Evaluation of the Pore Structure of Reservoirs Based on NMR T ₂ Spectrum Decomposition

Abstract

Nuclear magnetic resonance (NMR) T ₂ distributions can be employed to understand the geometry of pores, but they are not sensitive enough to determine pore connectivity. In this paper, the pore space in reservoirs is regarded as a series of different sizes of the combination of spherical pores and cylinder pores based on the pore network model and Sphere–Cylinder model. In terms of the optimized T ₂ inversion method, T ₂ spectral distribution is decomposed into two groups, sphere pore distribution and cylinder pore distribution. It is assumed that cylinder pore controls the connectivity and permeability of the reservoir and the sphere pore is the main place of fluid storage. The more cylinder pores that exist, the better the connectivity and permeability are. The data of eight core plugs are analyzed using this method, and the results are in accordance with those of core analysis, which proves the practicability of our research and provides a basis for the direct evaluation of reservoir pore structure using NMR T ₂ distribution.

Appendix 1

The conjugate gradient (CG) method is proposed by Hesteness and Stiefel for solving linear system of equations [27]. Here, we use this method to inverse the relaxation signal to obtain T ₂ distribution.

As mentioned previously, the observed relaxation signal is modeled as multi-exponential decay:

$$\begin{array}{*{20}c} {{\text{echo}}_{1} = \varphi_{1} e^{{{\raise0.7ex\hbox{${ - t_{1} }$} \!\mathord{\left/ {\vphantom {{ - t_{1} } {T_{2,1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,1} }$}}}} + \varphi_{2} e^{{{\raise0.7ex\hbox{${ - t_{1} }$} \!\mathord{\left/ {\vphantom {{ - t_{1} } {T_{2,2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,2} }$}}}} + \cdots + \varphi_{n} e^{{{\raise0.7ex\hbox{${ - t_{1} }$} \!\mathord{\left/ {\vphantom {{ - t_{1} } {T_{2,n} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,n} }$}}}} } \\ {{\text{echo}}_{2} = \varphi_{1} e^{{{\raise0.7ex\hbox{${ - t_{2} }$} \!\mathord{\left/ {\vphantom {{ - t_{2} } {T_{2,1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,1} }$}}}} + \varphi_{2} e^{{{\raise0.7ex\hbox{${ - t_{2} }$} \!\mathord{\left/ {\vphantom {{ - t_{2} } {T_{2,2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,2} }$}}}} + \cdots + \varphi_{n} e^{{{\raise0.7ex\hbox{${ - t_{2} }$} \!\mathord{\left/ {\vphantom {{ - t_{2} } {T_{2,n} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,n} }$}}}} } \\ { \cdots \quad \quad \quad \quad \quad \cdots \quad \quad \quad \quad \quad \quad \cdots \quad \quad \quad } \\ {{\text{echo}}_{m} = \varphi_{1} e^{{{\raise0.7ex\hbox{${ - t_{m} }$} \!\mathord{\left/ {\vphantom {{ - t_{m} } {T_{2,1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,1} }$}}}} + \varphi_{2} e^{{{\raise0.7ex\hbox{${ - t_{m} }$} \!\mathord{\left/ {\vphantom {{ - t_{m} } {T_{2,2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,2} }$}}}} + \cdots + \varphi_{n} e^{{{\raise0.7ex\hbox{${ - t_{m} }$} \!\mathord{\left/ {\vphantom {{ - t_{m} } {T_{2,n} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{2,n} }$}}}} } \\ \end{array}$$

(12)

where m and n are echo number and T ₂ bin set number, respectively; t _i (i = 1,2,…,m) is the measured time of i-th echo, which is the integer multiple of inter-echo spacing time T _E; T _2j (j = 1,2,…,n) is the bin set value with equal space after taking logarithm.

Equation (12) can be rewritten as:

$$\begin{array}{*{20}l} {b_{m \times 1} = A_{m \times n} x_{n \times 1} } \hfill \\ {\lg \left( {T_{2j} } \right) = \lg \left( {T_{2\hbox{min} } } \right) + \frac{j - 1}{n - 1}\left( {\lg \left( {T_{2\hbox{max} } } \right) - \lg \left( {T_{2\hbox{min} } } \right)} \right)} \hfill \\ {b_{m \times 1} = \left( {{\text{echo}}_{1} ,{\text{echo}}_{2} , \ldots ,{\text{echo}}_{m} } \right)^{T} ,\quad x_{n \times 1} = \left( {\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{n} } \right)^{T} } \hfill \\ {A_{m \times n} = \left[ {\begin{array}{*{20}c} {e^{{{{ - t_{1} } \mathord{\left/ {\vphantom {{ - t_{1} } {T_{21} }}} \right. \kern-0pt} {T_{21} }}}} } & {e^{{{{ - t_{1} } \mathord{\left/ {\vphantom {{ - t_{1} } {T_{22} }}} \right. \kern-0pt} {T_{22} }}}} } & \cdots & {e^{{{{ - t_{1} } \mathord{\left/ {\vphantom {{ - t_{1} } {T_{2n} }}} \right. \kern-0pt} {T_{2n} }}}} } \\ {e^{{{{ - t_{2} } \mathord{\left/ {\vphantom {{ - t_{2} } {T_{21} }}} \right. \kern-0pt} {T_{21} }}}} } & {e^{{{{ - t_{2} } \mathord{\left/ {\vphantom {{ - t_{2} } {T_{22} }}} \right. \kern-0pt} {T_{22} }}}} } & \cdots & {e^{{{{ - t_{2} } \mathord{\left/ {\vphantom {{ - t_{2} } {T_{2n} }}} \right. \kern-0pt} {T_{2n} }}}} } \\ \vdots & \vdots & \ddots & \vdots \\ {e^{{{{ - t_{m} } \mathord{\left/ {\vphantom {{ - t_{m} } {T_{21} }}} \right. \kern-0pt} {T_{21} }}}} } & {e^{{{{ - t_{m} } \mathord{\left/ {\vphantom {{ - t_{m} } {T_{22} }}} \right. \kern-0pt} {T_{22} }}}} } & \cdots & {e^{{{{ - t_{m} } \mathord{\left/ {\vphantom {{ - t_{m} } {T_{2n} }}} \right. \kern-0pt} {T_{2n} }}}} } \\ \end{array} } \right]} \hfill \\ \end{array}$$

(13)

The algorithm flow of conjugate gradient method can be summarized as:

• 1. Initializing: x ₀ ∈ R ⁿ, r ₀ = b − Ax ₀, d ₀ = A ^T r ₀, the maximum iterations k _max , the error constant err; k = 0, begin the iteration;

• 2. If ||r _k || < err, terminate the iteration and output x _k ; otherwise, go on;

• 3. α _k  = A ^T r ² _k−1 /Ad ² _k−1 , x _k  = x _k−1 + α _k d _k−1, r _k  = r _k−1 − α _k Ad _k−1;

• 4. β _k  = ‖A ^T r _k ‖²/‖Ar _k−1‖², d _k  = A ^T r _k  + β _k d _k−1;

• 5. k = k+1; go back to 2).