Renormalization group theory outperforms other approaches in statistical comparison between upscaling techniques for porous media

Determining the pressure differential required to achieve a desired flow rate in a porous medium requires solving Darcy's law, a Laplace-like equation, with a spatially varying tensor permeability. In various scenarios, the permeability coefficient is sampled at high spatial resolution, which makes solving Darcy's equation numerically prohibitively expensive. As a consequence, much effort has gone into creating upscaled or low-resolution effective models of the coefficient while ensuring that the estimated flow rate is well reproduced, bringing to the fore the classic tradeoff between computational cost and numerical accuracy. Here we perform a statistical study to characterize the relative success of upscaling methods on a large sample of permeability coefficients that are above the percolation threshold. We introduce a technique based on mode-elimination renormalization group theory (MG) to build coarse-scale permeability coefficients. Comparing the results with coefficients upscaled using other methods, we find that MG is consistently more accurate, particularly due to its ability to address the tensorial nature of the coefficients. MG places a low computational demand, in the manner in which we have implemented it, and accurate flow-rate estimates are obtained when using MG-upscaled permeabilities that approach or are beyond the percolation threshold.

Figure 1

A single tortuous channel is seen to cross from one horizontal end of the computational box to the other of a model coefficient that has crossed the percolation threshold. The leftmost column (panels a, b, and c) shows the “real” coefficient (resolution of $512\times 512$), and the three successive columns show coefficients decimated using MG (a1), (b1) and (c1), mean (a2), (b2) and (c2), and KK (a3), (b3) and (c3) methods to a size of $32\times 32$. Note that the color bars of the different panels are all different, and that, with successive upscaling, the magnitudes of the coefficients continue to decrease. The background permeability for the $a_{xx}$ and $a_{yy}$ coefficients is ${10}^{-6}$, whereas the background for the $a_{xy}$ coefficient is zero. The flow associated with this model is shown in Fig. 2.

Figure 2

The flow pattern associated with applying a pressure gradient in the $x$ direction, i.e., $\varphi (x=0,y)=1, \varphi (x=1,y)=0$, and impermeability conditions on the upper and lower boundaries, ${\partial \varphi /\partial y|}_{(x,y=0)}{=0=\partial \varphi /\partial y|}_{(x,y=1)}$. The overplotted contours show the “exact” coefficient $a_{xx}$ from the top-left panel of Fig. 1 to assist in identifying flow features. To obtain $\varphi$, we solve Eq. (14) using PETSc [16]. Nominally, for constant permeability, we have $\varphi =1-x$. To highlight the spatial complexity of the solution, we display $\varphi -1+x$. The choices that determine the size and orientation of the percolation channel (randomly assigned within allowed ranges) have an impact on the total flow rate. To avoid being overly sensitive to the associated finite-size effects, we perform a statistical survey over a broad range in sizes and orientations.

Figure 3

The statistical distribution of numerical errors associated with upscaling 500 models of permeability using three techniques: MG (panels a1, b1, and c1), KK (panels a2, b2, and c2), or straightforward averaging (panels a3, b3, and c3). In these tests, we explicitly set $a_{xy}=0$ at the highest resolution ($512\times 512$). The $x$ axis is the error in predicting the correct flow rate [Eq. (18)] and the $y$ axis is the probability of incurring that error. The original coefficient is sampled at $512\times 512$ resolution and subsequently decimated to resolutions of $256\times 256$ (row a), $128\times 128$ (row b), and finally $32\times 32$ (row c). With greater rates of decimation, MG introduces a nontrivial $a_{xy}$ coefficient, whereas KK and straight mean do not address this aspect. The models lie above the percolation threshold, with a single channel linking the two horizontal boundaries. The channel may assume a tortuous path. It is seen that although all the methods show a systematic theoretical bias toward overpredicting the flow rate, MG is the method that performs best.

Figure 4

Probability of incurring an error less than $x$ in calculating the flow rate [Eq. (18)] when decimating from $512\times 512$ to various resolutions for the three methods being considered here: MG (panel a), KK (panel b), and straight mean (panel c). This is concluded from a statistical survey of 500 randomly generated models that are above the percolation threshold. The off-diagonal coefficient $a_{xy}=0$ for the original model at the highest resolution of $512\times 512$. The upscaled coefficients obtained using MG have finite $a_{xy}$ and are therefore of higher accuracy than KK and the straight mean, which do not take these into account.

Figure 5

Probability of incurring an error less than $x$ in calculating the flow rate [Eq. (18)] when decimating from $512\times 512$ to $32\times 32$ for the three methods being considered here: MG, KK, and straight mean. This is concluded from a statistical survey of 500 randomly generated models that are above the percolation threshold. The off-diagonal coefficient $a_{xy}=0$ for the original model at the highest resolution of $512\times 512$. When decimating down to lower resolutions, only MG is able to address the tensorial nature of the coefficient and thereby introduces finite $a_{xy}$ at $32\times 32$, resulting in greater accuracy.

Figure 6

Probability of incurring an error less than $x$ in calculating the flow rate [Eq. (18)] when decimating from $512\times 512$ to $32\times 32$ for the three methods being considered here: MG (a), KK (b), and straight mean (c). The difference between this and Fig. 4 is that the off-diagonal coefficient $a_{xy}$ is now nonzero for the original model at the highest resolution of $512\times 512$. These results are concluded from a statistical survey of 500 randomly generated models that are above the percolation threshold. The upscaled coefficients obtained using MG and straight mean techniques have finite $a_{xy}$, performing better than KK, which does not take these into account and is therefore prone to significant error, even at $256\times 256$. In general, it is seen that the errors are systematically larger for all methods when finite off-diagonal permeabilities are included (compare the $x$ axes between this and Fig. 4).

Figure 7

Probability of incurring an error less than $x$ in calculating the flow rate [Eq. (18)] when decimating from $512\times 512$ to $32\times 32$ for the three methods being considered here: MG, KK, and straight mean. The difference between this and Fig. 5 is that the off-diagonal coefficient $a_{xy}$ is nonzero for the original model at the highest resolution of $512\times 512$. The results are concluded from a statistical survey of 500 randomly generated models that are above the percolation threshold. The upscaled coefficients obtained using MG and straight mean techniques have finite $a_{xy}$ and are therefore of higher accuracy than KK, which does not take these into account. In general, it is seen that the errors are systematically larger for all upscaling methods when finite off-diagonal permeabilities are included (compare with Fig. 5).