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Consulting
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Fractional Dimension Flow
The main geometric feature which distinguishes different flow dimensions is
the power law change in flow area with radial distance. This is second power
relationship for spherical flow, first power relationship for cylindrical flow,
and zero power (or constant) for linear flow. The dimension is simply the power
of the radial variation plus one. As pointed out by Doe and Geier (1991), power
law variability of hydraulic properties can also produce dimensional behavior.
We can combine the joint effects of area and property by considering the two
effects together as a conductance, which is the product of area and hydraulic
conductivity.
The dimension of the well test contains very fundamental and useful information
about the hydraulic geometry of fracture networks. One-dimensional flow may
indicate a single channel within a fracture, or a chain of channels forming
a linear network. Two-dimensional flow may indicate a single fracture normal
to the borehole, or a network of fractures that is confined to a planar zone,
such as a fracture zone or a highly fractured sedimentary bed. Three-dimensional
flow may indicate well-connected, a space-filling network of discrete fractures
or channels. Finally, non-integer dimensions will appear between these cases
where the fracture pattern does not fill a particular space, as in a fractal
or power-law network geometry.
Differently dimensioned flow systems have significantly different behavior.
In addition, since the systems are fractured, they can be both scale dependent
and heterogeneously connected. Research was carried out toward development of
procedures for analysis of fractional dimension type curve responses, using
Laplace transform solutions for the equation of fractional dimensional flow.
The main assumptions made in the course of developing the models for transient
rate and pressure behavior in a two-zone composite system are as follows:
- Transient Darcian flow takes place in the system, the near flow direction
is radial
- The ith zone is characterized by flow dimension ni
(i = 1 for the inner zone and i = 2 for the outer zone), where ni
is not necessarily an integer; the source well is an n1-dimensional
"sphere" projected through three-dimensional space
- The ith zone is characterized by hydraulic conductivity and specific
storage Ki and Ssi, respectively (i = 1 for the inner
zone and i = 2 for the outer zone)
- The system is infinite, and either a constant-rate or a constant-pressure
condition is imposed at the source well
- Wellbore/source storage capacity is non-negligible
The radial flow behavior of water in a two-zone composite system is governed
by the following equations (Barker, 1988):
 (5-1a)
and
 (5-1b)
respectively, where
 (5-1c)
In terms of the dimensionless variables, the initial and boundary conditions
become
 (5-2a)
 (5-2b)
 (5-2c)
 (5-2d)
and
 (5-2e)
where
 (5-2f)
 (5-2g)
and
 (5-2h)
Laplace transforms can be used to solve the system of partial differential
equations. The subsidiary equations are
 (5-3a)
and
 (5-3b)
After transforming the boundary conditions, Equations (2-3a) and (2-3b) are
solved simultaneously. The solutions in Laplace space are
 (5-4a)
and
 (5-4b)
where
 (5-5a)
 (5-5b)
 (5-5c)
 (5-5d)
and
 (5-5e)
Using equations 5-1 through 5-5 and the related type curves it is possible
to derive both transmissivity, storativity, and flow dimension as a function
of distance from the well bore from well tests. Of these, the flow dimension
as a function of distance may prove to be the most important for reservoir design,
since lower flow dimensions indicate that only a small portion of the reservoir
is being accessed.
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