P. W. J. Glover

Streaming potential in porous media: 1. Theory of the zeta potential

Electrokinetic phenomena are responsible for several electrical properties of fluid-saturated porous materials. Geophysical applications of these phenomena could include the use of streaming potentials for mapping subsurface fluid flow, the study of hydrothermal activity of geothermal areas, and in the context of earthquake prediction and volcanic activity forecasting, for example. The key parameter of electrokinetic phenomena is the ξ potential, which represents roughly the electrical potential at the mineral/water interface. We consider silica-dominated porous materials filled with a binary symmetric 1:1 electrolyte such as NaCl. When in contact with this electrolyte, the silica/water interface gets an excess of charge through chemical reactions. Starting with these chemical reactions, we derive analytical equations for the ξ potential and the specific surface conductance. These equations can be used to predict the variations of these parameters with the pore fluid salinity, temperature, and pH (within a /pH range of 6–8). The input parameters to these equations fall into two categories: (1) mineral/fluid interaction geochemistry (including mineral surface site density and surface equilibrium constants of mineral/fluid reactions), and (2) pore fluid /pH, salinity, and temperature. The ξ potential is shown to increase with increasing temperature and pH and to decrease with increasing salinity. The proposed model is in agreement with available experimental data. The application of this model to electric potentials generated in porous media by fluid flow is explored in the companion paper.

Nature of surface electrical conductivity in natural sands, sandstones, and clays

The electrical conductivity of rocks results from conduction through the bulk solution occupying the pores and from surface conduction occurring at the fluid/grain interface. The nature of the surface electrical conductivity of shaly and clean sands and sandstones is examined. Surface conduction is characterized by the specific surface conductance which is the sum of three contributions: (i) Conduction within the electrical diffuse layer, which makes a negligible contribution to the total specific surface conductance, (ii) Conduction in the Stern layer, which is shown to vary significantly with the salinity of the pore fluid at low salinities (10−6 to 10−3 mol 1−1), but becomes independent of salinity at higher salinities, (iii) A mechanism operating directly on the mineral surface, independent of salinity, and perhaps associated with proton transfer. At salinities higher than 10−3 mol 1−1 and at 25°C, the specific surface conductance of quartz and clays is equal to 8.9×10−9 S and 2.5×10−9 S respectively. Equations describing the influence of surface conductivity and microstructure upon the macroscopic electrical conductivity of sands, sandstones, and shales are also developed.

A modified Archie's law for two conducting phases

Abstract Many types of mixing model are used widely within the earth sciences to determine the electrical properties of porous media consisting of solid and fluid phases of known conductivities, volume fractions and distributions (i.e. phase connectivities). Most models are valid for two or more conducting phases. However, of the simple models only Archie’s law includes a variable term, the Archie cementation exponent m, that describes the connectivity of the phases. Unfortunately, Archie’s law is only valid for one conducting phase distributed within a non-conducting phase, which makes it inapplicable in instances where the rock matrix has a significant conductivity such as for clay-rich rocks and in calculations involving partial melting. More complex models exist which account for multiple conducting phases and control over phase conductivity. We have adapted the conventional Archie’s law to produce a simple modified Archie’s law that can be used with two conducting phases of any conductivity and any volume fraction, while retaining the ability to model variable connectivities within those phases that result from variations in their distribution. The modified model has two exponents (m and p) that describe the connectivity of each of the two phases. The exponents are related through an equation that depends also on the volume fractions of the two phases. The modified and the conventional versions of Archie’s law have been tested on a granular analogue porous medium with a conducting matrix and a pore space saturated with a range of saline fluids with different salinities and conductivities. The new model describes the experimentally determined electrical behaviour of the system extremely well, improving greatly on the conventional Archie’s law.

Permeability prediction from MICP and NMR data using an electrokinetic approach

Theaccuratemodelingofoil,gas,andwaterreservoirsdependsfundamentallyuponaccesstoreliablerockpermeabilitiesthatcannotbeobtaineddirectlyfromdownholelogs.Instead, a range of empirical models are usually employed.We propose a new model that has been derived analytically from electrokinetic theory and is equally valid for all lithologies. The predictions of the new model and four other common models Kozeny-Carman, Berg, Swanson, and van Baaren have been compared using measurements carried out on fusedandunfusedglassbeadpacksaswellason91rocksamples representing 11 lithologies and three coring directions. The new model provides the best predictions for the glass beadpacksaswellforallthelithologies.Thecruxofthenew modelistohaveagoodknowledgeoftherelevantmeangrain diameter,forexample,fromMICPdata.Hence,wehavealso predictedthepermeabilitiesof21NorthSeawellcoresusing all five models and five different measures of relevant grain size. These data show that the best predictions are provided by the use of the new model with the geometric mean grain size.Wehavealsoappliedthenewmodeltothepredictionof permeability from NMR data of a 500 m thick sand-shale succession in the North Sea by inverting the T2 spectrum to provide a value for the geometric mean grain size. The new model shows a good match to all 348 core measurements fromthesuccession,performingbetterthantheSDR,TimurCoates,HSCM,andKozeny-Carmanpredictions.

Grain-size to effective pore-size transformation derived from electrokinetic theory

Most permeability models use effective grain size or effective pore size as an input parameter. Until now, an efficacious way of converting between the two has not been available. We propose a simple conversion method for effective grain diameter and effective pore radius using a relationship derived by comparing two independent equations for permeability, based on the electrokinetic properties of porous media. The relationship, which we call the theta function, is not dependent upon a particular geometry and implicitly allows for the widely varying style of microstructures exhibited by porous media by using porosity, cementation exponent, formation factor, and a packing constant. The method is validated using 22 glass bead packs, for which the effective grain diameter is known accurately, and a set of 188 samples from a sand-shale sequence in the North Sea. This validation uses measurements of effective grain size from image analysis, pore size from mercury injection capillary pressure (MICP) measurement...

What is the cementation exponent? A new interpretation

In 2003, 182 billion barrels of oil reserves worth about US

A generalized Archie’s law for n phases

4.5 trillion were discovered worldwide (Johnson et al., 2004). Moreover, between 1950 and 2002 the total volume of reserves discovered has run to more than 1500 billion barrels for oil and 7.5 trillion cubic feet for gas (Bentley, 2002). More than half of these resources have already been produced and have driven the global economy for the last 50 years.

Ionic surface electrical conductivity in sandstone

Archies law has been the standard method for relating the conductivity of a clean reservoir rock to its porosity and the conductivity of its pore fluid for more than 60 years. However, it is applicable only when the matrix is nonconducting. A modified version that allows a conductive matrix was published in 2000. A generalized form of Archies law is studied for any number of phases for which the classical Archies law and modified Archies law for two phases are special cases. The generalized Archies law contains a phase conductivity, a phase volume fraction, and phase exponent for each of its n phases. The connectedness of each of the phases is considered, and the principle of conservation of connectedness in a three-dimensional multiphase mixture is introduced. It is confirmed that the general law is formally the same as the classical Archies law and modified Archies law for one and two conducting phases, respectively. The classical second Archies law is compared with the generalized law, which leads to the definition of a saturation exponent for each phase. This process has enabled the derivation of relationships between the phase exponents and saturation exponents for each phase. The relationship between percolation theory and the generalized model is also considered. The generalized law is examined in detail for two and three phases and semiquantitatively for four phases. Unfortunately, the law in its most general form is very difficult to prove experimentally. Instead, numerical modeling in three dimensions is carried out to demonstrate that it behaves well for a system consisting of four interacting conducting phases.

Synthetic rough fractures in rocks

Recent analyses of complex conductivity measurements have indicated that high-frequency dispersions encountered in rocks saturated with low-salinity fluids are due to ionic surface conduction and that the form of these dispersions may be dependent upon the nature of the pore and crack surfaces within the rock (Ruffet et al., 1991). Unfortunately, the mechanisms of surface conduction are not well understood, and no model based on rigorous physical principles exists. This paper is split into two parts: an experimental section followed by the development of a theoretical description of adsorption of ions onto mineral surfaces. We have made complex conductivity measurements upon samples of sandstone saturated with a range of different types and concentrations of aqueous solution with a frequency range of 20 Hz to 1 MHz. The frequency dependence of complex conductivity was analyzed using the empirical model of Cole and Cole (1941). The “fractal” surface models of Le Mehaute and Crepy (1983), Po Zen Wong (1987), and Ruffet et al. (1991) were used to calculate apparent fractal pore surface dimensions for samples saturated with different solution types and concentrations. These showed a pronounced decrease of apparent fractal surface dimension with decreasing electrolyte concentration and a decrease of apparent fractal dimension with increasing relative ionic radius of the dominant cation in solution. A model for ionic surface concentration (ISCOM I) has been developed as the first step in producing a rigorous physicochemical model of surface conduction in quartz-dominated rocks. The results from ISCOM I show that quartz surfaces are overwhelmingly dominated by adsorbed Na+ when saturated with NaCl solutions of salinities and pH found in actual geological situations. ISCOM I also shows that the concentration threshold for dominance of surface conduction over bulk conduction is aided by depletion of ions from the bulk fluid as a result of their adsorption onto the mineral surfaces as well as by changes in the ionic mobility in the surface conduction double-layer as the wetting solution becomes more dilute.

Electrical conductivity of the continental crust

Natural fractures serve a very important function in the transport of fluid through rocks, as well as in the flow of electrical charge and heat. The creation of numerically synthesized fractures can aid the study of the physical processes connected with fractures. Synthetic fracture is the term used to describe fractures that are created numerically in such a way that they share the same mean geometrical characteristics as specific natural fractures measured by profiling, by a process known as tuning. We have modified methods for producing synthetic rough surfaces whose geometric properties are tuned to mimic natural fracture surfaces in rocks in order to create synthetic fractures that are statistically identical to those found in rocks. One important such modification has been the incorporation of a method that allows the surfaces to be matched at long wavelengths and unmatched at short wavelengths, with the degree of matching varying smoothly in between, as it does for real fractures. We have compared numerically synthetic fractures created using the new method with fractures created using an existing technique that uses a mismatch wavelength as a sudden discontinuity between matched and unmatched behavior, as well as data from fractures in real rocks. This comparison has shown that the new technique provides much more realistic numerically synthesized fractures than previous methods. Synthetic fractures created with the new method have been used in normal closure and fluid flow modeling, and the results are reported in a companion paper.