Pore Fluid Compressibility Research Articles

On the Influence of the Matrix Compressibility in Two‐Phase Models

AbstractIn the theory of porous media the volume fractions are introduced as internal variables. Therefore, additional evolution equations for the volume fractions are required an general. In the present contribution, a binary model consisting of a compressible porous skeleton saturated by a compressible pore fluid is formulated. From this general model, the so‐called hybrid model and the incompressible model are derived. The different behavior of the models is illustrated in two examples.

Consolidation of Anisotropic Soil Deposits

A numerical procedure is presented to analyse consolidation problems involving anisotropic layered soils which contain incompressible as well as compressible pore fluid. The governing differential equations are reduced to ordinary differential equations by applying a Fourier transform along with a Laplace transform, and the solution is found in the finite element fashion without great computational efforts. The soil deposit is discretized by one-dimensional elements, and for each element, use is made of quadratic interpolating functions that implicitly enforce interelement continuity. A number of comparisons with existing solutions are shown in order to assess the accuracy of the present method. Finally, a parametric study is carried out to investigate the influence of elastic anisotropy on the consolidation processes caused by surface loading.

Theory for the effects of free gas in subsea formations on tidal pore pressure variations and seafloor displacements

Loading of the seafloor by regional‐scale pressure variations, such as those imposed by ocean tides, is supported by both the rock matrix and interstitial fluid. The nature of the partitioning of the support between the two depends primarily on the compressibility of the fluid and the compressibility and fluid‐transport properties of the rock matrix. In this paper, we examine theoretically the influence of free gas on pore fluid compressibility, on the nature of time‐dependent load partitioning, and on the consequent vertical rock deformation and seafloor displacement. An example is the gas trapped below deep‐sea gas hydrate. We have derived an expression for the steady state compressibility of pore fluid considering the influence of gas solubility in water. The effect of gas solubility is seen to be important at low, such as tidal, loading frequencies and thus must be included when observations of tidally induced pore fluid pressure variations or seafloor displacements are used to constrain gas content. For very low gas concentrations ng (much less than 0.1%), the steady state fluid compressibility can be enhanced by gas solution/dissolution over the loading cycle by several factors at high ambient pressure and more than an order of magnitude at low ambient pressures (< 5 MPa). At ng > 2%, the fluid compressibility increases sensitively with ng and greatly affects the tidal response of the pore fluid pressure regardless of the solubility. Thus, with careful experimental design, tidally induced pore pressure variations may be used to detect very small amounts of free gas and constrain the quantity if ng > 2%. This method is complementary to using acoustic velocity to constrain the quantity of free gas, which works well in the ng = 0.2–2% range. We have also given an expression for the vertical deformation of subsea formations and hence of the seafloor displacement under tidal loading. The presence of free gas enhances tidally induced seafloor displacement, but the maximum effect is limited by the compressibility of the matrix frame. Given relatively low frame compressibility, tidally induced seafloor displacement is small, of the order of 1 mm, which is presently difficult to detect at tidal frequencies.

Seismic image of a CO2reservoir beneath a seismically active volcano

Mammoth Mountain is a seismically active volcano 200 000 to 50 000 years old, situated on the southwestern rim of Long Valley caldera, California. Since 1989 it has shown evidence of unrest in the form of earthquake swarms (Hill et al. 1990), volcanic ‘long-period’ earthquakes (Pitt & Hill 1994), increased output of magmatic 3He (Sorey et al. 1993) and the emission of about 500 tonnes day −1 of CO2 (Farrar et al. 1995; Hill 1996; M. Sorey, personal communication, 1997), which has killed trees and poses a threat to human safety. Local-earthquake tomography shows that in mid-1989 areas of subsequent tree-kill were underlain by extensive regions where the ratio of the compressional and shear elastic-wave speeds VP/VS was about 9 per cent lower than in the surrounding rocks. Theory (Mavko & Mukerji 1995), experiment (Ito, DeVilbiss & Nur 1979), and experience at other geothermal/volcanic areas (Julian et al. 1996) and at petroleum reservoirs (Harris et al. 1996) indicate that VP/VS is sensitive to pore-fluid compressibility, through its effect on VP. The observed VP/VS anomaly is probably caused directly by CO2, and seismic VP/VS tomography is thus a promising tool for monitoring gas concentration and movement in volcanoes, which may in turn be related to volcanic activity.

Time-average equation revisited

Expressions that relate velocity to porosity and to pore-fluid compressibility are among the most important deliverables of rock physics. Such relations are used often as additional controls for inferring porosity from well logs, as well as in-situ indicators of pore fluid type. The oldest and most popular is the Wyllie et al. (1956) equation: [Formula: see text]where [Formula: see text] is the measured traveltime of a P-wave, [Formula: see text] is the traveltime expected in the solid-phase material, and [Formula: see text] is the traveltime expected in the pore fluid. It follows from equation (1) that [Formula: see text]where ϕ is porosity, [Formula: see text] is the measured P-wave velocity, and [Formula: see text] and [Formula: see text] are the P-wave velocities in the solid and in the pore-fluid phases, respectively.

Effect of pore fluid compressibility on localization in elastic-plastic porous solids under undrained conditions

Conditions are derived for band-shaped localization, i.e. for the existence of discontinuous bifurcations of the incremental strain field, in elastic-plastic porous materials, under undrained conditions. It is assumed that the pores are filled with nearly incompressible liquid/gas mixture, and the significant parameter is the bulk modulus of the pore fluid. The extreme situation is complete incompressibility, which would correspond to full saturation of a liquid with an infinite bulk modulus. In this case the surprising result is obtained that, for plane strain and cylindrical stress states, the critical bifurcation direction is always 45° to a principal stress axis regardless of the adopted yield criterion. It is also shown in the paper that incompressibility usually (although not generally) has a stabilizing effect. For example, when the flow rule is associated, the critical hardening modulus is always non-positive. A discontinuity in strain rate is always accompanied by discontinuous pore pressurerate across the characteristic surface, and the magnitude of this jump is related to the strength of the displacement jump. One restrictive assumption used in the analysis is that the gradient of the yield criterion and the flow direction are coaxial.A numerical evaluation of the critical bifurcation direction and the corresponding hardening modulus is given for the Mohr-Coulomb criterion, which is representative for the behavior of cohesionless granular materials such as soil.

A finite element model for the analysis of wave induced stresses, displacements and pore pressures in an unsaturated seabed II: Model verification

This second in a series of papers presents the verification of the finite element model WISPP (Wave Induced Stresses, displacements and Pore Pressures in an unsaturated seabed). Model verification is performed by comparing the numerical solutions to specific problems with the results given by known analytical solutions. The verification problems include wave induced pore pressures, displacements and stresses in saturated soils with both infinite and finite depth, unsaturated soils where the gas can exist in the form of large bubbles surrounded by a saturated matrix or where the gas exists as small bubbles within the soil pores thus combining with the water to produce a compressible pore fluid. Further numerical simulations are made to illustrate the behaviour of wave spectra on a two-layered soil system. This condition can produce shear stresses significantly higher than those produced in homogeneous sediment.

Excess pore pressure in a poroelastic seabed saturated with a compressible fluid

This paper presents an analytical investigation on the excess pore-fluid pressure in a finite seabed layer by taking into account the influence of a compressible pore fluid. The seabed layer is modeled as a poroelastic layer saturated with a compressible pore fluid and resting on a rough, rigid impermeable base. The surface of the poroelastic seabed layer is either completely pervious or completely impervious, and subjected to a normal surface traction induced by offshore structures. The paper presents analytical and numerical results to illustrate the time-dependent behaviour of excess pore pressure in the poroelastic seabed. The results demonstrate that the presence of a compressible pore fluid reduces the generation of excess pore pressure in the poroelastic seabed layer. Key words : excess pore pressure, poroelastic seabed layer, soil consolidation, compressible pore fluid, integral transforms.

Pore pressure buildup coefficient in synthetic and natural sandstones

We present laboratory measurements of Skempton's pore pressure buildup coefficient B for synthetic and natural sandstones, and use our results to estimate the unjacketed pore compressibility 1/ K ø . Measured values of B are always near unity for differential pressures below ∼2 MPa, regardless of porosity, grain composition, or the presence of small amounts of clay in the samples. We observed B near 0.7–0.8 for differential pressures of about 8–21 MPa in a clay-free synthetic sandstone. Similar values are expected for natural sandstones with low crack densities, whereas sandstones containing greater concentrations of microcracks apparently have values of B near 0.6. Our results suggest that the pore compressibility may have values close to the pore-fluid compressibility rather than the grain compressibility.

An absorbing boundary condition for wave propagation in saturated poroelastic media — Part II: Finite element formulation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation towards infinity. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. In Part I (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 411-21), a local absorbing boundary condition for wave propagation in saturated poroelastic media has been developed. In the present paper, this boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid. Spurious reflections for oblique incident waves on the absorbing boundary contribute to the solution errors. Therefore, a spectral element method, based on classical analytical solution techniques, is used to assess the accuracy of the finite element formulation.

An absorbing boundary condition for wave propagation in saturated poroelastic media — Part I: Formulation and efficiency evaluation

In a finite element formulation for dynamic soil-structure interaction, an absorbing boundary condition is needed to model wave propagation in an unbounded problem domain. When the soil is saturated, its dynamic behaviour can be modelled by means of Biot's poroelastic theory. Based on an analytical study of wave propagation in a saturated poroelastic medium, a frequency-dependent local absorbing boundary condition can be obtained at the expense of spurious reflections for oblique incident waves. Reflection curves are presented for all types of incident wave and varying dimensionless frequencies. Alternatively, the effective energy ratio allows the definition of a measure for the overall efficiency of the absorbing boundary condition. Based on these analytical investigations, good wave-absorbing capabilities of the absorbing boundary condition have been revealed. Encouraged by these results, the absorbing boundary condition is implemented in an irreducible finite element formulation for a compressible pore fluid, as discussed in Part II (Degrande, G. & De Roeck, G., Soil Dynamics & Earthquake Eng., 1993, 12(7), 423-32).

Response of fractured rock subject to fluid injection Part II. Characteristic behaviour

Distinct element modelling has been used to investigate the mechanics of naturally fractured reservoirs subject to a constant rate of fluid injection. Constant stress boundary conditions were imposed, with magnitudes typical of those pertaining at approximately 3 km depth in some sedimentary environments. Fluid leak-off through fracture walls and fracture dilation during shear were modelled. The progressive nature of the changes of stress and displacement occurring within the coupled fluid-rock system were examined for four combinations of uniform fracture geometry and applied stress. The effects of a uniform change of pore pressure on the directional nature of hydraulic conductivity in a fractured medium were characterised. The influence of variation of injection rate, pore fluid compressibility and leak-off rate was examined. Effects associated with contrasts of shear stiffness between fractures and blocks, and with inelastic deformation, were interpreted.

Withdrawal of a compressible pore fluid from a point sink in an isotropic elastic half space with anistropic permeability : Booker, J R; Carter, J P Int J Solid StructV23, N3, March 1987, P369–385

Withdrawal of a compressible pore fluid from a point sink in an isotropic elastic half space with anistropic permeability : Booker, J R; Carter, J P Int J Solid StructV23, N3, March 1987, P369–385

Analysis of pumping a compressible pore fluid from a saturated elastic half space

A solution method is presented for the consolidation of a saturated, porous elastic half space due to the pumping of a pore fluid at a constant rate from a point sink embedded beneath the surface. It is assumed that the saturated medium is homogeneous and isotropic with respect to both its elastic and flow properties. The soil skeleton is modelled as an isotropic, linear elastic solid obeying Hooke's Law while the pore fluid is assumed to be compressible with its flow governed by Darcy's Law. The solution has been evaluated for a soil with a value of Poisson's ration of 0.25 and for a number of different cases of pore fluid compressibility. It is demonstrated that this compressibility can have a significant influence on the rate of consolidation around the sink. The solutions presented may have application in practical problems such as the extraction of groundwater and other fluids from compressible geological media.

Withdrawal of a compressible pore fluid from a point sink in an isotropic elastic half space with anisotropic permeability

The complete solution is presented for the transient effects of pumping fluid at a constant rate from a point sink embedded in a saturated, porous elastic half space. It is assumed that the medium is homogeneous and isotropic with respect to its elastic properties and homogeneous but anisotropic with respect to the flow of pore fluid. The soil skeleton is modelled as an isotropic linear elastic material obeying Hooke's law while the pore fluid may be compressible with its flow governed by Darcy's law. The solution has been evaluated for a particular value of Poisson's ratio of the solid skeleton, i.e. 0.25, and the results have been presented graphically in the form of isochrones of excess pore pressure and surface profile for the half space. The solutions presented may have application in practical problems such as dewatering operations in compressible soil and rock masses and in the extraction of petroleum products from the crust of the earth.

Effect of pore-fluid compressibility on failure loads in elastic-plastic soil : Griffiths, D V Int J Num Anal Meth GeomechV9, N3, May–June 1985, P253–259

Effect of pore-fluid compressibility on failure loads in elastic-plastic soil : Griffiths, D V Int J Num Anal Meth GeomechV9, N3, May–June 1985, P253–259

Scattering in glass beads: Effects of frame and pore fluid compressibilities

Ultrasonic compressional wave phase velocities and attenuation have been measured as functions of frequency in packings of unconsolidated glass beads. External confining pressure was varied so as to modify the physical interactions between grains. Samples were studied when vacuum dry and when saturated with different pore fluids. Pressure/pore fluid combinations were chosen so as to minimize nonscattering loss mechanisms. In each experiment, phase velocity exhibited negative dispersion and attenuation was proportional to the fourth power of frequency. Increasing confining pressure (to 15 MPa) significantly reduced the scattering losses (by an order of magnitude at a given frequency in dry glass beads). Decreasing pore fluid compressibility produced a corresponding decrease in the scattering losses. This work shows that theoretical models based on suspensions are not adequate when the grains are interacting to form a solid frame. The present data, along with previous data for fused glass beads, suggest that the scattering strength is related to the contrast between the acoustic properties of the composite and of the grains, rather than to pore/grain or pore/composite contrasts.

Effects of pressure and fluid saturant on ultrasonic scattering in glass beads

Compressional wave phase velocity and attenuation have been measured in packings of unconsolidated glass beads as functions of external confining pressure. Samples were studied when vacuum dry and when saturated with different pore fluids. Pressure/pore fluid combinations were chosen to minimize nonscattering loss mechanisms. Phase velocity exhibits negative velocity dispersion and attenuation is proportional to the fourth power of frequency. Increasing confining pressure (to 15 MPa) significantly reduces the scattering losses (by an order of magnitude at a given frequency in dry glass beads). Decreasing pore fluid compressibility produces a corresponding decrease in the scattering losses. This work shows that theoretical models based on suspensions are not adequate when the grains are interacting to form a solid frame. The present data, along with previous data for fused glass beads, suggest that the scattering strength is related to the contrast between the acoustic properties of the composite and of the grains.

Consolidation analysis for partly saturated clay by using an elastic–plastic effective stress–strain model

AbstractThe theory of consolidation is extended to partly saturated clay soils, and formulated for finite element analyses. This formulation couples the effects of both stress and flow. It takes account of variations of this permeability of the soil and compressibility of the pore fluid with changes in void ratio, and the non‐linear stress–strain behaviour of soil. The Cam Clay model is revised to model the stress–strain behaviour of compacted soils. The compressibility of pore fluid is derived using Boyle's Law and Henry's Law, taking into account the effect of surface tension. An empirical equation is developed for permeability of pore fluid. An example of settlement of a footing on partly saturated soil is described and discussed.

Wave-induced pore pressures and effective stresses in a porous bed

A general theory for the pore pressures and effective stresses induced in a porous bed by ocean waves is developed. The pore fluid as well as the soil skeleton are considered compressible and the flow in the porous bed is assumed governed by Darcy's Law for an anisotropic medium. The general solution methods for a homogeneous bed of arbitrary thickness as well as for a horizontally layered bed are presented. Explicit solutions for pore pressures and effective stresses are given for an infinite homogeneous bed. The nature of these solutions' dependency on relative compressibility of pore fluid and soil skeleton, soil permeability and hydraulic anisotropy is examined in detail and illustrated by example calculations. It is found that an extremely simple form of the general solution is appropriate in most cases when the bed consists of soils finer than medium sand. The geotechnical implications of the wave-induced effective stresses are discussed in some detail. It is found that storm waves may induce stresses in cohesionless soils as well as in normally consolidated clays approaching or exceeding failure conditions. Une théorie général est développée pour les pressions interstitielle et les contraintes effectives produites dans une couche poreuse par les vagues de l'océan. Le fluide du pore ainsi que le squelette du sol sont considérés comme compressibles et l'écoulement dans le lit poreux est gouverné par la Loi de Darcy pour un milieu anisotrope. Les méthodes de solution générale sont présentées pour un lit homogène d'une épaisseur arbitraire ainsi que pour un lit à couches horizontales. Des solutions explicites pour des pressions interstitielles et des contraintes effectives sont données pour un lit homogène infini. La nature de la relation de ces solutions avec la compressibilité relative du fluide du pore et du squelette du sol, la perméabilité du sol et l'anisotropie hydraulique est examinée en détail et illustrée par des exemples de calculs. On constate qu'une forme extrêmement simple de la solution générale est appropriée dans la plupart des cas où le lit est constitué par un sol plus fin qu'un sable moyen. Les conséquences géotechniques des contraintes effectives produites par les vagues sont présentées en détail. On constate que les vagues de tempête peuvent produire des contraintes dans des sols non cohérents aussi bien que dans des argiles normalement consolidées approchant ou dépassant les conditions de rupture.