Thursday, November 16, 2017

Taking Advantage of Shear Waves

The introduction of the DSI Dipole Shear Sonic Imager tool, which measures shear waves in all type of formations, has stimulated development of new applications for measurement of shear wave speed , Vs, and its reciprocal slowness. Slowness measurements - shear, compressional and Stonely - can help improve identification of rock lithology and fluid content, and assist borehole and surface seismic interpretation. In addition, shear wave slowness provides the critical link in calculating rock shear stiffness and bulk compressibility, two quantities invaluable for planning fracturing operations and controlling production to avoid sanding. A new log of the pore-fluid bulk modulus Kf, based on DSI measurements, shows promise as an indicator of hydrocarbons in sandstone formations. 

Before the development of shear wave logging tools, shear waves created by traditional tools using monopole sources were measureable only in fast, or hard, formations. Consequently, geophysicist based their interpretation schemes largely on measurements of the more accessible compressional speed, Vp. To begin extending interpretation of both Vp and Vs measurements, We need to draw from three sources : Biot-Gassman theory, laboratory measurements of gas saturated, quartz sandstone and contact theory.

In  these equations, the composite density , the composite density is made of density contributions from water, hydrocarbon and rock. N is the rock frame shear modulus - shear modulus describes how a body deforms under a shear stress. Kb and Kp are the bulk moduli of the rock frame and pore space, respectively - bulk modulus measures a body's resistance to a change in volume under pressure. The Biot-Gassman theory also shows that Kp and Kb are related via pore-fluid bulk mdoulus Kf, the bulk modulus of the rock grains Km, and porosity f: 

Readily available shear wave mesurements from the DSI tool combined with Biot-Gassman relations gave researchers a handle on N, but they still needed to know more about Km, Kb and Kp for evaluating Kf. Laboratory data -ultrasonic measurements of shear and compressional velocities plus porosity measurements on gas saturated, pure quartz sandstones - yielded the necessary information on these elastic properties. Plots of Kb and N,calculated from the data, versus porosity reveal a nearly linear porosity dependence. In addition, the data establish that the ratio Kb/N is a constant 0.9 , independent of porosity. Finally, the lab data show that Km for quartz equals 36 Gigapascals (Gpa). 

The laboratory findings on Kb and N were consistent with Hertz-Mindlin contact theory, which models unconsolidated rock as a random packing of spherical, elastic grains. According to contact theory, any force applied to the rock is transmitted at the grain contacts, where one grain touches another. The grain contacts consequently govern the rock's bulk and shear moduli, and hence its compressional and shear velocities. 

Each grain contact can be thought of as two coil springs that move in directions tangential and normal to the contact surface. The stiffnesses of these contacts depend in part on the porosity of the rock because the grain contact area, and thus the force transmitted increases as porosity decreases.

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