Comparison of Symmetric and Asymmetric Schemes with Arithmetic and Harmonic Averaging for Fracture Flow on Cartesian Grids

Abstract

Performance of four finite-difference schemes for fluid flow in rough-walled fractures on regular Cartesian grids is evaluated numerically. The four schemes are an asymmetric scheme with arithmetic averaging, an asymmetric scheme with harmonic averaging, a symmetric scheme with arithmetic averaging, and a symmetric scheme with harmonic averaging. The schemes are compared with respect to their simulated hydraulic aperture and the mass balance error. 1320 flow simulations with different grid sizes, mean fracture aperture and root mean square (RMS)/mean aperture ratio are completed. The asymmetric scheme with arithmetic averaging arises naturally, without any extra assumptions about the correct transmissivity averaging procedure, when one uses second-order finite differences to approximate the generalized Laplace operator expanded as a derivative of a product. Hydraulic apertures obtained with harmonic averaging are found to usually be smaller than those obtained with arithmetic averaging, especially when the ratio of aperture RMS to the mean aperture is larger. The traditionally used asymmetric schemes are found to be superior to symmetric schemes in terms of mass balance accuracy.

Article Highlights

• Performance of four numerical schemes for fracture flow modelling on a Cartesian grid is studied numerically

• Schemes are compared w.r.t. their simulated hydraulic aperture and mass balance error on self-affine aperture maps

• Asymmetric schemes are found to be superior to symmetric schemes in terms of mass balance accuracy

Appendix: Derivation of Asymmetric Scheme with Arithmetic Averaging by Expanding the Generalized Laplace Operator

Consider a one-dimensional diffusion equation in heterogeneous media:

$$\frac{{\text{d}}}{{{\text{d}}x}}\left[ {c\frac{{{\text{d}}P}}{{{\text{d}}x}}} \right] = 0$$

(A1)

or, equivalently,

$$c\frac{{{\text{d}}^{2} P}}{{{\text{d}}x^{2} }} + \frac{{{\text{d}}c}}{{{\text{d}}x}}\frac{{{\text{d}}P}}{{{\text{d}}x}} = 0$$

(A2)

We aim to construct a second-order finite-difference approximation of Eq. (A2). A second-order approximation of the first term on the left-hand side in eq. (A2) can be achieved with the central difference:

$$c\frac{{{\text{d}}^{2} P}}{{{\text{d}}x^{2} }} = \Delta x^{ - 2} \left( {c_{i} P_{i + 1} - 2c_{i} P_{i} + c_{i} P_{i - 1} } \right) + O\left( {\Delta x^{2} } \right)$$

(A3)

We now aim to obtain a second-order accurate three-node approximation of the second term on the left-hand side of Eq. (A2), $\frac{{{\text{d}}c}}{{{\text{d}}x}}\frac{{{\text{d}}P}}{{{\text{d}}x}}$. Adding Taylor expansions of c(x + Δx)P(x + Δx) and c(x − Δx)P(x − Δx) yields:

$$\begin{aligned} &c\left( {x + \Delta x} \right)P\left( {x + \Delta x} \right) + c\left( {x - \Delta x} \right)P\left( {x - \Delta x} \right) \hfill \\ &\quad = 2c\left( x \right)P\left( x \right) + \Delta x^{2} \left[ {c\left( x \right)\frac{{{\text{d}}^{2} P\left( x \right)}}{{{\text{d}}x^{2} }} + 2\frac{{{\text{d}}c\left( x \right)}}{{{\text{d}}x}}\frac{{{\text{d}}P\left( x \right)}}{{{\text{d}}x}} + P\left( x \right)\frac{{{\text{d}}^{2} c\left( x \right)}}{{{\text{d}}x^{2} }}} \right] + O\left( {\Delta x^{4} } \right) \hfill \\ \end{aligned}$$

(A4)

Central difference approximation of the second derivatives on the right-hand side of Eq. (A4) yields the following approximation for $\frac{{{\text{d}}c}}{{{\text{d}}x}}\frac{{{\text{d}}P}}{{{\text{d}}x}}$:

$$\frac{{{\text{d}}c}}{{{\text{d}}x}}\frac{{{\text{d}}P}}{{{\text{d}}x}} = 0.5\Delta x^{ - 2} \left( {c_{i + 1} P_{i + 1} + 2c_{i} P_{i} + c_{i - 1} P_{i - 1} - c_{i} P_{i + 1} - c_{i} P_{i - 1} - c_{i + 1} P_{i} - c_{i - 1} P_{i} } \right) + O\left( {\Delta x^{2} } \right)$$

(A5)

Combining Eqs. (A3) and (A5) yields:

$$\frac{{\text{d}}}{{{\text{d}}x}}\left( {c\left( x \right)\frac{{{\text{d}}P}}{{{\text{d}}x}}} \right) = 0.5\Delta x^{ - 2} \left( {c_{i + 1} P_{i + 1} - 2c_{i} P_{i} + c_{i - 1} P_{i - 1} + c_{i} P_{i + 1} + c_{i} P_{i - 1} - c_{i + 1} P_{i} - c_{i - 1} P_{i} } \right) + O\left( {\Delta x^{2} } \right)$$

(A6)

It is straightforward to see that if we use a similar approximation for $\frac{{\text{d}}}{{{\text{d}}y}}\left( {c\left( y \right)\frac{{{\text{d}}P}}{{{\text{d}}y}}} \right)$, we recover Eq. (2) from the main text, with $c_{{i \pm {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},j}}$ and $c_{{i,j \pm {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}$ given by arithmetic averaging, Eq. (3). Hence, when using a modified form of the generalized Laplace operator given by Eq. (A2), asymmetric finite-difference scheme with arithmetic averaging arises naturally, without assuming any particular interpolation of c. The scheme derived from Eq. (A2) uses node-based P and c only.