Abstract Most wells are not produced under a single production condition in practice. Because the complexity caused by production condition changes is compounded, few studies have focused on wells under variable inner boundary conditions. This work proposes a semi-analytical model for multiple-fracture horizontal (MFH) gas wells under a two-stage inner boundary condition by applying source functions. The two-stage condition in this work represents the type of inner boundary condition change from the Dirichlet boundary condition to the Neumann boundary condition during production. An intermediate function is established to eliminate the effect whereby the initial pressure is not uniform. Pseudofunctions and Laplace transform are utilized to solve the model's mathematical problem. In addition, the accuracy of this model is fully verified by numerical examples. Finally, analysis of the characteristics of an MFH gas well under the two-stage condition is conducted. The results suggest that a low limiting pressure will lead to higher cumulative production. The initial rate mainly affects the production time and does not affect the final cumulative production. The initial rate should be adjusted based on the actual demand downstream and known fracture parameters, including the fracture half-length, fracture conductivity and fracture number. The limiting bottomhole pressure, initial rate and fracture parameters affect the formulation and optimization of the rational production allocation and production system for MFH wells. Moreover, combined with a variable-rate/pressure solution, this two-stage model can be expanded for more multistage production conditions. This work provides an approximation method for solving the nonlinear diffusivity gas equation under variable inner boundary conditions.

中文翻译：

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二级内边界条件下多裂缝水平气井的求解方法

摘要 在实践中，大多数井不是在单一生产条件下生产的。由于生产条件变化引起的复杂性是复杂的，很少有研究集中在可变内边界条件下的井上。这项工作通过应用源函数提出了两级内边界条件下多裂缝水平 (MFH) 气井的半解析模型。这项工作中的两阶段条件代表了生产过程中从 Dirichlet 边界条件到 Neumann 边界条件的内部边界条件变化类型。建立一个中间函数来消除初始压力不均匀的影响。利用伪函数和拉普拉斯变换来解决模型的数学问题。此外，数值例子充分验证了该模型的准确性。最后，对一口MFH气井在两阶段条件下的特征进行了分析。结果表明，较低的极限压力将导致较高的累积产量。初始速率主要影响生产时间，不影响最终累计产量。初始速率应根据下游的实际需求和已知的裂缝参数进行调整，包括裂缝半长、裂缝导流能力和裂缝条数。极限井底压力、初始速率和裂缝参数影响MFH井合理生产配置和生产系统的制定和优化。此外，结合可变速率/压力解决方案，这种两阶段模型可以扩展到更多的多阶段生产条件。这项工作为求解可变内边界条件下的非线性扩散率气体方程提供了一种近似方法。