the effect of porosity on hydraulic rock cutting

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 11, pp. 103~105. Pergamon Press 1974. Printed in Great Britain

The Effect of Porosity on Hydraulic Rock Cutting S. C. CROW*

This note amends the author's recent theory of hydraulic rock cutting. The rock is assumed to feed steadily under a water jet, which leaves a slot of uniform depth. The depth decreases as the feed rate exceeds an intrinsic speed c. The intrinsic speed embodies the interaction of cavitation, fracture, and permeability, and here it is shown that porosity must be included as well. Thus

kzo c ~lf#rg'

where q is the viscosity of the cutting fluid, k is the permeability of the rock, f is its porosity, ~o its shear strength, #~ its coefficient of internal friction, and g is a typical grain diameter. The intrinsic speed computed from measured properties of 1441keson sandstone agrees well with the value assumed for a best fit to rock cutting data in the original paper.

1. THEORETICAL REVIEW

The purpose of this note is to modify a theory of hydraulic rock cutting recently set forth by the author [1]. The mathematical form of the theory is unchanged, but an intrinsic speed for hydraulic rock cutting is redefined to include rock porosity as well as the various material properties introduced in [1]. The need for the change became apparent during experiments performed in the new UCLA Hydraulic Rock Cutting Laboratory.

The theory applies to the steady cutting operation illustrated in Fig. 1. A rock feeds leftward at a speed v under a jet nozzle. A high-speed water jet impinges on the rock at an angle 00 and swings round until the local angle 0 equals zero, at which point the slot has reached its terminal depth h. The problem is to find h in terms of feed rate, rock properties, and jet parameters. The relevant jet parameters are initial diameter do, total pressure Po, entry angle 00, and fluid viscosity r/. P0 is typi- cally 20,000 psi, corresponding to a water velocity of 1700 ft/sec.

The curvature of the jet stream induces a high average pressure p.~ at the interface between water and rock. The flow at the granular interface is strongly cavitating, and it was argued in [1] that the shear stress z under cavita- tional conditions should be proportional to the difference between p~ and the vapor pressure Pv of the water:

r = ~,~ ( p s - p,,). ( 1 )

i~,,. is a coefficient of Coulomb friction for a rough cavitating surface and was assigned the value 0"42 on the basis of preliminary data. The argument advanced in [ 1 ] was that increased pressure p.~ closes cavity bubbles and

* School of Engineering and Applied Science, University of Califor- nia, Los Angeles, California, U.S.A.

exposes more grains to direct water impact, thereby in- creasing z.

A rather different argument leads to the same friction law (1). The flow adjacent to the granular interface is turbulent, and the instantaneous surface pressure p may

NOZ ZLE~"~ do

Fig. 1. Steady slot cutting under a high-speed water jet.

depart significantly from its average value p~, depending on the intensity of eddy motion. Dimensional reasoning implies that the root-mean-square pressure fluctuation at the wall of a turbulent boundary layer is proportional to the mean shear stress:

( ( p _ ps)2> 1/2 = 2"3 r. (2)

The brackets denote a time average, and the coefficient 2-3 is taken from experiment [2]. Figure 2 illustrates the consequence of high stress in a cavitating turbulent boundary layer. In the upper part of the figure, the instantaneous wall pressure p(s) is plotted as a function of distance s along the rock surface. A vapor cavity forms wherever p(s) falls to Pv, as shown in the lower sketch. Eddies literally tear themselves apart. Vaporization

103

104 S.C. Crow

I

/

-~:~:i:~i:i~i~i:ii:iiiiii~i~iii:i~ii~i:i~i~i~i:i~i:iii:i~i:i~iii~i:i~i~i~iii~iii:i~i:iii~i:i~i:iii~ii~iiiiiii 'i:i'ii'i'i'iiiii~i~i'i:i'i'i'i;i~i~i~ ~,..~, ~,,~ ~,,,~, @~,~,,, ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::

:~!.~ i~!~ ........... ::::::::::::::::::::::::::::::: ...............

!iiiiiiiiiiiii ii! iiiiiiiiii{ iiiiiiiiiiiiiiiio

Fig. 2. Cavitation in a turbulent boundary layer. The upper plot shows the instantaneous pressure as a function of distance along the wall. The lower sketch shows the mean velocity profile and the formation

of cavities in low-pressure regions.

must inhibit turbulent momentum transfer, so it is reasonable to assume that the root-mean-square pressure fluctuation cannot be driven beyond some multiple M of the mean pressure difference (p~ - p,):

( (p _ p,)2, ~/2 _< M(p~ - p~), (3)

Conditions (2) and (3) yield an upper bound on the shear stress,

M

< 0"43 M(p~, - p,,),

where the equality should hold for an extremely rough surface. The Coulomb friction law (1) is recovered, with a friction coefficient

#w = 0.43 M.

The present argument does not provide a specific value for M, but the choice M = 1'0 corresponds to a friction coefficient very near the value 0"42 assumed for a best fit to the data in [1].

The hydrodynamic problem becomes one-dimensional if the jet is assumed to be slender, which means in particular that do/h is small. The friction law (1) and the momentum equations for a slender jet result in a simple integral formula for h:

r ,,e,,,.~o-oo) sin 0 h = 21~wdoPo dO. (4)

d o T

The shear stress z must suffice to induce continuous failure in the rock. The simplest criterion for failure is the Coulomb criterion discussed at length by Jaeger and Cook [3]:

Z = Z o + / # ( p ~ - p ) .

z0 is the shear strength under zero normal loading, #, is the coefficient of internal friction, p~ is the normal stress borne by the cutting surface, and p is the pore pressure

within the rock. Exactly where the failure takes place remains to be specified. A central assumption of [ l ] is that the rock is in a continuous state of incipient fracture one grain diameter g beneath the cutting surface. By a Taylor expansion of ( p , - p) at the cutting surface,

: = % + /~,9 \~n/,= o

where n is the local normal coordinate and is negative within the rock. Equations (4) and (5) yield a solution for h when the normal pressure gradient just beneath the cutting surface is known as a function of 0.

2. P E R M E A B I L I T Y A N D P O R O S I T Y

The intense pressure p~ forces water through the pores of the cutting surface and creates a thin precursor of saturated rock as shown in Fig. 3. The flow within the saturated region obeys Darcy's law, as explained in the classic book by Muskat [4]. The price of choosing coordinates in which the jet stream is steady, however, is that Darcy's law must be generalized for a moving porous medium, a situation apparently not contemplated in the literature on permeability. Darcy's law for a moving medium was stated in [1] as

k Vp = v - u , (6*)

r/

where v is the velocity of the medium, and u is the flow rate of the permeating fluid, u has the dimensions of velocity but must be interpreted as a volumetric flow rate per unit area. t/is the viscosity of the fluid, and k is the coefficient of permeability defined in accord with Muskat. The k used here corresponds to the quantity k' defined above equation (31) in [1] and has the advantage of being independent of the cutting fluid. Muskat's permeability convention will be used in subsequent papers from the UCLA Hydraulic Rock Cutting Laboratory.

Equation (6*) contains a subtle error. Suppose Vp = 0, in which case a saturated moving rock simply conveys passive fluid in its pores. Equation (6*) predicts that

U = V,

which would be true if u were a particle velocity subject to Galilean transformation from stationary to moving

Fig. 3. Precursor of saturated rock beneath the cutting surface.

The Effect of Porosity on Hydraulic Rock Cutting 105

coordinates. However u is not a particle velocity, but rather a bulk volumetric flow per unit area. The ability of the rock to convey fluid is proportional to the available pore volume divided by the bulk volume. That ratio is called porosity and is denoted by f i n [4]. Thus

u = f v

in a moving saturated rock under no internal pressure gradient. The correct generalization of Darcy's law for a moving porous medium is

k - V p = f v - u. (6) t/

The argument now follows[l] without further change. In coordinates fixed with respect to the jet, the interface between saturated and dry rock is steady, so the component of u normal to the interface is zero. Since the saturated precursor is thin, the component of u normal to the cutting surface is also nearly zero. Thus

k @ fv sin 0 - -

/7 n=O

after a little geometry. The failure criterion (4) takes the form

= ~o + ~ v sin 0, (7)

and solution (4) can be written

h = 2pw doPo f£0 e~,,.~O-Oo~ sin 0 % _v 1 + (v/c) sinodO' (8)

where c is the intrinsic speed for hydraulic rock cutting:

kZo c qfPrg" (9)

The only substantive change in the amended theory is that the ratio (k/J) replaces k. The maximum rate at which slot area can be created, for example, is predicted a s

2kdoPo (1 - (hv)~ ,x- ~ e-""°°), (10)

realized in the limit as (v/c) approaches infinity. Since k generally rises as a power o f f [5 ] , the incorporation of porosity in equations (7-10) moderates the predicted variation in cutting efficiency from one rock type to the next.

3. COMPARISON WITH EXPERIMENT

When the theory of hydraulic rock cutting was ori- ginally developed, no means was available to measure permeability. As a consequence, c could not be calcu- lated from (9), but the choice

c = 17-2 in./sec

brought solution (8) into line with preliminary data for slot depths cut into Wilkeson sandstone[l]. The mechanical properties corresponding to the chosen intrinsic speed were deemed reasonable, though enormous variation among sandstone permeabilities prevented a real test of equation (9).

Meanwhile the National Science Foundation has sponsored the new Hydraulic Rock Cutting Laboratory at UCLA, where each of the quantities appearing in (9) can be measured. The first rock type to be considered, of course, was Wilkeson sandstone. The author and his colleagues will soon report the experiments in detail [6], but a table of results will suit the purpose here:

TABLE 1. MEASURED PROPERTIES OF WILKESON SANDSTONE

to = 1620 psi p, = 1"23 g = 0'0063 in. k = 0"582 millidarcys f = 0.096

The millidarcy is a hybrid unit representing the permeability of a substance that permits a fluid of 10-2 c.g.s, viscosity (1 centipoise) to flow at a rate 10-3cma/sec/cm2 under a pressure gradient of 1 atm/cm. In the English units of [ l ] , the Wilkeson permeability yields a ratio

k _ 6-10 x 10 -6 ina/sec/in2 q psi/in.

for water at 20°C. Equation (9) and the properties listed in Table 1 give

c = 13.3 in./sec

as the intrinsic speed for Wilkeson sandstone, not far from the value 17"2 in./sec suggested by the preliminary jet cutting data. The agreement is impressive in view of the thousand-fold variation of permeability among different types of sandstone (cf. Table 8 of [43).

Porosity causes little formal change in the theory but has a large impact on the intrinsic speed computed from (9). The porosity of Wilkeson sandstone is 0.096, and values around 0"10 are probably typical. The computed intrinsic speed would be ten times too low if porosity were overlooked. The apparent discrepancy for Wilke- son sandstone led to the incorporation of porosity in the theory of hydraulic rock cutting.

Received 21 August 1973.

Acknowledgement--This work was supported by the National Science Foundation under Grant GI 37193 and by the UCLA School of Engineering and Applied Science. The NSF grant originated from the Division of Advanced Technology Applications, Research Applied to National Needs.

REFERENCES 1. Crow S. C. A theory of hydraulic rock cutting, lnt. J. Rock Mech.

Min. Sci. 10, 567-584 (1973). 2. Arndt R. E. A. and Daily J. W. Cavitation in turbulent boundary

layers. Cavitation State of Knowledge, pp. 64-86, ASME Fluids, Engineering and Applied Mechanics Conference (1969).

3. Jaeger J. C. and Cook N. G. W. Fundamentals of Rock Mechanics, Methuen, London (1969).

4. Muskat M. and Wyckoff R. D. The Flow of Homogeneous Fluids through Porous Media, McGraw-Hill, New York (1937).

5. Schedegger A. E. The Physics of FIow through Porous Media, Mac- millan, New York (1960).

6. Crow S. C., Lade P. V. and Hurlburt G. H. The mechanics of hydraulic rock cutting. To be presented at the Second International Symposium on Jet Cutting Technology, Cambridge, England (1974).

the effect of porosity on hydraulic rock cutting

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