Measurement of Pore Distribution and Compression Anisotropy by Nuclear Magnetic Resonance

Abstract

Coal pores and fractures vary in geometric structure, size distribution, and compressibility in different directions, referred to as the anisotropy of coal pores and fractures (APF). For high-dip coal seams, this anisotropy impacts coal reservoirs more severely owing to stress variations. A low-field nuclear magnetic resonance (NMR) test can detect the pore size distribution under different confining pressures. By changing the angle between the axis of the columnar sample and the coal bedding plane, the NMR test could express the evolution of the APF with stress. Pressurized NMR tests were performed to extrapolate the variation in the pore size distribution and compressibility with the dip angle. T₂ spectra of each columnar sample were divided into two peaks (P1 and P2) with T₂ = 2.5 ms as the boundary. By establishing a linear elastic model, the pore size distribution of P1 can be regarded as a superposition of the micropore deformation controlled by APF. Based on the predominant orientation of the pore structure, the anisotropy of the micropores was created before the formation was tilted. Most (80.1%, volume percentage) of the micropores had geometric spindles parallel to the bedding plane, and most were ellipsoids. Through stress and spectral analyses, we found that the seepage fractures were mostly (80.45%, volume percentage) occupied by the cleat system perpendicular to the bedding plane. The fracture compressibility was related to the normal stress received. The dip angle decomposed the confining pressure in the coalbed and changed its compressibility. As the pressure increased, the multi-scale effect narrowed the main fracture, but did not change other fractures, which reduced the anisotropy of fracture.

Appendices

Appendix

Settings of the Ellipsoid Model

Physical Setting

The space in the cylinder, except for the ellipsoid, was regarded as the coal matrix. Considering the matrix as a linear elastic formula material, based on the generalized Hooke’s law, we have:

$$\begin{aligned} {\boldsymbol{\sigma}} & = \frac{E}{{\left( {1 + \nu } \right)}}{\boldsymbol{\varepsilon}} + \frac{\nu E}{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}{\text{tr}}\left( {\boldsymbol{\varepsilon}} \right){\varvec{I}} \\ {\boldsymbol{\sigma}} & = \left[ {\begin{array}{*{20}c} {\sigma_{x} } & {} & {} \\ {} & {\sigma_{y} } & {} \\ {} & {} & {\sigma_{z} } \\ \end{array} } \right] \\ {\boldsymbol{\varepsilon}} & = \left[ {\begin{array}{*{20}c} {\varepsilon_{x} } & {} & {} \\ {} & {\varepsilon_{y} } & {} \\ {} & {} & {\varepsilon_{z} } \\ \end{array} } \right] \\ \end{aligned}$$

(14)

where ${\boldsymbol{\sigma}}$ is the stress, ${\boldsymbol{\varepsilon}}$ is the strain, and the angle indicates the direction; $\nu$ is Poisson's ratio and $E$ is Young's modulus. Under imitating laboratory conditions, pressure was applied on the side of the cylinder, and the vertical strain was limited to the two bottom surfaces, thus:

$${{\sigma }_{x}=\sigma }_{y}={P}_{c}$$

(15)

$${u}_{z}=0$$

(16)

where ${P}_{c}$ denotes the confining pressure, and ${u}_{z}$ is a component of the displacement field ${\varvec{u}}$, and

$$\begin{aligned} {\boldsymbol{\varepsilon}} & = \frac{1}{2}\left( {\nabla {\varvec{u}} + {\varvec{u}}\nabla } \right) \\ {\varvec{u}} & = \left[ {\begin{array}{*{20}c} {u_{x} } & {} & {} \\ {} & {u_{y} } & {} \\ {} & {} & {u_{z} } \\ \end{array} } \right] \\ \end{aligned}$$

(17)

The displacement field on the surface of the ellipsoid can be obtained using Eq. (17). The volume of the ellipsoid can be calculated as:

$$\begin{aligned} V & = \mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot} {\left( {\frac{{{\text{tr}}\left( {\left( {\mathbf{x} + \mathbf{u}} \right)^{2} } \right)}}{{{\text{tr}}\left( {\mathbf{x}^{2} } \right)}}} \right)^{{\frac{3}{2}}} {\text{d}}V_{0} } \\ \mathbf{x} & = \left[ {\begin{array}{*{20}c} x & {} & {} \\ {} & y & {} \\ {} & {} & z \\ \end{array} } \right] \\ \end{aligned}$$

(18)

$({V}_{0}-V)/{V}_{0}$ was used to measure the change in the pores (Eq. 11). In addition, the pore change degree per unit of confining pressure change (r) had a physical meaning similar to the linear fitting slope $k$ in the test results (Eq. 5). The difference between $r$ and $k$ is that one is a measure of the deformation of a single pore and the other is a measure of the entire sample.

Parameter Setting

The calculated results were controlled mainly by the following points: the mechanical parameters of the coal matrix (Young’s modulus $E$, Poisson’s ratio $\nu$), relative size of the matrix microelement (the ratio of the side length $L$ to the major axis $a$), shape (eccentricity), and pore major axis inclination ($\varphi$).

The empirical constants for the mechanical parameters were selected and adjusted appropriately based on the test results. The relative size of the matrix microelements affects the compressibility of pores (Li et al., 2019a, 2019b). The other parameters were fixed, gradually increasing $L/a$ and calculating the volume $V$ after pressurization. The value of $V$ tended to stabilize with an increase in $L/a$ (Fig. 16). The minimum value at which $V$ reaches stability can be used as the microelement size. The shape of the pores was controlled by the relative lengths of the three axes. To avoid the appearance of repeated shapes, $a\ge b\ge c$ was specified. In this case, $1\ge {e}_{ac}\ge {e}_{bc}\ge 0$. The inclination angle of the long axis of the pore affected the stress state. This inclination angle can be considered as the angle $\varphi$ between the long axis a and horizontal plane. However, it should be noted that the inclination angle here was different from the sampling dip angle $\theta$ in the test.

Figure 16

Volume $V$ after pressurization vs. the relative size of the matrix microelement $L/a$

16).

Derivation of Eqs. 11 and 12

Under the experimental conditions, the confining pressure was applied only in the radial direction, and the axial direction was fixed by the core holder. Therefore, the radial strain was equal, and the axial strain was 0. The strain components expressed by the confining pressure can be obtained as:

$$\left\{ {\begin{array}{*{20}l} {\varepsilon_{x} = \frac{1}{E}\sin^{2} \theta \left( {1 - \nu } \right)\left( {1 + 2\nu } \right)\left( {P_{c} - p} \right)} \hfill \\ {\varepsilon_{y} = \frac{1}{E}\left( {1 - \nu } \right)\left( {1 + 2\nu } \right)\left( {P_{c} - p} \right)} \hfill \\ {\varepsilon_{z} = \frac{1}{E}\cos^{2} \theta \left( {1 - \nu } \right)\left( {1 + 2\nu } \right)\left( {P_{c} - p} \right)} \hfill \\ \end{array} } \right.$$

(19)

where ${\varepsilon }_{x},{\varepsilon }_{y},{\varepsilon }_{z}$ are the 11,22,33 components of the strain tensor in the cleat coordinate system, where the xy plant is the bedding surface and the z-axis is perpendicular to the bedding plane. In this study, it was considered that the x direction represents the stress direction of the main crack (cleat), and the z direction represents the stress direction of the secondary crack. From the generalized Hooke's law, the relationship between the stress and strain can be expressed as:

$$\left\{ {\begin{array}{*{20}l} {\varepsilon_{x} = \frac{1}{E}\left( {\sigma_{x} - \nu \left( {\sigma_{y} + \sigma_{z} } \right)} \right)} \hfill \\ {\varepsilon_{y} = \frac{1}{E}\left( {\sigma_{y} - \nu \left( {\sigma_{x} + \sigma_{z} } \right)} \right)} \hfill \\ {\varepsilon_{z} = \frac{1}{E}\left( {\sigma_{z} - \nu \left( {\sigma_{x} + \sigma_{y} } \right)} \right)} \hfill \\ \end{array} } \right.$$

(20)

Combining the above two formulas, we can get:

$$\left\{ {\begin{array}{*{20}c} {\sigma_{1} = \left( {2\nu \cos^{2} \theta + \sin^{2} \theta } \right)P_{c} } \\ {\sigma_{2} = \left( {2\nu \sin^{2} \theta + \cos^{2} \theta } \right)P_{c} } \\ \end{array} } \right.$$

(21)

For volume changes,

$$V-{V}_{0}={V}_{p}-{V}_{p0}+{V}_{s}-{V}_{s0}$$

(22)

Dividing both sides of Eq. (22) by the initial volume and pressure, the left-hand side of the equation is approximated in the form of the compressibility, thus:

$$\frac{V-{V}_{0}}{{V}_{0}P}=\frac{{V}_{p}-{V}_{p0}}{{V}_{0}P}+\frac{{V}_{s}-{V}_{s0}}{{V}_{0}P}$$

(23)

Combined with the definition of volume ratio ${\xi }_{2}$ and the above formula, we get:

$$\frac{V-{V}_{0}}{{V}_{0}P}=\frac{{V}_{p}-{V}_{p0}}{\frac{{V}_{P0}}{{\xi }_{2}}\frac{{\sigma }_{1}}{{\mathrm{sin}}^{2}\theta }}+\frac{{V}_{s}-{V}_{s0}}{\frac{{V}_{s0}}{1-{\xi }_{2}}\frac{{\sigma }_{2}}{{\mathrm{cos}}^{2}\theta }}$$

(24)

On the premise that the compressibility of a crack is related only to its normal stress, the following definitions

$$\left\{ {\begin{array}{*{20}l} {C_{p} = - \frac{{V_{p} - V_{p0} }}{{V_{P0} \sigma_{1} }}} \hfill \\ {C_{s} = - \frac{{V_{s} - V_{s0} }}{{V_{s0} \sigma_{2} }}} \hfill \\ \end{array} } \right.$$

(25)

are brought into Eq. (23) to obtain:

$$C={C}_{p}{\xi }_{2}\left(2\nu {\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)+{C}_{s}(1-{\xi }_{2})\left(2\nu {\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta \right)$$

(26)