Fluid-saturated Porous Rocks Research Articles

Numerical modelling of fluids mixing, heat transfer and non-equilibrium redox chemical reactions in fluid-saturated porous rocks

Non-equilibrium redox chemical reactions of high orders are ubiquitous in fluid-saturated porous rocks within the crust of the Earth. The numerical modelling of such high-order chemical reactions becomes a challenging problem because these chemical reactions are not only produced strong non-linear source/sink terms for reactive transport equations, but also often coupled with the fluids mixing, heat transfer and reactive mass transport processes. In order to solve this problem effectively and efficiently, it is desirable to reduce the total number of reactive transport equations with strong non-linear source/sink terms to a minimum in a computational model. For this purpose, the concept of the chemical reaction rate invariant is used to develop a numerical procedure for dealing with fluids mixing, heat transfer and non-equilibrium redox chemical reactions in fluid-saturated porous rocks. Using the proposed concept and numerical procedure, only one reactive transport equation, which is used to describe the distribution of the chemical product and has a strong non-linear source/sink term, needs to be solved for each of the non-equilibrium redox chemical reactions. The original reactive transport equations of the chemical reactants with strong non-linear source/sink terms are turned into the conventional mass transport equations of the chemical reaction rate invariants without any non-linear source/sink terms. A testing example, for some aspects of which the analytical solutions are available, is used to validate the proposed numerical procedure. The related numerical solutions have demonstrated that (1) the proposed numerical procedure is useful and applicable for dealing with the coupled problem between fluids mixing, heat transfer and non-equilibrium redox chemical reactions of high orders in fluid-saturated porous rocks; (2) the interaction between the solute diffusion, solute advection and chemical kinetics is an important mechanism to control distribution patterns of chemical products in an ore-forming process; and (3) if the pore-fluid pressure gradient is lithostatic, it is difficult for the chemical equilibrium to be attained within permeable cracks and geological faults within the crust of the Earth. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear pressure and temperature waves propagation in fluid-saturated rocks

A numerical study for the simulation of rock deformation due to nonlinear temperature and pressure waves in fluid saturated porous rock is presented. The problem of an homogeneous, thermoelastic, and isotropic fluid-saturated matrix, lying over an aquifer in hyperthermal domain an limited by an upper free surface is considered. The equations of thermo-poro-elasticity, the energy balance and Darcy equation are employed to model the deformation induced by pressure and temperature gradients originating from the source at depth. The numerical approach adopted to solve the nonlinear problem is based on a cubic finite-element method for the spatial discretization and an implicit scheme for the temporal discretization. This work essentially provides a numerical tool for interpreting some geophysical phenomena, in particular for the study of ground uplift in volcanic areas such as Rabaul caldera, Long Valley caldera, and Campi Flegrei. Some experiments and results are presented.

Electrical Properties of Crustal and Mantle Rocks – A Review of Laboratory Measurements and their Explanation

The electrical properties of rocks and minerals are controlled by thermodynamic parameters like pressure and temperature and by the chemistry of the medium in which the charge carriers move. Four different charge transport processes can be distinguished. Electrolytic conduction in fluid saturated porous rocks depends on petrophysical properties, such as porosity, permeability and connectivity of the pore system, and on chemical parameters of the pore fluid like ion species, its concentration in the pore fluid and temperature. Additionally, electrochemical interactions between water dipoles or ions and the negatively charged mineral surface must be considered. In special geological settings electronic conduction can increase rock conductivities by several orders of magnitude if the highly conducting phases (graphite or ores) form an interconnected network. Electronic and electrolytic conduction depend moderately on pressure and temperature changes, while semiconduction in mineral phases forming the Earth’s mantle strongly depends on temperature and responds less significantly to pressure changes. Olivine exhibits thermally induced semiconduction under upper mantle conditions; if pressure and temperature exceed ~ 14 GPa and 1400 °C, the phase transition olivine into spinel will further enhance the conductivity due to structural changes from orthorhombic into cubic symmetry. The thermodynamic parameters (temperature, pressure) and oxygen fugacity control the formation, number and mobility of charge carriers. The conductivity temperature relation follows an Arrhenius behaviour, while oxygen fugacity controls the oxidation state of iron and thus the number of electrons acting as additional charge carriers. In volcanic areas rock conductivities may be enhanced by the formation of partial melts under the restriction that the molten phase is interconnected. These four charge transport mechanisms must be considered for the interpretation of geophysical field and borehole data. Laboratory data provide a reproducible and reliable database of electrical properties of homogenous mineral phases and heterogenous rock samples. The outcome of geoelectric models can thus be enhanced significantly. This review focuses on a compilation of fairly new advances in experimental laboratory work together with their explanation.

Time-Dependent Diffusion and Surface-Enhanced Relaxation in Stochastic Replicas of Porous Rock

Understanding the connection between pore structure and NMR behavior of fluid-saturated porous rock is essential in interpreting the results of NMR measurements in the field or laboratory and in establishing correlations between NMR parameters and petrophysical properties. In this paper we use random-walk simulation to study NMR relaxation and time-dependent diffusion in 3D stochastic replicas of real porous media. The microstructures are generated using low-order statistical information (porosity, void–void autocorrelation function) obtained from 2D images of thepore space. Pore size distributions obtained directly by a 3D pore space partitioning method and indirectly by inversion of NMR relaxation data are compared for the first time. For surface relaxation conditions typical of reservoir rock, diffusional coupling between pores of different size is observed to cause considerable deviations between the two distributions. Nevertheless, the pore space correlation length and the size of surface asperity are mirrored in the NMR relaxation data for the media studied. This observation is used to explain the performance of NMR-based permeability correlations. Additionally, the early time behavior of the time-dependent diffusion coefficient is shown to reflect the average pore surface-to-volume ratio. For sufficiently high values of the self-diffusion coefficient, the tortuosity of the pore space is also recovered from the long-time behavior of the time-dependent diffusion coefficient, even in the presence of surface relaxation. Finally, the simulations expose key limitations of the stochastic reconstruction method, and allow suggestions for future development to be made.

Effects of hot intrusions on pore-fluid flow and heat transfer in fluid-saturated rocks

We use the finite element method to solve coupled problems between pore-fluid flow and heat transfer in fluid-saturated porous rocks. In particular, we investigate the effects of both the hot pluton intrusion and topographically driven horizontal flow on the distributions of the pore-flow velocity and temperature in large-scale hydrothermal systems. Since general mineralization patterns are strongly dependent on distributions of both the pore-fluid velocity and temperature fields, the modern mineralization theory has been used to predict the general mineralization patterns in several realistic hydrothermal systems. The related numerical results have demonstrated that: (1) The existence of a hot intrusion can cause an increase in the maximum value of the pore-fluid velocity in the hydrothermal system. (2) The permeability of an intruded pluton is one of the sensitive parameters to control the pore-fluid flow, heat transfer and ore body formation in hydrothermal systems. (3) The maximum value of the pore-fluid velocity increases when the bottom temperature of the hydrothermal system is increased. (4) The topographically driven flow has significant effects on the pore-fluid flow, temperature distribution and precipitation pattern of minerals in hydrothermal systems. (5) The size of the computational domain may have some effects on the pore-fluid flow and heat transfer, indicating that the size of a hydrothermal system may affect the pore-fluid flow and heat transfer within the system.

Finite Element Modelling of Three-Dimensional Steady-State Convection and Lead/Zinc Mineralization in Fluid-Saturated Rocks

We use the finite element method to model three-dimensional steady-state convective pore-fluid flow and the relevant mineralization in fluid-saturated rocks when they are heated from below. In particular, we investigate lead/zinc precipitation and dissolution patterns caused by the reactive sulfide and sulfate fluids mixing in three-dimensional hydrothermal systems. The related numerical results demonstrated that: (1) The finite element method is useful in and applicable to the modelling of three-dimensional ore body formation and mineralization problems in fluid-saturated porous rocks; (2) The mixing of reactive sulfide and sulfate fluids is an important mechanism to generate high-grade lead/zinc deposits in three-dimensional hydrothermal systems in the upper crust of the Earth.

Stand for Investigation of Nonlinear Interaction between Physical Fields and Oil-Gas Stratum

The principal features are considered for the laboratory stand developed in the Institute of Mining of the Siberian Branch of the Russian Academy of Sciences. The stand is intended for testing the complex methods of wave action by physical fields on the productive strata of oil-and-gas deposits and investigating the nonlinear processes in fluid-saturated porous and fractured rocks.

Finite element modelling of temperature gradient driven rock alteration and mineralization in porous rock masses

We present finite element simulations of temperature gradient driven rock alteration and mineralization in fluid saturated porous rock masses. In particular, we explore the significance of production/annihilation terms in the mass balance equations and the dependence of the spatial patterns of rock alteration upon the ratio of the roll over time of large scale convection cells to the relaxation time of the chemical reactions. Special concepts such as the gradient reaction criterion or rock alteration index (RAI) are discussed in light of the present, more general theory. In order to validate the finite element simulation, we derive an analytical solution for the rock alteration index of a benchmark problem on a two-dimensional rectangular domain. Since the geometry and boundary conditions of the benchmark problem can be easily and exactly modelled, the analytical solution is also useful for validating other numerical methods, such as the finite difference method and the boundary element method, when they are used to deal with this kind of problem. Finally, the potential of the theory is illustrated by means of finite element studies related to coupled flow problems in materially homogeneous and inhomogeneous porous rock masses.

Analysis of pore-fluid pressure gradient and effective vertical-stress gradient distribution in layered hydrodynamic systems

A theoretical analysis is carried out to investigate the pore-fluid pressure gradient and effective vertical-stress gradient distribution in fluid saturated porous rock masses in layered hydrodynamic systems. Three important concepts, namely the critical porosity of a porous medium, the intrinsic Fore-fluid pressure and the intrinsic effective vertical stress of the solid matrix, are presented and discussed. Using some basic scientific principles, we derive analytical solutions and explore the conditions under which either the intrinsic pore-fluid pressure gradient or the intrinsic effective vertical-stress gradient can be maintained at the value of the lithostatic pressure gradient. Even though the intrinsic pore-fluid pressure gradient can be maintained at the value of the lithostatic pressure gradient in a single layer, it is impossible to maintain it at this value in all layers in a layered hydrodynamic system, unless all layers have the same permeability and porosity simultaneously. However, the intrinsic effective vertical-stress gradient of the solid matrix can be maintained at a value close to the lithostatic pressure gradient in all layers in any layered hydrodynamic system within the scope of this study.

The attenuation of stress waves in fluid saturated porous rock

The dynamic mechanical frequency spectrum and temperature spectrum measurements of dry and saturated sandstone with three different porosity are conducted by use of the viscoelastic spectrum instrument in the 0.01–100 Hz frequency region. The frequency responses of the attenuation and modulus at different temperature peaks are obtained. With increase of the porosity and the loss of the complex modulus, the attenuation in the saturated sandstones is increased, and the frequency dispersion is enhanced. The relation between the frequency spectrum and the temperature spectrum are also discussed.

On the theory of pressure and temperature nonlinear waves in compressible fluid-saturated porous rocks

Thermo-poro-elastic equations describing fluid migration through fluid-saturated porous media at depth in the crust are analyzed theoretically following recent formulations of Rice and Cleary (1976), McTigue (1986) and Bonafede (1991). In this study these ideas are applied to a rather general model, namely to a deep hot and pressurized reservoir of fluid, which suddenly enters into contact with an overlaying large colder fluid-saturated layer. In a one-dimensional idealization this system can be described by two nonlinear differential heat-like equations on the matrix-fluid temperature and on the fluid overpressure over the hydrostatic value. The nonlinear couplings are due to Darcy thermal advection and to the mechanical work rate. Here we first sketch nonlinear solutions corresponding to Burgers' “solitary shock waves”, which have recently been found valid for rocks with very low fluid diffusivity. Subsequently other nonlinear transient waves are discussed, such as “thermal” and “compensated” waves, which are found to exist for every value of the parameters present in the equations involved. One interesting aspect of these mechanisms is that the resulting time-scales are particularly small. Moreover, in order to figure out the system time-evolution and the role played by the fluid diffusivity/thermal diffusivity ratio, a mechanical similitude is proposed, which we treat both analytically and numerically. Although for realistic systems these solutions are somewhat idealized, they allow one to gain fundamental insight into fluid migration mechanisms in volcanic areas and in fault regions under strong frictional heating. As already discussed by McTigue, the theory is also of interest in studying areas of nuclear waste disposal. Furthermore such a theoretical study allows one to investigate the site at depth at which such nonlinear waves are generated.

Transient solutions for temperature and pressure waves in fluid-saturated porous rocks: G. Natale & E. Salusti, Geophysical Journal International, 124(3), 1996, pp 649–656

Transient solutions for temperature and pressure waves in fluid-saturated porous rocks: G. Natale & E. Salusti, Geophysical Journal International, 124(3), 1996, pp 649–656

Transient solutions for temperature and pressure waves in fluid-saturated porous rocks

SUMMARY In this paper, hot-fluid migration dynamics for a fluid-saturated porous rock is analysed theoretically. We first recall a Bonafede model implemented with a classical scale analysis; this allows us to discuss a fully non-linear formulation related to transient phenomena, which is solved analytically as a solitary shock wave. The resulting time-scales are particularly small, corresponding to intense short-time geothermal phenomena. A novel non-linear migration mechanism, related to Darcy convection, is also discussed.

Thermal conductivity of Hawaiian basalt: a new interpretation of Robertson and Peck's data : J Geophys Res V96, NB3, March 1991, P4125–4132

Robertson and Peck's data published in 1974 on the thermal conductivity of Hawaiian basalts are unique in the sense that the vesicularity of the samples used for the study covers a wide range of 2 to 98%. The data consist of the measurement results of thermal conductivity on 61 samples of basalt collected from Kilauea and other volcanoes in Hawaii. The thermal conductivity was measured, by the method of thermal stacking, on samples with their pores filled first with air and then with water. Thus the data are particularly suitable for evaluating the theoretical formula describing the thermal conductivity of fluid-saturated porous rock. In the formula for the thermal conductivity of a two-phase material the thermal conductivity of the solid matrix is an essential parameter, which Robertson and Peck determined by the wet cell method. The result of their determination was that they could not represent the data in the whole range of sample porosities with a single theoretical formula. This paper proposes that the thermal conductivity of the solid matrix can be estimated by extrapolating the data to zero porosity. For the extrapolation, either Fricke-Zimmerman's or Schulz's formula of thermal conductivity for a two-phase material having spheroidal inclusions can be used. In both models, for the results of extrapolation from the air-saturated and the water-saturated sample thermal conductivities to be identical, the spheroid's aspect ratio is assumed to be around 1/10. This is considered to be a combined effect of the spherical pores and the microcracks in the basalt samples. With the assumed aspect ratio of the spheroidal inclusions, either of the theoretical formulae agree with the data with sufficient precision.

Wave propagation through elastically-anisotropic fluid-saturated porous rocks : Digby, P J; Walton, K J Appl Mech (Trans ASME Ser E)V56, N4, Dec 1989, P744–750

Wave propagation through elastically-anisotropic fluid-saturated porous rocks : Digby, P J; Walton, K J Appl Mech (Trans ASME Ser E)V56, N4, Dec 1989, P744–750

Wave propagations through fluid saturated porous rocks : Walton, K; Digby, P J J Appl Mech (Trans ASME Ser E)V54, N4, Dec 1987, P788–793

Wave propagations through fluid saturated porous rocks : Walton, K; Digby, P J J Appl Mech (Trans ASME Ser E)V54, N4, Dec 1987, P788–793

Wideband acoustic response of fluid-saturated porous rocks: Theory and preliminary results using waveguided samples

The complex dilatational (C*12) and shear (C*44) frequency-dependent elastic constants of nonporous and porous solids are measured in a frequency range 5kHz to 1 MHz. The measurements are performed in a continuous wave acoustic transmission bridge using cylindrical samples. Using samples with free external surfaces produces waveguided modes, enabling experiments to be conducted circumventing problems of diffraction and wavefront spreading, which are difficult to correct over broad frequency ranges. The theory of waveguided elastic wave propagation in isotropic media with complex elastic constants [generalization of the Pochammer–Cree solution, Pochammer, J. Math. (Crelle) 81, 324 (1876)] is presented. For solids with frequency-independent complex elastic constants (constant Q), solution of the resulting dispersion relations reveals the appearance of specific dissipation peaks associated with waveguide geometry. The dispersion relations are utilized in the inversion of torsional shear and extensional experimental results from nonporous and porous solids, thus obtaining their complex frequency-dependent effective elastic constants. A great variation in the elastic constants of Plexiglas with frequency is demonstrated. Evidence of the effects of pore scattering and the reduction of matrix moduli due to water adsorption is shown for porous glass. The effects of adsorption on the effective elastic moduli of porous solids are quantified utilizing the concept of surface elastic constants [M. E. Gurtin and I. A. Murdoch, Arch. Rat. Mech. Anal. 57, 291 (1975)]. The resulting equation reveals the importance of the surface to volume ratio in determining the effective elastic constants.

Wave Propagation Through Fluid Saturated Porous Rocks

A sedimentary rock is modeled by a random packing of identical spherical particles. The connected pore space is filled with an inviscid, compressible fluid. A low-frequency expansion technique is used to calculate the effective wave speeds explicitly in terms of the microstructural properties of the rock considered. The effect of both the pore fluid and the initial confining pressure to which the rock is subjected can be included in the calculations.

Sample boundary effect in acoustic attenuation of fluid-saturated porous cylinders

It was shown previously, by using Biot theory, that there exists an attenuation in the low-frequency elastic modulus measurement on fluid-saturated porous rock cylinders caused by the open-pore boundary condition. This attenuation is associated with the energy dissipation of the viscous flow of the pore fluid in and out of an open-pore boundary of a fluid-saturated rock sample. For the extensional mode, the relaxation frequency of this attenuation is directly proportional to the permeability of the rock and inversely proportional to the viscosity of the pore fluid and the square of the sample radius. In this article, experimental studies that support the theoretical prediction of this sample boundary effect are reported. Experiments of varying saturation, sample radius, pore fluid viscosity, and rock permeability were carried out. The results, though consistent with the theoretical prediction, are not satisfactory in a quantitative sense because of the lack of an idealized sample. It was found, however, the procedure of jacketing the sample with an aluminum sleeve and measuring it at ambient pressure condition is not sufficient to remove the attenuation caused by the boundary effect completely.

Acoustic attenuation in fluid-saturated porous cylinders at low frequencies

Laboratory measurements on the elastic moduli and the acoustic attenuation of fluid-saturated porous rock cylinders are frequently affected by the boundary conditions of the rock sample, i.e., whether the curved surface of the cylindrical sample is exposed to air or properly sealed. In this paper, the analytical solutions of these problems for the extensional, torsional, and flexural modes are derived, and several numerical examples are computed. It was found that there exists an ‘‘artificial’’ attenuation caused by the open-pore boundary condition, whose relaxation frequency, in the case of extensional mode, is directly proportional to the permeability of the rock and inversely proportional to the viscosity of the pore fluid and the square of the sample radius. This attenuation exists in both the extensional and the flexural modes but not in the torsional mode.