Transient-inflow-performance modeling from analytic line-source solutions for arbitrary-trajectory wells

We present two easily computable, equally valid, semianalytic, single-phase, constant-rate solutions to the diffusivity equation for an arbitrarily oriented uniform-flux line source in a 3D, anisotropic, bounded system in Cartesian coordinates. With the addition of superposition, these become inflow solutions for wells of arbitrary trajectory. In addition, we produce analytic time derivatives for pressure-transient analyses (PTAs) of complex wells. If we extract solution components for 2D systems from the general solution, we can construct discrete complex-fracture-inflow and PTA capability for vertical, fully penetrating fractures, suitable for use as the basis solution in modeling complex phenomena, such as pressure-constrained production or development of fracture interference. For a 3D slanted well, the full characterization of dimensionless pressure over 10 decades of dimensionless time behavior can be produced in 1.5 seconds. With a fast-computing analytic solution for pressure anywhere in the system, we can also produce dense pressure maps at scalable resolution where any point could represent an observation well for convolution and enhanced interpretation. Likewise, the pressure derivative and the slope of the logarithmic temporal derivative of pressure can be mapped throughout to indicate local flow regime in a complex system. In particular, we compare and contrast the PTA signatures from symmetrical and asymmetrical horizontal, slanted, and diagonal line sources and examine when the behavior of a thin 3D reservoir collapses to the equivalent of a 2D fully penetrating fracture. Once the reservoir-thickness/length ratio reaches 1:100, all wells with the same projection onto the x–y plane are indistinguishable except for very early time, probably masked by wellbore/fracture-storage effects.