*Article*

#### **Development and Calibration of a Semianalytic Model for Shale Wells with Nonuniform Distribution of Induced Fractures Based on ES-MDA Method**

#### **Qi Zhang 1,2, Shu Jiang 1,2,\*, Xinyue Wu 1,2, Yan Wang 3,4 and Qingbang Meng 1,2**


Received: 22 June 2020; Accepted: 16 July 2020; Published: 20 July 2020

**Abstract:** Given reliable parameters, a newly developed semianalytic model could o ffer an e fficient option to predict the performance of the multi-fractured horizontal wells (MFHWs) in unconventional gas reservoirs. However, two major challenges come from the accurate description and significant parameters uncertainty of stimulated reservoir volume (SRV). The objective of this work is to develop and calibrate a semianalytic model using the ensemble smoother with multiple data assimilation (ES-MDA) method for the uncertainty reduction in the description and forecasting of MFHWs with nonuniform distribution of induced fractures. The fractal dimensions of induced-fracture spacing (*dfs*) and aperture (*dfa*) and tortuosity index of induced-fracture system (θ) are included based on fractal theory to describe the properties of SRV region. Additionally, for shale gas reservoirs, gas transport mechanisms, e.g., viscous flow with slippage, Knudsen di ffusion, and surface di ffusion, among multi-media including porous kerogen, inorganic matter, and fracture system are taken into account and the model is verified. Then, the e ffects of the fractal dimensions and tortuosity index of induced fractures on MFHWs performances are analyzed. What follows is employing the ES-MDA method with the presented model to reduce uncertainty in the forecasting of gas production rate for MFHWs in unconventional gas reservoirs using a synthetic case for the tight gas reservoir and a real field case for the shale gas reservoir. The results show that when the fractal dimensions of induced-fracture spacing and aperture is smaller than 2.0 or the tortuosity index of induced-fracture system is larger than 0, the permeability of induced-fracture system decreases with the increase of the distance from hydraulic fractures (HFs) in SRV region. The large *dfs* or small θ causes the small average permeability of the induced-fracture system, which results in large dimensionless pseudo-pressure and small dimensionless production rate. The matching results indicate that the proposed method could enrich the application of the semianalytic model in the practical field.

**Keywords:** history matching; semianalytic model; unconventional gas reservoirs; multistage fractured horizontal wells; fractal theory
