Numerous mathematical models have been proposed to describe the complex nature of gas transport in naturally fractured, Nano-porous shale gas reservoirs (NNSGR's). However, a critical assessment of the literature reveals that the anomalous nature of gas transport in these systems are not adequately captured in those mathematical models. To this end, we extend the transient linear double porosity (DP) model established in the literature for scenarios where the underlying diffusion in the stimulated reservoir volume (SRV) is non-classical (i.e. anomalous). Furthermore, the contribution of gas desorption, and its subsequent diffusion to the fracture network in the SVR is considered.
The mathematical model developed describes gas transport into a horizontal well located in the center of a closed rectangular NNSGR by employing a time fractional flux relationship, the classic continuity equation, a transient interporosity source term between both overlapping media, as well as the multi-layer BET adsorption model. The time fractional flux relationship portrays a power law decay trend as compared to the exponential decay trend demonstrated in the Darcy-based mathematical models. The Laplace transform method is utilized to handle the resulting mathematical model to obtain semi-analytical expressions for the dimensionless flowrate and cumulative gas production at the wellbore for a constant flowing pressure inner boundary condition at the producer well. Subsequently, qualitative and quantitative validation of these outputs with existing models in literature are investigated.
The developed model converges to the established DP model, for the very limiting case when the diffusion behavior approaches Darcy type flux. The sensitivity of the diffusion exponent and phenomenological coefficients in the SRV were analyzed. Of great significance, the parametric study demonstrates that the developed transient anomalous linear dual porosity (ADP) model proffers the reservoir engineer more flexibility to capture the complexities and the power law trend observed in these complex systems in the field.
The mathematical model presented in this study will find wide spread applications in predicting the performance of horizontal wells producing in NNSGR's where analog studies reveal high spacing aspect ratio in the reservoir.
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