3.2.1. NMR Theory

NMR can effectively reveal the important information about pore structure of rock, based on the T2 transverse relaxation time [15]. The total transverse relaxation T2 time is associated with three relaxation members: bulk relaxation T2B, surface relaxation T2S, and diffusion of pore fluid T2D, described as [40].

$$\frac{1}{\text{T}\_2} = \frac{1}{\text{T}\_{2\text{B}}} + \frac{1}{\text{T}\_{2\text{S}}} + \frac{1}{\text{T}\_{2\text{D}}},\tag{1}$$

Generally, the bulk relaxation and diffusion relaxation are usually ignored when magnetic field is uniform. In this case, T2 can be related directly to pore size: [16].

$$\frac{1}{\text{T}\_2} \approx \frac{1}{\text{T}\_{2\text{S}}} = \rho \frac{\text{S}}{V} = \rho \frac{a}{r} \tag{2}$$

where ρ (μm/s) is the transversal surface relaxation rate; *S* (μm2) and *V* (μm3) are the surface area and fluid volume of pore space, respectively; the surface/volume ratio (*S*/*V*) is a function of pore radius *r* (μm), and *a* is the pore shape factor (*a* = 3 for spherical pore, while *a* = 2 for tubular pore).

Thus, the T2 distribution under fully brine-saturated conditions can be converted to the curve of pore size distribution by Equation (2), with the help of ρ. Details of this method have been shown in the previous studies [41].

#### 3.2.2. Multifractal Methods Based on NMR T2 Distributions

Numerous studies have introduced the algorithm of multifractal theory in detail [17,25,42,43]. In this study, the popular box counting method [31] was employed for the implement of multifractal algorithm on the basis of the 100% water-saturated T2 distributions of the samples. T2 distributions are split into *N* square boxes of size *r*, here *r* = 2*m* (*m* = 0, 1, 2 ... ). The probability mass distribution function *Pi*(*r*) of the *i*th box could be represented as

$$P\_i(r) = \frac{M\_i(r)}{\sum\_{i=1}^{N(r)} M\_i(r)},\tag{3}$$

where *Mi* is the pore volume in the ith box and &*<sup>N</sup>*(*r*) *i*=1 *Mi*(*r*) is the total porosity. If porosity has a multifractal distribution, and then *Pi*(*r*) has a power exponent relationship to *r*, as follows:

*Pi*(*r*) ∝ *r*α*<sup>i</sup>* , (4)

where α*i* is the singularity strength for boxes [23]. Furthermore, the number of boxes with a similar α value is defined as *N*α(*r*), by the relationship:

$$N\_a(r) \propto r^{-f(a)},\tag{5}$$

where *f*(α) is a multifractal or singularity spectrum, expressing the fractal dimension of boxes with similar values of α [19]. Furthermore, *f*(α) could reach its maximum value when the following conditions are met:

$$\frac{\mathrm{d}\mathrm{f}(\alpha)}{d\alpha(q)} = 0,\tag{6}$$

In order to accurately acquire the distribution properties, the partition function is expressed as:

$$X(q,r) = \sum\_{i=1}^{N(r)} P\_i^q(r) \propto r^{\pi(q)},\tag{7}$$

where *q* is a moment expressing the contribution to *<sup>X</sup>*(*q*, *r*) of boxes with diverse *Pi*(*r*), which is commonly defined as [−10, 10]; When *q* < 0, *<sup>X</sup>*(*q*, *r*) represents the density probability of the area with low concentration of porosity; when *q* > 0, *<sup>X</sup>*(*q*, *r*) denotes the density probability of the area with high concentration of porosity [44]. Moreover, τ(*q*), known as mass exponent, could be depicted by:

$$\pi(q) = -\lim\_{r \to 0} \frac{\log \sum\_{i=1}^{N(r)} P\_i^q(r)}{\log r},\tag{8}$$

The generalized multifractal dimension *Dq*, another way to characterize singularity, which can be defined as:

$$D\_{\emptyset} = \frac{\pi(q)}{q - 1},\tag{9}$$

On the other hand, the α(*q*), *f*(α) can also be determined from τ(*q*) with the Legendre transformation, respectively [43]:

$$a(q) = \frac{d\tau(q)}{dq},\tag{10}$$

$$f(a) = qa(q) - \pi(q),\tag{11}$$

Generally, four sets of parameters, such as *Dq*, α(*q*), *f*(α) and Δα (=α*max*<sup>−</sup>α*min*), are commonly applied to characterize pore structure heterogeneity. The more complex and nonhomogeneous pore structure corresponds to the larger value of *Dq* and Δ<sup>α</sup>.
