Finite-Wellbore Solution: Cylindrical Inner Boundary, Wellbore Storage, and Early-Time Pressure Transient Analysis

The finite-wellbore solution is the analytical solution to the radial diffusivity equation, the partial differential equation that governs pressure diffusion in a porous reservoir, obtained when the inner boundary condition at the well is treated as a cylinder of finite radius rather than as an idealized line source of zero radius. In the classic and most widely used pressure-transient model, the line-source solution, the well is collapsed to a mathematical line and the reservoir is assumed to extend inward to zero radius; this assumption yields the familiar exponential-integral solution and, at later times, its logarithmic approximation that underlies conventional semilog well-test analysis. The line-source idealization is mathematically convenient and accurate at the moderate to late times that dominate most build-up and drawdown interpretation, but it breaks down at very early time and very close to the wellbore, because no real well has zero radius. The finite-wellbore solution corrects this by imposing the flow-rate or pressure condition on the actual cylindrical wall at the wellbore radius rw, so the reservoir is modeled only outward from r equal to rw and the near-well pressure field is represented physically rather than as a singularity. The resulting solution is expressed in terms of Bessel functions in the Laplace domain and is numerically inverted, most commonly with the Stehfest algorithm, to give bottomhole pressure as a function of time. In practice the distinction matters most when interpreting early-time data, when modeling wellbore storage and skin together with the formation response, and when the dimensionless time based on wellbore radius is small, meaning the pressure transient has not yet propagated far enough for the line-source approximation to hold. Modern pressure-transient analysis software builds the finite-wellbore solution into its catalog of inner boundary conditions alongside wellbore storage coefficient C and the van Everdingen-Hurst skin factor s, so that an analyst matching a Montney or Duvernay multistage-fractured horizontal build-up can honor the true near-well geometry instead of forcing a line-source fit through the earliest data. For gas wells the finite-wellbore solution is typically combined with pseudo-pressure and pseudo-time transforms to handle the strong pressure dependence of gas viscosity and compressibility, and Green's-function formulations extend it to bounded cylindrical reservoirs and to complex well geometries. In Western Canadian Sedimentary Basin (WCSB) tight-gas and shale operations, where wellbore radius is on the order of 0.1 m (about 0.33 ft) and reservoir permeability can be in the microdarcy to nanodarcy range, the early-time transient governed by the finite-wellbore condition can persist for hours or days, making the choice between line-source and finite-wellbore inner boundaries a material decision in estimating permeability, skin, and ultimately fracture half-length and stimulated reservoir volume that feed AER Directive 040 and reserves evaluations.

Line Source Versus Finite Wellbore in Practice

The line-source solution is the workhorse of build-up analysis because at the times most tests reach, typically hours after a long flow period, the two solutions are indistinguishable and the simpler exponential-integral form is sufficient. The finite-wellbore solution earns its keep when the interpreter must trust early-time data, for example in a short, storage-dominated WCSB tight-gas build-up where the line-source fit would force an artificial skin to compensate for the missing near-well geometry. Honoring the finite radius lets storage, skin, and formation permeability separate cleanly rather than trading off against one another in the regression.

Bounded Reservoirs and Green's-Function Extensions

Real WCSB wells drain finite volumes, so the unbounded finite-wellbore solution is often paired with an outer boundary, either a no-flow circle for a depletion analysis or a constant-pressure circle for support. Green's-function and source-and-sink techniques combine the finite cylindrical inner boundary with these outer conditions and with multi-region models, such as a stimulated inner region around a fractured horizontal contrasted with an unstimulated outer matrix. These composite models let analysts of Montney pads estimate both near-well stimulation quality and the surrounding tight-matrix permeability from a single shut-in.

Related Terms

The finite-wellbore solution sits within the broader framework of pressure-transient theory. The Diffusivity Equation is the governing partial differential equation it solves, balancing fluid compressibility, viscosity, permeability, and porosity. Wellbore Storage is the early-time effect that the finite-wellbore inner boundary helps separate from true formation response, and Skin Factor quantifies near-well damage or stimulation that the solution resolves once geometry is honored. Pressure Transient Analysis is the interpretation discipline that selects among line-source, finite-wellbore, and bounded models to extract reservoir parameters.

Real-World WCSB Scenario: Montney Tight-Gas Build-Up at Dawson

An operator running a 96-hour shut-in on a multistage-fractured Montney horizontal near Dawson, British Columbia, records a build-up dominated for the first several hours by wellbore storage in a well with a 0.108 m (0.354 ft) radius. An initial line-source fit returns an implausibly large negative skin and a permeability that conflicts with offset core, because the early data violate the zero-radius assumption. Switching to a finite-wellbore inner boundary, combined with pseudo-pressure and pseudo-time and a composite inner-stimulated outer-matrix model, costs nothing extra in a modern interpretation package licensed at roughly 25,000 CAD per year.

The revised match separates storage, skin, and a microdarcy matrix permeability cleanly, yielding a fracture half-length consistent with the completion design and a defensible stimulated reservoir volume. That interpretation feeds the AER and BCER deliverability filings and the type-well used to book the pad's reserves, demonstrating that the inner boundary condition is not an academic nicety but a direct lever on booked value.