低滲透率地層中地層流體流態以線性流為主，並可持續很長一段時間。Duong 法被證實適用於線性流為主的地層，可提供正確的產率預測與最終可採蘊藏量評估。然而，Duong 法會高估偽穩態流期間之產率，因此並不適用於偽穩態流期間。傳統Arps 產率遞降曲線適用於偽穩態流期間，因此對於低滲透率地層流體流態為線性流為主，晚期為偽穩態流，可將產率資料進行分段分析。即是以Duong法應用於線性流期間，結合Arps產率遞降曲線於偽穩態流期間。線性流結束時間可使用Wattenbarger等人線性流理論進行估算，並以此線性流結束時間做為混合Duong法與Arps 產率遞降曲線的轉換時間。
本研究的主要目的是決定適當的線性流結束時間，並將此線性流結束時間應用於混合Duong法與Arps 產率遞降曲線之產率預測分析。本研究修正Wattenbarger 等人的解析解，以及經驗解 (包含Duong法與晚期Arps 指數型遞降曲線)，並由黃金分割搜尋法得修正線性流結束時間。將修正線性流結束時間應用於混合型產率遞降曲線。使用解析解產生之產率生產資料，考慮地層為緻密砂岩氣層或頁岩氣層，因此假設滲透率範圍為10-1 md至10-4 md。分析結果顯示當地層滲透率很小(k<10-4 md)，使用修正線性流結束時間或Wattenbarger 等人的線性流結束時間，兩者預測結果均與解析解差異不大。然而，當地層滲透率介於10-1 md至10-3 md分析結果顯示使用Wattenbarger等人的線性流結束會高估產率與最終可採蘊藏量，需使用修正線性流結束以避免產生較大的最終可採蘊藏量估算誤差。最後，並以此方法對Barnett 頁岩氣層與Bossier緻密砂岩氣層進行現場案例分析。

Duong’s method of forecasting production and estimated ultimate recovery (EUR) in low permeability reservoirs with a long-term linear flow has been verified by several authors. However, Duong’s method overestimates future production during boundary-dominated flow. It is reasonable to combine Duong’s method and the Arps decline relations for fractured wells exhibited linear flow followed by boundary-dominated flow, because the Arps decline relation has better prediction for boundary-dominated flow. For wells that have not reached boundary-dominated flow, Wattenbarger et al.’s linear flow theory is frequently used to determine the duration of linear flow, and the end of linear flow time (telf) can be used to estimate when to switch from Duong’s method to the Arps decline relation.
This paper focuses on the availability of the end of linear flow time determined by linear flow theory with a hybrid forecasting method, which combines Duong’s method and the Arps exponential relation for tight sand/shale gas reservoirs. To obtain the proper end of linear flow time, modified Wattenbarger et al.’s analytical solution and modified empirical solutions (Duong’s method combines the Arps exponential relation) was derived. The golden section search method is used to find the modified end of linear flow time that minimizes the difference between modified analytical and empirical solutions.
Then, the hybrid forecasting method was used to forecast production of synthetic production data generated from an analytical solution. A number of cases were tested to verify the efficacy of this method for forecasting production. To account for tight sand/shale gas reservoirs, reservoir permeability ranging from 10-1 md to 10-4 md was considered. The results indicated that the hybrid forecasting method used with the modified end of linear flow time is theoretically accurate for production forecast and EUR estimation. In low-permeability reservoirs with a permeability ranging from 10-1 md to 10-3 md, this method provides a significant improvement in EUR estimation. Finally, application of this method to field examples from Barnett shale gas and Bossier sand gas were also presented.

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