Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid-solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical nonDarcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.
It is known that rock anisotropy can significantly influence the phase and energy velocities of an elastic wave, as well as its reflection/transmission (R/T) coefficients. As a result, it can distort the velocity analysis of seismic-reflection data. In this work we present a velocity analysis for seismic-reflection data based on the available anisotropic rock parameters. We analyzed the created errors on time-depth relation of the seismic-reflection data in neglecting rock anisotropy and/or neglecting the difference between energy velocity and phase velocity, including the case of wide-angle reflection. The calculated results show that the effect of rock anisotropy on time-depth relation of seismic-reflection data is dependent not only on the values of anisotropic parameters, but also on the space arrangement of both source and receiver-array. For all studied cases (weak, moderate or strong anisotropy), we found that the effect of rock anisotropy on time-depth relation could not be neglected. Nevertheless, for the case of weak anisotropy, the energy velocity may be replaceable by the phase velocity to obtain a very good approximation on time-depth relation. Consequently, the seismic-reflection data processing algorithm for numerical computations can be simplified. rock anisotropy, phase velocity, energy velocity, time-depth relation Citation: Fa L, Castagna J P, Zeng Z W, et al. Effects of anisotropy on time-depth relation in transversely isotropic medium with a vertical axis of symmetry.
It has been widely accepted in hydraulic fracturing that the higher the pressurization rate, the higher the breakdown pressure. However, linear elastic fracture mechanics (LEFM) suggests the opposite. In an effort to clarify this "injection rate paradox," three controlled laboratory hydraulic fracturing experiments were conducted using different injection rates. These tests showed that the higher the injection rate, the lower the breakdown pressure. Further investigations indicated that the elastic- and poroelastic- models were not able to properly predict the observed breakdown pressures. Using Griffith's energy balance concept, an LEFM-based breakdown pressure model was derived which gives satisfactory predictions. Introduction Breakdown pressure is the peak pressure corresponding to the initiation of a hydraulic fracture (HF). It represents an important factor for the determination of in-situ stresses via micro-HF tests. Hubbert and Willis1 derived the elastic formula for the breakdown pressure in the case of impermeable materials. By considering the poroelastic effect, Haimson and Fairhurst2 derived a similar expression for the case of permeable formations. Those two approaches defined the upper and lower bounds of the breakdown pressures. But in both formulae, the influence of injection rate is ignored. Haimson and Fairhurst3 published their experimental observations on the effect of pressurization rate on hydraulic fracturing breakdown pressures in porous, permeable hydrostone. They found that the breakdown pressures increased systematically with the pressurization rates. Zoback et al.4 tested the effect of borehole pressurization rate on the breakdown and fracture initiation pressures using Ruhr and Weber sandstones; both rock types having a permeability ranging from 0.1 to 1.0 md. The authors concluded that the pressurization rate had a marked influence on the breakdown pressure: the higher the pressurization rate, the higher the breakdown pressure. Solberg et al.5 experimentally investigated the injection rate influence on the fracture mechanisms and the permeability of the induced fractures under geothermal conditions using Westerly granite. It was found that under high injection rates, tensile fractures were induced; otherwise, shear fractures were generated. The hydraulic fractures induced under intermediate injection rate, on the other hand, had permeability an order of magnitude higher than that in the other two limiting cases. In addition, the experiments showed that higher injection rates resulted in higher breakdown pressures. Weijers et al.6 investigated the induced fractures in horizontal wells that were drilled in the direction parallel to the minimum horizontal stress. Depending on the injection rates, three resulting geometries were observed:initiation of an axial fracture along the wellbore together with transverse fractures located in the preferred plane (relatively low treating pressures);gradual fracture reorientation (intermediate treating pressures); and,multiple fractures and stepwise reorientation (high treating pressures). Clearly, the pressurization rate not only influences the breakdown pressure, but also the fracturing mechanisms and the eventual permeability of the induced fractures. By defining a dimensionless pressurization rate, Detournay and Cheng7 proposed a breakdown pressure formula that included the effects of the pressurization rate, the diffusion distance and the microfracture length. It is interesting to point out that all three formulae were based on continuum mechanics, while the hydraulic fracturing is definitely a fracture mechanics phenomenon. In addition, most above-mentioned experiments were carried out using constant pressurization rate, while hydraulic fracturing treatments are usually injection rate controlled. Even though it is true that the higher the injection rate, the higher the pressurization rate, the two are not linearly correlated8.
TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractA series of laboratory experiments were conducted to investigate the influence of overburden and in-situ stresses on non-Darcy gas flow behavior in Dakota sandstone. Nitrogen was flooded through cylindrical core in a triaxial core holder under specific condition of temperature at 100ºF, with core outlet pore pressure at 500 psi, axial and radial stress from 2,000 to 10,000 psi, and. nitrogen reservoir pump pressure at 2,000 psi with pump flow rates from 25 to 10,000 cc/hr at 80ºF. Permeability and non-Darcy coefficient were determined using Forchheimer's method. It was found that with the increase of overburden and in-situ stresses, permeability decreases while non-Darcy flow coefficient increases. Average effective normal stress and shear stress were used to quantitatively express the influence of overburden and in-situ stresses. It was found that average effective normal stress has a good linear relationship with both permeability and non-Darcy flow coefficient. In contrast, average shear stress did not appear to influence the permeability and non-Darcy coefficient.
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