- In this module, we will delve into Gassmann's equation. First we will discuss why we use the Gassmann equation. We need to identify the assumptions and its derivation which leads to an appreciation of its limitations. Next we need to review elasticity and the factors that affect seismic waves, because at the core of Gassmann is averaging the elastic moduli. Porous rock is fairly complicated, so I'd like to introduce a visual model to illustrate how the Gassmann equation is derived. Finally, we can look at a typical application of Gassmann. So why should we concern ourselves with the Gassmann equation? It provides the means to calculate the effect of different fluids in the rock, on the seismic response. Specifically, it relates the bulk modulus of the rock to its porosity, porous rock frame, mineral matrix, and fluid properties. It is theoretically sound, making it a better choice than using empirically derived relationships. Gassmann can be applied into interpreting a seismic anomaly to decide if it could be fluid related, or the calibration of a 4D seismic response that is caused by saturation changes. Gassmann assumes that the rock is homogeneous and isotropic. This is a common assumption for many reservoir rocks. This assumption is violated if the rock is composed of mineral grains with large differences in the elastic moduli. A rock made up of oriented anisotropic minerals, like a shaley sand, is not isotropic. A rock with multiple pore types, as can be found in some carbonates, may not need this assumption. The second assumption is the rock has connected pores, allowing free movement of the fluids. This assumption will break down in rocks with low porosity, or those that have poorly connected pores. A passing seismic wave asserts a stress, or a pressure change on the fluid in porous rock. Gassmann is valid only at frequencies that allow the pore pressure to equalize over a length scale that is much greater than the dimension of the pores and much less than the wavelength of the seismic. For rocks with reasonable porosity and connected pores, this is typically not an issue. It could be problematic reconciling velocity measurements made at very high frequencies on core plugs, for example. Seismic response to changes in velocity and/or density of the Earth. In a homogeneous, isotropic, elastic medium, the p velocity is dependent on the density of the rock and the elastic property's bulk modulus, indicated by K, and shear modulus, indicated by mu. The s velocity is only related to the density in the shear modulus. Velocity and density are fairly easy to grasp, but let's review the other elastic properties, the bulk and shear moduli. Imagine a small cube of rock within the Earth. A passing seismic wave exerts a small stress on that cube of rock. The stress is indicated by sigma. Some of the stress will be exerted perpendicular to the faces of the cube, and some will be exerted tangential to the cube, or parallel to a face. The stress perpendicular to the cube will compress the rock ever so slightly, changing its volume. The bulk modulus is the ratio of applied stress to the fractional volume change. The bulk modulus is a measure of the compressibility of the rock, or to use a more common word, let's call this the stiffness of the rock. The stress tangential to the cube will change its shape slightly. This strain is indicated by epsilon. The shear modulus is a ratio of the applied tangential stress to the amount of strain. The shear modulus is a measure of the rigidity of the rock. A reservoir rock consists of grain material arranged in a porous frame and filled with a fluid. To understand the property of the rock as a whole, we need to consider each of the components. The Gassmann equation describes how to average these components. Let's return to a simple cube of porous rock that has a volume of 1. Let's imagine that we can push to the back the fraction of the rock that provides its stiffness and rigidity. Let's call this fraction beta. This component is composed of the solid grains of the rock that have a density of rho sub zero and a bulk modulus of K sub zero. What remains are two components. An obvious one is the porosity which contains the fluid that is in the rock. The porosity is indicated by phi, and the fluid in this space has a density of rho sub fluid, and a bulk modulus of K sub fluid. For most porous rocks, there's another component. This component consists of solid grains that contribute to its density but do not contribute to its stiffness or rigidity. These floating grains make up the remaining fraction of 1 minus porosity minus beta. The solid grains have a density of rho sub zero, and a bulk modulus of K sub zero. Since porous rock is composed of a mixture, we need to know how to mix them properly. Let's define a rock where there are only two components: one is the solid grain material, which is difficult to compress, in this region, and two is the fluid. The density of the mixture is simply a weighted average, using the fraction of each component, indicated by f, as the weight. Mixing the bulk modulus is more complex. Let's look at the extremes. Imagine that the stress is acting vertically on this cube. The rigid, stiff rock component will determine how much the cube will compress. Both component one and two will experience an equal amount of strain. For equal strain, the bulk modulus of the mix will be the Voigt Average, or a weighted, arithmetic average, where the weight is a fraction of each component. The stress could just as well be acting horizontally on the cube. In this situation, the weaker fluid component determines how much the cube is compressed, resulting in equal stress or pressure in each component. For this case, the bulk modulus of the mix is the Reuss Average, a weighted harmonic average. Since the stress from the passing seismic wave will be acting in all directions, a final bulk modulus of the mixture will be a combination of these two laws. Finally, there's a tangential shear stress. Again, the rigid component will determine how much the cube deforms. The Voigt Average is appropriate. But we know that fluids cannot support a shear stress. A fluid will not contribute to the shear modulus of the mixture. Only the rigid component of the rock determines the shear modulus. Therefore, a conclusion is that the shear modulus is independent of the fluid. Now let's put together what we have learned. The density of the flud-saturated rock is simply the porosity times the density of the fluid plus 1 minus porosity times the density of the solid grains. The stiff, rigid component of the rock will determine how much it is compressed, and all components will have identical strain. So we start with the Voigt Average. The bulk modulus of the saturated rock then becomes beta times the bulk modulus of the solid grains plus 1 minus beta times the bulk modulus of the remaining components. The remaining components are the fluid and the unconsolidated grains. As the rock is compressed, these components will experience equal stress, and therefore, the Reuss Average must be used. The portion of the rock that is under equal stress is 1 minus beta, so the normalized fractions of the remaining components become porosity divided by 1 minus beta for the bulk modulus of the fluid, and 1 minus porosity minus beta divided by 1 minus beta for the bulk modulus of the unconsolidated solid grains. Putting these two equations together, the bulk modulus of the saturated rock is shown. This is one form of the Gassmann equation. The first term, beta times the bulk modulus of the solid grains, is equivalent to the bulk modulus of the dry rock frame. I will leave it up to you to show that this equation is identical to other forms of Gassmann equation. Since the shear modulus is insensitive to fluids, the shear modulus of the saturated rock is equivalent to the shear modulus of the dry rock frame. Now we can determine the effect of replacing a water saturated rock with hydrocarbons. Typical hydrocarbons, that is, oil and gas, are less dense than water, therefore the density of a hydrocarbon-saturated rock will be less than that of a water-saturated rock. Hydrocarbons are more compressible than water, so they have lower bulk moduli. From the bulk modulus equation, we can see that using a smaller fluid bulk modulus for the hydrocarbons will result in a hydrocarbon-saturated bulk modulus that is smaller than a water-saturated rock. Here's that result for varying saturations of water and hydrocarbons. In the crossplots, density and bulk modulus increase upwards. 100% water saturation is on the right, and 100% hydrocarbon saturation is on the left. Results using typical oil and gas are shown. But what happens to the p velocity when adding hydrocarbons? Remember the shear modulus is constant. From the equation it can be seen that as density decreases with added hydrocarbons, the p velocity would increase. But the bulk modulus decreases with added hydrocarbons, and that would make the p velocity decrease, so which one dominates? Density or bulk modulus? It may not be obvious. Here is a plot of the p velocity. Oil is well behaved, with p velocity decreasing fairly steadily with increasing oil saturation. Gas is quite different. With just a little gas, the bulk modulus decreases rapidly, causing a large initial decline in p velocity, but as gas saturation continues to increase, the bulk modulus is fairly constant. Here, the decreasing density causes the p velocity to increase with increasing gas saturation. One thing that can be concluded is that for typical hydrocarbons, the density and p velocity of hydrocarbon saturated rock will be lower than that of a water-saturated rock. This results in lower acoustic impedance for hydrocarbon saturated rocks. What is the effect of hydrocarbons on the s velocity? Since the shear velocity for a given rock sample is constant, the s velocity is only dependent on the density. Since the density decreases with increasing hydrocarbon saturation, the s velocity will increase, and gas-saturated rock will have a larger s velocity than oil-saturated rock. In this module we showed that Gassmann is thoretically robust for computing the effect of fluids in a porous rock. We discussed the assumptions and noted several pitfalls where the theory may break down, namely low-porosity rocks, shaley sands, and some carbonates. We quickly reviewed elasticity and its effect on seismic waves. We illustrated a porous rock as separate components to understand how the moduli should be averaged, thus deriving Gassmann's equation, and finally, we discussed the effect of hydrocarbons on the elastic properties of a porous rock.