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Institute of Geophysics of the CAS, v. v. i.
Petrophysical analysis is an approach based on measurements of rock physical properties and aiming to describe their relation with the tested rock chemical composition and structure. Our team applies laboratory measurements of rock porosity, permeability, thermal conductivity and elastic and magnetic properties and supports the petrophysical data by microstructural analysis, analog modelling and other techniques of structural geology and petrology in the scope of our research topics.
We run an appartus for multidirectional measurements of P-wave velocity in many directions on spherical samples at confining pressures up to 400 MPa and a kappabridge for analysis of anistoropy of magnetic susceptibility. Within the Institute we cooperate with the Department of Geomagnetism for detailed analyses of rock magnetic properties and with the Department of Geothermics for measurement of thermal conductivity by optical scanning. For measurements of pore space volume, pore size distribution and permeability we cooperate with the petrophysical lab of the ENSG at the University of Lorraine providing mercury porosimetry and nitrogen permeametry.
Ultrasound pulse transmission
We perform laboratory measurements of P-waves velocity (VP) on spherical samples of rocks described in Pros & Babuška (1968), see also Machek et al., 2007 and Staněk et al., 2013. The main advantage of using a spherical sample, as opposed to using an orthogonal or multiple cylindrical samples is the possibility to continuously evaluate the spatial distribution of VP within the studied material. The apparatus consists of a pressure vessel containing the sample, which is placed in a special positioning unit, a pulse generator, a pulse acquisition unit and an oscilloscope (Fig. 1a).
Figure 1. (a) Equipment for the laboratory study of P-wave anisotropy on spherical samples under confining pressure of up to 400 MPa, (b) sample set up (T - transmitter, R - receiver) and (c, d) grid of measurement directions with a step of 15°.
For placement into the positioning unit, two pinions are glued to the poles of the spherical sample. After mounting the sample into the pressure vessel, this enables rotation of the sample around its vertical axis. Furthermore, a rectangular frame containing two spring-hold piezoelectric transducers, with eigenfrequency of 1 MHz, allows symmetrical rotation of the transducers around a horizontal axis (Fig. 1b). This setup of the sample enables measurement of P-wave traveltime in any direction, with exception of an area of 15° as a result of interference with the pinions. Usually we measure VP in 132 directions with a step size of 15° (Figs. 1c and d), at confining pressure levels of 0.1, 10, 20, 50, 100, 200 and 400 MPa. The VP measurements are performed both during stepwise pressurization and depressurization of the sample.
During the experiments P-wave traveltimes are continuously logged using a PC. The measured traveltimes are corrected for any time delay caused by the epoxy layers covering the sphere and by the acquisition elements of the apparatus. The thickness of the epoxy is calculated by subtraction of diameter of the sample before and after epoxy application, typically being 0.05 mm. The delay of the acquisition elements is calibrated by measuring a standard metal sample with known traverse time. The VP is then calculated using the distance given by the sphere diameter.
The method is based on calculation of mercury volume intruded into the sample pore network as a function of external pressure applied on the mercury. In contact with air or in vacuum, mercury is a not-wetting liquid (contact angle exceeding 90°) for most of common solid materials and therefore cannot be spontaneously absorbed by the pores of the solid itself, because of surface tension. However, this resistance to penetration can be won by applying an external pressure, which depends on the pore size. The relation between the pore size and the pressure applied on mercury, assuming the pore is cylindrical, is based on the Young-Laplace equation. For the purpose of experimental porosimetry this was first expressed by Washburn (1921), who stated that the pressure required to force mercury to enter an evacuated capillary pore is related to the capillary radius by the equation:
where r is pore throat diameter, σ is the mercury surface tension, θ is contact angle between the liquid and the solid and p is pressure. Hence, the Washburn equation shows that the pore radius is inversely proportional to the applied pressure. Though in almost all natural porous substances no cylindrical pores exist, the Washburn equation is generally used to calculate the pore size distribution starting from the data obtained by mercury porosimetry.
Mercury porosimeters are highly standardised and widespread analytical instruments. Apart of a pressure vessel, pressure transducer, vacuum and high pressure pumps and a pressure intensifier, the essential part of the setup is a penetrometer which is a sample-holder accommodating the specimen intruded by mercury during the analysis. A schematic drawing of a penetrometer containing a porous sample can be seen in Fig. 2a.
Figure 2. Mercury porosimetry technique basics. (a) - schematic description of a penetrometer showing the partial volumes used for calculation of sample density, (b) charts of cumulative porosimetric curves showing the three basic phases of analysis and (c) the linked terms of total and free porosity and (d) chart of incremental porosimetric curve calculated form the cumulative one; (b, c and d modified after Rosener & Géraud 2007).
The penetrometer consists of two factory-joined parts: a glass bulb for the sample placement and a plastic stem coated by a thin metal layer. During analysis, the volume of mercury in the penetrometer stem changes due to progressively pressurized oil entering the stem and expulsing the mercury into the penetrometer bulb. Taking into account that the mercury and the stem coating are electrical conductors, whereas the oil and the plastic wall of the stem are dielectric, the replacement of the mercury by the oil induces change of capacitance of the stem. This change of the stem electric capacitance is measured by the apparatus during analysis and used as a proxy to calculate the volume of mercury that has moved into the bulb. After corrections for volumetric changes related to compressibility of the penetrometer and mercury, the volume of mercury present within the porous sample is estimated as a function of mercury pressure.
The goal of the permeability measurements is to evaluate the intrinsic permeability of the examined rock specimen. Intrinsic permeability represents a mobility of fluid within porous medium and is solely related to pore geometry of the material (porosity, pore shape and pore size distribution), and is independent of fluid quality. It was first defined by Darcy (1856) as a proportional constant between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance:
where Q is total discharge (m3 s-1), k is intrinsic permeability (m2), A is cross-sectional area to flow, pup-pdown is pressure drop (Pa) over the distance L (m) and μ is the pore fluid viscosity (Pa.s).
The Darcy’s law considers an ideal (uncompressible) fluid. In the actual experiment Nitrogen gas (compressible fluid, pV = constant) is used and the average gas permeability is expressed as (Scheidegger 1974):
where ka is apparent permeability as defined by Klinkenberg (1941).
Klinkenberg (1941) discovered that permeability to gas is relatively higher than that to water, and he interpreted this phenomenon as “slip flow” between gas molecules and solid walls. To quantify the intrinsic permeability, i.e. permeability proper to the material and independent of the fluid, correction has to be made to account for the gas slippage effect. Gas molecules collide each other and to pore-walls during traveling through the porous medium. When the pore radius approaches to the mean free path of the gas molecules the frequency of collision between the gas molecules and the solid walls rises. Therefore this additional flux due to the gas flow at the wall surface becomes effective to enhance the flow rate. This phenomenon is called Klinkenberg effect, and its effect is expressed as follows:
where b is constant characteristic for the pore space geometry and pm is the mean pressure. In practice, the examined intrinsic permeability k (and also the geometric constant b) are defined from permeability tests at several mean pressures by extrapolating the ka to infinite mean pressure given that if pm →∞, then (1+b/pm) → 1 and therefore ka →k.
The used experimental setup is schematically shown in Fig. 3.
Figure 3. Scheme of setup for permeability measurements using nitrogen as flow medium (modified after Rosener 2007).
The essential parts of the setup are represented by a pressure vessel, 3 differential-pressure transducers, a thermal gas flow-meter, a source of pressurised gas (gas bottle in this case) and high pressure tubing, fitting and valves. Both the confining and the flow media is pure Nitrogen gas.
For preparation of the experiment a cylindrical sample is set into a perforated steel tube equipped by a rubber sleeve on its internal side. The sample assembly is then installed into a pressure vessel in a manner allowing application of confining pressure on the cylinder mantle and independent application of gas pressure on the cylinder bases. After installing the sample into the pressure vessel, confining pressure of max 4 MPa (40 bar) is applied and the circuit of confining pressure is closed. Then, upstream pressure is increased and kept constant in order to create a constant pressure gradient across the sample. Gas flow across the sample is thus induced. When steady-state flow is reached (usually after a short period of time on the order of seconds or tens of seconds), the actual upstream and downstream pressures and the flow rate are recorded. In the next step the upstream pressure is increased and after reaching the steady state flow all the values are recorded again. One continues in this manner in order to record flow rates for at least 3 (usually 5 to 10) mean pressures, i.e. the average of upstream and downstream pressures.
Thermal conductivity (TC) is the ability of material to conduct thermal energy, i.e. the heat, without macroscopic displacement of the solid mass. At microscopic scale in crystalline materials (rocks, metals) the heat is transferred by exchange of kinetic energy by vibrations of the crystalline lattice. According to the second law of thermodynamics, differences in temperature in an isolated system tend to equilibrate over time with a preferred direction of progress opposite to the energetic potential, i.e. from hotter to colder region. In such a scheme the TC represents the material proportional constant of the heat transfer rate as defined by the Fourier’s law:
Where Q is heat flux (W), λ is the thermal conductivity (W m-1 K-1), A is cross-sectional area (m2) and (T1-T2) is the temperature drop (K) over the distance l (m).
The optical scanning setup is shown on a photograph in Fig. 4.
Figure 4. Photograph of optical scanning setup for thermal conductivity measurements.
A laser heat source and two infrared radiometers for measurements of sample temperature are placed on a mobile platform that moves at a constant speed relative to samples and reference standards placed on the table. In this manner, temperatures of the sample(s) and of the reference standard(s) are measured before and after the heating by the laser heat source. Given that the heating power, the heat incidence area, the platform movement speed as well as the distances between the radiometers and the heat source are known and constant throughout the scanning, the TC at every examined point along the scanning line (typically each mm) is calculated from the difference of temperatures measured before and after the heat source passage and by comparison to the reference material of known TC. The working surface of the sample has to be coated by a black matte layer prior to analysis in order to minimize the influence of varying optical reflection coefficients of the natural sample surface. Measurements are carried out on either plane or cylindrical surfaces of dry samples.
Darcy, H., 1856. Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris.
Klinkenberg, L. J., 1941. The permeability of porous media to liquids and gases, Drilling and Production Practice, American Petroleum Inst., 200–213.
Pros, Z. & Babuška, V., 1968. An apparatus for investigating the elastic anisotropy on spherical rock samples, Stud. geophys. Geod., 12, 192–198.
Rosener, M., 2007. Petrophysical study and numerical modelling of heat transfer effects between rock and fluid in the Soultz-sous-Forêts geothermal project., University of Strasbourg, Strasbourg.
Rosener, M., and Géraud, Y., 2007. Using physical properties to understand the porosity network geometry evolution in gradually altered granites in damage zones, edited by C. David and M. Le Ravalec-Dufin, pp. 175-184.
Scheidegger, A. E., 1974. The physics of flow through porous media, University of Toronto Press.
Washburn, E. W., 1921. The Dynamics of Capillary Flow, Physical Review, 17(3), 273-283.