A Nonparametric Approach for Determining NMR Relaxation Distributions

Abstract

Nuclear magnetic resonance (NMR) and imaging (MRI) can provide unique information about fluid distributions, flow, transport properties, and morphology within permeable media. The mathematical description of NMR relaxation of fluids in permeable media is one key element used to develop such information. We introduce and evaluate nonparametric regression theory to determine continuous relaxation distribution functions from NMR/MRI experiments. Unlike other methods, ours is based on determining the best estimate of the distribution function from experimental data. Our method is robust and does not require user interventions, so it is particularly valuable for accurately determining fluid distributions from MRI experiments. Such information is essential for determining porosity, permeability, and fluid saturation distributions, as well as permeable media morphologies.

Appendices

Appendix 1: Modeling Relaxation in Heterogeneous Media

In this appendix, we summarize the relaxation model developed by Zimmerman and Brittin (1957) and highlight some of the key equations. We have adopted some changes in nomenclature, inspired in part by a recent review of their work (Callaghan 2011). We also review the Brownstein and Tarr (Brownstein and Tarr 1977) model and indicate additional assumptions typically made to describe morphology.

The observed spins are distributed within $n_p$-phases having different characteristic relaxation times. The probability that a spin chosen at random is in the ${i\hbox {th}}$ phase is $P_i$. Over the course of the observation time, the spins occupy different phases due to molecular diffusion. The process is assumed to be Markovian. In the absence of relaxation, the Kolmogoroff equation (Seinfeld and Lapidus 1974) governs $P_{ij}(t)$, the conditional probability that a spin, initially in the $i$th phase, is in the $j$th phase at a later time $t$:

$$\begin{aligned} \frac{\hbox {d} P_{ij}(t)}{\hbox {d} t} = -\frac{1}{\tau _j} P_{ij}(t) + \sum _k P_{ik}(t) \frac{1}{\tau _k} p_{kj} \end{aligned}$$

(65)

where $1/ \tau _j$ is the probability per second that a spin in the ${j\hbox {th}}$ phase leaves the ${j\hbox {th}}$ phase, and $p_{kj}$ is the conditional probability that if a spin leaves the ${k\hbox {th}}$ phase it will transfer to the ${j\hbox {th}}$ phase.

An additional term is introduced to account for relaxation, and the state is redefined as $I_{ij}(t)$:

$$\begin{aligned} \frac{\hbox {d} I_{ij}(t)}{\hbox {d} t} = -\frac{1}{\tau _j} I_{ij}(t) - \frac{1}{T^{(j)}} I_{ij}(t) + \sum _k I_{ik}(t) \frac{1}{\tau _k} p_{kj} \end{aligned}$$

(66)

where $1/T^{(j)}$ is the relaxation rate of phase $j$. This equation can be written in matrix form as:

$$\begin{aligned} \frac{\hbox {d} \, \mathbf{I}}{\hbox {d}t} = - \mathbf{I}\, \mathbf{F}\end{aligned}$$

(67)

where

$$\begin{aligned} F_{ij} = \delta _{ij} \left[ \frac{1}{\tau _j} + \frac{1}{T^{(j)}} \right] -\frac{p_{ij}}{\tau _i} \end{aligned}$$

(68)

and $\delta _{ij}$ denotes the delta function. The solution to Eq. (67) is:

$$\begin{aligned} \mathbf{I}(t) = \exp {\left[ - \mathbf{F}\, t \right] }. \end{aligned}$$

(69)

The relaxation function is:

$$\begin{aligned} r(t)&= \sum _{i,j} I_{ij}(t) P_i \end{aligned}$$

(70)

$$\begin{aligned}&= \mathbf{u}^T \mathbf{I}\, \mathbf{v}\end{aligned}$$

(71)

for $\mathbf{u}= \left[ \begin{array}{cccc} P_1&P_2&\ldots&P_n \end{array} \right] ^T$ and $\mathbf{v}= \left[ \begin{array}{cccc} 1&1&\ldots&1 \end{array} \right] ^T$.

Suppose that the eigenvalues of $\mathbf{F}$ are distinct. [If not, arbitrarily small changes in coefficients in the matrix $\mathbf{F}$ can guarantee distinct eigenvalues (Franklin 1968).] Then:

$$\begin{aligned} \exp {\left[ - \mathbf{F}\, t \right] } = \mathbf{S}\exp {\left[ - \mathbf{D}\, t \right] } \, \mathbf{S}^{-1}, \end{aligned}$$

(72)

where the diagonal matrix $\mathbf{D}$ comprises the eigenvalues of $\mathbf{F}$ and the columns of $\mathbf{S}$ are the corresponding eigenvectors. The relaxation function becomes

$$\begin{aligned} r(t)&= {\mathbf{u}}^T \mathbf{S}\exp {\left[ -\mathbf{D}\,t\right] } \, \mathbf{S}^{-1} \mathbf{v}\end{aligned}$$

(73)

$$\begin{aligned}&= \sum _{i=1}^{n_p}a_i \exp {\left( -d_i t\right) }. \end{aligned}$$

(74)

This development shows that the relaxation function can be represented as a sum of exponentials. It is noted, however, that the relaxation rate $d_i$ and corresponding strength (or relative amount of fluid) $a_i$ are actually apparent values, in that they may not represent properties associated with any of the specific phases (and corresponding spatial regions). This is illustrated next in an example with two phases. This means that while Eq. (74), or the corresponding continuous distribution form (see Eq. 8), is suitable for determining the intrinsic intensity, it may not be directly pertinent for discerning morphology.

1.1 Two-Phase Situation

Suppose there are two phases. The solution for the parameters in the relaxation model can be readily calculated from the parameters in the stochastic representation.

For this case, the relaxation function is

$$\begin{aligned} r(t) = a_1 \exp {\left( -d_1 t\right) }+a_2\exp {\left( -d_2 t\right) }. \end{aligned}$$

(75)

We can form $\mathbf{F}$ and determine the eigenvalues $(d_i)$ and coefficients $(a_i)$:

$$\begin{aligned} 2 d_1&= \frac{1}{T^{(1)}} + \frac{1}{T^{(2)}}+ \frac{1}{\tau _1}+\frac{1}{\tau _2}-\left[ \left( \frac{1}{T^{(2)}}-\frac{1}{T^{(1)}}+\frac{1}{\tau _2}- \frac{1}{\tau _1}\right) ^2+\frac{4}{\tau _1 \tau _2} \right] ^{\frac{1}{2}}\nonumber \\ 2 d_2&= \frac{1}{T^{(1)}} + \frac{1}{T^{(2)}} + \frac{1}{\tau _1} + \frac{1}{\tau _2} + \left[ \left( \frac{1}{T^{(2)}} - \frac{1}{T^{(1)}} + \frac{1}{\tau _2} - \frac{1}{\tau _1} \right) ^2 + \frac{4}{\tau _1 \tau _2} \right] ^{\frac{1}{2}} \end{aligned}$$

(76)

and

$$\begin{aligned} a_1&= \frac{1}{\hbox {d}_2 - \hbox {d}_1} \left[ \hbox {d}_2 - \frac{1}{P_1}{T^{(1)}} + \frac{1}{P_2}{T^{(2)}} \right] \nonumber \\ a_2&= \frac{1}{\hbox {d}_1 - \hbox {d}_2} \left[ d_1 - \frac{1}{P_1}{T^{(1)}} + \frac{1}{P_2}{T^{(2)}} \right] . \end{aligned}$$

(77)

Note that the apparent relaxation rates and strengths depend on the corresponding properties associated with both phases. We note that the eigenvalues are, in fact, distinct.

1.2 Limiting Cases

Depending on the relative magnitudes of relaxation and exchange rates, two limiting cases can be identified. In the slow-exchange regime, for which $T^{(j)} \gg \tau _i$, the weights of the relaxation terms are given by the probabilities that a spin will be found in the corresponding phase, so that:

$$\begin{aligned} r_{se}(t) = \sum _{i=1}^{n_p} P_i \exp {\left[ - \frac{t}{T^{i}} \right] }. \end{aligned}$$

(78)

In this case, the model is again multi-exponential. However, note that the relaxation times and coefficients now correspond to actual values for the respective phases.

For the two-phase situation (see Eqs. 76 and 77), it is easy to show that under the slow-exchange limit:

$$\begin{aligned} d_1&= \frac{1}{T^{(1)}}\end{aligned}$$

(79)

$$\begin{aligned} d_2&= \frac{1}{T^{(2)}}\end{aligned}$$

(80)

$$\begin{aligned} a_1&= P_1\end{aligned}$$

(81)

$$\begin{aligned} a_2&= P_2. \end{aligned}$$

(82)

In the fast-exchange regime, for which $\tau _i \gg T^{(j)}$, Zimmerman and Brittin used asymptotic analysis to show that the system decay can be approximated with a single exponential:

$$\begin{aligned} r_{fe}(t) = \exp {\left[ - \frac{t}{T^{av}} \right] }, \end{aligned}$$

(83)

where

$$\begin{aligned} \frac{1}{T^{av}} = \sum _{i = 1}^{n_p} \left( \frac{P_i}{T^{i}} \right) . \end{aligned}$$

(84)

1.3 Brownstein and Tarr Relaxation Model

Brownstein and Tarr (1977) took a different approach in developing a relaxation model for fluids in permeable media. They used a continuum formulation and incorporated diffusion into the Bloch equations. They assumed that the relaxation rate was uniform within each of two regions: bulk and surface. The surface region represents a small volume immediately adjacent to the solid, in which relaxation is enhanced, as compared to the bulk. Assuming this region is vanishingly thin, relaxation associated with the surface region can be represented as a boundary condition to the Bloch equations. Using an asymptotic analysis, they showed that, under fast-exchange conditions, the relaxation within a closed region corresponding to a “pore” can be represented with a single exponential having the relaxation rate:

$$\begin{aligned} \frac{1}{T^{av}} = \frac{1}{T^{b}} + \frac{\rho S}{V}, \end{aligned}$$

(85)

where $T^b$ is the relaxation rate associated with the bulk fluid, $\rho$ is the surface relaxivity, and $S/V$ is the surface-to-volume ratio of the pore.

This forms the basis for many investigations of pore-size distributions in permeable media. By assuming that individual pores are in the fast-exchange regime, while neglecting exchange among pores, the system is represented by Eq. (78). Determination of the relaxation times and coefficients provides the relative number of nuclei having a given relaxation time. If the surface relaxivity is known, the surface-to-volume ratio for the ${i\hbox {th}}$ pore can be calculated:

$$\begin{aligned} \left( \frac{S}{V} \right) _i = \left( \rho T^{(i)} \right) ^{-1}. \end{aligned}$$

(86)

This follows from Eq. (85) by neglecting the bulk relaxation rate. The pore-size distribution can thus be determined by scaling the relaxation distribution (Liaw et al. 1996).

It should be noted that the Brownstein and Tarr representation is in fact equivalent to the two-component situation for the Zimmerman and Brittin representation. Using the fast-exchange representation for the relaxation for a pore (Eq. 84):

$$\begin{aligned} \frac{1}{T^{av}} = \frac{P_b}{T_b} + \frac{P_s}{T_s} \end{aligned}$$

(87)

where subscripts $b$ and $s$ refer to properties for the fluid phases associated with the bulk and surface regions, respectively. Define $\eta$ as the thickness of the surface layer. Then

$$\begin{aligned} \frac{1}{T^{av}} = \frac{\left[ 1 - \eta \left( \frac{S}{V} \right) \right] }{T_b} + \frac{\eta }{T_s} \left( \frac{S}{V} \right) . \end{aligned}$$

(88)

Define the surface relaxivity (Uh and Watson 2004):

$$\begin{aligned} \rho = \frac{\eta }{T_s}. \end{aligned}$$

(89)

In the Brownstein and Tarr development, the volume of the surface region is vanishingly small in comparison with the bulk region, and thus Eq. (85) follows.

Appendix 2: Linearly Constrained Quadratic Minimization Problem

In this section, the minimization problem with equality constraints is converted to an unconstrained problem to obtain an explicit expression for the solution. Consider a linearly constrained, quadratic minimization problem:

$$\begin{aligned} \min _{\mathbf{c}} J(\mathbf{c}) = || \mathbf{w} - \mathbf{B} \mathbf{c} ||^2 \end{aligned}$$

(90)

subject to

$$\begin{aligned} \mathbf{G}' \mathbf{c} = \mathbf{0}. \end{aligned}$$

(91)

Here, $\mathbf B$ is a $n \times n_s$ matrix, and $\mathbf w$ and $\mathbf c$ are, respectively, $n \times 1$ and $n_s \times 1$ column matrices (or vectors).

Suppose all dependent constraints are eliminated so that $\mathbf{G}'$ is a $u \times n_s$ matrix with full row rank ($u < n_s$). $\mathbf{G}'$ can be factorized to obtain

$$\begin{aligned} \mathbf{G}' = \left[ \mathbf{L}:\mathbf{0}\right] \mathbf{Q}, \end{aligned}$$

(92)

with a nonsingular $u \times u$ lower diagonal matrix $\mathbf{L}$ and $n_s \times n_s$ orthonormal matrix $\mathbf{Q}$.

We partition $\mathbf{Q}^{T}= [\mathbf{Q}_1^T ~:~ \mathbf{Q}_2^T ~]$, where $\mathbf{Q}^{T}_{1}$ consists of the first $u$ columns of $\mathbf{Q^T}$. Post-multiplying Eq. (92) by $\mathbf {Q}^T$, it follows that

$$\begin{aligned} \mathbf{G}' \mathbf{Q}^{T}_{1} = \mathbf{L} \end{aligned}$$

(93)

and

$$\begin{aligned} \mathbf{G}' \mathbf{Q}^{T}_{2} = \mathbf{0}. \end{aligned}$$

(94)

$\mathbf{Q}^{T}_{1}$ provides a basis for the subspace spanned by the rows of $\mathbf G$, while $\mathbf{Q}^{T}_{2}$ provides a basis for the null space.

The parameter vector can be expressed as

$$\begin{aligned} \mathbf{c} = \mathbf{Q_1}^T \mathbf{c}_1 + \mathbf{Q_2}^T \mathbf{c}_2, \end{aligned}$$

(95)

with $\mathbf{c}_1 \in {\fancyscript{R}}^u$ and $\mathbf{c}_2 \in {\fancyscript{R}}^{n_s - u}$. It follows from Eqs. (91) and (93)–(95), that

$$\begin{aligned} \mathbf{L}\mathbf{c}_1 = \mathbf{0}. \end{aligned}$$

(96)

Since $\mathbf L$ is nonsingular, it follows that $\mathbf{c}_{1} = \mathbf{0}$ and

$$\begin{aligned} \mathbf{c} = \mathbf{Q}_2^T \mathbf{c}_2. \end{aligned}$$

(97)

We can thus calculate the solution to the equality constrained problem by solving the following unconstrained problem:

$$\begin{aligned} \min _{\mathbf{c}_2} J(\mathbf{c}_2) = || \mathbf{w} - \mathbf{B} \mathbf{Q}_2^T \mathbf{c}_2||^2. \end{aligned}$$

(98)

Equation (98) implies that the degrees of the freedom of the performance index is reduced from $n_s$ of $\mathbf c$ to $n_s-u$ of $\mathbf{c}_2$ due to the equality constraints. If $(\mathbf{B}\mathbf{Q}_2^T)^T (\mathbf{B}\mathbf{Q}_2^T)$ is not singular, the estimate of the parameter vector $\mathbf{c}_2$ is derived explicitly as (Beck and Arnold 1977)

$$\begin{aligned} \hat{\mathbf{c}}_2 = \left[ (\mathbf{B}\mathbf{Q}_2^T)^T (\mathbf{B}\mathbf{Q}_2^T) \right] ^{-1} (\mathbf{B}\mathbf{Q}_2^T)^T \mathbf{w}. \end{aligned}$$

(99)

Then, $\hat{\mathbf{c}}$ is calculated as

$$\begin{aligned} \hat{\mathbf{c}} = \mathbf{Q}_2^T \left[ (\mathbf{B}\mathbf{Q}_2^T)^T (\mathbf{B}\mathbf{Q}_2^T) \right] ^{-1} (\mathbf{B}\mathbf{Q}_2^T)^T \mathbf{w}. \end{aligned}$$

(100)

It can be shown that the estimate of the following minimization problem

$$\begin{aligned} \min _{\mathbf{c}} J(\mathbf{c}) = ||\mathbf{w} - \mathbf{B} \mathbf{Q}_2^T \mathbf{Q}_2 \mathbf{c} ||^2 \end{aligned}$$

(101)

gives the same solution of Eq. (100) provided that $\mathbf c$ and $\mathbf{c_2}$ satisfy the relation in Eq. (97) and $(\mathbf{B}\mathbf{Q}_2^T \mathbf{Q}_2)^T (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)$ is not singular. The estimate of $\mathbf c$ for the minimization problem of Eq. (101) is

$$\begin{aligned} {\hat{\mathbf{c}}} = \left[ (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)^T (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)\right] ^{-1} (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)^T \mathbf{w}. \end{aligned}$$

(102)

Multiplying Eq. (102) by $\mathbf{Q}_2(\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)^T (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)$, it follows that

$$\begin{aligned} \mathbf{Q}_2 (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)^T (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2) {\hat{\mathbf{c}}} = \mathbf{Q}_2 (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)^T \mathbf{w}. \end{aligned}$$

(103)

Since $\mathbf{Q}^T$ is orthonormal, $\mathbf{Q}_2 \mathbf{Q}_2^T$ and $\mathbf{Q}_2^T \mathbf{Q}_2$ are $(n_s-u)\times (n_s-u)$ and $n_s \times n_s$ identity matrices, respectively. Therefore,

$$\begin{aligned} \mathbf{Q}_2 (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2)^T = (\mathbf{B}\mathbf{Q}_2^T\mathbf{Q}_2 \mathbf{Q}_2^T)^T = (\mathbf{B}\mathbf{Q}_2^T)^T. \end{aligned}$$

(104)

It follows from Eqs. (103) and (104) that

$$\begin{aligned} (\mathbf{B}\mathbf{Q}_2^T)^T (\mathbf{B}\mathbf{Q}_2^T) \mathbf{Q}_2 \hat{\mathbf{c}} = (\mathbf{B}\mathbf{Q}_2^T)^T \mathbf{w}. \end{aligned}$$

(105)

Multiplying Eq. (105) by $[(\mathbf{B}\mathbf{Q}_2^T)^T (\mathbf{B}\mathbf{Q}_2^T)]^{-1}$ and subsequently multiplying $\mathbf{Q}_2^T$, it follows that

$$\begin{aligned} \mathbf{Q}_2^T \mathbf{Q}_2 \hat{\mathbf{c}} = \mathbf{Q}_2^T [(\mathbf{B}\mathbf{Q}_2^T)^T (\mathbf{B}\mathbf{Q}_2^T)]^{-1} (\mathbf{B}\mathbf{Q}_2^T)^T \mathbf{w}. \end{aligned}$$

(106)

This is the same as Eq. (100) since $\mathbf{Q}_2^T \mathbf{Q}_2$ is an identity matrix.

If

$$\begin{aligned} \mathbf{w} = \left[ \begin{array}{c} \mathbf{y} \\ \mathbf{0} \end{array} \right] \end{aligned}$$

(107)

and

$$\begin{aligned} \mathbf{B} = \left[ \begin{array}{c} \mathbf{A} \\ \lambda ^{1/2} \mathbf{M} \end{array} \right] , \end{aligned}$$

(108)

the unconstrained minimization problem (Eq. 101) and the corresponding estimate of $\mathbf c$ (Eq. 102) can be written, respectively, as

$$\begin{aligned} \min _\mathbf{c} J(\mathbf{c}) = \left\| \left( \begin{array}{c} \mathbf{y} \\ \mathbf{0} \end{array} \right) - \left( \begin{array}{c} \tilde{\mathbf{A}} \\ \lambda ^{1/2} \tilde{\mathbf{M}} \end{array} \right) \mathbf{c} \right\| ^2 \end{aligned}$$

(109)

and

$$\begin{aligned} \hat{\mathbf{c}} = \left[ \left( \begin{array}{c} \tilde{\mathbf{A}} \\ \lambda ^{1/2} \tilde{\mathbf{M}} \end{array} \right) ^T \left( \begin{array}{c} \tilde{\mathbf{A}} \\ \lambda ^{1/2} \tilde{\mathbf{M}} \end{array} \right) \right] ^{-1} \left( \begin{array}{c} \tilde{\mathbf{A}} \\ \lambda ^{1/2} \tilde{\mathbf{M}} \end{array} \right) ^T \left( \begin{array}{c} \mathbf{y} \\ \mathbf{0} \end{array} \right) , \end{aligned}$$

(110)

where $\tilde{\mathbf{A}}$ and $\tilde{\mathbf{M}}$ are defined as $\tilde{\mathbf{A}} = \mathbf{A}\mathbf{Q}_2^T\mathbf{Q}_2$ and $\tilde{\mathbf{M}} = \mathbf{M}\mathbf{Q}_2^T\mathbf{Q}_2$, respectively. These equations are rewritten as

$$\begin{aligned} \min _\mathbf{c} J = \parallel \mathbf{y} - \tilde{\mathbf{A}} \mathbf{c} \parallel ^2 + \lambda \parallel \tilde{\mathbf{M}} \mathbf{c} \parallel ^2 \end{aligned}$$

(111)

and

$$\begin{aligned} {\hat{\mathbf{c}}}_{\lambda } = [\tilde{\mathbf{A}}^{T} \tilde{\mathbf{A}} + \lambda \tilde{\mathbf{M}}^T \tilde{\mathbf{M}}]^{-1} \tilde{\mathbf{A}}^{T} \mathbf{y}. \end{aligned}$$

(112)