For the unconventional reservoir, triple-porosity models are widely applied to take the macro-fracture, micro-fracture and matrix system into consideration. However, the models are usually built based on the assumption of sequential flow from matrix to micro-fracture to macro-fracture, which will result in inaccuracy of production evaluation. Although a quadri-linear flow model (QFM) has been proposed to consider the simultaneous flow from matrix into micro-fracture and macro-fracture. It is relatively complicated to solve the model with the Laplace transform and numerical inversion. In this paper, a new analytical solution for the QFM is derived.

In order to simplify the problem, the matrix flow is divided into two parts: one feeding the macro-fracture and the other feeding the micro-fracture. Then, four partial differential equations (PDEs) are obtained to express the transient linear flow in different media. The PDEs are transformed into ordinary differential equations (ODEs) by integration bypassing the Laplace transform and numerical inversion. Finally, a rate vs. time solution in real-time space is derived.

The results are validated by typical analytical models. While the micro-fracture system is neglected, the results agree well with the dual-porosity model. While ignoring flow between the matrix and macro-fracture, the results agree with the triple-porosity model. What’s more, according to the output parameters from the new model, one can infer the ratio of pore volume of different media and even the ratio of flow from matrix to the micro-fracture and to the macro-fracture simultaneously. The model is also applied to analyze the field production data. After identifying the flow regime, the solution can match well with the data and the model parameters can be obtained. Through the parameters, we can make production forecast accurately.

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