CUBIC FLOW LAW AND PORTABLE PACKER TESTS

J Dowd

Steady state flow of an imcompressible liquid in a fracture, isothermal conditions: Stokes Law

Where: : Driving Force and : Viscous Resistance Force

STOKES LAW

For flow under uniform gradient between two smooth platesNo slip conditionIntegration of Stokes Law yields:

Where parallel to flow; perpendicular to flow; velocity parallelb is aperture

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

SYSTEM OF PARALLEL CRACKS

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

• 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:

DETERMINING B IN THE FIELD

APERTURE/HYDRAULIC CONDUCTIVITY/POROSITY

Fracture Spacing: 2.5m

• 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

DEVIATION FROM CUBIC LAWBRINKMAN EFFECT

SINGLE HOLE PACKER TESTS

QUADRUPLE PACKER

• 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

• Gov eqn:

SINGLE HOLE PACKER TESTS

FLOW APPROXIMATIONSRadial flow Prolate Spheroidal

• 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)

INJECTION TESTS

• Prolate spheroidal (ellipsoidal)

Solution for a line source (L) and constant Q (Hvorslev, 1951):

Where:

INJECTION TESTS

• 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

SLUG TEST

SLUG TEST METHODS

• 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

• Solution given by Cooper et al. (1967) and Bredehoeft and Papadopulos (1980)

Where:

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

TYPE CURVES