Integrated LSM-DFN Modeling of Naturally Fractured Reservoirs: Roughness Effect on Elastic Characteristics

Abstract

Crack micro-geometries and tribological properties have a significant influence on the elastic characteristics of naturally fractured reservoirs. Numerical simulation as a promising approach for this issue still faces some challenges. For this purpose, we develop an integrated numerical scheme accounting for roughness effects by coupling a modified lattice spring model (LSM) with discrete fracture networks (DFNs). Complex fracture networks presented by DFNs are automatically extracted based on the gradient Hough transform algorithm (GrdHT). Smooth joint logic (SJL) is employed to avoid the artificial roughness effect from numerical discretization. Improved constitutive laws are also implemented in the modified LSM to calculate the realistic normal force-displacement from rough contact deformation. We validate this presented framework through theoretical solutions. It shows the potential for reconstructing actual structural attributes and quantitatively investigating the fracture attributes and microscale surface roughness effects on elastic characteristics.

Appendices

Appendix A: Material Properties and Lattice Node Parameters

In two dimensions, the basic idea in setting up the spring network models is based on the equivalence of strain energy stored in a unit cell (Fig. 19) with the area ${A_{{\mathrm{cell}}}}$ of a network (Ostoja-Starzewski, 2002)

$$\begin{aligned} {U_{{\mathrm{cell}}}} = {U_{{\mathrm{continuum}}}}, \end{aligned}$$

(A1)

where the energies of the cell and its continuum are equivalent:

$$\begin{aligned} {U_{{\mathrm{cell}}}}= \; & {} \sum \limits _{n = 1}^6 {\frac{1}{2}\left( {{K_{\mathrm{n}}}{{\left( {u_{\mathrm{n}}^i} \right) }^2} + {K_{\mathrm{s}}}{{\left( {u_{\mathrm{s}}^i} \right) }^2}} \right) } , \end{aligned}$$

(A2)

$$\begin{aligned} {U_{{\mathrm{continuum}}}}= \; & {} \frac{1}{2}{\varvec{\varepsilon }} \cdot {\mathbf{C}} \cdot {\varvec{\varepsilon }}, \end{aligned}$$

(A3)

in which superscript i in Eq. (A2) stands for the ${i^{{\mathrm{th}}}}$ interaction. $u_{\mathrm{n}}^{ij}$ and $u_{\mathrm{s}}^{ij}$ are the distance increments in the normal and tangential directions, respectively. Then, the fourth-order effective stiffness tensor ${\mathbf{C}}$ can be derived by

$$\begin{aligned} {C_{ijkl}} = \frac{{{\partial ^2}{\omega _{{\mathrm{cell}}}}}}{{\partial {\varepsilon _{ij}}\partial {\varepsilon _{kl}}}}. \end{aligned}$$

(A4)

Therefore, we can obtain the expressions for the two Lamé parameters $\lambda$ and $\mu$ of isotropic elastic media:

$$\begin{aligned} \lambda= \; & {} \frac{{\sqrt{\mathrm{3}} }}{4}\left( {{K_{\mathrm{n}}} - {K_{\mathrm{s}}}} \right) . \end{aligned}$$

(A5)

$$\begin{aligned} \mu= \; & {} \frac{{\sqrt{\mathrm{3}} }}{4}\left( {{K_{\mathrm{n}}} + {K_{\mathrm{s}}}} \right) . \end{aligned}$$

(A6)

in which ${K_{\mathrm{n}}}$ and ${K_{\mathrm{s}}}$ are the normal and shear spring stiffness, respectively. For a two-dimensional (2D) issue, we also have

$$\begin{aligned} \lambda= \; & {} \frac{{E\nu }}{{1 - {v^2}}}, \end{aligned}$$

(A7)

$$\begin{aligned} \mu= \; & {} \frac{E}{{2\left( {1 + \nu } \right) }}, \end{aligned}$$

(A8)

where E and $\nu$ are Young’ modulus and Poisson’s ratio, respectively.

Fig. 19

A triangular lattice with a hexagonal unit cell

Appendix B: Commonly Used Effective Medium Theories

In general, we describe the anisotropic media by the models of transverse isotropy (TI). Elasticity matrices present the elasticity tensor components, depending on the spatial orientation of TI models. Here, we start with the most straightforward fracture geometry, a single set of parallel fractures with a given orientation, that has been extensively studied in the past (Hudson, 1980, 1981; Cheng, 1993). In three dimensions, only five independent constants can entirely specify a TI elastic material. Therefore, the stiffness matrix of the TI media with the symmetry axis of the 3-axis is given by

$$\begin{aligned} \left( {\begin{array}{*{20}{c}} {{C_{1111}}}&{}{{C_{1122}}}&{}{{C_{1133}}}&{}0&{}0&{}0\\ {{C_{1122}}}&{}{{C_{1111}}}&{}{{C_{1133}}}&{}0&{}0&{}0\\ {{C_{1133}}}&{}{{C_{1133}}}&{}{{C_{3333}}}&{}0&{}0&{}0\\ 0&{}0&{}0&{}{{C_{{\mathrm{1}}3{\mathrm{1}}3}}}&{}0&{}0\\ 0&{}0&{}0&{}0&{}{{C_{{\mathrm{1}}3{\mathrm{1}}3}}}&{}0\\ 0&{}0&{}0&{}0&{}0&{}{{C_{1212}}} \end{array}} \right) , \end{aligned}$$

(B1)

where ${C_{1212}} = \frac{1}{2}\left( {{C_{1111}} - {C_{1122}}} \right)$.

Hudson (1980, 1981) developed the first- and second-order expansions for the effective moduli of a crack-induced TI medium using a scattering approach with the small-aspect-ratio assumption in a series of papers. He gave the results as follows:

$$\begin{aligned} C_{ijkl}^* = C_{ijkl}^0 + C_{ijkl}^1 + C_{ijkl}^2, \end{aligned}$$

(B2)

$C_{ijkl}^*$, $C_{ijkl}^0$, $C_{ijkl}^1$, and $C_{ijkl}^2$ are the effective elastic stiffness, the matrix stiffness, and the first- and second-order corrections, respectively. The first-order corrections are given by

$$\begin{aligned} {C_{ijkl}}^1 = - \frac{{C_{r3ij}^0C_{s3kl}^0}}{\mu }\varepsilon {U_{rs}}\left( 0 \right) , \end{aligned}$$

(B3)

and the second-order corrections are expressed by

$$\begin{aligned} {C_{ijkl}}^2 = \frac{1}{\mu }C_{ijpq}^1C_{rskl}^1{\chi _{pqrs}}, \end{aligned}$$

(B4)

where for the dry cracks, the expressions are

$$\begin{aligned} {U_{11}}\left( 0 \right)= \; & {} \frac{{16\left( {\lambda + 2\mu } \right) }}{{3\left( {3\lambda + 4\mu } \right) }}, \end{aligned}$$

(B5)

$$\begin{aligned} {U_{33}}\left( 0 \right)= \; & {} \frac{{4\left( {\lambda + 2\mu } \right) }}{{3\left( {\lambda + \mu } \right) }}, \end{aligned}$$

(B6)

$$U_{{kl}} \left( 0 \right) = 0\left( {k \ne l} \right);$$

(B7)

$$\begin{aligned} {\chi _{ijkl}}= \; & {} \frac{1}{{15}}\left( {\delta _{ik}}{\delta _{jl}}\left( {4 + \frac{\mu }{{\lambda + 2\mu }}} \right) \right. \nonumber \\&\left. - \left( {{\delta _{kj}}{\delta _{il}} + {\delta _{ij}}{\delta _{kl}}} \right) \left( {1 - \frac{\mu }{{\lambda + 2\mu }}} \right) \right) , \end{aligned}$$

(B8)

in which $\varepsilon$ is the crack density. Cheng (1993) used the Padé approximation to solve the divergent phenomenon when the expressions are in a power series, as shown in Eq. (B2), namely,

$$\begin{aligned} C_{ijkl}^* = C_{ijkl}^0\frac{{1 - {a_{ijkl}}\varepsilon }}{{1 + {b_{ij}}\varepsilon }}, \end{aligned}$$

(B9)

where

$$\begin{aligned} {b_{ijkl}}= \; & {} - \frac{{C_{ijkl}^2}}{{C_{ijkl}^1\varepsilon }}, \end{aligned}$$

(B10)

$$\begin{aligned} {a_{ijkl}}= \; & {} - \frac{{C_{ijkl}^1}}{{C_{ijkl}^0\varepsilon }} - {b_{ijkl}}. \end{aligned}$$

(B11)