Semi-analytical modeling of transient rate behaviour of a horizontal well with multistage fractures in tight formations considering stress-sensitive effect
Introduction
With the depletion of conventional reserves, unconventional resources, such as tight oil and shale gas, has attracted more and more attention and become the major energy resources in recent years (Wang and Yu, 2019). Horizontal drilling and hydraulic fracturing have been key technologies to successfully and efficiently develop tight formations since the generated multistage hydraulic fractures can greatly improve oil production rate. However, such designed fractures do not usually lead to good well performance as expected due to the composite effect of non-Darcy flow, stress-sensitive permeability, and non-uniform hydraulic fracture properties (Barree and Conway, 2009; Zhang and Yang, 2014; Ji et al., 2017; He et al., 2018; Luo et al., 2018). How to dynamically and accurately quantify the reservoir properties and accurately evaluate well performance in a tight formation, thus, has become a challenging task under such complex conditions.
Both pressure transient analysis (PTA) and rate transient analysis (RTA) have been widely regarded as effective methods to estimate reservoir properties and evaluate well performance by analyzing the bottomhole pressure responses and production profiles, respectively (Cinco-Ley et al., 1978; Agarwal et al., 1999; Nejadi et al., 2015; Xue et al., 2019). Usually, the pressure data for PTA can be obtained from well test measurements which will last for hours or even days. However, the PTA technique is found to be not satisfactory in tight formations due to its very slow and long transient response which makes this technique with short testing time not able to detect the main radial flow regimes and boundary conditions (Cheng et al., 2009; He et al., 2018). In contrast, the RTA technique may not be applicable to estimate the reservoir parameters by using the daily production rates and pressure responses in formations with moderate and high permeability due to the fact that the low-resolution acquired data cannot be used for further analysis without details (i.e., transient flow data) (Ilk et al., 2010). Nevertheless, the RTA technique is more attractive in tight formations with long-term transient flow regime, and thus the daily production data may be sufficient to conduct a competent analysis. Besides, the production data are available in great quantities which can provide useful information about the late-time flow regimes and the boundary conditions.
For the production decline analysis method, Arps (1945) presented three types of production decline curves (i.e., exponential decline, hyperbolic decline, and harmonic decline) by using the rate-time relationship; however, this method is limited to the boundary dominant flow and is not suitable for the transient flow analysis. Fetkovich (1980) extended the Arps type curves to the transient flow period and generated new type curves by combining the transient rate decline curve and Arps decline curves. Both Arps and Fetkovich methods, nevertheless, are not available to solve the variable wellbore pressure problem. Blasingame et al. (1989, 1991) generated decline type curves by adopting the pressure-normalized rate and the material balance time in order to deal with the variable rate and variable wellbore pressure problem. In order to reduce the noise of the field production data and the ambiguity associated with the inherent analysis, Blasingame et al. (1991) introduced the rate integral function and rate integral derivative function, which have been widely used in production decline analysis (Wang et al., 2012; Wei et al., 2016; Johnson and Jamiolahmady, 2018). However, it is usually assumed that the fracture conductivity and matrix permeability remain constant during the production (Wei et al., 2016, 2017; Zhao et al., 2016).
Although numerous efforts have been made to take the stress-sensitive permeability into account on the transient pressure responses (Berumen and Tiab, 1997; Wang et al., 2017, 2018a; Jiang et al., 2019a; b), few attempts have been made to investigate the effect of variable permeability on Blasingame type curves. Once the transient pressure solution in the Laplace domain has been obtained, it is natural for us to use the classical relationship between transient pressure and transient rate proposed by Van Everdingen and Hurst (1949). Accordingly, the transient rate solution in the Laplace domain at a constant wellbore pressure can be obtained by using the formula once the transient pressure solution at a constant rate is known. Although this method can readily be applied to generate the transient rate solution without considering the stress-sensitive effect (Wang et al., 2012, 2018b; He et al., 2018; Qin et al., 2019), it cannot be directly employed when stress-sensitive permeability has been taken into account. In fact, Van Everdingen and Hurst (1949) derived the formula based on the Duhamel's principle, which is only applicable to the linear equations (Ren and Guo, 2018). As such, the transient rate solutions in previous literature (Zeng et al., 2015; Li et al., 2017; Huang et al., 2018b; Lu et al., 2019) obtained directly from the classical relationship with considering stress-sensitive effect are not reliable and will lead to incorrect interpretation from the type curve matching method.
Luo et al. (2018) developed a semi-analytical model in the Laplace domain for a vertical well intercepted by a stress-sensitive fracture at constant wellbore pressures, but only stress-sensitive effect in fractures has been considered. By proposing a multi-linear flow model for a multi-fractured horizontal well (MFHW) with consideration of stress-sensitive effect, Ji et al. (2017) obtained the transient rate solution by adopting the perturbation method to deal with the nonlinearity caused by stress-sensitivity. Huang et al. (2018a) presented a modified tri-linear flow model for an MFHW with considering stress-sensitive effect, the Pedrosa substitution and perturbation methods were employed to eliminate the nonlinearity and obtain the transient rate solution in the Laplace domain. Ren and Guo (2018) generalized the relationship between the transient rate solution with consideration of stress-sensitive effect and the transient pressure responses without stress-sensitive effect, improving the model proposed by Van Everdingen and Hurst (1949) and making it available to take stress-sensitive effect into account. However, it is generally assumed the same permeability modulus between the matrix and fracture subsystems in order to successfully obtain the solutions which may lead to inaccurate estimation of the reservoir and fracture properties. Obviously, it is not physically reasonable to assume the same value of permeability modulus for matrix and proppant-filled fractures due to the different characteristics for these two subsystems. In fact, hydraulic fractures have a broader range of permeability modulus values, depending on the type of proppants and their properties (Berumen and Tiab, 1997; Wang et al., 2017).
In this paper, a novel coupled matrix-fracture flow model is developed and validated to examine the different stress-sensitive effects in matrix and fracture on transient production rate. More specifically, a slab source function in the Laplace domain was developed to describe the transient flow from matrix to the hydraulic fracture subsystem, while new solutions were obtained to describe the fluid flow in the fractures with considering the stress-sensitive fracture conductivity. A perturbation method is employed to eliminate the nonlinearity caused by stress-sensitivity of the matrix subsystem, while, for the hydraulic fractures, a semi-analytical method is introduced to obtain the solutions. The model has been validated and the effects of matrix permeability modulus, fracture permeability modulus, and initial fracture conductivity on production decline curves have been examined, analyzed, and discussed. New Blasingame type curves with consideration of different stress-sensitivities of matrix and fracture subsystems have been obtained. Finally, a field case has been analyzed with the newly developed model in this work.
Section snippets
Theoretical formulations
In this study, a hydraulically fractured horizontal well is located in a box-shaped tight formation (see Fig. 1a), with the main assumptions listed as follows:
- (1)
The closed rectangular reservoir is isotropic and homogeneous with uniform thickness and porosity . A slightly compressible single-phase fluid is assumed with constant viscosity ;
- (2)
The proppant-filled hydraulic fractures are fully penetrated in the box-shaped reservoir and equally spaced with length , the fractures are identical and
Model validation
To verify the newly developed semi-analytical model, the transient rate responses obtained in this paper are compared with solutions in the previous literature (Luo et al., 2018) and the results from a numerical simulator (Kappa, Version 5.0).
Results and discussion
In this section, dimensionless production decline curves and new Blasingame type curves are generated to identify the flow behaviour of a hydraulically fractured horizontal well with and without considering stress-sensitive effect. The effect of parameters including the dimensionless matrix permeability modulus, dimensionless fracture permeability modulus, and initial dimensionless fracture conductivity on the type curves are analyzed.
Case study
The case study is a horizontal well with multistage fractures located in the Pembina Cardium field, which was fractured in 10 stages with an average fracture spacing of 130.8 m (Clarkson and Pedersen, 2010, 2011). The 430-day production data of the well are plotted in Fig. 12a, other main parameters are listed in Table 2. As can be seen, it took the fractured well about 21 days to be produced under constant wellbore pressure condition. Fig. 12b presents a good agreement between the production
Conclusions
A semi-analytical model has been successfully developed in the Laplace domain to examine the stress-sensitive effect (both in matrix and fracture) on the transient rate behaviour for a multi-fractured horizontal well in a tight formation. The new Blasingame type curves can amplify the small difference in the flow process caused by stress-sensitive effect and help us easily quantify and analyze the transient rate behaviour. The existence of stress-sensitivity will decrease the production rate,
CRediT authorship contribution statement
Liwu Jiang: Conceptualization, Methodology, Formal analysis, Writing - original draft. Jinju Liu: Validation. Tongjing Liu: Resources, Supervision. Daoyong Yang: Conceptualization, Supervision, Funding acquisition, Resources, Writing - review & editing.
Declaration of competing interest
The authors declare no competing financial interests.
Acknowledgements
The authors acknowledge a Discovery Development Grant, a Discovery Grant, and a Collaborative and Research Development (CRD) Grant awarded to D. Yang from the Natural Sciences and Engineering Research Council (NSERC) of Canada and the financial support from the DM Oilfield High-Tech Ltd.
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