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Cubic Flow Law and portable packer tests

Cubic Flow Law and portable packer tests. J Dowd. Stokes Law. Steady state flow of an imcompressible liquid in a fracture, isothermal conditions: Stokes Law Where: : Driving Force and : Viscous Resistance Force. For flow under uniform gradient between two smooth plates

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Cubic Flow Law and portable packer tests

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  1. Cubic Flow Law and portable packer tests J Dowd

  2. Stokes Law Steady state flow of an imcompressible liquid in a fracture, isothermal conditions: Stokes Law Where: : Driving Force and : Viscous Resistance Force

  3. For flow under uniform gradient between two smooth plates No slip condition Integration of Stokes Law yields: Where parallel to flow; perpendicular to flow; velocity parallel b is aperture

  4. varies parabolically from 0 at edge to max in middle and the average seepage velocity is Where K is fracture hydraulic conductivity: k k is fracture intrinsic permeability: and , f = Lomize’s roughness coeff

  5. System of parallel cracks q = volume flux = Where: Thus: D; The permeability of parallel fractures is proportional to , known as the cubic flow law

  6. Determining b in the field • Isolate one or more fractures with packers • Obtain D between packers (core or tv) • Inject fluid between packers; Interpret measured flow rate and pressure data as radial flow to obtain “effective” K and k • Assume f = 1/12 • Compute effective hydraulic aperture:

  7. Aperture/Hydraulic conductivity/porosity Fracture Spacing: 2.5m

  8. Deviation from cubic lawBrinkman effect • Flow with permeable walls, modified Darcy’s Law: • Where = Brinkman term, which accounts for shear near rock interface • Brinkman term in terms of velocity: • , , • NB: cubic flow law underestimates flux

  9. Single hole packer tests

  10. Quadruple Packer

  11. Single hole packer tests • Two categories • Injection tests (water injected at constant head) • Slug tests (hydraulic head instantaneously increased or decreased) • Standard methods of analysis assume homogeneous, isotropic conditions • Goveqn:

  12. Flow approximations Radial flow Prolate Spheroidal

  13. Injection Tests • Injection Tests • Transient: Rarely used because of difficulty in accurately measuring flow • Steady-State: Performed after injection flow rate stabilizes • Radial flow pattern • Prolate spheroidal (ellipsoidal)

  14. Injection Tests • Prolate spheroidal (ellipsoidal) Solution for a line source (L) and constant Q (Hvorslev, 1951): Where:

  15. Slug test • Conventional (gravity) slug • Standing column open to atm • Column subject to instantaneous step change • Pressure slug • Test interval isolated from the atm • Head in interval increased by injecting a small volume of fluid • Difference between the two methods in rate of recovery

  16. Slug test methods

  17. Conventional method • Water must flow out of column before head change is registered; recovery relatively slow • Pressure-slug method • Response governed by compressibility effects; recovery relatively fast • Mathematical theory for both methods similar

  18. Solution given by Cooper et al. (1967) and Bredehoeft and Papadopulos (1980) Where:

  19. where: , , , are the zero- and first-order Bessel functions of the first and second kinds.

  20. Type curves

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