Theory

Multiexponential relaxation and local diffusion cells distribution

  
Uniform PENalty inversion algorithm (UPEN)

Posters

Multiexponential relaxation 
and local diffusion cells distribution
 

  
The basic phenomenon used to interpret this effect is the diffusion of a water molecule from the centre of a pore to the walls, where the relaxation is enhanced. The strength of the surface effect is usually represented by the parameter ρ (having the dimensions of a velocity). If diffusion is sufficient to maintain the nuclear magnetization nearly constant over the pore space, including at the surfaces, and if bulk relaxation is negligible (conditions usually satisfied in porous rocks), the relaxation time is inversely proportional to S/V and the simple relationship 1/T_1,2 = ρ S/V  will apply. Here the subscripts 1 and 2 of the relaxation time stand for the longitudinal (T₁) and transverse (T₂) components of the nuclear magnetization, respectively, of ¹H nuclei.

 
What will happen in real permeable porous media, made of networks of interconnected pores, of different shapes, and with distributions of characteristic lengths?
  

Relaxation represent an average over regions that can be much larger than an “individual pore”. If  diffusion is fast enough to maintain magnetization nearly constant within distances of diffusion in a local relaxation time, that is, within the local diffusion cell, the local relaxation rate is proportional to an  S/V averaged over this local volume. It is often useful to regard the local V/S as an operational definition of “pore size”. 
  

If the pore-space is heterogeneous in ρ and/or S/V only on a scale shorter than local diffusion lengths, which implies uniform local diffusion lenghts and local relaxation times, the relaxation of the sample will be single-exponential. Otherwise, the distribution of local S/V will be reflected by a multiexponential relaxation [J. Appl. Physics 1996, 3656-3664].


The model of local relaxation averaging over local diffusion cells was well validated [Magn. Reson. Imaging, 12, 1994, 191-195] by an experiment of progressive desaturation of an initially fully-saturated rock sample.  If we saturate a sandstone sample with water, we get a distribution covering about three decades, from a ms to one second. After progressive desaturation by centrifugation of the sample we have a clear visualization of the desaturation process: most of the water leaves the larger pores, so we observe a progressive vanishing of the amplitudes at high T₁ values. The  progressive shift of some of the signal to shorter T₁ values reflects less mixing by diffusion between adjacent pores of different sizes, one or both of which is partly emptied. Furthermore the larger pores have probably lost a larger fraction of fluid. Thus, both small and large pores may have relaxation time decreased.
   

UPEN
 

• Inversion from signal time to relaxation times

• Errors of Fit, Noise, Quality of Fit

• Resolution of Two Equal Lines

• Linewidth due to the Noise

• Significance Criterion for Improvement in Fit

Inversion from signal time to relaxation times

The function to minimize is the squared error of fit plus a penalty term. Usually some smoothing is implemented in the distributions by means of a penalty  function in order to avoid excess variation. We have proposed an algorithm (Uniform-PENalty, UPEN)) to obtain distributions of relaxation times, tested on rocks and other porous materials, as well as artificial data, where we could see distributions more than three decades wide. UPEN uses a smoothing coefficient varying with the relaxation time and determined by iterative feedback in such a way that the smoothing penalty, rather then the coefficient, is roughly uniform.

UPEN: Uniform PENalty inversion of multiexponential decay data

The first line is weighted (error of fit)²; the second is the penalty.
B_i is the weighting factor, 
g_k the computed amplitude,
t_i is data time, 
T_k is relaxation time, 
s_i is signal amplitude. 
A_k is amplitude penalty coefficient (=0 for times covered by data),
C_k is curvature penalty coefficient (with negative feedback applied to the penalty through c_k).

UPEN tends to avoid more maxima and minima in the distribution than demanded by the data, minimizing the appearance of separate populations in the computed distributions to the extent permitted by the data.

This is an example of a carbonate rock saturated by water. UPEN gives a sharp line that is not broadened more than is consistent with the noise and in the same distribution shows a tail decades long without breaking it up into several peaks not required by the data to be separate. On the contrary, other solutions obtained by fixed smoothing coefficient or by singular-value decomposition (with substantially the same rms error of the fit) in any case tend to broaden the major peak  and break the tail into more peaks, which might be interpreted as physically meaningful resolved compartments.

Errors of Fit, Noise, Quality of Fit
  

The E_i 's are the errors of fit.  For T₂ data, signal amplitude s_i may represent a window average of B_i echoes.  R_r (R for noise: Rumore, Rauschen) is an estimate of individual-echo error of fit, using a compromise weighting to give equal weights at long and short times, rather than Σ(B_iE_i)²/ΣB_i to give the best statistical estimate of the ensemble average echo noise R.  This compromise weighting is used for all the R's.

If there is a slowly-varying error of fit in addition to random noise, one may get a better estimate of the noise from R_v (v for variation), computed from differences of the E 's for second-nearest neighbors, since a non-random slowly-varying error of fit would not greatly affect these differences.

R_rv = ln ( R_r / R_v )

Resolution of Two Equal Lines

Linewidth due to the Noise

Significance Criterion for Improvement in Fit