User:Dsmolk1/Petrophysical seismic modeling in unconsolidated sediment

Petrophysical seismic modeling is the use of mathematical expressions and simplifying assumptions to model or predict the response of the earth. In order to model unconsolidated porous media, typically the bulk modulus and the density of the grain, the skeleton, and the pore fluid and the fractions of each are considered in a mathematical formula. Petrophysical seismic modeling can be used to determine values for amplitude versus offset and attenuation factors, S-wave velocity values from P-wave velocities and P from S, the fluid bulk modulus to determine water or oil and porosity.

Porous media, by their nature, are very complex. A general formula for porous media does not exist, although there are many different models that assume different properties of the media and have certain limitations. Within these models, there are two subsets of assumptions. There are models where the frequency is sufficiently low that the passing wave does not affect the media, like in field seismic. Then there are high frequency models that take the affect of the passing wave into account, like in sonic logs or physical modeling.

Basic Assumptions for Modeling
The main simplifying assumption is reducing the media that is being modeled to a macroscopic level, rather than examining the microscopic heterogeneity, and thereby creating an effective elastic modulus. To achieve this, many models (e.g. Biot, Gassman) that rely on an effective elastic modulus use an average value for the minerals that constitute the grains. Most effective moduli models also require the wavelength used to inspect the media to be much greater than the heterogeneities within the media. It is important to note that dry rock is not the same as gas-saturated rock. Dry rock is rock under zero confining pressure. Also, inelastic effects such as friction and viscosity are neglected in all the models.

Contact Stiffness and Effective Elastic Moduli
In order to describe the effective elastic moduli of a package of unconsolidated sediment, the grain-to-grain interactions must be considered. This interaction is described by the stiffness between two grains in contact, which for this purpose are assumed to be identical spheres. The contact stiffness is measured in both the normal and tangential directions (Fig. 1). In the normal direction contact stiffness, $$S_n$$, between the two spheres is defined as the ratio of the increment in a normal confining force to the shortening of the radius of the sphere. In the tangential direction, stiffness, $$S_t$$, is defined as the ratio of the increment of a tangential force to the displacement of the grains relative to the center. These properties can be used along with the coordination number, C, sphere radius, R, and porosity, φ, to estimate the effective bulk and shear moduli of a random packing of spheres.

$$K_{eff}=\frac{C (1- \phi)}{12 \pi^2 R} S_{n}$$

$$G_{eff}=\frac{C (1- \phi)}{20 \pi R} (S_{n}+ 1.5 S_{\tau})$$

Coordination Number (vs. Porosity)
The coordination number, C, describes how many contacts each grain has with other grains. This can be useful in many petrophysical models that rely on the interactions at the grain level to describe the physical response of the overall package.

Hertz-Mindlin
The Hertz-Mindlin model for predicting dry effective elastic moduli in unconsolidated sediment is based on work done by Hertz (1882) and Mindlin (1949) on the behavior of elastic bodies in contact. Spheres are often used to model an ideal grain, so for simplicity this model assumes that the grains are identical spheres with a random packing. Several other assumptions are made within this model. First, in solving for the normal and tangential stiffnesses, Mindlin (1949) assumes that the compressional force is applied, followed by a subsequent tangential force. Again, for simplicity, Mindlin assumes that there is no slip along the contact surface between the grains. For the sake of estimating the effective bulk moduli error is negligible.

It is demonstrated by Mindlin that the normal and tangential stiffnesses, Sn and St, respectively, are

$$S_{n}=\frac{4 G a}{1- \nu}$$

$$S_{t}=\frac{8 G a}{2 - \nu}$$

According to Hertz, the radius of the contact area, a, and the normal displacement, z, are defined by the following equations, which also take into account grain radius, R, Poisson’s ratio, ν, of the grain material, and the shear modulus of the grain, G.

$$a=\sqrt[3]{\frac{3 F R}{8 G} (1 - \nu)}$$,                                                       $$z=\frac{a^2}{R}$$

With a confining pressure, P, acting upon the sediment package, the confining force, F, acting on the grains is

$$F=\frac{4 \pi R^2 P}{C (1- \phi)}$$

Which then gives

$$a=R \sqrt[3]{\frac{ 3 \pi (1 - \nu)}{ 2 C (1 - \pi) G} P}$$

Solving for the normal and tangential stiffnesses we get

$$K_{eff}=\sqrt[3]{\frac{C^2 (1 - \phi)^2 G^2}{18 \pi^2 (1- \nu)^2} P}$$

$$G_{eff}=\frac{5 - 4 \nu^3}{2 - \nu} \sqrt[3]{\frac{3 C^2 (1- \phi)^2 G^2}{2 \pi^2 (1- \nu)^2} P}$$

This can then be extended to predict the seismic velocities of p-waves and s-waves traveling through the dry package. The velocity of a compressional wave through an isotropic, homogeneous media is then given by

$$V_{p}=\sqrt{\frac{K_{eff} + \frac{4}{3}G_{eff}}{\rho}}$$

$$V_{s}=\sqrt{\frac{G_{eff}}{\rho}}$$

Biot Theory
In two separate papers in published in 1956 , Biot derived theoretical formulas for predicting the frequency-dependent velocities of saturated rocks in terms of their dry rock properties for low frequencies and high frequencies. To do this he incorporated standard rock and fluid seismic properties of interest (listed below), as well as considering the impact of the passing wave on the fluid using Darcy’s law

$$k_eff$$, effective bulk modulus of the frame $$k_s$$, shear modulus of the frame $$k_r$$, bulk modulus of the minerals making up the grains $$k_f$$, the bulk modulus of the pore fluid $$\phi$$, the porosity $$\rho_f$$, fluid density $$\rho$$, the mineral density

Pressure wave Equation

$$ H \nabla^2 \phi_s - C \nabla^2 \phi_f = {\frac{(\partial ^2 (\rho \phi_s - \rho_f \phi_f))}{(\partial t^2 )}}$$

$${C \nabla^2 \phi_s - M \nabla^2 \phi_f} ={\frac{\rho_f (\partial^2 \phi_s)}{(\partial t^2 )} - c \frac{(\rho_f \partial \phi_f)}{(\beta \partial t^2 )} - \frac{(\eta F \partial \phi_f)}{(k_s \partial t)}}$$

Where:

$$H = {\frac{(k_r-k_b)^2}{(D - k_b )}+k_b + \frac{3}}$$ $$C = {\frac{(k_r (k_r- k_b))}{(D - k_b )}}$$ $$M = {\frac{k_r^2}{(D - k_b )}}$$ $$k_b$$, bulk modulus of the porous frame $$k_r$$, bulk modulus of the solid material $$k_f$$, bulk modulus of the fluid

and

$$\phi_s = {A e^{i(\omega t - k r)}}$$ $$\phi_s = {B e^{i(\omega t - k r)}}$$

are substituted to find the equation for the pressure wave.

Shear wave equation

$$\mu \nabla^2 \psi_s = \frac{\rho (\partial^2 \psi_s)}{(\partial t^2 )}- \frac{(\rho_f \partial \psi_f)}{\partial t}$$

$$ 0 = \frac{(\rho_f \partial \psi_f)}{\partial t} - \frac{c (\rho_f \partial \psi_f)}{\beta \partial t}-\frac{(\eta F \partial \psi_f)}{(k_s \partial t)} $$

$$ v_s=\sqrt{\frac{\mu }{\rho-(\rho_f^2)/m'}}$$

Also, the Biot coefficient can be used to determine the bulk modulus if the dry frame is known:

$$k = k_ma (1 - \beta)+ \beta^2 M$$

$$\frac{1}{M}= \frac{(\beta - \phi)}{k_ma} + \frac{\phi}{k_fl}$$

Biot found there was a critical frequency, $$f_c$$, which determines the frequency range where the forces of the elastic wave are dominated by viscous effects or by inertial effects. When $$f<f_c$$, the high-frequency model is appropriate. $$ f_c=\frac{\phi \eta }{(2 \pi \rho_fl k)}$$ Of particular interest, Biot theory predicts two types of propagating waves, those of the first kind, where the elastic wave travels in phase through the grains and the fluid, and those of the second kind, where the elastic wave travels out of phase through the grains and the fluid. The first recorded evidence of this wave is in an experiment in 1980 by Plona. The experiment consisted of several different types of media (e.g. sintered glass and porous steel) held in a water-filled tank so the transmitters and the receivers did not have to be coupled to the media, which would prevent the slow wave from being observed. The fluid surrounding the observed media also ensured that only compressional waves would be recorded. The three expected arrivals, a fast compressional and shear wave and a slow compressional wave, were observed.

In order for the theory to be applicable, several assumptions must be taken into account. As with most modeling, the wavelength must be large relative to the macroscopic elementary volume, or large with respect to the heterogeneous elements within the media. In line with the wavelength, it is assumed that the matrix is elastic and isotropic. The displacements must be small for both fluid and solid phase, which is generally satisfied in lab and field experiments where values are generally less than 10^-6. The media must be fully saturated. The frequency must be low enough such that it does not impact the media it is passing through. Last, there is an absence of coupling, thermomechanical or otherwise. Rise in V with F: inertial forces increase simultaneously, less mass with the frequency as less fluid moves with higher frequency and increasing dissipation with the square of the angular frequency.

Gassmann’s Relations
In a 1951 paper by Gassmann, a formula for relating the dry bulk modulus to the saturated bulk modulus through (for derivations, see Zimmerman, 1991.) where he created an effective modulus by: $$\frac{k_{sat}}{(k_0 - k_{sat})} = \frac{k_{dry}}{(k_0 - k_{dry})} + \frac{k_{fl}}{(\phi (k_0 - k_{fl}))}$$ where: $$K_{dry}$$, the effective bulk modulus of the dry rock $$K_{sat}$$, the effective bulk modulus of the rock with the pore fluid $$K_0$$, the bulk modulus mineral material $$K_{fl}$$, effective bulk modulus of the pore fluid $$\phi$$, the porosity $$\mu_{dry}$$, effective shear modulus of dry rock $$\mu_{sat}$$, effective shear modulus of rock with pore fluid

the mineral and fluid bulk moduli through the bulk modulus of the grain material and the bulk modulus of the fluid. It is assumed that the fluid in the pore space does not impact the shear modulus of sediments. There are examples, however, that show that shear wave velocity changes with pore fluid. All of the assumptions under the Gassmann model are much like those of the Biot low-frequency model. The frequencies must be low enough such that the wave does not influence the pore pressure in the media, the media must be isotropic, all the minerals must have the same bulk and shear moduli, and the rock must be completely saturated.

The Biot coefficient can be calculated from measured velocities using dry rock and/or water saturated rocks by

$$ a \beta_p^2 + b \beta_p  + c = 0 $$

where:

$$ a =\frac{(4 \mu_{ma} k_{fl})}{3}$$ $$ b = \rho k_{fl} V_p^2 - k_{ma}  [k_{fl}  (1 + \phi)- k_{ma} \phi] - \frac{4 \mu_{ma}  [k_{fl}  (1 + \phi)- k_{ma} \phi]}{3}$$ $$c = \phi (k_{fl} - k_{ma} )(- \rho V_p^2 + k_{ma} + \frac{(4 \mu_{ma})}{3})$$ $$ \beta S = \frac{(1 - \rho V_s^2)}{\mu_{ma}}$$ where: $$k_{ma}$$ denotes the bulk modulus of the mineral grains $$k_{fl}$$ denotes the bulk modulus of the fluid $$\phi$$ is the porosity $$\rho$$ is the density $$V_p$$ is the compressional wave velocity $$V_s$$ is the shear wave velocity $$\beta$$ is the Biot coefficient

Bachrach,Ran and Amos Nur.
High-resolution shallow-seismic experiments in sand, Part I: Water table, fluid flow, and saturation. Geophysics, Vol. 63, No. 4 (July-August 1998); P. 1225-1233.

Bachrach and Nur carried out a shallow, high resolution refraction and reflection seismic survey on Moss-Landing Beach in sandy Monterey Bay, California in an attempt to monitor the water table level as it varied with a tidal cycle. The setup included a seismic line of five receivers placed 0.2 m apart, and placed parallel to the baseline about 50 cm below high tide. Water table measurements were taken directly every 30 minutes from a control well, located 1.5 meters from the line. The experiment began during low tide, and data was collected until the geophones were under water during high tide. Measurements resumed as soon as the water receded from the geophones and the experiment was concluded when the sea level reached low tide again. During the course of the experiment the tide varied by about 1.65 m, and the water table as measured in the control well varied by about 1 m.

The apparent depth to the water table was calculated from the refration data, assuming a two-layer model and a continuous velocity profile, and plotted against the measured depth of the water table from low tide to high tide. The water table refractor not consistent with the observed water table changes from the control well. This can be explained by looking at the effect of water saturation on seismic velocities. Biot-Gassmann theory predicts the influence of saturation on p-wave velocity. In the Biot-Gassmann model, the p-wave velocity of dry sand starts at around 170 meters per second. It is observed that as water saturation increases from 0 to about 90% the seismic velocity is lowered. This is due to increased density from the increased water saturation. It is apparent from the equation for p-wave velocity that an increase in density would cause a decrease in velocity. However, beyond a saturation of approximately 90% the water actually creates a stiffening of the rock or sediment package that has a greater effect than that of density and velocity begins to increase. This concept gives a possible explanation for the results seen in the experiment by Bachrach and Nur. As the water table rose due to higher tide levels, the sediment became more and more saturated causing a velocity draw down effect. The saturation level must have never gotten high enough to cause a stiffening of the grain package, and thus no increase in velocity was observed.

Bachrach et al.
High-resolution shallow-seismic experiments in sand, Part II: Velocities in shallow unconsolidated sand. Geophysics Vol. 63, No. 4 (July-August 1998); P. 1234-1240.

Bachrach et al. used Hertz-Mindlin predictions of a velocity profile for varying coordination numbers for a beach sand and to compare the theoretical model with the collected field data. The field data was found to follow the trend predicted by the model but quantitatively did not match the model.

Myung W. Lee
Myung W. Lee 2006 Lee found that elastic velocities depend primarily on the Biot coefficient, which decreases as the differential pressure and the degree of consolidation increase and the porosity decreases. An in-situ Biot coefficient could be calculated from P-wave velocities and porosity from which he predicted an s-wave velocity. It was also found that this method works well for both consolidated and unconsolidated sediment. In his results, he uses corrected biot coefficients derived from P-wave velocities of water saturated sediments measured at various pressures and found that it is an accurate method for predicting s-wave velocities.