Probabilistic Decline Curve Model
Despite the simplicity and easy application of DCA methods as compared to other production forecast tools such as analytical methods, semi-analytical methods, and numerical methods, DCA methods have some shortcomings due to their nature of the deterministic inversion procedures. These shortcomings include the strong dependency on the smoothness of the production data. If there are many outliers in the production data, the determination of the unknown parameters in DCA models could become difficult.
This difficulty becomes worse in shale gas reservoirs, as the production data of shale gas reservoirs are more complex and more random than those of conventional reservoirs. Common disadvantages of deterministic inversion methods include the following : (1) The nonlinear minimization requires explicit information about model derivatives; (2) no explicit information is provided about the uncertainty of the interpretation results; (3) inversion-based interpretation results are deeply biased by the initial search point in model space. To minimize these disadvantages and prevent large errors in deterministic DCA models, various probabilistic DCA methods can be used to forecast the production in shale gas reservoirs based on the limited production data available. Although this review focuses on deterministic DCA models, probabilistic DCA methods are still briefly reviewed below so that interested readers could use this as a reference list.
Bayes’s Theorem  has been widely used for uncertainty quantification [59,60,61,62]. Gong et al.  introduced a Bayesian probabilistic methodology using Markov chain Monte Carlo (MCMC) coupled with the standard Metropolis–Hasting (MH) algorithm [63,64]. The Arps decline curve model was applied to quantify the uncertainty in production forecasts and reserve estimates. Applying the new method to 197 Barnett shale gas wells, the authors concluded that their methodology can quantify the uncertainty in hindcasted production with a narrower P90–P10 interval and higher computational efficiency than the modified bootstrap method , which was originally proposed by Jochen and Spivey .
A Bayesian machine learning method with the MCMC coupled with the MH algorithm was proposed in Fulford et al. , and was used to forecast production in shale wells by utilizing the transient hyperbolic model introduced by Fulford and Blasingame . However, the behavior of b and D parameters as a function of time is only valid for homogeneous reservoirs with equal fracture half-length and spacing, not suitable for heterogeneous reservoirs with varying fracture half-length and spacing .
In addition, the standard MH algorithm has a lower acceptance ratio  than other MCMC algorithms such as delayed rejection , adaptive Metropolis , and the combination of delayed rejection and adaptive Metropolis . A good summary and discussion of the advanced MCMC algorithms can be found in Goodwin . To improve the MH algorithm, Yu et al.  proposed a new probabilistic approach with the Bayesian methodology combined with MCMC sampling and a FDC decline curve model to increase the efficiency and reliability of the uncertainty quantification.
Table 1. Summary of all DCA models in this review.
The authors used an adaptive Metropolis algorithm to replace the MH algorithm and applied their model to multiple gas wells in Fayetteville shale. Recently, de Holanda et al.  constructed a physics-based DCA model accounting for linear flow and material balance in horizontal multi-stage hydraulically fractured wells. In that work, the model was applied to a large dataset in a workflow that incorporates heuristic knowledge into the history matching and uncertainty quantification by assigning weights to rate measurements. The uncertainty quantification was performed via a Bayesian approach with hindcasts, and applied to 992 gas wells from the Barnett shale.
Comparisons of DCA Models with Field Data
The above reviewed eight DCA models are popular candidates to be used for shale gas production forecasting. However, different models provide different predictions for production data from various shale formations. In this section, several comparison studies are illustrated to show the differences among these DCA models.
Kanfar and Wattenbarger  compared five DCA methods explained in our study: Arps, PLE, SEPD, Duong, and LGM models. These models were compared by matching field data from Barnett shale and Eagle Ford shale. The field data are Barnett shale gas Well 314 and Eagle Ford shale gas Well 204.
Figure 4 shows the matching results for these five models, which shows that for Well 314 in Figure 4a, the PLE and LGM methods have close results while Arps returns the most optimistic estimate and SEPD is the lowest estimate. For Well 204 in Figure 4b, the SEPD method has the lowest estimate and Arps has the highest estimate. From these applications, the authors concluded that: if a well is within a linear flow regime, the most accurate EUR result could be obtained by using Arps, PLE, or Duong; for a well within the bilinear flow, the best DCA models are the PLE or Duong model; if the well production shows boundary-dominated flow, the best DCA model is the PLE method.
Figure 4. Comparisons of four DCA models : (a) Barnett shale gas well 314 results comparison; (b) Eagle Ford shale gas Well 204 comparison.
Joshi and Lee  compared three DCA models—Arps, SEPD, and Modified Duong—by matching field data from Barnett Shale. Figure 5 illustrated that, for two cases in Figure 5a,b, the Arps model overestimated the production compared to historical data, while SEPD and Modified Duong provided acceptable results.
Figure 5. Comparison results for two well groups from Barnett shale , where EOH stands for the end of history used in the matching, and EOP is the end of production data. In both cases 36 months of historical data were used for matching. (a) Comparison results for an 81-well Denton County group; (b) comparison results for a 127-well Wise County group.
Here we also studied the differences of Arps, Duong, FDC, and SEPD by matching the Fayetteville shale gas production data with these four models. The comparison results of the flow rate and EUR are plotted in Figure 6. They show that Arps returns the most optimistic estimate, and SEPD returns the most pessimistic one, while FDC and Duong return similar intermediate matching results.
Despite challenges to shale gas reservoir modeling due to complicated transport mechanisms and the existence of fracture networks, oil companies have not slowed down on shale gas investment and production using horizontal well drilling and hydraulic fracturing techniques. Because building a reservoir model usually requires drilling test wells and performing well logging measurements, many small oil companies may not have the budget to do so. Even for large oil companies, building a reservoir model is not worthwhile for evaluation of small-scale oil fields. In those cases, comprehensive numerical simulation methods are likely impractical. In order to forecast the production of these reservoirs, DCA is one of the most convenient and practical techniques.
Figure 6. Comparison results for one well from Fayetteville shale. The red dots are the original field data. (a) The matching results for monthly flow rate; (b) the EUR results of DCA models.
With the rapid increase in shale gas production over the past 30 years, there have been numerous production data for shale gas reservoirs. Many different DCA models have been constructed to model the shale gas production rate, from the classical Arps to the latest FDC model; each has its advantages and shortcomings. In practice and in all existing commercial DCA software, most of these eight DCA models are implemented and open to be used. They are also extensively compared by clients, shown in this review.
Most of the deterministic DCA models are empirical and lack a physical background so that they cannot be used for history matching of the reservoir properties. Up until now there have been only a few studies that probed the relationships between DCA parameters and reservoir properties, such as permeability, porosity, fracture length, and fracture widths. Zuo et al.  discovered that, for shale gas wells in the Fayetteville formation, the DCA parameters are all within a certain small range. For example, for Fayetteville shale, parameter α defined in Equation (28) lies in the range from 0.65 to 0.75, which indicates that this parameter might have some relationship with the physical parameters.
Zhang et al.  showed that some recommended empirical coefficients can be determined for specific shale formations such as Eagle Ford, which may help, particularly in the early stages of development when little production data are available. In the future, once more information about the fractures and reservoirs becomes available, more detailed investigations about the links between DCA parameters and reservoir properties can be performed.
Nowadays, with the advancement of machine learning and data analysis techniques, the combination of deterministic DCA models with machine learning techniques could also be an interesting future trend of DCA model applications.
Because of their simple mathematical expressions, fast computation speed, and low dependence on the geophysical data of the reservoirs, DCA models have been widely applied to to forecast shale gas production using data. As the simplest technique for shale gas production forecasting, DCA models do not need specific reservoir or fracture data input. DCA models do not require the construction of a complicated reservoir model. Using only the production data, DCA models can return accurate history matching with the production data by different type curves and forecast the EUR using interpolated curves. In this study, eight popular DCA models for shale gas reservoirs are reviewed, including their origins, formulations, and the types of reservoirs they fit. Their advantages and disadvantages have also been clearly presented.
As one of the earliest DCA models, Arps is designed for boundary-dominated flows but most shale gas reservoirs rarely reach the boundary-dominated flow regime, so the original Arps curve is not appropriate to simulate the shale gas reservoir production. If used directly on shale gas production, Arps tends to overestimate EUR.
To be compatible with two different flow regimes (i.e., the transient flows and the boundary-dominated flows), two decline rates can be used for different time periods. By doing so, the modified Arps curve and MHD could return a good match for shale gas production by fitting the unknown parameters. However, the determination of the critical decline rate D * relies on empirical methods.
The SEPD model has a better performance for transient flows than for boundary-dominated flows. Because its expression has a finite limit, it has the advantage of providing a bounded value of EUR. The disadvantage is that SEPD requires a sufficiently large set of production data in order to obtain good determination of the unknown parameters. With few production data, SEPD usually return the lowest EUR.
The modified Duong model is more accurate for linear flow and bilinear flow than other DCA models proposed before. However, if the production history is shorter than 18 months, the Duong model could return unreliable results for the EUR forecast. Most of the time, the Duong model overestimates the total EUR.
For shale gas reservoirs with a single well produced over a sufficiently long time, the SEPD model can be applied. As the SEPD model, LGM returns a finite estimate of EUR because of the constraints of carrying capacity K as well as the vanishing production rate at infinity time. SEPD is suitable for extremely low-permeability shale gas reservoirs and could provide less EUR than the Arps model. It usually yields similar results to the PLE model.
By considering both the early and late production profile, the EEDCA model does not require a switch from a transient model to a boundary-dominated flow model. It could result in smooth decline profiles. EEDCA model can be applied for shale gas reservoirs with both early and late production data.
Originating from the anomalous diffusion phenomena with a rigorous physical background, the FDC model is constructed based on the anomalous diffusion kernel function. It has historical dependence effects and could match the long tail behavior of the shale gas production well. For shale gas wells close to each other, the FDC model could have very similar parameter values.
To decrease the error caused by noisy data and better use the new production data, many probabilistic DCA models can be constructed to increase the accuracy of the deterministic DCA methods. When sufficient data are available, probabilistic DCA models can be applied to return reliable probability ranges.
The biggest disadvantage of all eight DCA models is that most of them are empirical without physical background with exception being FDC for some physical background. There are potential relationships between DCA parameters and reservoir properties, which need to be further investigated when more accurate field measurement of reservoir and fracture properties are available.
This work has been supported by the National Natural Science Foundation of China (Grant number 51741906), the State Key Laboratory of Hydroscience and Engineering (2018-KY-02), the Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering (sklhse-2018-E-01), the Tsinghua University Initiative Scientific Research Program [2014z21041], and the Key Laboratory of Fluid and Power Machinery (Xihua University), the Ministry of Education [szjj-2017-100-1-004]. The authors thank the anonymous reviewers for their valuable comments.
Lei Tan, Lihua Zuo, and Binbin Wang wrote the paper. All authors contributed equally to this work.
Conflicts of Interest
The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
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