The Propagation of Depletion — The Inclusion of Inertia in the Derivation of the Diffusivity Equation
Authors: Goldsberry Fred , Fair Chris , Nguyen Don , Zhumagulova Venera 
Published: November 2020

Abstract
When flow is initiated from a well, a physical shock front is created in the reservoir, beginning at the completion. This shock front moves out into the reservoir as a Mechanical Wave with constant intensity and with velocity decreasing vs. the square of the distance. This is known as infinite acting radial flow (aka ‘transient flow’) and is very well understood (at least at the producing well’s location). The capillary shockwave represents a moving boundary to the pressure decay field that forms around the well (Hurst 1934). This active region of decay behind the moving shock front will adopt a near semi-steady state flow regime in the reservoir, as required by the second law of thermodynamics. This depletion region is composed of radial capillaries that produce flow to the well and create pressure or elastic depletion in the reservoir. The cone of influence or region of pressure depletion bounded by the shock front around the well is the active reservoir at any given time. Eventually, the capillary shockwave boundary condition at the radius of investigation will either encounter all of the reservoir boundaries, or will continue to grow into an aquifer. Energy is redistributed by a faster moving inertial wave that passes through the open capillaries. Each capillary has finite rupture strength due to the same initiating capillary pressure that produces the bounding shockwave (Goldsberry 1998, 2000). Hence the cone of influence can be recognized as a structure composed of radiating capillaries behind a moving boundary condition.
Each capillary is considered an individual contributor to the flow to the well and each capillary has a contribution to flow that is in proportion to its volume (Goldsberry 1998). When a capillary reaches a sealing boundary, it ceases to grow and contributes less to the flow of the well than the capillaries that are still growing. Since the well is generally controlled by a choke, the demand upon the cone of influence is constant. This creates an imbalance in the second law of thermodynamics. In order to reconcile this imbalance, the choke-well interaction then produces a secondary cone of influence bounded by its own capillary shockwave boundary condition.
The goal of this paper is to describe the primary shockwave boundary condition and the inertial waves that produce pressure redistribution within the cone of influence. Traditionally, the inertial term is ignored in order to create a simplified diffusion equation model. It is not zero. Furthermore, in the classic diffusivity derivation (Earlougher 1977), the capillary entry pressure is assumed to be small and hence neglected. It is also not zero. By excluding these terms in the classic derivation, the result is a simplified model that works for infinite acting radial flow at the producing well location, but does not match observation well data, nor producing well data after the first boundary is encountered. If both inertia and capillary entry/threshold pressure are included in the derivation for the diffusivity equation for transient flow, a model that is based upon a capillary structure emerges, which predicts (and matches real field data) a depletion region that grows predictably with time, with distinct boundary contacts and arrivals of primary and secondary shock fronts.