Fundamentals of Fluid Flow in Porous Media


Chapter 3

Diffusion Coefficient: Measurement Techniques

Computer-Assisted Tomography

Computer-assisted Tomography (CAT) scanning using X-ray has been extensively used in research laboratories around the world for reservoir rock characterization and fluid flow visualization.

Principles of CAT Scanning and Processing

X-rays lose their energy as they pass through a medium, and this reduction depends on the density of the substance and the path length through that substance. CAT is based on emitting x-rays from a source which revolves around the object in consideration while one-dimensional projections of attenuated x-rays are collected by a detector on the other side of the source. These projections are collected as the sample travels through the scanner longitudinally and are used to reconstruct a two-dimensional image of the object.

Intensity values of attenuated x-rays are collected from small volumetric elements, called pixels. These elements are typically 0.40×0.40 mm in area and 3 cm in depth (along the direction of the x-ray beam) for a second generation CAT scanner. Once these elements are all assigned an intensity values after a complete radial and longitudinal scan, these data are processed by a computer. This processing constitutes the major part of the CAT. The inlet intensity and the outlet intensity are related through the following relationship:
Inlet and Outlet Intensity Relationship


I = The intensity remaining after the X-ray passes through a thickness (kV),

Io = The incident X-ray intensity (kV),

µ = Linear attenuation[2] coefficient,

L = Path Length.

This relationship applies only for a narrow mono-energetic beam of x-ray photons which travels across a homogeneous medium. If the medium in consideration is heterogeneous, the above equation holds true while replaced by the line integral of the linear attenuation coefficients. The modified form is:
Equation 3-44

The following equation relates the linear attenuation coefficients to the number stored in computer (known as the CT numbers or CTn),
CT number


CTn = CT number,

μi = x-ray linear attenuation coefficient of the object scanned,

μw = x-ray linear attenuation coefficient of water.

Linear attenuation coefficient (µ) is a function of the bulk density and the effective atomic number of the sample, given by:
Linear attenuation coefficient is a function of bulk density and effective atomic number


ρ = Bulk oil density (kg/m3)

a = Energy-independent coefficient called Klein- Nishina coefficient

b = Constant

Z = Effective atomic number of the sample

E = Mean photon energy (kV)

When exposing a medium to x-rays, gathering the exiting x-rays from the medium (Figure 3‑22), and averaging the intensity at each cross section, a transmitted intensity vs. elevation curve can be constructed (Figure 3‑23). The resulted curve could be converted to a density curve. According to the relation between the x-ray intensity and density.

Schematic View of CAT Scanning Using X-Ray
Figure 3-22: Schematic View of CAT Scanning Using X-Ray

A series of calibration tests for liquid and solid samples of known densities are performed in order to correlate the CT numbers (as the intensity of the detected x-ray), generated by the scanner, to densities. In Figure 3‑23 the calibration curves for liquid samples and liquid/solid samples are shown respectively. Using these calibration curves, the densities of the scanned samples can be back calculated.

Calibration Curves for the CAT Scanning, (a) Liquid Calibration Curve, (b) Liquid-Solid Calibration Curve
Figure 3-23: Calibration Curves for the CAT Scanning, (a) Liquid Calibration Curve, (b) Liquid-Solid Calibration Curve[3]

In contrast to the refractive index method, this method has the ability to operate with opaque solutions such as bitumen + pentane.


[1] L. Song, A. Kantzas, J. Bryan, 2010.

[2] In physics, attenuation (in some contexts also called extinction) is the gradual loss in intensity of any kind of flux (X-Ray) through a medium.

[3] D. Salama and A. Kantzas, 2005.


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