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Thin-Walled Hollow Shafts

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In analyzing a thin-walled hollow shaft subjected to torsional loading, a segment with width dx is isolated for examination. Despite its equilibrium state, this segment faces torsional shearing forces at its ends. These forces are quantitatively described by the product of the longitudinal shearing stress on the segment's minor surface and the area of this surface, leading to the concept of shear flow. This shear flow is consistent throughout the structure, indicating a uniform distribution of...

Major Losses in Pipes

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Deformation in a Circular Shaft

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One of the distinctive characteristics of circular shafts is their ability to maintain their cross-sectional integrity under torsion. In other words, each cross-section continues to exist as a flat, unaltered entity, simply rotating like a solid, rigid slab. To understand the distribution of shearing stress within such a shaft, consider a cylindrical section inside this circular shaft. This section has a length of L and a radius of R, with one end fixed. The radius of the cylindrical section is...

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Consider the elastic torsion formula, which applies to a circular shaft with a consistent cross-section. This formula assumes that the shaft's ends are loaded with rigid plates firmly attached. However, in many cases, torques are applied to the shaft through mechanisms like flange couplings or gears, which are connected by keys inserted into keyways. This application method modifies the stress distribution near the point of torque application, causing it to deviate from the distributions...

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Updated: Jun 20, 2026

Mechanical Expansion of Steel Tubing as a Solution to Leaky Wellbores

Mechanical Expansion of Steel Tubing as a Solution to Leaky Wellbores

Published on: November 20, 2014

General steady-state shape factor for a partially penetrating well.

Vitaly A Zlotnik¹, David Goss, Glenn M Duffield

¹Department of Geosciences, University of Nebraska-Lincoln, 214 Bessey Hall, Lincoln, NE 68588-0340, USA. vzlotnik1@unl.edu

|September 8, 2009

Summary

We derived a general formula for aquifer shape factors in anisotropic conditions. This analytical solution improves accuracy for partially penetrating wells in groundwater studies.

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Area of Science:

Hydrogeology
Fluid dynamics in porous media
Geotechnical engineering

Background:

Partially penetrating wells are common in hydrogeological investigations.
Accurate shape factors are crucial for analyzing well tests.
Existing semi-empirical methods have limitations in accuracy and applicability.

Purpose of the Study:

To derive a general, closed-form analytical expression for the steady-state shape factor.
To provide a unified and accurate solution for partially penetrating wells in uniform anisotropic aquifers.
To offer a tool applicable to various subsurface flow scenarios.

Main Methods:

Development of a general analytical solution for steady-state flow.
Derivation of a closed-form expression for the shape factor.
Validation against existing semi-empirical methods and numerical simulations (implied).

Main Results:

A simple, accurate, and general analytical expression for the shape factor.
The derived formula uniformly represents the shape factor across a full range of parameters.
The new expression meets or exceeds the accuracy of previous semi-empirical methods.

Conclusions:

The presented analytical solution offers a significant advancement for analyzing partially penetrating wells.
This general shape factor is directly applicable to slug tests and injection/extraction tests.
The findings support more accurate characterization of aquifer hydraulic conductivity and improved design of remediation systems.