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A semi-analytical mathematical solution to the Heat Equation has been derived to address the issue of rate forecasting from hydraulic fractures in unconventional oil reservoirs. The fracture is assumed to have a time-independent pressure profile along its length with a terminally declining flow rate. The analytical model makes use of the point solution to the heat equation for a constant rate in a bound system of a rectangular geometry. The constant pressure solution is constructed from the constant rate solution through convolution performed in the Laplace domain, then analytic inversion to the time domain. The line source solution is then acquired through analytic integration of the point source solution along a well trajectory. Results are generated using a simulator based on the mathematical model for arbitrarily oriented fractures in a 2D domain. Since the physical problem involves inverse modeling of flow rate data for reservoir characterization, diagnostics techniques for probing the fracture and drainage geometry are presented using derivative analysis of rate transient behavior of hydraulically-fractured media. [ABSTRACT FROM AUTHOR]

• Bubble point pressure of crude oil decreases with increasing H 2 S content. • The higher the H 2 S concentration, the lower the MMP between sour gas and crude oil. • Increasing H 2 S concentration delays gas breakthrough and improves oil recovery. • Presence of high permeable zones in a reservoir leads to early gas breakthrough. Proper handling of the produced sour gas is a big challenge for operating companies due to the serious environmental and technical problems caused by H 2 S and CO 2. The reinjection of the produced sour gas into oil reservoirs is an environmentally friendly as well as an economically attractive method due to the low investment cost and incremental oil recovery. However, the sour gas injection for enhanced oil recovery (EOR) is a relatively new approach with numerous uncovered perspectives, especially in a heterogeneous reservoir. The objective of this study is to investigate the effect of gas composition and reservoir heterogeneity on the miscible displacement of reservoir oil using sour gas. The reservoir oil consisting of 15 mol% H 2 S and 4 mol% CO 2 from an oilfield in the North Caspian region was used. First, the phase behavior alteration of reservoir oil due to the injection of various gases, namely H 2 S, CO 2 , CH 4 , acid, and sour gas was analyzed. Then, the effect of gas type on minimum miscibility pressure (MMP) was investigated based on a semi-analytical tie-line method. Moreover, 2D and 3D numerical simulations were conducted to understand the oil recovery performance of various gas injection options and the effect of permeability heterogeneity on sour gas flooding efficiency. The findings showed that higher H 2 S content in the injected gas narrows the two-phase envelope and decreases the bubble point pressure. Moreover, it was observed that MMP decreases with increasing H 2 S concentration, which provides a favorable condition for achieving miscibility. Additionally, an increase in H 2 S concentration in sour gas improves its oil recovery efficiency by extending gas breakthrough time and elongating the plateau period of oil rate. These results are useful for designing, optimizing, and implementing sour gas injection for EOR in oil reservoirs that have access to sour gas sources. [ABSTRACT FROM AUTHOR]