Summary
This paper presents a general and unified equation for flowing temperature prediction that is applicable for the entire range of inclination angles. The equation degenerates into Ramey's equations for ideal gas or incompressible liquid and into the Coulter and Bardon equation, with the appropriate assumptions. This work also proposes an approximate method for calculating the Joule-Thomson coefficient for black-oil models.
Introduction
Flowing temperature distribution often is predicted with different methods for pipelines and wellbores. The Ramey method usually is used for predicting wellbore temperature distribution. This method rigorously incorporates the complex process of transient heat transfer between the wellbore and the reservoir. Ramey's method, however, is limited to either ideal gas or incompressible liquid flow. The Coulter and Bardon equation commonly is used for pipeline temperature prediction. A more rigorous thermodynamic behavior of the flowing fluid is taken into account, incorporating the Joule-Thomson coefficient. Although the Coulter and Bardon equation originally was derived for gas flow, it also is used for single-phase liquid or two-phase flow. This equation is limited, however, by the assumptions of steady-state heat transfer with a constant-temperature environment and horizontal flow.
Summary
The flow toward hydraulic fractures is visualized at high resolution using a newly developed analytical streamline simulator that is based on complex potentials. Drainage contours show progressive fluid recovery from the stimulated rock volume (SRV). The method plots streamlines, time-of-flight contours, velocity-field contours, and pressure distribution around fractured wells. Independent simulations with a commercial reservoir simulator confirm that visualizations with complex potentials are accurate, and that the latter method provides high-resolution images of the pressure and flow fields around individual fractures. Contours for the drained rock volume (DRV) that are based on particle-velocity tracking outline the actual region drained by a well through its fractures. First, matrix drainage by two-fracture and three-fracture clusters is studied in detail. Flow-separation surfaces between two clustered fractures (with equal length and flux) are always straight, creating planes of symmetry between adjacent drainage regions. Clusters of three fractures develop curved-flow-separation surfaces, convex toward the inner fracture. For fracture spacing less than four times total fracture length, drainage of the central region of the three-fracture clusters slows down because of flow interference, which confirms earlier findings that production gains become insignificant above certain fracture length/spacing ratios. Next, the analysis shows the flow field, drainage contours, velocity contours, and pressure distribution for a horizontal, synthetic well with 11 transversal, kinked fractures. A final section shows a brief example of application to a field case.
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