Gassmann Relations Research Articles

Gassmann's equation and fluid‐saturation effects on seismic velocities

Gassmann's (1951) equations commonly are used to predict velocity changes resulting from different pore‐fluid saturations. However, the input parameters are often crudely estimated, and the resulting estimates of fluid effects can be unrealistic. In rocks, parameters such as porosity, density, and velocity are not independent, and values must be kept consistent and constrained. Otherwise, estimating fluid substitution can result in substantial errors. We recast the Gassmann's relations in terms of a porosity‐dependent normalized modulus Kn and the fluid sensitivity in terms of a simplified gain function G. General Voigt‐Reuss bounds and critical porosity limits constrain the equations and provide upper and lower bounds of the fluid‐saturation effect on bulk modulus. The “D” functions are simplified modulus‐porosity relations that are based on empirical porosity‐velocity trends. These functions are applicable to fluid‐substitution calculations and add important constraints on the results. More importantly, the simplified Gassmann's relations provide better physical insight into the significance of each parameter. The estimated moduli remain physical, the calculations are more stable, and the results are more realistic.

Velocity‐porosity relationships, 1: Accurate velocity model for clean consolidated sandstones

We use numerical simulations to derive the elastic properties of model monomineralic consolidated sandstones. The model morphology is based on overlapping spheres of a mineral phase. We consider model quartzose and feldspathic sands. We generate moduli‐porosity relationships for both the dry and water‐saturated states. The ability to control pore space structure and mineralogy results in numerical data sets which exhibit much less noise than corresponding experimental data. The numerical data allows us to quantitatively analyze the effects of porosity and the properties of the mineral phase on the elastic properties of porous rocks. The agreement between the numerical results and available experimental data for clean consolidated sandstones is encouraging.We compare our numerical data to commonly used theoretical and empirical moduli‐porosity relationships. The self‐consistent method gives the best theoretical fit to the numerical data. We find that the empirical relationship of Krief et al. is successful at describing the numerical data for dry shear modulus and that the recent empirical method of Arns et al. gives a good match to the numerical data for Poisson's ratio or Vp/Vsratio of dry rock. The Raymer equation is the best of the velocity‐porosity models for the water‐saturated systems. Gassmann's relations are shown to accurately map between the dry and fluid‐saturated states.Based on these results, we propose a new empirical method, based solely on a knowledge of the mineral modulus, to estimate the full velocity‐porosity relationship for monomineralic consolidated sands under dry and fluid‐saturated states. The method uses the equation of Krief et al. for the dry shear modulus and the empirical equation of Arns et al. for the dry Poisson's ratio. Gassmann's relations are applied to obtain the fluid‐saturated states. The agreement between the new empirical method, the numerical data and available experimental data for dry and water‐saturated states is encouraging.

Reflection and transmission of waves at a fluid/porous‐medium interface

We study the wave properties at a fluid/porous‐medium interface by using newly derived closed‐form expressions for the reflection and transmission coefficients. We illustrate the usefulness of these relatively simple expressions by applying them to a water/porous‐medium interface (with open‐pore or sealed‐pore boundary conditions), where the porous medium consists of (1) a water‐saturated clay/silt layer, (2) a water‐saturated sand layer, (3) an air‐filled clay/silt layer, or (4) an air‐filled sand layer. We observe in the frequency range 5 Hz–20 kHz that the fast P‐wave and S‐wave velocities in the four porous materials are indistinguishable from the corresponding frequency‐independent ones calculated using Gassmann relations. Consequently, for these frequencies we would expect the reflection and transmission coefficients for the four water/porous‐medium interfaces to be similar to the ones for corresponding interfaces between water and effective elastic media (described by Gassmann wave velocities). This expectation is not fulfilled in the case of an interface between water and an air‐filled porous layer with open pores. A close examination of the expressions for the reflection and transmission coefficients shows that this unexpected result is because of the large density difference between water and air.

The effect of fluid pressure on wave speeds in a cracked solid

In a porous or cracked elastic solid, the effective stress (defined in terms of the loads applied to the solid part of the outer boundary) and effective strain (defined in terms of the displacements at the solid part of the outer boundary) occurring in small-amplitude deformations are connected by a linear relation along with the pressure within the fluid occupying the pores and cracks. We derive here a formula of this kind for a static system in which enough time is allowed for pressure to be equalized throughout the fluid (on the assumption that all pockets of fluid are connected in some way). The formula depends on the overall stiffnesses relating stress to strain for the same material with the fluid removed (dry or empty cracks and pores). For undrained conditions where no fluid is allowed to enter or leave the body, the pressure is directly related to the effective stress and strain, and the Gassmann relations are obtained relating the stiffnesses for an isotropic material in dry and undrained conditions. For an anisotropic material, the Brown–Korringa relations are recovered. Externally imposed stresses and fluid pressure distort the material structure and influence the wave speeds of elastic waves. The main way in which this occurs is in changing the aspect ratios of flat cracks, the most compliant part of the microstructural geometry. This effect on the wave speeds is studied here both in terms of crack closure, with corresponding changes in crack number density, and in variations in crack aspect ratios. The principal way in which the latter influences the wave speeds is through the fluid incompressibility factor in the formula for the properties of materials with connected cracks. An increase in aspect ratio of the cracks is equivalent to a reduction in the bulk modulus of the fluid. This effect is apparent in the limits of both high frequencies, when the material behaves as if the cracks were isolated, and low frequencies, when undrained conditions apply.

Seismic pore space compressibility and Gassmann's relation: G. Mavko & T. Mukerji, Geophysics, 60(6), 1995, pp 1743–1749

Seismic pore space compressibility and Gassmann's relation: G. Mavko & T. Mukerji, Geophysics, 60(6), 1995, pp 1743–1749

Exact results for generalized Gassmann’s equations in composite porous media with two constituents

Wave propagation in fluid‐filled porous media is goverened by Bilot's equations of poroelasticity. Gassmann's relation gives an exact formula for the poroelastic parameters when the porous contains only one type of solid constituent. The present paper generalizes Gassmann's relation and derives exact formulas for two elastic parameters needed to describe wave propagation in a conglomerate of two porous phases. The parameters were first introduced by Brown and Korringa when they derived a generalized form of Gassmann's equation for conglomerates. These elastic parameters are the bulk modulus [Formula: see text] associated with changes in the overall volume of the conglomerate and the bulk modulus [Formula: see text] associated with the pore volume when the fluid pressure [Formula: see text] and confining pressure (p) are increased, keeping the differential pressure [Formula: see text] fixed. These moduli are properties of the composite solid frame (drained of fluid) and are shown here to be completely determined in terms of the bulk moduli associated with the two solid constituents, the bulk moduli of the drained conglomerate and the drained phases, and the porosities in each phase. The pore structure of each phase is assumed uniform and smaller than the grain size in the conglomerate. The relations found are completely independent of the pore microstructure and provide a means of analyzing experimental data. The key idea leading to the exact results is this: Whenever two scalar fields (in our problem [Formula: see text] and [Formula: see text]) can be independently varied in a linear composite containing only two constituents, there exists a special value γ of the increment ratio for these two fields corresponding to an overall expansion or contraction of the medium with no change of relative shape. This fact guarantees that a set of consistency relations exists among the constituent moduli and the effective moduli, which then determines all but one of the effective constants. Thus, [Formula: see text] and [Formula: see text] are determined in terms of the drained frame modulus K and the constituents’ moduli. Because the composite is linear, the coefficients found for the special value of the increment ratio are also the exact coefficients for an arbitrary ratio. Since modulus K is commonly measured while the other two are not, these exact relations provide a significant advance in our ability to predict the response of porous materials to pressure changes. It is also shown that additional results (such as rigorous bounds on the parameters) may be easily obtained by exploiting an analogy between the equations of thermoelasticity and those of poroelasticity. The method used to derive these results may also be used to find exact expressions for three component composite porous materials when thermoelastic constants of the components and the composite are known.