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Test_1.2004 - 3/D and a point sink at(300 200 800 with...

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PGE 323 Test#1 Open notes February 20, 2004 Pope 1. (40%) The diffusivity equation for single-phase flow written in Cartesian coordinates is given by Equation (1.5-3) of the course notes. (a) Derive this equation from first principles. Most of the derivation is given on page 24, so you just need to fill in the detailed steps that are missing and explain what you do. (b) Starting with Equation (1.5-3), show how to get Equation (1.5-46). State all the required assumptions and show how you use these assumptions to derive this simplified form of the diffusivity equation. 2. (10%) Starting with Equation (1.7-58) for the flow potential of multiple line sources and sinks, derive Equations (1.7-10) and (1.7-11) for the x and y components of the velocity vector. 3. (50%) All of the assumptions we made to derive the solution of the diffusivity equation for point sources and sinks apply to this problem. There is a point source at (0,0,0) with rate 2 m 3 /D, a point sink at (-1000, -500, -100) with rate -1 m
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Unformatted text preview: 3 /D and a point sink at (300, 200, 800) with rate -1 m 3 /D in an isotropic reservoir with a permeability of 1 D and a porosity of 0.1. The distances are in meters for the (x,y,z) coordinates. (a) Write the particular expression for the flow potential for this problem. (b) Write the particular expressions for the three velocity components. (c) Calculate the x component of the velocity at location (100, 100, 100) in m/s. (d) Calculate the change in potential between the source and one of the sinks if the viscosity is 1 cp. Use a radius of 0.1 m for both the source and sink. (e) Suppose now that there is an impermeable boundary located at z=1000 m. Give the coordinates of the required image wells to model this no flow boundary and show that the flux in the z direction is in fact zero at this boundary with these image wells. Make a sketch showing the location of both the real and image sources/sinks and the boundary....
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