Discrete wellbore and fracture productivity modeling for unconventional wells and unconventional reservoirs

Newly developed, generalized analytic solutions to the heat equation for arbitrary 3D well trajectory in anisotropic media are demonstrated to solve benchmark horizontal- and slanted-well productivity problems with unprecedented speed and accuracy. Arbitrary well trajectory is constructed as an assemblage of spatially integrated, linear well segments, as opposed to a distribution of numerically integrated point sources, to provide advantages in both computational speed and accuracy in singularity handling. Production from each arbitrarily oriented segment is reduced to a combination of purely analytic expressions and rapidly convergent, exponentially damped infinite sum approximations. With offered flexibility in cell boundary conditions, the expressions can yield standalone well-productivity estimates for complex wells or serve as the basis for advanced well equations, if integrated within a numerical reservoir simulator. Transients are also computed with analytical integrations in time, thus requiring no time marching. The breakthrough speed and accuracy in productivity assessment open possibilities for advanced well-testing and reservoircharacterization methods. We further demonstrate the usefulness of analytic methodology with several time-dependent, discrete fracture problems for shale gas production with typical Barnett conditions, allowing direct use of complex fracture patterns, such as those interpreted from microseismic mapping. In addition to uniform-flux and uniform-pressure modeling options, a new analytic model is introduced that is capable of modeling both time-dependent material transport between matrix and a stimulated zone and the interplay between a well and fracture. We illustrate our solution method with Barnett fracturedwell examples from the literature. With optional effects such as gas desorption and stress-dependent fracture conductivity as easy add-ons, we can produce full-operational-life production forecasts for shale or tight gas reservoirs from discrete, complex fracture patterns along with reservoir-pressure mappings in a matter of minutes on common PC platforms.