A semi-analytical approach to nonlinear multi-scale seepage problem in fractured shale gas reservoirs with uncertain permeability distributions

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To investigate the influence of the inhomogeneity and uncertainty of porosity and permeability on gas production, a mathematical model for shale gas nonlinear seepage problem with random distributions of inhomogeneous equivalent porosity and permeability is established. Based on the statistical characteristic of the artificial fracture density distribution and incompletely controllable hydraulic fracturing technology, the random distribution models of continuous inhomogeneous equivalent porosity and permeability are proposed to simulate the uncertainty of the porosity and permeability distribution in SRV. Coupled with multi-scale flow and adsorption effects, a mathematical model of shale gas nonlinear seepage problem is presented to address the inhomogeneity and uncertainty of equivalent porosity and permeability. By Boltzmann transformation and local homogenization approximation, a semi-analytical method and the corresponding explicit iterative scheme are developed.Simulation results match well with field data, which verifies the validity of the presented mathematical model and approach. The morphology of the SRV has significant influence on gas production, followed by the uncertainly of the permeability distribution and the hydraulic fracture half-length. When the horizontal width reaches its maximum in the front part of the macro-fracture zone, shale gas production reaches its peak and increases by 60%. The uncertainty of the permeability distribution is resulted from the deviation of the hydraulic fracture. As the deviation angle increases from 0° to 10°, shale gas production drops by 16%. The greater the gas production, the bigger the drop rate. When the hydraulic fracture half-length is 50 m, gas production reaches its peak, however it drops by 13% if the hydraulic fracture half-length increases to 70 m.