La relación con el tempo de la música

Solutions to the Navier-Stokes equations and Laplace equation were calculated using the KD- 18-03 pore geometry for different sample sub-volumes in three orthogonal directions. Figure 4.1 shows the flow diagrams as solutions to the Navier-Stokes equations and electrical current density diagrams as solutions to the Laplace’s equation from a KD-18-03 subsample. Figure 4.1 The sample size shown in Figure 4.1 is (30 voxel)3_{, (23.4 µm)}3 _{with a CMT resolution of} 0.78 µm. The streamline is the velocity field and the streamline color is the velocity magnitude. The surface mesh is displayed in grey. These solutions will be discussed in greater detail in the next section.

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Figure 4.1 Flow diagrams and electrical current density diagrams. A, B and C, show solutions to the Navier-Stokes equations in the x-direction, y-direction and z- direction respectively. D, E and F, show solutions to Laplace’s equations in the x-direction, y-direction and z-direction respectively. The sample size is 30 x 30 x 30 voxel (23.4 x 23.4 x 23.4 µm). The streamlines show the Current Density Magnitude and Velocity Magnitude. The red color represents the high values while the blue color represents the low values.

A summary of the calculated porosity, resistance, resistivity, formation factor and cementation exponent are listed in Table 4.2. Appendix B illustrates the scaling required to convert model calculated results to those for a sample with the given dimensions. The example, used in Appendix B, is Model 1-y5.

Multiple simulations were run in order to investigate the effects of varying model resolution. The difference between the inlet and outlet mass flux was used as an estimate of the model error. [Model 2] has the highest mass flux difference of 8.3%. This high mass flux difference is due to using a coarse mesh with 1,899,423 equations. A finer mesh could not be applied since

x z

y A B C

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topography errors were generated in COMSOL. The majority of the models have a difference of 2.0% or less. The total porosity refers to all the porosity in the isolated subsample while the effective porosity refers to the porosity that is connected to the boundary by a pore network. The lower resolution calculations, such as [Models 1-y1] to [Model 1-y4], were used to assess whether the permeability was a function of the mesh and so only the permeability and error was recorded.

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Table 4.2 Calculated Properties for the KD-18-03 Subsample

__________________________________________________________________________________________________ Model Permeability Total Effective Resistance Resistivity Formation Cementation Figures

Porosity Porosity Factor Exponent (mD) (%) (%) (Ω∙𝑚𝑚 voxel) (Ω ∙ 𝑚𝑚) (F) (m) __________________________________________________________________________________________________ 1-y1 2.63 - - - - 1-y2 2.99 - - - - 1-y3 3.23 - - - - 1-y4 3.54 - - - 1-y5* 3.40 19.7% 18.5% 0.98 29.3 29.3 2.00 4.5 & 4.6 1-z1 2.26 - - - 1-z2 2.65 - - - 1-z3 2.86 - - - 1-z4 3.19 - - - 1-z5* 3.27 19.7% 18.5% 1.00 30.08 30.1 2.02 4.7 & 4.8 1-x5* 3.53 19.7% 18.5% 0.94 28.36 28.4 1.98 4.9 & 4.10 2-z1 0.41 - - - 2-z2 0.60 - - - 2-z3 0.67 - - - 2-z4* 0.67 17.5% 11.8% 1.75 87.26 87.26 2.09 4.11 & 4.12 2-x1* 0.67 17.5% 11.8% 1.78 89.20 89.20 2.10 4.13 & 4.14 2-y1* 0.37 17.5% 11.8% 2.70 135.06 135.06 2.29 4.15 & 4.16 3-y1* 0.10 12.2% 5.8% 6.19 309.43 309.43 2.02 4.17 & 4.18 4-y1* 0.87 14.2% 12.0% 2.27 68.00 68.00 1.99 4.19 & 4.20 4-x1* 0.22 14.2% 12.0% 2.74 82.16 82.16 2.08 4.21 & 4.22 __________________________________________________________________________________________________ Formation Resistivity Factor, F; Cementation Exponent, m;

*Models are plotted and further explained in the referenced Figures.

Similar permeabilities in orthogonal flow directions indicate that the Midale Marly KD-03-18 subsample is nearly isotropic.

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A summary of the computational model settings used for computing the simulations is shown in Table 4.3. The times required to complete the simulations were recorded. This time does not include time allocated for mesh generation. For complex mesh cases, the 3D mesh generation computation took longer than the simulation time. All simulations were completed using COMSOL 4.3. The relative repair tolerance for the STL file was 10-6 for all simulations. The number of equations listed in Table 4.3 correspond to the Navier-Stokes and Laplace PDE solutions.

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Table 4.3 Computational results for the KD-18-03 Subsample

_____________________________________________________________________________ Model Equations Time RAM Error Mesh Figures

(#) (min) (GB) (%) Setting

_____________________________________________________________________________ 1-y1 56,536 1.2 4.7 23.0 Boundary Layer -

1-y2 86,176 1.2 4.6 13.8 Boundary Layer - 1-y3 198,672 1.9 4.8 11.1 Boundary Layer - 1-y4 772,704 6.5 7.7 3.1 Boundary Layer - 1-y5* 4,332,920 28.7 23.8 1.7 Boundary Layer 4.5 & 4.6 1-z1 56,636 0.9 4.6 24.6 Boundary Layer - 1-z2 86,176 1.0 4.7 14.8 Boundary Layer - 1-z3 198,672 1.6 5.1 8.7 Boundary Layer - 1-z4 772,704 5.5 7.3 3.4 Boundary Layer - 1-z5* 4,332,920 28.1 23.9 2.0 Boundary Layer 4.7 & 4.8 1-x5* 4,332,920 27.3 24.0 0.9 Boundary Layer 4.9 & 4.10 2-z1 156,364 5.3 6.0 - Boundary Layer -

2-z2 621,152 8.0 7.0 - Boundary Layer - 2-z3 678,804 9.2 7.0 - Boundary Layer -

2-z4* 1,899,423 198.6 18.0 3.8 Boundary Layer 4.11 & 4.12 2-x1* 1,899,423 152.4 18.0 8.3 Boundary Layer 4.13 & 4.14 2-y1* 1,899,423 137.6 18.3 5.9 Boundary Layer 4.15 & 4.16 3-y1* 5,061,699 35.1 24.0 0.2 Free Tetrahedral 4.17 & 4.18 4-y1* 1,158,653 6.6 8.2 0.5 Free Tetrahedral 4.19 & 4.20 4-x1* 1,158,653 6.5 8.6 0.1 Free Tetrahedral 4.21 & 4.22 _____________________________________________________________________________ *Models are plotted and further explained in the referenced figures.

Random Access Memory, RAM;

Models 1 and 2 used a multigrid iterative solution and a coupled iterative solution Models 3 to 4 used a coupled iterative solution

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Table 4.4 shows the results for permeability, formation resistivity factor and cementation exponent. The new calculated results for the KD-03-18 Marly (M0) subsample vary by less than one order of magnitude and are similar to the experimentally derived values of Glemser (2007). Table 4.4 Results for Permeability, Formation Resisitivity Factor and Cementation

Exponent

__________________________________________________________________________________________ Permeability Formation Resistivity Cementation

(mD) Factor Exponent

__________________________________________________________________________________________

Marly Dolostones* 1-150 - -

Ruska Liquid Permeameter** 7.7 - -

CMT** 7.73 11.71 1.87

CMT (this study) 0.1 to 3.53 29.3 to 309.43 1.99 to 2.10 __________________________________________________________________________________________ *from Churcher, et al. (1994); Glemser (2007)

**from Glemser (2007)

The permeability, formation resistivity factor and cementation exponent were calculated in this study using the natural geometry of the CMT data and the 3D Navier Stokes Equations. The calculated permeability ranges from 0.1 to 3.53 mD as shown in Table 4.2. The formation resistivity factor ranges from 29.3 to 309.43 and the cementation exponent ranges from 1.99 to 2.10.

The cementation exponent can also be calculated using all of the data from the slope of the best fitting line through a plot of the log of the formation factor vs. the log of the effective porosity (Figure 4.2). This method does not require the assumption that a=1. All of the formation resistivity factor data and effective porosity data in Table 4.2 were used to calculate the cementation exponent. The red dot, in Figure 4.2, represents the formation resistivity factor and porosity calculated by Glemser (2007). This plot is shown in Figure 4.2.

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Figure 4.2 Log of Formation Resistivity Factor (F) vs. Log of Effective Porosity (%) The cementation exponent, also the slope of the line, in Figure 4.2, is calculated to be m = 2.11. The cementation exponent was estimated by Glemser (2007), using equation 1.5, to be m = 1.87. Considering that I completed simulations on some smaller subsamples, due to finite computer memory available, comparison between the cementation exponent results are reasonably close.

The extrapolated linear trend line is close to Glemser’s (2007) formation resistivity factor of 11.71. The equation of the trend line was used with an effective porosity of 31.7% to calculate a formation resistivity factor of 10.32.

The value of ‘a’ can be calculated using Archie’s Law, as shown in Equation 1.7. Table 4.5 shows the calculated value of ‘a’ using the total and effective porosity.

y = -2.1097x - 0.0387 R² = 0.934 0 0.5 1 1.5 2 2.5 3 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 Lo g o f Fo rm at io n Re sis tiv ity Fa ct or (F)

Log of Effective Porosity