The significant temperature difference between the fractured and non-fractured regions during the stimulation fluid flow-back period can be very useful for fracture diagnosis. The recent developments in downhole temperature monitoring systems open new possibilities to detect these temperature variations to perform production logging analyses. In this work, we derive a novel analytical solution to model the temperature signal associated with the shut-in during flow-back and production periods. The temperature behavior can infer the efficiency of each fracture. To obtain the analytical solution from an existing wellbore fluid energy balance equation, we use the Method of Characteristics with the input of a relevant thermal boundary condition. The temperature modeling results acquired from this analytical solution are validated against those from a finite element model for multiple cases. Compared to the warm-back effect in the non-fractured region after shut-in, a less significant heating effect is observed in the fractured region because of the warmer fluid away from the perforation moving into the fracture (after-flow). Detailed parametric analyses are conducted on after-flow velocity and its variation, flowing, geothermal, and inflow temperature of each fracture, surrounding temperature field, and casing radius to investigate their impacts on the wellbore fluid temperature modeling results. The inversion procedures characterize each fracture considering the exponential distribution of temperature based on the analytical solutions in fractured and non-fractured regions. Inflow fluid temperature, surrounding temperature field, and after-flow velocity of each fracture can be estimated from the measured temperature data, which present decent accuracies analyzing synthetic temperature signal. The outputs of this work can contribute to production logging, warm-back, and wellbore storage analyses to achieve successful fracture diagnostic.
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