_{1}

^{*}

A fractal model containing n-series of spheres and ellipses with corresponding radii, and surrounded by a thin film of adsorbed water has been developed. This model is more suitable to real sedimentary rocks because it not only allows for varying grain sizes and number of fractions, but also takes into consideration the influence of double electrical layers on the physical properties of sediments. Ellipsoid fractal models provide mathematical proof of the phenomenon of supercapillary conductivity observed in laboratory measurements. Using the parameters of the fractal models several petrophysical parameters can be calculated, namely resistivity porosity, and permeability. In a case study, layers with different resistivity were identified on a section taken from a TDEM survey, and porosity and salinity of water bearing layers were estimated using the fractal model. Estimated permeability using the fractal model showed good agreement with other methods.

For many years determining the resistivity of layers was deemed a significant result of applied EM geophysics. However, problems intended to be solved by geophysics have become more complex. Now electromagnetic methods are aimed at defining other petrophysical parameters such as porosity, permeability, polarization parameters, oil and gas saturation etc. Therefore a new model for mathematical modeling of petrophysical parameters is required.

There are a number of factors that must be taken into account when creating this model. It is well known that resistivity decreases with increasing porosity of sediments. However, components filling the pore spaces of rocks considerably influence the petrophysical parameters of rocks. In addition, rocks/sediments can be regarded as two component aggregates, i.e. matrix and fluid, where fluid can be water, gas and/or a mixture of them. Rocks or sediments can also be composed of three component aggregates (matrix-clay-fluid/gas). Moreover, for many cases the influence of double electrical layers (DEL) surrounding grains of sediments must be taken into account when calculating resistivity.

There are numerous experimental works looking at the interdependence of porosity and resistivity of sediments of different kinds and different ages. However, there is insufficient application of these theoretical considerations to the interpretation of electromagnetic data and physical modelling of data. Traditionally for many years the empirical Archie’s law [

where is a coefficient that is dependent on the kind of sediments and can be in the range 0.4 -> 1, φ is the porosity of sediments (dimensionless unit) and m is an index of tortuosity of the pore channels which can be in the range 1.3 - 2.2 or greater [

Another way to calculate resistivity that shall be used for this study is by way of mathematical modelling of petrophysical properties using matrix (fractal) models. It appears that the first person to introduce the fractal model in petrophysics in 1948 was A. S. Semenov [

Usually the solid part of sediments consists of a mixture of polymineral grains. They are naturally dielectric with high resistivity values reaching up to 10^{−}^{11} Ωm and more (ρ_{1}). The second component of sediments is fluid or a mixture of material filling the pore spaces, such as fluid and clay (ρ_{2}). More seldom (in the case of ore) the second component is presented by conductors or semi-conductors with very low resistivity. It follows then that the resistivity of sediments/rocks (ρ_{1,2}) depends on the values of ρ_{1} and ρ_{2}; on the relative volume of components V_{1} and V_{2}, whereV_{2} = (1 − V_{1}); and on the form and distribution of dielectrics and conductors in the solid. The dependence of ρ_{1,2} on ρ_{1} and ρ_{2}can be presented as [

where P_{P} is a parameter of porosity (or structural factor) which is the ratio of the resistivity of the sediment to the resistivity of the material filling the pore spaces.

The dependence of ρ_{1,2} on ρ_{1} and ρ_{2} were studied both experimentally and theoretically for different kinds of rocks/sediments. For the theoretical investigation, the solids were presented as simplified matrix models. The matrix models can be regarded as uniformly distributed packages of identical sizes or different sizes. The packages can be spheres, cubes, ellipsoids, cuboectaedres, rectangles, planes and other geometrical figures. The pore space is filled by a second component, such as water, clay, water/clay, or metal. _{1,2} using different matrix models.

It must be noted that only the model introduced by Semenov take into account different sizes of solid particles. However even this model does not take note of the existence of double electrical layers (DEL), which are known to be associated with the interface between minerals and pore fluids.

The DEL can be regarded as consisting of two regions, namely an inner region or dense part where ions are adsorbed onto the solid surface through electrostatic and Van der Waal’s forces (the Stern layer); and an outer or diffuse region where ions are under the combined influence of the ordering electrical and disordering thermal forces (the Gouy layer). Therefore, two planes can be distinguished in the DEL, namely an inner plane, which is the surface of the solid, and an outer diffuse plane.

The thickness of the dense part of the DEL is approximately σ_{0del} = 10^{−9} [_{0del} then the electrolyte is concentrated. Otherwise if

_{1,2} using different two component matrix models, assuming and is low (after Kobranova, 1984, [

^{*}where k and l are coefficients of form, which change from k → 0 and l →for avery oplate ellipsoid of rotation, up to k →and k → −1 for avery prolate ellipsoid of rotation. For spherical grains k = l = −2.

the electrolyte is regarded as diluted. For a 1:1 valence electrolyte the boundary between concentrated and diluted is accepted as C = 0.1 N. In the case of concentrated electrolytes the electrokinetic phenomena is suppressed. A characteristic thickness of the diffusion part of the DEL σ_{del} can be described as [

where ε is dielectric permeability, ε_{0} is the electrical constant, R is the gas constant, T is the absolute temperature, F is the Faraday number, z is the valence of the ions and C is the concentration of electrolyte (mol). Note the thickness of the DEL σ_{del} indicates the distance where the transfer number of cations C_{c} and anions C_{a} are noticeably different. The thickness of the DEL ranges from a few nanometres up to hundreds of micrometres. If the concentration of the electrolyte NaCl is 0.01 N (0.56 g/l), the thickness of the DEL is equal to 3e−9 m. If the concentration of the electrolyte NaCl is 0.001 N the thickness of the DEL is equal to 1e−8 m.

For this study, the most applicable model for clastic sediments is the model introduced by A. S. Semenov. This model contains n-series of spheres of different sizes. Let us assume the resistivity of the matrix is very high and the second component of the model is water with resistivity ρ_{f}. Following [_{0} and the rest of the volume φ_{0} = (1 ‒ V_{0}) is occupied by a series of smaller grains. The second fraction of smaller grains will occupy a volume (1 ‒ V_{0})· V_{0}. The third fraction of even smaller grains will occupy a volume (1 ‒ V_{0})^{2}·V_{0}, etc. The rest of the volume (1 ‒ V_{0})^{n} will be equal to the porosity coefficient of the sedimentφ. Obviously the coefficient of porosity φ will be determined by the fraction of spherical grains n. The resistivity of this model is equal to:

Let us consider the case for one unit of the medium - a cube with size L = 1, and include into this cube a sphere with radius r_{1} = L/2 = 0.5 (_{1}. This sphere will be regarded as a main grain (grain of the first fraction). The rest of the volume (porosity coefficient) will then be equal to

. (The most widely used physical unit of porosity is percent; however it can also be presented as a dimensionless unit. Therefore, the dimensionless unit can refer both to porosity and the coefficient of porosity). This model is very often used for description of sediments structure although the porosity of a package of identically sized spheres described by Maxwell’s formula (

The third fraction occupies the volume Grains of this fractionare located on the corners of the cubes which include the grains of the second fraction (_{3} = 0.1503. It follows then that the total volume of the solid is equal to V_{1+2+3} = 0.8918 and the volume of the pore space is equal to φ = 0.1082.

Continuing in this way, we obtain the parameters of the fractal model which are presented in

The table shows the dependence of the porosity coefficient (porosity) on the number of fractions of different size (in the solid. When these calculations are applied to a model containing grains of such a size such that the thickness of the DEL is negligible compared to the radii of all spheres, the effective and dynamic porosity of the sediment will equal the total porosity.

However, all spheres are surrounded by planes of double electric layers. The number of fractals composing the solid can be reduced if the size of the first grade becomes smaller. In this case the volume of the double electrical layers will be more considerable. It is easy to calculate the volume of the DEL as the difference between the volume of the solid surrounded by films of adsorbed water and the volume of the pure solid. The thickness of adsorbed water τ_{aw} for a salinity of 0.1 mol/l (0.56 g/l) has been accepted as 10^{−}^{8} m (1e−5 mm) [

The influence of the DEL increases with the amount of clay particles (radius of clay particle is defined as 0.05 mm and less).

Using (2) and (3) as the equation below describes the porosity parameter P_{P} for a fractal model composed of spheres of different sizes as [

where φ_{0} is an initial porosity, meaning the porosity of the matrix created by the main sediment grains only and φ is the volume of the sediment. _{0} initial poros-

ity. In the first case P_{P}(φ) is presented as the fractal model. Obviously, with φ_{0} = 0.4764 as the maximum possible porosity of the matrix represented by a package of identical spheres. For a hexagonal package or a package of identical cubes the initial porosity φ_{0} will be less. _{P}(φ) when φ_{0} = 0.6. In this case the grains are not packed densely and the mixture is presented as a colloid, however both curves are located very close to each other.

It is therefore possible to use the fractal model to calculate the porosity parameter P_{P}. In the case where the pores have two components (matrix-fluid/gas, non-clay sediments) and the amount of DEL is negligible, the resistivity can be calculated using (2) and (4):

where ρ_{f}_{+g} is the resistivity of the material filling the pore space (fluid and gas), ξ is the ratio of water occupying the pore spaces of the sediment, and σ_{f} is the specific

conductivity of the fluid. If the pore space is occupied by a mixture of water, gas and hydrocarbons, the resistivity of the material filling the pore space can be calculated as follows:

where ξ_{1} and ξ_{2} are the ratio of water and hydrocarbons occupying the pore spaces of the sediment respectively. However, if the DEL can not be neglected the resistivity of rocks can be calculated using (2), but taking note:

where σ_{free} is the specific conductance of the free solution (or mixture of components filling the pore spaces) and σ_{del} is the specific conductance of DEL.

The specific conductivity of the DEL can be calculated using the Poisson-Boltzman distribution. The density of charges for a symmetrical binary electrolyte is defined by the difference between the concentration of anions and cations [

where C_{0} is the salinity of the electrolyte in free solution, C_{c} and C_{c} are the salinities of the cations and anions, z is the valence of the ions, and r is the distance from the centre of a pore to its wall. Function Φ(r) is the psi-potential and is described by the Poisson-Boltzman equation. In the inner plane of the DEL Φ(r_{1}) = Φ_{1}, (see _{1} is the coordinate of the outer plane, and the centre of cylindrical coordinates is located in the centre of the pore. Note that the value of Φ_{1} is very close to the ζ(zeta) potential, and is constant for specific kinds of sediments and must be given a priori. Experiments to measure psi-potentials exist, and the values of Φ_{2} can be calculated theoretically. Studying ceramic diaphragms, Fridrikh-sberg and Barkovskaya [_{2}~170 - 195 mV so we use a value of 180 mV (and consequently Φ_{1} ≈ ζ ≈ 100 mV). For carboniferous sediments (limestones) numerous experiments showed that ζ~30 - 40 mV and the influence of the DEL will in this case be much smaller.

Numerical integration of the Poisson-Boltzman equation is a very difficult procedure. However as was shown by Kormiltsev [_{1}) increases very fast with distance from the wall of the solid. Therefore, for large pores the potential distribution in the diffuse part of the DEL can be found using the equation:

where, , , The solution of (10) is:

The average conductance of an electrolyte-filled pore is the ratio of conductivity of a pore to its surface area [

_{0} = 0.01 and 0.005 mol/l, ρ_{f} = 23.5 and 11.2 Ohmm accordingly), with filled pores of difference size and taking into account the DEL. Porediameters in therange of 1e−4 - 2e−6 mm are common for natural adsorbents such as clays, clayish limestone, a shed tuffs etc. [

The calculated specific resistivity of DEL ρ_{del} = 1/σ_{del} can be accepted as 2.45 - 2.5 Ohmm and 4.90 - 4.95 Ohmm (_{0} = 0.01 and 0.005 mol/l accordingly).

It should be noted that Equation (11) only applies for relatively large pores, however, the resistivity of the DEL of a very thin pore does not differ more than 10% compared to the value obtained for large pores.

A fractal can be composed not only of spheres but also of ellipsoids, which is important for clays and shields as they are characterized by planar structures (

C_{0} = 0.01 mol/l

In this case the resistivity of packages of identical ellipsoids of rotation can be calculated using the following equation proposed by I. K. Ovichinnikov [

where is the parameter of porosity of a model containing oblate ellipsoids of one size, and l is a coefficient of form. Based on previous assumptions, we can construct a fractal model contains n series of rotated ellipsoids of different sizes (

As described above, the first fraction is represented by the biggest grains occupying volume V_{0}, with the rest of the volume φ_{0} = (1 ‒ V_{0}) occupied by a series of smaller grains. The second fraction of smaller grains will occupy a volume (1 ‒ V_{0})·V_{0}, etc. The resistivity of the fractal ellipsoid model is then equal to:

where is the parameter of porosity. _{Pell_l} for different coefficients l. P_{P} for the spherical fractal model is also shown.

In

Note that the amount of DELs in the ellipsoid fractal model is higher than in the spheroid fractal model, as one unit of sediments contains considerably more oblate ellipsoids than spheres.

Lithological classification determines the size of clay particles to be approximately 10 µm or less in diameter.

different diameters of the main ellipse of 0.002 mm and 0.02 mm. It can be seen in

where is a coefficient showing the influence of a DEL on the resistivity of the pore space filling. Equation (17) can be written as:

The resistivity of sediment ρ increases with increasing P_{P} but decreases with increasing α. For a specific structural construction of grains (planes) making up sediments it can happen that α > P_{P}. Then the resistivity of the sediment becomes lower than when water saturates the pore spaces. The ellipsoid fractal model shows the conditions when α > P_{P} will be satisfied. For example, consider a case where the diameter of clay particles is about 0.002 mm and the sediment contains particles of two sizes. The porosity is equal to 0.227, water salinity is 0.01 mol/l and all pores are filled by DEL with resistivity 2.5 Ohmm (Figures 4 and 11). The calculated resistivity of the clay is The resistivity of water is 11.6 Ohmm, therefore the resistivity calculated for the clay in this example is lower than the resistivity of free water.

The theoretical value of the resistivity of sediments composed of three components (water, clay and solid matrix) can be calculated using equation [

where P_{P Ps+Av} is parameter of the porosity of the solid matrix (psammito-aleuritic fraction) (Equation (5)), ρ_{cl} is the resistivity of clay, φ_{Ps+Av} is the porosity of the solid (psammito-aleuritic fraction), k_{Vcl} is the volume of the clay in the unit volume of sediments.

The resistivity of clay can be calculated using the following equation:

where, ξ is the volume of pores occupied by free electrolyte, (1 ‒ ξ) is the volume of pores occupied by the DEL (for fractal model see _{Pcl} is the porosity of clay that can be determined using the fractal model. can be determine using (13).

The dependence of permeability on porosity for simple matrix models is well known [

and

where d_{k} is the diameter of spherical pores, and φ = π/4 for cubic packing and θ = 60˚ for rhombic packing (θ is the angle between the centre of the grains for different levels). Equation (21) demonstrates the dependence of permeability k_{P} on the type of capillary packing, and that the permeability k_{P} is proportional to the pore diameter (to the power of two) and porosity of the model φ.

Let us consider a more complex model, such as a cylindrical sample with a surface area S and length l (_{i}, and length l. One filtering channel is located inside each unit. The surface area of each filtering channel ω_{ki} within the unit is changeable and the length of the filtering channel is equal to l_{ki}.

This is a classical Karman-Kozeny model. In this case, the permeability k_{P} of the model is equal to [

where φ_{eff} is the coefficient of effective (dynamic) porosity, S_{FV} is the surface of the specific filtering channel in the unit, T_{H} = l_{k.av}/l is the hydraulic tortuosity, f is the Kozeny constant which falls in between 2 and 3 and is usually close to 2.5. The Karman-Kozeny model is regarded as a model closer to real sediments. In this model

the permeability k_{P} is proportional to the cube of the effective (dynamic) porosity φ_{eff}, and inversely proportional to the specific surface of a filtering channel and the hydraulic tortuosity, both to the power of two, as well as the Kozeny constant f, which is characterized by the form of the pore channels.

The pore channels are filled with free water and adsorbed water. The DEL occupies a volume φ_{del}, and the thickness of the film of adsorbed water is τ_{av}. The final equation for permeability in the Karman-Kozeni model is equal to [

Since f ≈ const, T_{H} = const, τ_{av} ≈ const for each sediment type, k_{P} depends on φ and φ_{del}. The dependences φ and φ_{del} for a fractal model was shown in _{av} = 1e−5 mm). An increasing amount of adsorbed water leads to a decreasing permeability for the sediment. The sediments with permeability less than k_{P} < 0.01 μm^{2} (<0.01 D) are impermeable. Therefore rocks for which the size of the main grain is less than 1e−5 - 5e−5 m (5 - 10 micron) (clays) are impermeable. However, there is not direct relation between porosity and permeability as the amount of DEL governs the permeability of rocks. Whereas on the other hand, the amount of DEL depends on grain size of rocks.

In 21^{st} century, interpretation of EM data cannot be presented asa comparison of resistivity data any more. There are many publications where the relation between resistivity, porosity, resistivity and permeability has been found statistically. However, even within one type of rocks, the dependence of resistivity on porosity is often different. Moreover the relationship between resistivity and porosity (or permeability) usually is displayed as a

large cloud of measured values. Statistical method is simply a set of measurements in the laboratory (or boreholes data) without “looking inside pore of pores” and without taking a note of the grain sizes and amount of DEL. A fractal model is a useful tool for the interpretation of electromagnetic data. It can be used to determine porosity, salinity of water, permeability and other electrophysical parameters such as the diffusion coefficient C_{D} and volume of D. Below we shall present some examples of electromagnetic interpretation using the fractal model.

Time domain electromagnetic method (TDEM) has been used to search for fresh water in the Orenburg region, Russia. TDEM soundings were located along several profiles. In this area, there are several wells that have been electrically logged. Examples of logging diagrams are shown in

For the calculation of porosity some a prioi parameters have to be taken into account, namely the amount of clay contain k_{Vcl} (25%) and salinity of water (0.03 g/l). According to the lithological description, the sandstones contain 25% - 35% clay material. The salinity of the fresh drinking water is recorded as 0.25 - 0.3 g/l. Using these parameters in (19) the porosity of sandstone φ_{Ps+Av}_{ }has been estimated. The P_{P Ps+Av} is parameter of the porosity of the solid matrix and it has been calculated using (5). For calculating theresistivity of clay the fractal

model has also been used. As the porosity of clay is in general approximately 50% and the size of the main grains < 10^{−}^{8} m [_{f} is equal to:

where F is the Faraday constant [9.64955·10^{−}^{4} K/mol], z is the valence, Μ is the mobility of ions (in m^{2}/Vs) and C is the salinity of cations and anions (in mol). Subscripts “a” and “k” indicate anions and cations.

The complete nomogram showing water resistivity’s dependence on salinity and temperature is presented by Itenberg [

A few layers with different resistivities were identified in the cross-section and a comparison with borehole data and borehole loggings showed good agreement between the logs and TDEM resistivity (

The TDEM method has been used to define a fresh water aquifer in the polar oil and gas fields of Zapolyarnoe, in the Tumen region, Russia [

using (19) where the parameter of porosity is presented as fractal. The salinity of water with a resistivity 6.5 - 8 Ohmm was estimated as 2 - 3 g/l (square box in

The geological log from a well is summarized below:

0 - 45 dark grey loam, frozen with rare layers of pure ice; 45 - 75 m layering, loam and sandy loam, the sediments are unfrozen within the interval of 62 - 68 m, with a layer of water saturated sand; 75 - 80 m dark grey clay, frozen. With regards to the ground water, the daily pumping rate was 12 - 17 m^{3}/s, and the subsequent chemical analysis resulted in a concentration of 3 g/l. The taste of water is slightly salty, and unfortunately this ground water does not fit the drinking water standard. However, the interpretated TDEM data using (10) and the fractal model show a good agreement with drilling data.

Lastly we have selected a number of well characterized sandstone samples for study, although we focus here on measurements made on the Berea, Tennessee, Coconino and Island Rust sandstones. Measurements of samples and analysis have been provided by Baker [

Berea sandstone is of Mississippian age, quarried in Berea, Ohio. The major components of this sandstone are quartz, quartzite lithic fragments and feldspar. The mean grain size of 0.3 mm; the porosity of the Berea sandstone sample is 19%. Coconino is an Early-Permian age sandstone of windblown origin and comes from one of the Grand Canyon formations. The major components of this sandstone are quartz, feldspar, and some lithic fragments. The sandstone porosity is connected with a 0.5 mm mean grainand is equal to 11%. Tennessee Sandstone is of Pennsylvanian age and is part of the Crab Orchard formation. The major components of this sandstone are quartz, feldspar, and some litic fragments. The main grain is 0.2 mm with non-interconnected pore spaces. Porosity is 6%.

Island Rust sandstone is composed mainly of surrounded to angular quartz, with a porosity of 14%. The main grain size is 0.2 mm with pores that are not well connected.

Pore size distribution and permeability of each sample have been measured using the mercury invasion capillary pressure method (MICP), which is the most precise method for study internal structure of rocks. The permeability of the investigated samples measured by MICP are given in the

Another way of predicting permeability was proposed by Berg [

Zadorozhnaya and Maré [

The main pore radius has been estimated using both MICP and mathematical modelling. Pore size distribution for Berea, Tennessee and Island samples obtained by MICP and MM are very similar and the main radii of pores are the same for both method. However, we did not obtained a good agreement between MICP and MM for the Coconino sample, therefore an estimation of permeability has been provided for both of these methods. In

which involves the measurement of samples in the laboratory and mathematical modelling of the pore size distribution.

A fractal model containing n-series of spheres with corresponding radii and surrounded by a thin film of adsorbed water tested in this study. This model is more suitable to sedimentary rocks because it not only allows for varying grain sizes and number of fractions, but also takes into consideration the influence of the double electrical layers on the physical properties of sediments. It is also shown that if sediments contain plated particles (for example clay) the resistivity of these sediments can be considerably lower than would be expected, due to a decreasing porosity parameter and more importantly, due to the increasing amount of DEL. Ellipsoid fractal models provided mathematical proof of the phenomenon of supercapillary conductivity observed by laboratory measurements. Using the parameters of the fractal model several petrophysical parameters can be calculated, namely resistivity porosity, and permeability. Mathematical modelling of petrophysical properties can be done to enhance the interpretation of TDEM data. In a case study, layers with different resistivity were identified on a section from a TDEM survey. The results were compared to borehole data and borehole loggings and showed good agreement between the log and TDEM resistivity. In addition, using the given (calculated) resistivity of layers, the porosity and salinity of water bearing layers has been estimated using the fractal model. Estimated permeability calculated using the fractal model showed good agreement with the additional methods (MICP method, Berg’s relationship and mathematical modelling), thus allowing permeability to be determined.