Equation 5 represents a general form of gassmanns equation gassmann, 1951. Han and batzle, 2004 have discussed the formulations, strength and limitations of the gassmann fluid substitution. Numerical examples biotgassmann and wyllie equations. Gassmann s equations gassmann s equations relate the bulk modulus of a rock to its pore, frame, and fluid properties. Carcione 2 osservatorio geofisico sperimentale, po box 2011, opicina, 34016 trieste, italy. The pseudo k dry is not measured, but we can estimate using the hill 1963 average of the effective bulk modulus of the rock matrix including pore fluids in.
For a given porosity, points with low values practical gassmann fluid substitution in sandshale sequences rob simm. Fortunately, gassmanns equation is free of assumptions about pore geometry, although the pore system must be connected and fluids must be moveable. Parameters for the application of gassmann equation generally come from well and core data, plus empirical equations. The sensitivity of v p on gassmann equation parameters is attained by performing monte carlo simulations.
Graduate level problems and solutions igor yanovsky 1. Using keystrokes to write equations in microsoft office. Gassmann theory applies to nanoporous media gor 2018. The isotropic gassmann s equations are often used to assess the. For example, a timevarying current j on a linear antenna generates a circulating. The gassmann equation, first described by fritz gassmann, is used in geophysics and its. Basically, from the elasticplastic point of view the seismic. Gassmanns equation for fluid substitution petrowiki.
A constitutive equation and generalized gassmann modulus for. Gassmanns equations provide a simple model to estimate fluid saturation effect on bulk modulus. Modeling the sonic log response from biot gassmann equations an alternate and more rigorous approach is the biot gassmann equation set. Create the worksheets you need with infinite algebra 1.
A rigorous derivation of the equations for the clamped. The gassmannburgers model to simulate seismic waves at. Gassmanns equation are not well understood, in general practice, no constraints are placed on input parameters and there is no quality control of the results. In this section, we will derive a new rock physics template based on the pore space stiffness approach. In this study, we have analyzed the complexity of pore structure and its interaction with pore fluid. A generalized biotgassmann model for the acoustic properties. The bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the plate thickness. This is commonly performed using gassmanns equation gassmann, 1951. Thus, it is naturally to consider the associated initial value problem y00x fx,yx,y0x. Saturatedrock moduli can be obtained from estimated v p and v s values, while reservoir density, in turn, can be calculated from sonic logs and. A look into gassmanns equation nawras alkhateb, chorus heavy oil consortium, department of geoscience, university of calgary nawras. Even at a modest porosity of 15%, changes can be substantial. Your instructor tim barker tim barker received a b. Based on the mixture theory in two porosities 9, warren and root 36 extended the biots poroelasticity into the double porosity model 8,10,11,16,29, 30, 31,37.
The fluid modulus can be calculated according to the tait. The values of v p and v s are, in general, estimated from fullwave sonic logs. Geological survey scientific investigations report 20085196, 14 p. Summary by describing the influence of the pore fluids on seismic properties, we attempt to provide a comprehensive understanding of gassmanns equation. Effective porosity for gassmann fluid substitution fuyong yan, dehua han and qiuliang yao, rock physics lab, university of houston. Gassmann and krief have both developed models relating compressional and shear acoustic velocities to bulk moduli of the solids and contained liquids. At insitu conditions, pore fluids are often multiphase mixtures. However, gassmann s equations 1951 are by far the most widely used relations to calculate seismic velocity changes caused by different fluid saturation. In particular, problems arise in automated analysis in which results are usually taken at face value. Select multiple pdf files and merge them in seconds.
Elastic properties of saturated porous rocks with aligned. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. Sensitivity analysis of gassmanns fluid substitution. The worksheet allows a fluid replacement based on gassmann s equation. Gassmanns equation and fluidsaturation effects on seismic velocities. Table 1 from a tutorial on gassmann fluid substitution. In the clayfree case in which g 1, equation 14b is identical to the formula. This implies that the sand and clay particles are mixed homogeneously, forming, in effect, composite grains, which in turn form the rock matrix. Gassmann is the eponym for the gassmann triple and gassmanns equation.
Differential equations department of mathematics, hkust. Deriving the gassmann equation we can combine these two fundamental equations to arrive at the simplest form of the gassmann equation. Gassmann equations and the constitutive relations for. Substitution of these parameters into the gassmann equation provides predictions consistent with measured data. The essential input for the application of the gassmann equation is the bulk and shear. The velocity of sound in a rock is related to the elastic properties of the rockfluid mixture and its density, according to the wood, biot, and gassmann equations.
The coupon code you entered is expired or invalid, but the course is still available. Gassmann 1951 developed an equation for fluid substitution that relates the bulk modulus of the saturated rock to the porous space, pore fluid properties, mineral composition and the frame of the rock dvorkin et al. The bulk modulus of a saturated rock is given by the low frequency gassmann theory gassmann, 1951 as a tutorial on gassmann fluid substitution. Gassmann s equation is widely used in rock physics because of its relative simplicity and limited data requirement. Free mathematical physics books download ebooks online. A generalized biotgassmann model for the acoustic properties of shaley sandstones 1 jose m. Partial differential equations of mathematical physics pdf 105p this note aims to make students aware of the physical origins of the main partial differential equations of classical mathematical physics, including the fundamental equations of fluid and solid mechanics, thermodynamics, and. Comparison of the modified biotgassmann theory and the. The next step was the calculation of the compressional and shear wave velocities for brine saturated samples using. The specific surface area is an important parameter related to water retention in reservoir rocks. Comparison of the modified biotgassmann theory and the kuster.
Ravazzoli4 abstract we derive the timedomain stressstrain relation for a porous medium composed of n. The dye will move from higher concentration to lower. K 0 lines is an indicator of relative rock stiffness, or rock compliance xu and white, 1995. As shown by russell and smith 2007, equation 2 allows us to model k dry at different porosities. Bulk modulus and density of the fluid are k f and,respectively. Gassmann equation can be lower or higher than measured velocities. Porosity is the most important factor controlling sonic velocity but our data document that pore type, pore fluid compressibility and variations in shear modulus due to saturation are also important factors for velocities in carbonate rocks. Application of gassmanns equation for laboratory data from. The gassmann 1951 theory predicts the bulk modulus of the fluidsaturated porous medium from the known bulk moduli of the solid matrix, the frame, and the pore fluid and assumes that the shear modulus of sedimentary rocks is not affected by fluid saturation.
Crains petrophysical handbook modeling biot gassmann. This assumes that the chemical interactions between the fluids and the solids are always the same or have the same consequences for rigidity. A constitutive equation and generalized gassmann modulus for multimineral porous media jose m. The gassmann equation, first published in the 1950s, relates compressional acoustic velocity to bulk moduli of the empty rock frame, solids and fluids, density and porosity. Gassmann effective porosity rocks, thus we can assume gassmann equation work and use equation 1 to invert the effective porosity. Dec 28, 2017 the solid bulk modulus can be estimated from adsorption. Two step functions, properly positioned, can be summed to give a solution for finite layer placed between two semiinfinite bodies. Gassmann s equations assume that the shear modulus of the saturated rock is the same as the shear modulus of the dry rock, for all saturating fluids.
Practical gassmann fluid substitution in sandshale sequences. Notes systems of linear equations system of equations a set of equations with the same variables two or more equations graphed in the same coordinate plane solution of the system an ordered pair that is a solution to all equations is a solution to the equation. Pdf gassmanns equation and fluidsaturation effects on seismic. Gassmanns equation fluid substitution relates the elastic moduli of fluidsaturated rock k s, m s to those of dry porous rock k d, m d. Equations of brown and korringa 1975 were used by cardona 2002 to develop a model for fluid substitution in fractured porous media, which is similar to the one presented.
The upper part of the crust shows generally brittle behaviour while deeper zones, including the mantle, may present ductile behaviour, depending on the pressuretemperature conditions. When using gassmanns equations at partial saturation, one has to calculate the effective bulk. Gassmanns 1951 equations relate the bulk and shear. Lakeland community college lorain county community college modified by joel robbin and mike schroeder university of wisconsin, madison june 29, 2010. However, the latter is valid only when the solid matrix is homogeneous brown and korringa 1975. Then, substituting 12 back into either 10 or 11 gives one form of the result commonly known as gassmanns equation for the bulk modulus. Fluid substitution is an important part of the seismic rock physics analysis e. So depending on what exactly you are searching, you will be able to choose ebooks to suit your own needs.
Aug 25, 2017 gassmann with the symmetry of voigts class nr. The analysis usually quoted for addressing this problem in regions of partial melt have normally used some type of classical effective medium theory, which is most appropriately used for systems in which the inclusions are both. A gassmannconsistent rock physics template cseg recorder. A look into gassmanns equation search and discovery.
Identifying patchy saturation from well logs short note. Introduction gassmann s relations are receiving more attention as seismic data are. Model equations and excel worksheets sciencedirect. Pdf a short tutorial on the derivation of gassmanns equations is provided. Gassmanns equations are used for pore fluid identification from. A comparison of the krief and gassmann equation general. A constitutive equation and generalized gassmann modulus. Less wellknown is that gassmann 1951 also published a general equation for. When the diffusion equation is linear, sums of solutions are also solutions. Gassmann s equation how does the constant c 1 depend on the fluid bulk modulus in a region of partial melt. The pressure equation is parabolic only in the vertical direction.
They used irreducible water saturation to quantify the pore fluids that cannot be driven out by hydraulic force. Apr 02, 2018 download gassmann s fluid substitution carry out calculations and generate graphs with seismic velocity changes using an automatic plotting mode and employing the gassmann s equation method. Dec 28, 2017 the agreement is further improved by substituting into the gassmann equation the fluid compressibility obtained as a function of the solvation pressure using the tait. Here is an example that uses superposition of errorfunction solutions. Seismic waves can be used to detect these conditions on the basis of reflection and transmission events. The coupon code you entered is expired or invalid, but. Summary by describing the influence of the pore fluids on seismic properties, we attempt to provide a comprehensive understanding of gassmann s equation. Pdf different forms of gassmann equation and implications. Download gassmann s fluid substitution carry out calculations and generate graphs with seismic velocity changes using an automatic plotting mode and employing the gassmann s equation.
Identifying patchy saturation from well logs jack dvorkin, dan moos, james packwood, and amos nur department of geophysics, stanford university introduction gassmann s 1951 equations relate the elastic bulk k sat and shear g sat moduli ofa fullysaturated rock to those of the dry rock frame kdry and gdry, respectively,porosity. Numerous assumptions are involved in the derivation and application of gassmanns equation. Overall, we can conclude that the saturated moduli of the nanoporous glass are consistent with the predictions of the gassmann theory. Download gassmanns fluid substitution carry out calculations and generate graphs with seismic velocity changes using an automatic plotting mode and employing the gassmann s equation method. If k m1 0, that is, if the medium is a suspension of solid particles in the. Gassmann s equation gassmann s equations provide a simple model for estimating the. When combined with biots analysis 1961, compressional and shear velocities can be determined.
Physics 0506 the most general applications of bernoullis equation. Gassmann s equation is the zero frequency limit of more general equations presented by biot and geertsma. Pdf application of gassmanns equation for laboratory data from. The fraction of total volume occupied by fluid is the porosity, which is assumed to be uniform on some appropriate length scale. Importance of gassmann equation for predicting effect of different fluid saturations on the wave speed are is discussed in two tutorials berryman, 1999.
With respect to elasticity all directions perpendicular to z are equivalent transversely isotropic symmetry, love 1944, page 160 and the elasticity is determined by five constants c1, c2. Isotropic, homogeneous, elastic, monomineralic medium. Consider a porous medium whose connected pore space is saturated with a singlephase viscous fluid. To do this, note first that equation 2 can be rewritten as. Gassmann is the eponym for the gassmann triple and gassmann s equation. Murnaghan equation at the solvation pressure in the pore. Each parameter was modeled as a random variable, whose probability density function pdf corresponds to a gaussian function, characterized by a mean value. Gassmanns e quation how does the constant c 1 depend on the fluid bulk modulus in a region of partial melt. Tran and others published different forms of gassmann equation and implications for the estimation of rock properties find, read and cite all the research you need.
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