Property voxelization

The first part of the present chapter describes how three-dimensional seismic facies can help characterizing the reservoir. Other techniques are available, as described in the chapters [29, 36]. These facies, also called geobodies, represent some typical geological features, which in turn can be linked to lithologies, thereby giving a more detailed and better structural description of the reservoir. The approach adopted here for linking three-dimensional seismic facies to lithologies is explained later in the property voxelization step. 

Property Voxelization. Today commercial simulator software usually require as input values of porosity and permeability at each grid cell of the simulation grid. In the classical workflow this is done in the property population step. The most common methods used are geostatistical approaches such as kriging or co-kriging [11], stochastic simulation methods [16], or Bayesian... 

A comparison of the distribution of porosity values for both wells shows relatively higher values for the well G15, but both are slightly bimodal with their main mode at 35%. On the other hand, the distribution of permeability values of both wells are more differentiated the well G21S shows a wider range with a clear bimodal distribution with a main mode at 8500 mD and a secondary mode around 2500 mD. The well G15 is slightly bimodal with a main mode at 8500 mD and a minor one at 5500 mD. These changes in the porosity and permeability values for each seismic facies and their respective position within the field have been taken into account during the property voxelization step. 

It is noteworthy that the use of three-dimensional seismic facies information during the property voxelization step is not unique. It would have been possible to use a rock physical transform (such as Han s relations for shaley sandstones) giving porosity as a function of P-wave velocity and volume of... 

A property voxelization step allowed to populate the model with rock properties based on (1) empirical models linking porosity and permeability to acoustic impedance, or (2) empirical models and additional constraints given by the calibration of the seismic facies with well data. 

Molecular calculations provide approaches to supramolecular structure and to the dynamics of self-assembly by extending atomic-molecular physics. Alternatively, the tools of finite element analysis can be used to approach the simulation of self-assembled film properties. The voxel4 size in finite element analysis needs be small compared to significant variation in structure-property relationships for self-assembled structures, this implies use of voxels of nanometer dimensions. However, the continuum constitutive relationships utilized for macroscopic-system calculations will be difficult to extend at this scale because nanostructure properties are expected to differ from microstructural properties. In addition, in structures with a high density of boundaries (such as thin multilayer films), poorly understood boundary conditions may contribute to inaccuracies. 

Generally, all analytical methods can be applied for VOls or on the voxel level to calculate the parametric images. These images reflect certain properties of the tracer within their spatial distribution. The parametric images can be helpful to support diagnostic or therapeutic decisions. Figure 2 demonstrates a comparison of the different parametric imaging methods. 

In a practical implementation, the domain on which the phase volume functions are specified is typically a cubic grid of Nx x Ny x Nz voxels, which corresponds to real dimensions of Lx — hNx, Ly - hNy, and Lz — hNz, where h is the voxel size. We will further call this region of real space the computational unit cell. The relationship between the unit cell and the multiphase medium of interest depends on the absolute dimensions of the medium and on the spatial resolution at which the medium is represented (feature dimensions). The unit cell can either contain the entire medium and some void space surrounding it, as in the case of virtual granules described in Section IV.D below, or be a sample of a much larger (theoretically infinite) medium, as in the case of transport properties calculation, described in Section II.E below. 

When CRs are in extra vascular volume, they occupy extra cellular space, and are designated as CRo (outside) where they directly bind with extra cellular water. 1H2O0. Since the transverse relaxation time, T is an intensive property of F O, its CR induced change depends directly on the molar ratio of CRo to H2O0. Thus measurement of T allows the direct determination of the concentration of CR (i.e.) [CR]o in the space in which CR is distributed. Further, because most of the water is intracellular ( F Oj), the change in the tissue l- O T from the entire voxel by CR allows the determination of the kinetics of water movement across the cell membrane. The kinetics is characterized by the average lifetime of a water molecule inside a cell, x. and is inversely proportional... 

The observed tissue time-density curve represents a combination of the effects of the AIF and the inherent tissue properties. Thus, to fit the model, the effects of the AIF on the tissue concentration curve must be removed using a mathematical process known as deconvolution to derive R(t), which is dependent only on the hemodynamic properties of the voxel under consideration. R(t) demonstrates an abrupt (indeed, instantaneous) rise, a plateau of duration equal to the minimum transit time through the tissue of interest, and then decay towards baseline. It was shown by Maier and Zierler [110] that the tissue concentration curve can be represented as the CBF multiplied by AIF convolved with R(t) ... 

The sensitivity of a small volume dv (voxel) within a material is a measure of how much this volume contributes to the total measured impedance (Geselowitz 1971), provided that the electrical properties (e.g., resistivity) are uniform throughout the material. If, in this case, the resistivity varies within the material, the local resistivity must be multiplied with the sensitivity to give a measure of the volume s contribution to the total measured resistance. 

For the dynamic lung impedance model to be useable in Finite Difference Method or Finite Element Method impedance signal simulations, the dynamic tissue sample model is discretized into volume data. At first 3D data with 35 x 35 x 35 voxel resolution is prepared from each of the 40 time frames. This allows for easy import into MATLAB or COMSOL based calculation. The volume data includes percentage of blood vessels (blood) for each of the 35 X 35 X 35 X 40 voxels. It can readily be transformed into electric/dielectric properties for each voxel with tissue data available on the internet. But data can also be exported with arbitrary resolution depending on calculation-simulation requirements. The simulations are run separately for each of the 40 time-frames to get full frequency characteristic of impedance measurement across the tissue sample. Finally we can get 40 frequency characteristics—one for each time-frame and to see a dynamic electrical impedance signal on a certain frequency, we just need to plot the impedance value at the chosen frequency from the 40 time-frames. 

Usually, each voxel is characterized by two different values an opacity value that defines the light quantity that crosses the voxel, and a shading value that represents the reflection or diffusion properties of the tissue. 

Quantitative MRI is possible by calculating the real T1 and T2 figures from the T1 and T2 weighted acquisitions, using the Bloch equation of MRI physics. Multi-modal MRI scans can be exploited for tissue classification when different MRI techniques are applied to the same volume, each voxel is measured with a different physical property, and a feature space can be constructed with the physical units along the dimensional axes e.g. in the characterization of tissue types in atherosclerotic lesions with Tl, T2 and proton density weighted acquisitions, fat pixels tend to cluster, as do blood voxels, muscle voxels, calcified voxels, etc., see Figure 9.8. 

On the other hand, the object that is to be analyzed with dual-energy techniques has to have properties that allow a diagnostically useful differentiation. In order to quantify the spectral behavior of different materials, a Dual-Energy Index can be calculated independently from the mere CT density as the relation of attenuations of the same voxel divided by its mean attenuation at the different tube potentials (Eq. 5.1) ... 

A voxel model then defines a voxelized voxel grid, i.e., a voxel grid associated with numeric values representing some measurable properties or independent variables of the real phenomenon or object residing in the unit volume represented by the voxels [20]. These properties or variables can be of different types sampled data, computed data, simulation results etc. 

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