THE BETA FACTOR AS A FUNCTION OF SATURATION

Ellen Angelina Hjelleset (1995)

ABSTRACT

Procedures for immobile liquid saturation and the effect of an immobile liquid saturation on 
high-velocity flow through Berea and Bentheimer sandstone cores have been analyzed.  The 
cores were examined for  saturations in the range of 0 - 30 percent pore volume.  Glycerol was 
used as the immobile liquid due to its physical properties of low viscosity and low vapor 
pressure.  The cores were saturated with a solution of destilled water and the desired weight 
volume of glycerol.  The drying process was recorded as liquid weight variations as a function 
of time.  The drying process was dominated by an exponential decline until a critical weight 
was reached.  In the next period, the weight remained almost unchanged.  This indicates that 
the water is vaporized by diffusion and the glycerol is left in the cores.  

The existence of the Mei and Auriault regime as a transition regime from Darcy´s to the 
Forchheimer flow was also studied.  This is, to the best of our knowledge, the first experiment 
that supports the theory of Mei and Auriault.  Finally, the high velocity coefficient, b, as a 
function of immobile saturation was measured.  Even though the number of different 
saturations were small, a dip in b  was found for Sw»10 percent.  A fit to a second order 
polynomial suggests that the minimum of b could be found for Sw»5.0 percent.

1.  INTRODUCTION

Forchheimer introduced in 1901 a modified equation of the Darcy´s law to account for high 
velocity flow effects.  This modified equation has since been referred to as the Forchheimer 
equation.  One of the parameters which was introduced in the equation, the high velocity 
coefficient, b, is a function of pore geometry.  Thus, b  for a core with a liquid saturation is 
supposed to differ from b of an unsaturated core.  In some cases, it is important to evaluate b 
for saturated cores accurately, to better estimate the well productivity.

Today, b is normally estimated from previously developed equations for single-phase flow.  
However, the correlations that exist in the literature are not always applicable.  The 
correlations are typically simple power laws, and do not fit the actual behavior of b.  For rocks 
different from those the correlations were developed for, the measured b may differ 
significantly from the b estimated by the correlations, due to differences in the pore geometry.  
Even for the rocks that were used to develop the correlation, the estimated value of b may be 
an order of magnitude different from the value estimated by the correlation.  From laboratory 
measurements we know that b generally increases with increasing saturation, except for 
saturations less than 10 percent, where Gewers and Nichol have noticed a decrease of b.

In this project we will evaluate b as a function of water saturation on cores with different 
permeability.  To investigate the effect of immobile saturation systematically, it is important to 
have controlled saturation procedures.  Therefore, the choice of saturation liquid is important.  
Primary, it is important that the saturation phase stays immobile during the flow measurements, 
which requires a high-viscous saturation liquid.  Secondly, it is important that the amount of 
the saturation liquid does not change during flow measurements.  This may to some extend be 
avoided by keeping the vaporization low, so that the saturation liquid should have a low vapor 
pressure.  Finally, the saturation liquid should be easily injected into the cores.  For this reason 
it should be readily soluble in a liquid, preferably water.  All these specifications are fulfilled for 
glycerol, and we will therefore in our experiments saturate the cores with a solution of water 
and glycerol.  

The second objective is to perform laboratory measurements on the preparated cores at 
different levels of immobile saturations.  We will essentially pay our attention to saturations 
below 30 percent liquid saturation in order to eventually confirm that b may decrease with 
increasing saturation for sandstones.  We will also perform a more detailed analysis to observe 
departures from the Forchheimer equation in the weak inertia flow regime.  In addition to the 
experimental part and data analysis, this project also includes a literature review of the effect of 
liquid saturation on b.

6.  CONCLUSIONS

1. The saturation procedure of Gewers and Nichol appeared to be very successful and can be 
recommended for further researching.
 
2. The drying procedure follows an exponential decline, which is in accordance with the theory 
of drying.  Thus, we may estimate the characteristic drying time to obtain a desired 
saturation of the cores.
 
3. The effect of liquid saturation on the b factor differs from that indicated by the effective 
permeability and the dry-core correlations in the literature.  In particular, Geertsma´s 
correlation was examined, and appeared to give remarkable deviations form our laboratory 
measurements.  The results were also compared to the results of Gewers and Nichol, but we 
did not observe such a noticeably deviation from the correlations as Gewers and Nichol 
measured.
 
4. At low flow rates, the results supports the theory of Mei and Auriault (1991).  To the best 
of our knowledge this is the first experimental verification of the Mei and Auriault regime, 
and the subsequent transition to the Forchheimer regime.
 
7.  REFERENCES

1.  J. Geertsma, Estimating the coefficient of inertial resistance in fluid flow through porous 
media, SPE October 1974, p.447

2.  E.V. Evans, R.D. Evans, The Influence of an immobile or mobile liquid saturation upon 
non-Darcy compressible flow of real gases in propped fractures, SPE 15066, p.182

3.  S.W. Wong, Effect of Liquid Saturation on Turbulence factors for gas-liquid systems, 
Technology, Oct. -Nov. 1970

4.  C.E. Avila, R.D. Evans, The effect of temperature and overburden pressure upon the non-
darcy flow coefficient in porous media, Proc. 27th u.s. Symp on rock mechanics, Univ. of 
Alabama 1986, p. 628

5.  K.C. Khilar, H.S.J. Fogler, Colloid and Interface Sci., (1984) p. 214

6.  Luffel, D.L., Herrington K.L., and Walls, J.D., Effect of drying on Travis Peak cores 
containing fibrous illite, SPE rock Physics Assocs.

7.  deWaal, J.A. et al., Petrophysical core analysis of sandstones containing delicate illite, 
The log Analyst (Sept. - Oct. 1988), p. 317-31

8.  Gewers, C.W.W, Nichol, L.R., Gas turbulence factor in a microvugular carbonate», 
Technology, April-June, 1969

9.  R.H.Perry, D.W.Green, J.O.Maloney, Perry´s chemical engineers´handbook, Law Hill 
Book Company, 6.th edition, p. 20-9 - 20-11

10.  Mei and Auriault, The effect of weak inertia on flow through a porous medium, J. Fluid. 
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11.  Wodie and Levy, Correction non linear de la loi de Darcy, Mechanique des sols et 
milieux poreux (1991)

12.  F.A.L. Dullien, Porous media, fluid transport and pore structure, Acadmic Press, 2nd 
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