Anthony F. Gangi

Variation of whole and fractured porous rock permeability with confining pressure

Phenomenological models have been devised to determine the variation with pressure of the permeability of whole and fractures porous rock. For whole porous rock, the permeability variation with pressure is based upon the Hertzian theory of deformation of spheres. This model gives a permeability variation with pressure given by k(P) = k0 {1 − C0[(P + P1)/P0]sol23}4 where k0 is the initial permeability of the loose-grain packing, C0 is a constant depending upon the packing (and is of the order of 2), P1 is the ‘equivalent pressure’ due to the cementation and permanent deformation of the grains and P0 is the effective elastic modulus of the grains (and is of the order of the grain material bulk modulus). The permeability variation of a fracture (or fractured rock) with confining pressure is determined by using a ‘bed of nails’ model for the asperities of the fracture. Its functional dependence is k(P) = k0 [1 − (P/P1)m]3 where ko is the zero pressure permeability, P1 is the effective modulus of the asperrities (and is of the order of one-tenth to one-hundredth of the asperity material bulk modulus) and m is a constant (0 < m < 1) which characterizes the distributions function of the asperity lengths. The above expression assumes a simple power-law variation for the asperity-length distribution. More complicated asperity-length distributions can be used, but the data quality and the fracture-to-fracture variability does not warrant the use of such distributions. A comparison of experimental data with the theoretical curves shows good correlation between the two and gives reasonable values for the constants k0, P1 and m.

Deriving the equations of motion for porous isotropic media

The equations of motion and stress/strain relations for the linear dynamics of a two‐phase, fluid/solid, isotropic, porous material have been derived by a direct volume averaging of the equations of motion and stress/strain relations known to apply in each phase. The equations thus obtained are shown to be consistent with Biot’s equations of motion and stress/strain relations; however, the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls. The stress/strain relations of the theory correspond to ‘‘quasistatic’’ stressing (i.e., inertial effects are ignored). It is demonstrated that using such quasistatic stress/strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size.

Permeability of Wilcox shale and its effective pressure law

The permeability of illite-rich shale from the Wilcox formation has been measured as a function of effective pressure for bedding-parallel flow of 1 M NaCl pore fluid. Permeability k decreases from ∼300×10−21 m2 to 3×10−21 m2 as effective pressure Pe is increased from 3 to 12 MPa; these values confirm that shales form effective barriers to fluid transport in sedimentary strata over extended geologic times. The variation of k with Pe for Wilcox shale is given by k = k0 [1 − (Pe/P1)m]3, where P1 = 19.3 (±1.6) MPa and m = 0.159 (±0.007). The value of k0 for Wilcox shale is of the order of 10−17 m2 and may vary among samples by as much as 70%. Effective pressure is given in terms of the external confining pressure Pc and internal pore pressure Pp by Pe = Pc − χPp, where χ = 0.99 (±0.06). While our measurements yield χ = ∼1 for shale with a clay content of ∼45%, others have reported χ values for clay-bearing sandstones that rise from ∼0.75 to 7.1 with increasing clay content (from 0 to 20%). The trends between χ and clay content revealed by these comparisons imply that the value of χ depends upon the relative distributions of compliant clay minerals and other stiffer minerals. These values of χ also suggest that effective pressures within interbedded sandstones and shales may differ, even at the same equilibrium Pc and Pp conditions.

Permeability of illite-bearing shale: 1. Anisotropy and effects of clay content and loading

direction relative to bedding, clay content (40–65%), and effective pressure Pe (2– 12 MPa). Permeability k is anisotropic at low Pe; measured k values for flow parallel to bedding at Pe = 3 MPa exceed those for flow perpendicular to bedding by a factor of 10, both for low clay content (LC) and high clay content (HC) samples. With increasing Pe, k becomes increasingly isotropic, showing little directional dependence at 10–12 MPa. Permeability depends on clay content; k measured for LC samples exceed those of HC samples by a factor of 5. Permeability decreases irreversibly with the application of Pe, following a cubic law of the form k = k0 [1 � (Pe/P1) m ] 3 , where k0 varies over 3 orders of magnitude, depending on orientation and clay content, m is dependent only on orientation (equal to 0.166 for bedding-parallel flow and 0.52 for flow across bedding), and P1 (18–27 MPa) appears to be similar for all orientations and clay contents. Anisotropy and reductions in permeability with Pe are attributed to the presence of crack-like voids parallel to bedding and their closure upon loading, respectively. INDEX TERMS: 5114 Physical Properties of Rocks: Permeability and porosity; 5139 Physical Properties of Rocks: Transport properties; 5112 Physical Properties of Rocks: Microstructure; 1832 Hydrology: Groundwater transport; KEYWORDS: permeability, shale, connected pore space

Primary Migration by Oil-Generation Microfracturing in Low-Permeability Source Rocks: Application to the Austin Chalk, Texas

Fracturing of low-permeability source rocks is induced by pore-pressure changes caused by the conversion of organic matter to less dense fluids (oil and gas); these fractures increase the permeability and provide pathways for hydrocarbon migration. An equation for the pressure change is derived using four major assumptions. (1) The permeability of the source rock is negligibly small (0.01 µd; 10-20 m2) so that the pore-pressure buildup by the conversion is much faster than its dissipation by pore-fluid flow. (2) The stress state is isotropic so that horizontal and vertical stresses are equal. The source rock fails when the pore pressure equals the overburden pressure. (3) The properties of the rock, organic matter, and fluids remain constant during oil generation. This assumption is valid when the change in depth (i.e., pressure and temperature) is small. (4) Only two reaction rates are required for the conversions, a low-temperature reaction rate for the kerogen/oil conversion (E approx. = 24 kcal/mol, A approx. = 1014/m.y.) and a high-temperature reaction rate for oil/gas conversion (E approx. = 52 kcal/mol, A approx. = 5.5 ´ 1026/m.y.). The equations for generation rate and pressure change are applied to the Austin source rock by adjusting the several variables to fit geochemical data, core saturations, and observed levels of oil and gas production. This application demonstrates that the equations are easily applied in calculating depths of primary migration for low-permeability source rocks.

An asperity-deformation model for effective pressure

Abstract Variations of the mechanical and transport properties of cracked and/or porous rocks under isotropic stress depend on both the confining pressure ( P c ) and the pore-fluid pressure ( P p ). To a first approximation, these rock properties are functions of the differential pressure, P d = P c − P p ; at least for low differential pressures. However, at higher differential pressures, the properties depend in a more complicated way upon the two pressures. The concept of effective pressure, P e , is used to denote this variation and it is defined as P e ( P c , P p ) = P c − n ( P c , P p ) P p . If n = 1 (and therefore, is independent of P c and P p ), the effective pressure is just the differential pressure. We have used an asperity-deformation model and a force-balance equation to derive expressions for the effective pressure. We equate the total external force (in one direction), F c , to the total force on the asperities, F a , and the force of the fluid, F p , acting in that same direction. The fluid force, F p , acts only on the parts of the crack (or pore-volume) faces which are not in contact. Then, the asperity pressure, P a , is the average force per unit area acting on the crack (or grain) contacts P a = F a /A=F c /A−F p /A= P c − (1 −A c /A)P p , where A is the total area over which F c acts and A c is the area of contact of the crack asperities or the grains. Thus, the asperity pressure, P a , is greater than the differential pressure, P d , because P p acts on a smaller area, A − A c , than the total area, A . For elastic asperities, the area of contact A c and the strain (e.g., crack and pore openings) remain the same, to a high degree of approximation, at constant asperity pressure. Therefore, transport properties such as permeability, resistivity, thermal conductivity, etc. are constant, to the same degree of approximation, at constant asperity pressure. For these properties, the asperity pressure is, very accurately, the effective pressure, P c . Using this model, we find that the dynamic (undrained) elastic modulus ( M cr ) of saturated cracks (rocks) at low effective pressure is given by M cr = (1 − P p A′ f )M a + (1 − A f )M f = − w dP c /d w , where M a is the (dry-matrix or crack-) asperity modulus, M f is the fluids modulus, A f is the fractional area of contact ( A f = A c / A ), A ′ f =d A f /d P a and w is a measure of the crack or pore openings. This simple model accounts for the dependence of the rock modulus (and elastic velocity) on: (1) the elastic properties of the fluid, (2) the elastic properties of the dry rock, and (3) the pore-fluid and confining pressures. Explicit expressions depend upon the choice of the asperity- (or grain-) deformation models and their contact distribution functions. The effective pressures for transport properties are different than the ‘effective pressure’ for the mechanical properties. Calculated results based on the ‘bed-of-nails’ model having power-law (or fractal) asperity-height distribution functions can be fitted quite well to experimental data with a minimum of fitting parameters.

Transmission and reflection of Rayleigh waves by wedges

Experimental observations have been made of the transmission and reflection of Rayleigh waves by wedges. Results are reported for Rayleigh waves in aluminum wedges. It is observed that the wave shapes of the transmitted and reflected waves differ from that of the incident wave and depend on the angle of the wedge. The change of shape is attributed to an interference between part of the incident wave‐form and the radiation from a line source placed at the vertex. A procedure is given for the calculation of the partition between the two terms.

Surface amplitudes of reflected body waves

The components of the surface motions of a plane free surface are computed for the incidence of plane body waves.

Gas generation and overpressure: Effects on seismic attributes

Drilling of deep gas resources is hampered by high risk associated with unexpected overpressure zones. Knowledge of pore pressure using seismic data, as for instance from seismic-while-drilling techniques, will help producers plan the drilling process in real time to control potentially dangerous abnormal pressures. We assume a simple basin-evolution model with a constant sedimentation rate and a constant geothermal gradient. Oil/gas conversion starts at a given depth in a reservoir volume sealed with faults whose permeability is sufficiently low so that the increase in pressure caused by gas generation greatly exceeds the dissipation of pressure by flow. Assuming a first-order kinetic reaction, with a reaction rate satisfying the Arrhenius equation, the oil/gas conversion fraction is calculated. Balancing mass and volume fractions in the pore space yields the excess pore pressure and the fluid saturations. This excess pore pressure determines the effective pressure, which in turn determines the skeleton bulk moduli. If the generated gas goes into solution in the oil, this effect does not greatly change the depth and oil/gas conversion fraction for which the hydrostatic pressure approaches the lithostatic pressure. The seismic velocities versus pore pressure and differential pressure are computed by using a model for wave propagation in a porous medium saturated with oil and gas. Moreover, the velocities and attenuation factors versus frequency are obtained by including rock-frame/fluid viscoelastic effects to match ultrasonic experimental velocities. For the basin-evolution model used here, pore pressure is seismically visible when the effective pressure is less than about 15 MPa and the oil/gas conversion is about 2.5% percent.

Transient and steady-state deformation of synthetic rocksalt

Abstract Synthetic rocksalt with a porosity less than 2.5% and an average grain size of about 0.35 mm was made by warm-pressing at 100°C, 150 MPa and for 15 min. Cylinders of this material, 25 mm in diameter by 50 mm long, were deformed at strain rates of 0.1 ksec −1 at confining pressures of 20, 50, 100 and 200 MPa and at temperatures of 100, 200 and 300°C. The resulting stress-time data show transient-stress behavior before steady-state stress occurs. Very little variation in the stress/time data occurs for the above confining pressures at a constant temperature. Many of the tests reach steady state at 10% strain, where all the experiments were terminated. The differential stress at 10% varies from about 22 MPa at 100°C to about 6 MPa at 300°C. These “strengths” are slightly less than those measured by Heard (1972) (also on synthetic, polycrystalline rocksalt) and are similar to those measured by Wawersik and Hannum (1980) and by Hansen and Mellegard (1980) in modified creep tests on coarse-grained, natural rocksalt under similar pressure and temperature conditions. Activation energies computed from these steady-state stresses vary between 7.5 and 25 kcal/mole and are consistent with those obtained by Heard (1972) and by Parrish and Gangi (1977, 1981). The stress/time data were fit using one- and two-mechanism, transient-stress functions which assume independent mechanisms or processes. The rms errors decrease from about 1 MPa for the one-mechanism fits to about 0.2 MPa for the two-mechanism fits, indicating at least two mechanisms are operative in these tests. Similar one- and two-mechanism fits were made to creep tests performed by W. Wawersik et al. on New Mexico bedded rocksalt. Similar improvements in the fits were obtained for those tests that lasted long enough so that the effect of a second mechanism could be noted. It was found that the “steady-state” strain rate found in creep tests could be interpreted as the beginning of another mechanism. This raises the question of whether the “steady-state” phenomena exist at all or whether it is just an approximation to a mechanism with a time constant that is long compared to the length of the test.