EQUATION PREDICTS EFFECTIVE GAS PERMEABILITY

I.M. Kutasov
Consultant
Los Angeles

An equation has been developed to calculate effective gas permeability, k,. Unlike other permeability correlations, this equation implicitly takes into account the effective closure stress.

The interpretation of porosity, irreducible water saturation, and effective gas permeability from log data is generally based on equations that simulate laboratory measurements of these parameters.' In most of the permeability equations, the permeability is expressed as a function of porosity and irreducible water saturation.

In hydraulically fractured gas wells, the fracture conductivity dependence on effective closure stress cannot be assessed theoretically.

Required are empirical relations based on extensive proppant conductivity measurements over a wide range of conditions.2

A semitheoretical equation is proposed here for relating the effective gas permeability, (kg), to:

• Porosity,F

• Irreducible water saturation, (Siw)

• Implicitly (through the empirically defined value of gas permeability at Siw) to the effective closure stress.

Our analysis of the results of experimental and analytical investigations 3 4 5 has shown that the following formula can be used as a generalized permeability equation:

kg =

ki (1 - Siw) 3 (1 - f) 2

--------------------- (1)

[1 - f ( 1 - siw)] 2

or

kj = kix; ki = kg/x (2)

or

In kg = In ki + In x (3)

x =(1 - siw) 3 (1 - f) 2

--------------------- (4)

[1 - f (1 - siw)]2

where:

ki Is the permeability at Siw = o

If kg is plotted vs. x on a log-log plot, linear regression of the data defines a line with slope m - 1 that intercepts the y axis at ki (Fig. 1).

The results of calculations (Table 1) confirm the conclusion that fracturing-fluid residue may reduce the proppant conductivity by 90% or more, depending on the fracturing-fluid type and quality, and that the effects of liquid saturation should be considered when designing and evaluating propped hydraulic fracture treatments for wells that produce significant quantities of water.2 3 For high-permeability formations, the permeability of two proppants 3 is shown in Table 2. The experimental value of ki is used.

The comparison is computed and measured values of the effective gas permeability indicates a good match.

To get a better agreement between kge (experimental gas permeability) and kg (Equation 1) linear regression analysis of the data should be employed to determine the value of ki (Fig. 1).

In a low-permeability sandstone reservoir (Table 3) with a water saturation range of 0.38-0.50, it was assumed that formation water saturation derived from logs is a good approximation of irreducible water saturation.' The average value of ki 0.02968 md was calculated and the value of kg was obtained from Equation 5:

kg = 0.02968 x x (5)

The values of kge and kg are in satisfactory agreement. However, additional experimental investigations are needed to verify the application of the proposed equation for low-permeability formations.

REFERENCES

1. Kukal, G.C., and Simons, K.E., "Log Analysis Techniques for Quantifying the Permeability of Submillidarcy Sandstone Reservoirs," SPE Formation Evaluation, December 1986, pp. 609-22.

2. Davies, D.R., and Kuiper, T.O.H., "Fracture Conductivity in Hydraulic Fracture Simulation," Journal of Petroleum Technology, May 1988 pp. 550-552.

3. Evans, E.Y., and Evans, R.D., "influence of an Immobile or Mobile Saturation on Non-Darcy Compressible Flow of Real Gases in Propped Fractures," Journal of Petroleum Technology, October 1988 pp. 1343-51.

4. Fand, R.M., Kim, B.Y.K., Lam, A.C.C., and Phan, R.T., "Resistance to the Flow of Fluids Through Simple and Complex Porous Media Whose Matrices are Composed of Randomly Packed Spheres," Journal of Fluids Engineering, September 1987, pp. 268-74.

5. Olovin, B.A., "Gas Permeability of Frozen Coarse Rocks," Problems of Geocryology, "Nauka," Moscow, 1988, pp. 120-123.

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