Analysis of Development Methods for Gravel Envelope Wells - 3
2.1.1 Mathematical Model of Jetting
For the mathematical, the Darcy equations are used to consider a flow
field generated in a porous medium by injecting a flow Q over a circular surface of
radius c. This is a favorable representation of a jet since, with any high speed jet,
a significant fraction of flow would bounce from the screen and filter pack.
The well wall is modeled as a filter pack of hydraulic conductivity
k1 and thickness T (which is equal to the difference in the screen and filter
pack radii) and a formation of hydraulic conductivity k2. In actuality the
surface of the formation may well have a hydraulic conductivity much less than
k2 as a consequence of drilling wall cake. The physical configuration is as
shown in Figures 1 and 6.
6
Mathematically, this problem is identical to finding the flow of heat
into a layered medium, with heat being applied over a circular area of radius c and the
surrounding medium being kept at a constant reference temperature. The solution is given
for a homogeneous medium (no layers) by Carslaw and Jaeger* (1959), and this serves as a
starting point for the solution of the problem with two layers of different conductivity.
Details of the solution are given in Appendix A. It is in
the form of an integral and involves the ration of filter pack thickness to jet radius
(T/c), velocity of the jet vj, distance from the jet impact point on the
screen, and the ratio of the hydraulic conductivities of the formation and filter pack.
Numerical evaluation of the solution is possible using a digital computer so that flow
fields can drawn in both the filter pack and formation.
Figure 7 shows graphs of radial and tangential
velocities into and along the formation/ filter pack interface for the case when ratio of
jet radius to filter pack thickness is 1 to 12 (e.g., ½-inch diameter jet into
3-inch filter pack). Velocities are given as a fraction of jet discharge velocity
vj, where vj is the jet flow Q divided by jet area
c2. Pak radial
velocity into the formation is less than 1/2000th of the original jet
velocity vj and peak tangential velocity along the formation is only
1/5000th of the original jet velocity. In other words, very little jet
energy propagates into the formation. This is confirmed by the results appropriate to
the problem when there is no filter pack. In this case, an exact solution for the peak
radial velocity can be found (see Carslaw and Jaeger, 1959) and it indicates a magnitude
of the order (c/T)3 vj at depth T into the formation when c/T is
small.
Figure 7
>When the ratio of filter pack thickness to jet radius is 28 (7-inch
filter pack and ½-inch diameter jet), velocities are even lower. Computations show
a peak tangential velocity at the interface of the filter pack and formation of only
1/59000th of the jet velocity.
The results make it clear that very little jet flow energy will
penetrate more than a few jet diameters into the filter pack, and very little flow will
be generated at the filter pack/ formation interface. Recall that this solution was
based on Darcy flow equations and a presumption that all jet flow would enter the filter
pack. Given that, at high jet flow velocities, friction will be greater and a fraction
of the jet flow will bounce off the well screen and filter pack, the induced velocities
will be lower than those predicted in the above analysis. The above results are, of
course, only valid when the integrity of the filter pack is maintained. In the event of
disruption of the pack to a degree that the jet directly impacts the formation, these
conclusions no longer will be valid, and recourse is made to a laboratory test model.
* See References
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