An Improved Method for Reconstructing the Digital Core Model of Heterogeneous Porous Media

Abstract

The heterogeneous pore space of porous media strongly affects the storage and migration of oil and gas in the reservoir. In this paper, the cross-correlation-based simulation (CCSIM) is combined with the three-step sample method to reconstruct stochastically 3D models of the heterogeneous porous media. Moreover, the two-point and multiple-point connectivity probability functions are used as vertical constraint conditions to select the boundary points of pore and matrix, respectively. The heterogeneities of pore spaces of four rock samples are investigated, and then our methods are tested on the four samples. Quantitative comparison is made by computing various statistical and petrophysical properties for the original samples, as well as the reconstructed model. It was found that the results from CCSIM-TSS are obviously better than that from CCSIM. Finally, the analysis of the distance (ANODI) was used to measure of the variability between the realizations of the four rock samples. The results demonstrated that the results from CCSIM-TSSmp are better than that from CCSIM-TSStp as the complexity of connectivity and heterogeneities of pore spaces increase.

Appendices

Appendices

1.1 A. Two-Point Correlation Function (ACF)

The void–void (pore–pore) ACF is defined by

$$\begin{aligned} \hbox {ACF}(r)=\frac{E\left\{ {\left[ {I\left( u \right) -\phi } \right] \left[ {I(u+r)-\phi } \right] } \right\} }{\phi -\phi ^{2}} \end{aligned}$$

(A-1)

where the averaging is over all locations u within the volume, and I(u) is an indicator function such that $I(u) = 1$, if u is in the pore space, and $I(u) = 0$ otherwise. The porosity is simply $\varphi =E\left\{ {I(u)} \right\}$, $E\left\{ {I(u)} \right\}$ is the mean of I(u) over all locations u within the volume.

1.2 B. Multiple-Point Connectivity Probability Functions

The multiple-points connectivity probability is proposed by Krishnan and Journel (2003). It is a measure of the connectivity of the system within an image. The truncated tolerance cone is defined, as shown in Fig. 16. Consider a seed location u in channel: $I (u) = 1$ (u is in the pore space). Within the defined tolerance cone, we would like to connect u through a channel path to another maximally distant location u is in the pore space $u'$, that distance being measured along the axis of the cone.

More precisely, the calculation of the multiple-points connectivity probability proceeds as:

Loop through all nodes u of the study area.

• If I $(u) = 0$, set $d\hbox {max} = 0$ and go to next node u.

• If I $(u) = 1$, i.e., location u belongs to the pore space, call MATLAB function bwselect to determine within the tolerance cone with apex at u, the set (body) of all locations $u'$ belonging to the pore space, i.e., I $(u') = 1$. Loop through all these locations $u'$ and determine the maximum distance dmax to the apex u. Increment the n proportions of continuous (and curvilinear) channel paths of length $n\vec {h}\le d\hbox {max}$. $\vec {h}$ is a unit vector in any given direction.

It should be emphasized that the tolerance angle is $\pm \, 22.5^{\circ }$, the bandwidth is 10 pixels during calculating the curvilinear directional connectivity in this paper.

Fig. 16

The truncated tolerance cone. bandw is the bandwidth, controlling deviation from the central target direction at larger distances. Maxlag is the maximum distance