Measurement and Testing 45
temperature fluctuations, and it is possible to measure permeability variations during progressive loading and microcrack development without interrupting the loading sequence. The method also makes it possible to determine whether the pressure cycling retains the rock in the elastic regime, or whether any permanent pore collapse has occurred.
Fig. 1 shows an example of data acquired using this method and Fig. 2 shows the gas permeability response to effective pressure cycling in the elastic regime of a typical quartz/clay- dominated Jurassic shale (collected from the intertidal zone at Runswick Bay, Yorkshire, England) of 7% porosity. Initial bedding-parallel permeability at low effective pressure is high, and would correspond to that measured by the GRI method. Increasing effective pressure for the first time provokes a rapid permeability reduction, after which further pressure cycles define a reproducible relationship between effective pressure and log permeability given by
(3) in which effective pressure Peff is given in psi, k is in mD. Thus the permeability decreases by about
×10 for an effective pressure change of 10000 psi, and is too large a change to be ignored in reservoir evaluation. The pore pressure is assumed to be fully effective. This is not the only possible representation of the experimental data, and is not necessarily the best, but it is the most commonly assumed form in previous literature, which is why it is used here for illustration. Fig. 2 also shows high degree of permeability anisotropy displayed by shales. Permeability along the layering is ×300 higher than across the layering. The origins of this anisotropy remain unclear.
In the transport equation (1), k is now a function of p, and this makes the equation once again non-linear. Kikani & Pedrosa7
the definition of fluid compressibility; (4) Permeability then varies exponentially with pore pressure according to (5) where ki and pi (pressure) sensitivity in gas reservoirs (Franquet8
are initial values. This is a useful formulation for demonstrating the effect of stress ). The transport equation (1) can be linearised
once more using a modified definition of pseudopressure, m’(p) to replace p in (1) so that it can describe the flow of a real gas through a stress-sensitive formation:
(6) proposed the concept of a permeability modulus γ, by analogy with
Figure 3: Comparative production rate decay with time for stress-independent (γ = 0) and stress-dependent (γ = 0.00024) reservoirs at constant bottom hole pressures (Pwf) of 4000 and 9000 psi, for an overburden pressure
of 10000 psi. The effect of stress-dependence of permeability is more marked at smaller bottom hole pressure.
hence
(7) Figure 4: Variation of permeability as a fraction of the initial value ki
Influence of stress-dependent permeability on reservoir behaviour
To illustrate the effect of stress-dependent permeability on the behaviour of a gas reservoir, finite difference numerical solutions to eqn.(7) for appropriate initial and boundary conditions were obtained using program GASSIM (Lee & Wattenbarger9
), for formation-linear flow in an infinite-acting
dry gas reservoir of 100 ft thickness, at 140 ºC, containing a single hydraulic fracture extending 400ft on either side of a vertical production hole. Porosity is 0.075 and initial permeability is 0.00034 mD, corresponding to eqn.(3) at zero effective pressure. Overburden pressure is 10000 psi.
Drawing down fully the gas pressure reduces the permeability by about one order of magnitude (γ = 0.00024). Fig. 3 shows how flow decreases with time for two constant downhole pressures (Pwf
= 4000 psi and 9000 psi), more rapidly for the case of stress-dependent than stress-independent permeability. Fig.4 shows how the reduction of permeability propagates into the reservoir as the gas pressure is progressively reduced. Fig.5 shows how stress-sensitive permeability impacts on total production after 225 and 2000 days for different fixed downhole pressures. The greatest production is always at the lowest Pwf, but it may not be practicable to maintain this at a low value.
Well test results can appear similar in form whether formation permeability is stress-sensitive or not, but if the analysis of results does not take into account stress-sensitivity then erroneous inferences will be made of permeability, gas in place and productivity to be expected (Franquet et al. 2004). This illustration is a simple one to demonstrate the effects of stress-dependent permeability. The range of stress-sensitivities of permeability that can be displayed, how they depend upon porosity and the microstructural arrangement and elasticity of the component mineral phases in different shales, any effects of non-hydrostatic stresses, and the influence of partial liquid saturation, are issues that have barely begun to be touched upon in laboratory measurements.
Acknowledgements
Prof. R. Wattenbarger (Texas A & M University) kindly provided the code modifications to GASSIM version 6 to permit stress-sensitive reservoirs to be simulated. References
1
Al-Hussainy, R., Ramey, H.J., Crawford, P.B. 1966. The flow of real gases through porous media. Journal of Petroleum Technology, 237, 624-636
2
Brace, W.F., Walsh, J.B., Frangos, W.T., 1968. Permeability of granite under high pressure. Journal of Geophysical Research, 73, 2225-236
3
Kranz, R. L., Saltzman, J. S., Blacic, J. D., 1990. Hydraulic diffusivity measurements on laboratory rock samples using an oscillating pore pressure method, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 27, 345– 352
4
Fischer, G. J., 1992. The determination of permeability and storage capacity: Pore pressure oscillation method. In: Fault Mechanics and Transport Properties of Rocks, edited by B. Evans and T.-F. Wong, 187– 212, Academic Press, San Diego, Calif.
5 6
Figure 5: Illustration of the effect of stress-dependent permeability (γ = 0.00024) on cumulative production (millions standard cu. ft.) after 225 and 2000 days for a range of constant bottom hole pressures, compared
to stress-insensitive reservoir (γ = 0). Overburden pressure = 10000 psi, initial gas pressure = 9800 psi.
Faulkner, D. R., Rutter, E. H., 1998. The gas permeability of clay-bearing fault gouge at 20°C. Geological Society, London, Special Publications, 147, 147-156, doi:10.1144/GSL.SP.1998.147.01.10
Bernabé Y., Mok, U., Evans, B., 2006. A note on the oscillating flow method for measuring rock permeability. International Journal of Rock Mechanics & Mining Sciences, 43, 311–316
7
Kikani, J., Pedrosa, O. A., 1991. Perturbation analysis of stress-sensitive reservoirs. SPE Formation Evaluation, 379-386
8
Franquet, M., Ibrahim, R.A., Wattenbarger, Maggard, J.B., 2004. Effect of pressure-dependent permeability in tight gas reservoirs, transient radial flow. Petroleum Society of the Canadian Institute of Mining, Metallurgy and Petroleum, Canadian International Petroleum Conference 2004, Calgary, Alberta, Canada, 2004 Paper 2004-089, 1-10
9
Lee, J., Wattenbarger, R. A., 1996. Gas reservoir engineering. Society of Petroleum Engineers Textbook Series v. 5, Richardson, Texas, 349pp
AUGUST / SEPTEMBER 2013 •
WWW.PETRO-ONLINE.COM with distance from hydraulic fracture as a result
of gas pressure drawdown in a stress-sensitive reservoir after 225 and 2000 days for a constant downhole pressure Pwf. In a stress-insensitive reservoir the permeability would remain constant at the initial value 0.00034 mD.
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