General poroelastic model for hydraulic fracturing

Roegiers, J. C. ; Cui, L. ; Roegiers, J. C.
Organization: Society for Mining, Metallurgy & Exploration
Pages: 3
Publication Date: Jan 1, 1999
X. Huang (1997) recently suggested a poroelastic model for simulating the hydraulic fracturing breakdown pressure. His paper began with a discussion questioning Haimson and Fairhurst's (H&F) model. He claimed that the H&F model failed to lead to the Hubbert and Willis (H&W) model as[ a --> 0.] Huang also tried to explain why the H&F model could only work for special types of rock conditions. He pointed out that one possible reason could be that the Terzaghi's effective-stress concept had been adopted. The H&F model (Haimson and Fairhurst, 1969) was derived under the conditions that the borehole wall is fully penetrated (pp = pi) and a drained state is realized (steady pore-pressure field). Cui et al. (1997a) demonstrated that, under drained conditions, the total stresses in the penetrating poroelastic model (identical to the H&F model) degenerate into their counterparts in the elastic model (identical to the H&W model) as [a -- 0.] However, the effective-stress conditions are different for both of these models, because different pore-pressure conditions at the borehole wall were adopted. In the H&W model, p = po was assumed, i.e., the pore pressure field is not disturbed; but pp = pi was assumed in the H&F model. Assuming that Terzaghi's effective-stress controls tensile failure (that was the hypothesis adopted in both the H&F and the H&W models), only the following two special drained poroelastic cases may degenerate into the H&W model for very small a: • when the pore pressure at the borehole wall remains at the same level as the virgin pore pressure for a penetrating model and when the borehole wall is simply impermeable (i.e., the nonpenetrating model, Cui et al., 1997b). Therefore, simply setting a = 0 in the H&F model generally does not lead to the same problem described by the H&W model. On the other hand, when the porepressure boundary conditions do not correspond to the ones in the H&W model, a degeneration of the poroelastic model to the H&W model as [a – 0] should be questionable. The pore pressure at the borehole wall is generally dependent on the injection-fluid pressure and the penetrating conditions at the borehole wall, such as the existence of a filter-cake. For an impermeable wall, pp is independent of Pi, and it is basically an unknown [(how¬ever, pp -- po as t --oo).] For a fully permeable wall, pp is the same as Pi. Between these two extremes, pp should be a function of p; and the permeable condition of the borehole wall, which may be dependent of the leak-off coefficient cf. (the range of cf is from 0 to 1). Theoretically, for a rock with low permeability, a penetrating borehole wall is still possible. For saturated porous materials with very low values of a, poroelasticity shows that a pore pressure will be built up due to the stress concentration subjected to a nonhydrostatic in situ stress field (Cheng et al., 1993). This phenomenon is known as the Skempton effect. The variation of the pore pressure may be evaluated by [AP = 3 B(A rr + 06ee + A (Y,,)] (1) where B is the Skempton pore pressure coefficient. This pore pressure variation dissipates as time increases (it is totally gone as the drained state is approached). The rate of the dissipation mainly rely on the permeability of the formation. The dissipation is very slow for tight formations because their permeability is very low, and it is fast for rocks of high permeability. According to our analyses, the time period for this process could be from seconds (for sandstone's) to a couple of days (for stales). One possible reason that the H&C model did not agree well with the experimental results for rocks with low permeability might be that the time interval between the application of the loading and the fluid injection had not been long enough for the dissipation of the Skempton effect. The effective-stress law basically defines how much pore pressure contributes to the total stress. The difference between Terzaghi's effective stress and Biot's effective stress is that 100% of the pore pressure contributes to the total stress in Terzaghi's definition, while only a certain portion of the pore pressure ((ap) goes to the total stress in Biot's definition. Therefore, when the pore pressure at the boretole wall (pp) is determined according to Biot's effective-stress law, app in the total tangential stress is attributed to the pore pressure. In other words, Terzaghi's effective tangential stress is expressed
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