in this paper to solve the problem. Core idea of this method is obtaining the pressure change corresponding to a reasonable simulated productivity well test system based on a numerical model, which is built based on production data analysis. Through establishing numerical model with the parameters of reservoir and well derived from production data, the bottom hole flowing pressure data of productivity well test for fractured horizontal well can be obtained by simulating the process of modified isochronal well test. The binomial deliverability equation and openhole capacity of gas well could be determined with the rate and pressure obtained. The practical application shows that this method has advantages of strong pertinence, convenient operation, reliable result and timesaving. Different from conventional horizontal well, wellbore of fractured horizontal well in tight gas reservoir is always cased or contributes relatively litter gas compared to the fractures. Research shows that as a result of high velocity, additional drawdown due to turbulence flow within and around the fracture cannot be neglected. Thus, Darcy’s flow, which describes velocity as a linear function of the pressure gradient and neglects inertial forces, is not suitable for fractured well in tight gas reservoir. The characteristics of fluid flow can be described with Forchheimer law [4]. 
(5) 
According to highvelocity nonDarcy flow theory, a factor Dq_{sc} similar to skin factor can be used to describe the degree of highvelocity nonDarcy flow. Where, D is called nonDarcy flow coefficient. In this paper, nonDarcy flow in reservoir as well as in fractures is considered. 
(6) 
(7) 
Where, velocity coefficient β in reservoir and fracture can be expressed as Equatation. (8) and Equatation. (9), respectively [3]. 
(8) 
(9) 
It can been conclude from Eq. (6~9) that β and D are permeability– dependent (including permeability of formation and fracture). For stresssensitive reservoir and fractures, permeability of formation and fractures would change with pressure, which leads to the pressuredependent (stresssensitive) nonDarcy flow coefficient. Contrary to the conventional method, approach in this paper facilitates the consideration of highly inertial flow and accounts also for permeability (pressure) dependency of nonDarcy flow coefficients. In this case, due to the complexity of porous flow model and stresssensitivities, obtaining parameters and openhole capacity with a unique analytical equation turn out to be difficult. Therefore, Infinitesimal method theory can be used in obtaining formation and fracture parameters and thus determining productivity equation. A production history is divided into several segments, each with minor change in reservoir and fracture parameters. So, parameters including formation permeability, fracture halflength, fracture conductivity and nonDarcy flow coefficients can be treated as parameters with fixed value for each stage. By doing this, not only parameters and openhole capacity needed for each stage can be obtained, but also the change rule of parameters and deliverability equation is investigated. Thus, the binomial deliverability equation of fractured horizontal well could be expressed as follows. 
First stage 
(10) 
Second stage 
(11) 
Nth stage 
(12) 
By this analogy, binomial deliverability equations for different stages could be obtained. With infinitesimal analysis method, influence of formation/fracture parameters and nonDarcy flow coefficient change caused by stresssensitivity on gas well productivity is quantified. 
To build models for different pressure, reservoir and fracture parameters as well as the pressuredependence of them needed be determined. For tight gas reservoir, as a result of stress sensitivities, the formation permeability, fracture parameters (halflength and conductivity) and nonDarcy flow coefficients would change with formation pressure. Thus, the accuracy of historymatch would be reduced when a unique set of parameters are used. Therefore, with reasonable time of division, segmented history match method can be used to obtain parameters of each stages and analysis the relationship between them and pressure. Before segmented history match, based on the production data, flowregime should be identified. Then PDA methods including straightline and typecurve method modified for stresssensitive reservoir are used to provide an initial historymatch input. Due to the parameters change caused by stresssensitivity, dynamic parameters need to be adjusted to reach a perfect match. The following is a brief description of workflow. 
(1) Compile production and associated flowing pressures, fluid properties, initial reservoir pressures, volumetric reservoir information (reservoir thickness, porosity), wellbore and completion/stimulation data together with the initial value of dynamic parameters obtained by straightline and typecurve. Based on the information above, an initial numerical model is built for historymatch. 
(2) An overall historymatch for the whole production history data is conducted to determine an approximate range of parameters so that to provide reference for segmented historymatch. A deviation between matched curve and actual curve is always obvious in this step. The deviation is caused by stresssensitivity. 
(3) Based on the overall historymatch, segmented historymatch is conducted for each stage. For this step, a reasonable time division is very important to get an accurate result. It should be noted that, nonDarcy flow coefficients used for historymatch should satisfy the relationship between this coefficients and reservoir/fracture parameters. 
(4) If transient well test are conducted to the well, the value obtained from welltest analysis should be used to validate the results. 
(5) After completing the historymatch for every segment, dynamic parameters for each segment are determined; the pressuredependencies of them are also quantified. 
(6) Based on the change rule, value of dynamic parameters for every pressure could be determined, which is essential to forecast the deliverability equation. 
With accurate parameters of each stage obtained by segmented historymatch method, deliverability equations of each stage can be determined by modified isochronal well test simulation. The following workflow is recommended: 
(a) For a certain pressure, calculate μ and Z with Lee and Gonzalez method, Beggs and Brill method, respectively. 
(b) Based on the pressuredependence of formation and fracture parameters obtained with (1) ~ (6), formation permeability and fracture parameters (halflength and conductivity) are determined; velocity coefficient and nonDarcy flow coefficient can also be calculated. 
(c) Compile dynamic parameters obtained above together with static parameters, numerical simulation model of fractured horizontal well for the pressure can be built. 
(d) Based on the model, deliverability and openhole capacity for the pressure are determined with modified isochronal well test simulation. 
(e) If actual productivity well test are conducted to the well, the value obtained should be used to validate the results. 
Depending on studies above, pressuredependence of formation/ fracture parameters are obtained, based on which, dynamic deliverability equation for different pressure could be forecasted. 
ResultsExample Application 
The application of this method is illustrated with a field example of Sulige gas reservoir in Ordos Basin. SuA is a typical fractured horizontal well completed in tight gas reservoir, which is multifractured in 5 stages, the healtotoe perforated length of the well is 552 m. Firstly, production data (rate and flowing pressure) and information provided by well logging, core experiment and other technologies are collected (Table 1). Production data analysis is conducted to provide initial value for historymatch together with logging and core experiment results, based on which, a numerical simulation model is built as shown in Figure 1. 
Then, an overall historymatch for the whole production history data is conducted to determine an approximate range of parameters so that to provide reference for segmented history match. A deviation between matched curve and actual data is obvious in the result of overall history match a unique set of parameters, which is caused by significant stresssensitivities in reservoir and fractures. Based on the overall history match, segmented historymatch is conducted for each stage. The parameters determined by segmented historymatch method are shown in Table 2. 
Based on the pressuredependence of formation/fracture parameters, the relationships between stresssensitive parameters and average formation pressure are investigated. The results show that exponential form could be used to describe the change of formation permeability and fracture parameters. Furthermore, the fracture conductivity drops faster than formation permeability when average formation pressure declines. However, in this case, nonDarcy flow coefficients in both fracture and formation increase, which caused additional drawdown and lead to negative effect on well production performance. Based on the relationship, the value of each parameter for certain pressure could be forecasted (Figures 2 and 3) and applied for deliverability equation determination. 
With modified isochronal well test simulation, binomial deliverability equation coefficients and openhole capacity of each stage are determined (Table 3). 
At the first segment of well life, an actual productivity well test was conducted to SUA, the equation obtained by the test analysis was P_{r}^{2} P_{wf}^{2}=23.58q_{g}+0.176q_{g}^{2}, While another test conducted during the fifth segment shows the equation was P_{r}^{2} P_{wf}^{2}=21.35qg+0.256q_{g}2, Both of these are similar to the results obtained with the method proposed in this paper. Therefore, modified isochronal well test simulation method is a useful and economical method to determine productivity of fractured horizontal well in tight gas reservoir. 
With this method, we forecasted deliverability for different pressure for SuA, the results is shown in Figure 4. It can be concluded that with the decrease of average formation pressure, the value of coefficient a keeps decreasing, while the value of coefficient B shows an opposite character. 
Conclusion 
Based on the results of the present study, the following conclusions are studied. 
• Segment historymatch method can be used to determine change rule of dynamic parameters. With this method, stresssensitive nonDarcy flow coefficients in both reservoir and fracture could be researched. 
• Modified isochronal well test simulation presented in this paper is a useful and economical way to determine deliverability equation and openhole, validated by actual productivity well test, the result obtained by modified isochronal well test simulation is accurate. 
• Due to the existence of pressuredependent gas properties and stress sensitivity, deliverability equation would not be constant during the well life. While stresssensitive nonDarcy flow coefficients are taken into consideration, the change rule of deliverability equation would be much more complex. 
Acknowledgements 
This work was supported by the National Basic Research Program of China (973 program, grant No.2011CB707302), Chinese National Major Science and Technology Projects (2011ZX05016006 and 2011ZX05009004001), The Specialized Research Fund for the Doctoral Program of Higher Education of China (No 20120007120007). 
References 

Table 1  Table 2  Table 3 
Figure 1  Figure 2  Figure 3  Figure 4 