This work presents a workflow that can be used to analyze and forecast time/rate data of wells in low-and ultralow-permeability reservoirs. The key component of the workflow is the application of diagnostic plots to guide the analysis and obtain model parameters for a given time/ rate relation. Once model parameters are obtained, the production profile is extrapolated to yield the estimated ultimate recovery (EUR) at a specified time limit or abandonment rate.
Introduction
The starting point for any discussion of decline-curve analysis for unconventional reservoirs must be an understanding that no simplified time/rate model can accurately capture all elements of the performance behavior. In addition, no time/rate model can be expected to provide a completely unique forecast of future performance or prediction of EUR. It is important to be both realistic and practical when attempting to characterize production performance from systems where the permeability is on the order of 10–500 nd, the reservoir flow system is complex, and, although the induced-hydraulic-fracture system enables (and dominates) the production performance, there is only the most rudimentary understanding of the flow structure in the fracture systems.
It is essential that these conditions be established as a starting point. Not to do so will inevitably lead the analyst to interpretations based on incorrect assumptions as well as significant bias. The authors assert that reasonable production forecasts and predictions of EUR can be made, but not in isolation, not solely looking at the data and the selected time/rate model. The analyst must consider the nature of the resource and the significant uncertainty in the ability to apply simple time/rate relations to a very complex reservoir system.
As an attempt to better represent the general character of time/rate production data for a multistage-fractured horizontal well in an ultralow-permeability reservoir, numerous authors have developed time/rate relations using certain specific bases to represent a particular scenario. These developments include the following time/rate relations:
- Power-law exponential model
- Stretched exponential model
- Logistic growth model
- Duong model
Each relation has its own strengths, and, at this time, each of these models can be described only as empirical; there is no direct link with reservoir-engineering theory other than through analogy. For example, the stretched exponential model is essentially an infinite sum of exponentials, so the concept of adding the rigorous exponential decline to some limit could be thought to define this model. The power-law exponential model is essentially the same as the stretched exponential model (except for a constraining variable). At this point, we must assume that the proposed models are essentially empirical in nature, and generally center on a particular flow regime or characteristic behavior.
Field-Case Data
The complete paper focuses on three different shale-gas plays in North America. Field A is a formation composed of siltstone and dark gray shale, with dolomitic siltstone in the base and fine-grained sandstone toward the top. The formation of interest is a highly unusual, approximately 400- to 500-ft-thick package of continuous gas-charged siltstone with very small clay content. The formation is slightly overpressured, with pressure gradients of approximately 0.50–0.65 psi/ft.
Time/Rate-Analysis Relations
The basic definitions and diagnostic functions for time/rate analyses and a complete summary of the time/rate-analysis relations are given in the complete paper.
These relations are formulated as diagnostic relations and are used to make long-term rate projections and predictions of EUR. As a matter of process, any given relation is calibrated against the historical rate and cumulative data by use of a diagnostic approach and the model extrapolations are made only from the end of the data (not the body of the data). This approach ensures that all extrapolations/projections are based on the actual (not model-based) cumulative production.
Diagnostics and Characteristic Time/Rate Behavior
This section presents characteristic time/ rate performance from six wells from Field A. The primary objective of this effort is to demonstrate time/rate behavior of the wells with diagnostic plots without performing analysis corresponding to the play. Diagnostic plots used are the reciprocal of the loss ratio (D) and time (t), Arps decline exponent (b) and t, beta function (β) and t, and production rate/cumulative gas production (q/Gp) and t. Diagnostic plots have significant importance in our applications because these plots provide direct insight into our understanding of decline behavior.
For example, a straight-line trend of the continuously evaluated D-parameter [i.e., D(t)] vs. t on log-log scale could indicate power-law behavior that would yield the power-law exponential (or stretched exponential) function when the ordinary differential equation is solved for the rate function. Furthermore, from the continuous evaluation of the b-parameter, it is possible to verify the hyperbolic behavior. A constant b-parameter trend [i.e., b(t)=constant] suggests hyperbolic rate-decline behavior; as such, it is possible to establish the value of the b-parameter in the hyperbolic equation. In addition, a constant β-derivative trend verifies power-law flow regimes such as linear or bilinear flow. These diagnostic functions involve differentiation of time/rate data, and, therefore, errors and inconsistencies associated with the data are amplified in the derivative functions, which may prevent the analyst from establishing a unique interpretation.
The diagnostic plot of q/Gp and t provides significant diagnostic value because it does not include any numerical differentiation and it serves as a complementary diagnostic tool to the other diagnostic plots.
From another point of view, diagnostic plots are particularly useful while performing time/rate analysis. Each time/rate relation has more than two model parameters, and it is generally difficult to establish the values directly from production-rate data. In particular, the log[D(t)] vs. log(t) plot is used to establish the power-law exponential and stretched exponential model parameters because these parameters are related to the slope and intercept values on this log-log plot.
The general procedure for time/rate analysis is to use the diagnostic plots and calibrate the parameters of each model simultaneously until an optimum (visual) match is achieved. This procedure ensures consistency in the analysis and prevents the nonuniqueness associated with simply matching a single variable.
Fig. 1 presents the time/rate behavior of six wells producing in Field A. Shallower decline behavior and dominantly power-law-type flow regimes are observed throughout the production history. When data are plotted on the log[qg/ Gp]-vs.-log(t) plot (Fig. 2), almost all wells exhibit almost identical behavior.
The log[D(t)]-vs.-log(t) data are presented in Fig. 3 (left axis), and it is observed that certain (but not major) differences exist in the slope values of these wells—which could be related to production characteristics. The log[b(t)]-vs.-log(t) data are presented in Fig. 3 (right axis), and these data suggest that the hyperbolic relation could be applicable to model time/rate data because the b(t) trend exhibits a very gradual decrease with time and a constant b-value in the 2–3 range could reasonably be assumed. Fig. 4 presents the log(β-derivative)-vs.-log(t) trend, and a stabilization of data with time is seen, which suggests that power-law-type flow regimes are being established. In conclusion, the diagnostic interpretation of time/rate behavior of wells in Field A was concluded with the remark that time/rate behavior is being dominated by power-law-type flow regimes.
It is vitally important that the analyst realize that the diagnostic analysis of production data is a necessary step. Although some of the conclusions are qualitative, the diagnostic analysis of multiple data functions ensures a degree of impartiality in the data analysis and helps at least qualify the uncertainty in the data, which will likely ensure that the relevant time/rate models are isolated and that analyses/interpretations are not attempted that are not justified by the quality or nature of the given production data. It is critical that data diagnostics always be performed as part of the data analysis.
Application of the Time/ Rate Models to Long-Term Production Data
This section presents a (relatively) long-term production-data example to investigate the model behavior of the rate-decline equations considered in this paper. This field example consists of a tight gas well from east Texas (permeability values are estimated to be approximately 7.0 μd) with more than 7 years of production. For this case, we demonstrate our diagnostic interpretation procedure for matching data and performing forecasts.
All of the matches of production data with each of the time/rate models are performed simultaneously by calibrating model parameters. Each of the models matches the data for the entire production history. In particular, when the log[qg/Gp]-vs.-log(t) plot is used, the models can approximate the data trend to a considerable extent (this rendering tends to force the impression of a linear relationship, which may not be the case). Therefore, the differences in EUR will be dictated by the long-term model behavior. This is where the differences between the time/rate models begin to emerge.
Duong’s model is based on the linear behavior of the (qg/Gp)-vs.-t data trend (on a log-log scale); whereas the power-law exponential, the stretched exponential, and the logistic growth models exhibit nonlinear behavior. This difference in behavior dictates that the EUR estimates from Duong’s model should (almost always) be higher than those for the other models. On the other hand, when a terminal decline is imposed on the modified-hyperbolic relation, deviations from the linear trend are readily evident. The modified-hyperbolic and power-law exponential have specific terms that limit the overestimation of EUR; the Duong model does not. It is worth noting that, for methods that use a terminal decline, the prescribed value of the terminal decline is generally an arbitrary number and is often based on a company’s policies or the analyst’s experience.
In the log[D(t)]-vs.-log(t) and log[b(t)]-vs.-log(t) data trends for this case, very strong linear behavior of the computed log[D(t)]-vs.-log(t) trend is observed, confirming the applicability of the power-law exponential time/rate model. It can be argued that the latest-time data are affected by the numerical-differentiation algorithm and, therefore, can be considered as artifacts. Nevertheless, each of the models matches the data trends in its own fashion. The b(t) trend does appear to be decreasing with time (with the noted artifact near the endpoint); and an average b-value can be inferred from the data behavior.
Time/Rate Analyses
This section presents the time/rate analyses for each well from a given shale play using each of the models specified in this study. It is worthwhile to note that each of the matches produced in this study are based uniquely on the authors’ interpretation of the model behavior. Different matches with different EUR values can be obtained with similar probability.
It can be suggested that wells in Field A exhibit power-law-type flow regimes. The basis for this observation is mainly the signature on the time/rate plot and the near-constant character (at intermediate and late times) exhibited by data on the log[β(t)]-vs.-log(t) plot. And almost all of the models match the entire production history. Differences in model behavior are observed at late times in the forecasts. Generally, EUR values from the power-law exponential and stretched exponential relations are very similar (as should be expected). These models, along with the results from the logistic growth model, tend to provide conservative estimates across all wells. The Duong model and the modified-hyperbolic model always yield the highest EUR predictions. A 5% terminal decline rate is used for the modified-hyperbolic relation.
In this particular case, the logistic growth model appears to provide the most conservative EUR values (quite comparable to those of the power-law exponential and stretched-exponential time/rate models).
Interpretation of Results
The authors’ interpretation of results is provided by presenting comparison plots of the EUR values predicted by each model. The power-law exponential model results were chosen as the reference results, and EUR values from different models were compared with respect to those from the power-law exponential model. This approach should identify any correlations or inconsistencies that might exist between models.
For Field A, the EUR values from the Duong model are consistently higher than the results from the power-law exponential model. A comparison between the power-law exponential model and the logistic growth model shows the EUR predictions to be similar, with the exception of a single case. The next comparison considers the power-law exponential model and the modified-hyperbolic model. Some consistency in predicted EUR values is seen, but a couple of outliers suggest that the modified-hyperbolic relation will always predict higher EUR values compared with the power-law exponential model. Finally, a comparison of the power-law exponential model and the stretched exponential model reveals essentially identical results, somewhat as expected because these relations have essentially the same mathematical formulation. Almost identical results are seen because these two equations are essentially the same relations, and wells in Field A are not (yet) in the boundary-dominated flow regime after only a few years of production.
Conclusions
- The D/t-and-b/t diagnostic plot should be the primary diagnostic used to establish the well/ reservoir character.
- The qg/Gp-vs.-t diagnostic plot is an excellent data check, and should be incorporated into diagnostic analyses; however, the expectation of a completely linear trend is optimistic.
- The β-derivative/t diagnostic plot is useful for establishing the existence of power-law flow regimes.
This article, written by Editorial Manager Adam Wilson, contains highlights of paper SPE 162910, “Practical Considerations for Decline-Curve Analysis in Unconventional Reservoirs—Application of Recently Developed Time/Rate Relations,” by V. Okouma, SPE, Shell Canada Energy; D. Symmons, Consultant; N. Hosseinpour-Zonoozi, SPE, and D. Ilk, SPE, DeGolyer and MacNaughton; and T.A. Blasingame, SPE, Texas A&M University, prepared for the 2012 SPE Hydrocarbon Economics and Evaluation Symposium, Calgary, 24–25 September. The paper has not been peer reviewed.