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It's very long, I know. But please, don't be discouraged from going through this question if you can help me. The question is at the second to the last paragraph, but you need to read through to understand where I am coming from. Thank you.

I am currently doing some simulations in digital rock physics (using Comsol Multiphysics software). The compression energy density in the solid can be calculated from $\frac{1}{2}K\Delta^2$; where $\Delta$ is the total dilatation (or fractional change in volume), and $K$ is the bulk modulus parameter of the solid. When I integrate this equation over the volume of the solid, I get the energy. And when I divide the result by the total volume, I get the energy density.

Naturally, if the total volume is changed, the result of the energy density should remain the same, provided the porosity and the material parameters (like the bulk and shear moduli) are the same for both rock volumes. However, I find that this is not so with my models. I have tried using simple 3D models; e.g. I built concentric spheres having the inner sphere as the fluid region and the outer sphere as the solid region. Then I ran the simulation and compared the energy density result with that of another model, which I set up in the same way, but having a different volume. Mind you, these two models both have the same porosity.

My question then is, why am I not getting the same result? Is my energy density equation wrong in the first place (although I doubt that, as that has been of old from Hooke's law for isotropic media).

Thank you for your time. And, if you can, please suggest possible solutions I can try in order to fix this. I have tried everything in my knowledge base.

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  • $\begingroup$ Not a specialist, but just a thought: you say Hooke's law applies to isotropic media, and you say you have a solid and a fluid in your sphere. Is your sphere really isotropic with two different states of matter? $\endgroup$ – Jean-Marie Prival Feb 18 at 9:47
  • $\begingroup$ @Jean-MariePrival, yes, I have set up the model in such a way as to simulate isotropic stresses and strains. I achieved this by applying the Dirichlet and Neumann boundary conditions on the outer walls and pore walls, respectively. $\endgroup$ – Somto Feb 18 at 10:05
  • $\begingroup$ What is causing the dilation in your model? Is it a constant value that you specify, or generated by, for instance, gravitational compression? $\endgroup$ – M Juckes Feb 18 at 21:58
  • $\begingroup$ It is constant. $\endgroup$ – Somto Feb 19 at 1:53
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Eureka! I and my prof figured it out. We found something interesting - that the energy density actually depends on a number of internal variables, namely, the displacements of the outer and inner walls, and porosity (which we already knew). So, when we specified the outer displacement to always be a fraction of the radius of the outer sphere, that seemed to fix the problem.

Thanks to all who attempted to help me out! :)

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