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Eccentric Helical HB Creeping-Flow FEM Calculator

Solve all three velocity components and cross-sectional pressure in an eccentric annulus using regularized HB rheology and mixed finite elements.

Inputs

Results

Calculate to view linked results. Default values are examples, not recommended operating inputs.

Calculation sequence

  1. Supply offset normalized by radial clearance; zero is concentric and values up to 0.7 retain an open narrow gap. The inner wall rotates around its own displaced centre. The mixed finite-element system solves two cross-sectional velocities, axial velocity and pressure together, with no slip on both cylinders and one pressure gauge.
  2. Use a regularized HB invariant rather than declaring exact yielded/unyielded cells. Refine the annular triangular mesh and compare volume flow and torque; separately increase the Papanastasiou parameter. A returned solution must satisfy the nonlinear residual, weak incompressibility and boundary-reaction mechanical power balance. Refinement failures are errors, not warnings attached to unconverged values.
  3. The velocity plot samples the narrow and wide radial gaps. Axial and transverse coupling is retained through the viscosity invariant, and work from the rotating wall is extracted from nodal reactions, not a concentric torque formula. Concentric Newtonian analytical results and the separate coupled HB quadrature tool provide independent limiting-case checks; neither constitutes measured eccentric-drilling validation.
  4. Calculations remain local in a dedicated WebAssembly worker. The existing self-hosted scientific engine is about 33 MB raw on first use; the envelope additionally loads the pinned CoolProp engine and eccentric flow loads a 178 KB FEM package. Nothing is downloaded until calculation. No composition, heat history or measurements are sent to a calculation server. Save retains canonical input values; file backup can be shared with colleagues, while CSV/PDF retain samples, assumptions and sources. Browser storage alone can be lost.

Worked check

The default case uses 40% eccentricity, K=2 Pa.s^0.8, yield stress 1 Pa, 0.1 rad/s rotation and 150 Pa/m axial drive. Inputs are illustrative, not field data. Set eccentricity to zero and compare flow/torque with the concentric coupled solver at the same rheology; set n=1 and yield stress to zero for the independent Newtonian limit.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

  • Fully developed, isothermal creeping regularized HB, stationary outer cylinder. No exact plug-boundary claim, no turbulent correction or full Taylor stability analysis.
  • A bounded two-dimensional cross-section with three velocity components, not a full three-dimensional drillstring/cuttings CFD simulation. No axial wall translation, wall slip, beds, viscoelasticity or thixotropy.
  • Mesh, regularization and nonlinear tolerances are enforced. Cases that exceed the refinement or Reynolds gates fail; the tool does not promise to solve every HB parameter combination.

Implemented equations

Regularized three-component Stokes system

−∇⊥p+∇⊥ ⁣⋅(2ηD)+Gez=0∇⊥ ⁣⋅u⊥=0η=K(γ˙2+ϵ2)(n−1)/2+τy1−e−mγ˙γ˙\begin{gathered}\begin{aligned}-\nabla_\perp p&+\nabla_\perp\!\cdot(2\eta D)\\&+G\mathbf e_z=0\end{aligned}\\\nabla_\perp\!\cdot\mathbf u_\perp=0\\\begin{aligned}\eta&=K(\dot\gamma^2+\epsilon^2)^{(n-1)/2}\\&\quad+\tau_y\frac{1-e^{-m\dot\gamma}}{\dot\gamma}\end{aligned}\end{gathered}

Sources and maintenance

Published by PetroCalcHub Team. Implementation and source checks: . This does not claim independent expert review.

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