Representation of multiscale heterogeneity via multiscale pore networks



[1] Developing a better understanding of single-/multiphase flow through reservoir rocks largely relies on characterizing and modeling the pore system. For simple homogeneous rock materials, a complete description of the real pore structure can be obtained from the pore network extracted from a rock image at a single resolution, and then an accurate prediction of fluid flow properties can be achieved by using network model. However, for complex rocks (e.g., carbonates, heterogeneous sandstones, deformed rocks), a comprehensive description of the real pore structure may involve several decades of length scales (e.g., from submicron to centimeters), which cannot be captured by a single-resolution image due to the restriction of image size and resolution. Hence, the reconstruction of a single 3-D multiple-scale model of a porous medium is an important step in quantitatively characterizing such heterogeneous rocks and predicting their multiphase flow properties. In this paper, we present a novel methodology for the numerical construction of the multiscale pore structure of a complex rock from a number of CT images/models of a carbonate sample at several length scales. The success of this reconstruction relies heavily on image segmentation, pore network extraction and stochastic network generation, which are provided by our existing software system, referred to as Pore Analysis Tools (PAT). Specifically, the statistical description of pore networks of 3-D rock images at multiple resolutions makes it possible for us to: (a) construct an arbitrary sized network which is equivalent in a specified domain, and (b) integrate multiple networks of different sizes into a single network incorporating all scales. Using multiscale networks of carbonate rocks generated in this manner, two-phase network modeling results are presented to show how the resulting flow properties are dependent on inclusion of information from multiple scales. These outcomes reinforce the importance of capturing both geometry and topology in the hierarchical pore structure for such complex pore systems. The example presented reveals that isolated large-scale (e.g., macro-) pores are mainly connected by small-scale (e.g., micro-) pores, which in turn determines the combined effective petrophysical properties (capillary pressure, absolute and relative permeability). It is also demonstrated that multi- (three) scale networks reveal the effects of the interacting multiscale pore systems (e.g., micropores, macropores, and vugs) on bulk flow properties in terms of two-phase flow properties.

1. Introduction

[2] Pore-level network modeling has been extensively applied over the past decade to simulate multiphase flow properties using input pore networks extracted from digital representations of the pore space (i.e., CT images or models). Although simulations of this type may be considered as an ideal a priori approach to the calculation of various petrophysical properties (e.g., permeability, relative permeability, etc.), two significant problems remain for this method. First, there is an issue of constructing a digital representation of the rock fabric using either 3-D micro-CT imaging or some 2-D → 3-D reconstruction method which is both large enough to be representative whilst being sufficiently detailed to resolve fine pore detail. Even if we are able to achieve such digital rock images, they may be too large for us to be able to actually carry out multiphase simulations in them at a scale that is “representative” of the rock continuum. Thus, we encounter a trade-off between image size and resolution (i.e., one image large volume, low resolution and the other small volume, high resolution). For homogeneous porous media with a relatively narrow range of pore sizes, this approach may be manageable but if the rock exhibits complex heterogeneity at multiscales (e.g., from < micrometer to millimeter to centimeter scale), then the process will not be feasible. Carbonate rocks, which contain a very significant fraction of the world's remaining oil reserves, have pore systems that contain pores of size less than 1 µm up to voids of the order of ∼1 cm (called vugs) and fractures (Figure 1a). The complexity of carbonate rock pore systems leads to observed permeability ranges for these materials that range over several orders of magnitudes for a given porosity value (Figure 1b; see Jiang et al. (in review) for a more complete analysis of this issue).

Figure 1.

For a carbonate out-crop sample a pore size distribution (a) and the porosity-permeability relationship (b) illustrate a complex pore structure with a wide range of pore size from nanometer up to centimeter and distinct connectivity—well-connected micropores while rarely touching vugs.

[3] An attractive approach, which has been extensively researched in several scientific fields, is to construct model pore systems represented by an idealized network of nodes (sometimes identified as “pores”) connected by bonds (sometimes called “pore throats”). These nodes and bonds have some geometrical properties and additional properties such as wettability, pore geochemistry, etc. which are related to their multiphase flow behavior [Blunt et al., 2002]. This network approach may address the second problem discussed above (i.e., no appropriate approach to reveal the multiscale structure from several data sets at different scales), since carrying out flow calculations in such simplified systems are much less computationally expensive and may be carried out on a physically representative volume. For simpler more homogeneous rocks having a single pore system, such as Berea sandstone, two-phase flow dynamics (e.g., relative permeabilities) can be modeled either (1) by solving the basic equations (Navier-Stokes) directly on the detailed geometry using a finite-difference scheme or the lattice-Boltzmann method [Manwart et al., 2002; Vogel et al., 2005] or (2) by using extracted pore network when the structure is more complicated, such as for many carbonate rocks.

[4] For extremely heterogeneous porous media (e.g., carbonate rocks), then the problem of constructing a pore level model at the representative scale, must be solved by including information from each length scale of the multiscale system. This may be achieved by (1) building a pore level network model at each relevant length scale which holds the information from that particular length scale, and then (2) superimposing these images to create a combined network at a suitable longer length scale. An example of the image superposition approach is seen in the work of Okabe and Blunt [2007]. These workers proposed a two-step approach to combine different types of carbonate images: microtomography at the resolution of tens microns to resolve large pores and vugs, with statistically reconstructed high-resolution images for smaller features. Ghous et al. [2008] used microcomputed tomography and focused ion beam microscopy to describe 3-D imaging of macropores and micropores in carbonate core samples respectively. However, tomographic images (or reconstructed models) require a trade-off between voxel size and sample size. Thus, it is impossible to capture the details of tight and heterogeneous rocks such as carbonate materials for large core samples (at least for a typical volume, for example, several millimeters in diameter) at the submicron scale because current computer processing capacity is unable to acquire a 3-D rock image of several millimeters in diameter at submicron scale (e.g., the number of voxels > 10,0003). As a direct consequence of this limitation, the value of commonly used network extraction methods is limited by the size of 3-D images which can be handled. To account for the connectivity of microporosity to vugs, Moctezuma et al. [2003] have developed a dual network model to incorporate information on the primary (matrix) and the secondary (vugs) porosity. Also, more realistically, Biswal et al. [2007, 2009] presented a stochastic geometrical modeling method for reconstructing three dimensional pore scale microstructures of multiscale porous media. All of these techniques have made good progress, but they are still unable to adequately capture the multiscale pore structure in a representative image volume.

[5] Despite the significant advances of recent years, the construction of digital carbonate rocks with several orders of magnitude variation in pore sizes still poses a very significant challenge. In this paper, a novel method to generate a pore network of arbitrarily large size (in theory) is proposed. In this approach, the multiscale pore structure and pore interconnection representing the model are based on the statistical information obtained from pore networks at several scales, rather than superimposing the rock images themselves. Building on our method for generating equivalent stochastic networks [Jiang et al., 2011], a novel method is presented here which integrates pore networks from images/models at various length scales. After a large amount of effort on multiscale reconstruction, image superposition and artificial network for small-scale pore systems, the aim with this multiscale network is to be able to carry out accurate predictions of petrophysical and multiphase flow properties with these representative pore networks. It is the network topology and the details of the local pore geometry which ultimately control many macroscopic phenomena associated with fluid flow responses. This multiscale network approach circumvents the imaging hardware limitations because, unlike voxel-representation data sets, the handling of pore networks does not require such a large amount of computational power and can be performed on personal computers.

2. Construction of Multiscale Pore Network

2.1. Methodology and Workflow

[6] In outline our new approach is as follows: (1) First, we extract individual pore networks from images at distinct length scales. (2) Second, we quantify and statistically characterize these original networks. The statistical information from the pore network at each scale is used to generate, in domains of arbitrary volume, stochastic networks that are equivalent to the original single-scale network in terms of pore geometry and topology, as well as petrophysical properties [Jiang et al., 2011]. (3) Using the stochastic networks generated, we integrate two networks created in the same domain into a single two-scale network by characterizing the cross-scale connection structure between the two networks. (4) Multiscale networks are generated by repetition of this method. The schematic workflow for two-scale network reconstruction is shown in Figure 2.

Figure 2.

Workflow to generate a two-scale network. Note that for the micropores a stochastic network is generated (steps 2 and 3) in a domain with the size of the macropore network.

2.2. Pore Network Extraction

[7] For a single-scale binarized image or model of a porous medium, we take the traditional approach of building a node (pore)/bond(throat) model where the bonds connect the nodes together. In the process of network extraction, larger pore regions are identified as nodes, which are usually junctions of adjacent pore channels, while narrow regions along pore channels are partitioned as bonds.

[8] In this work, we extract the pore network using the Pore Analysis Tools (PATs) described previously [Jiang et al., 2007; Jiang, 2008], involving the following steps:

[9] (1) Tiny pores which are isolated and “floating” solid elements are removed from the binary image.

[10] (2) A thinning algorithm (which is Euclidean distance based) is then applied which preserves the topology of the network skeleton (medial axis).

[11] (3) Nodes are identified as the skeleton junctions and the other skeletal voxels are the bonds which join these nodes; sometimes these are referred to as “node backbones”.

[12] (4) The node backbones are then removed from the medial axis and we then proceed to isolate the 26 connected components from the remaining skeleton voxels as individual bond backbones. The topology is preserved in the resulting structure and in addition the problem referred to a “snow balling” is avoided [Sheppard et al., 2005].

[13] (5) Radially symmetric nodes and (straight) bonds with a constant cross section are assumed.

[14] (6) Measure the geometrical properties of the nodes and bonds through digital image analysis. For use, in network flow simulations [Ryazanov et al., 2009], a node is described by its radius (inscribed or hydraulic), its volume, its shape factor and its coordination number, while a bond is described by its radius, volume, length and shape factor.

[15] (7) Finally, the network is idealized by removing bonds between nodes which are redundant and then representing dead-end pores as virtual nodes.

2.3. Network Quantification and Stochastic Network Generation

[16] After the pore network has been extracted from the rock image of volume V, we characterize its geometry and topology (Step 2 in Figure 2) in terms of probability distribution functions (PDF) of individual parameters and correlation functions between network parameters [Jiang et al., 2011]. This includes external correlations between the property of a node on the one hand and that of a bond on the other. In addition to linear and nonlinear correlation functions (e.g., regression equations), for weaker correlations (determined by the correlation coefficient), we use a combination of multiple conditional probability distributions (CDF) of a parameter for intervals of values of the correlated parameter.

[17] Commonly the (mean) node coordination number and its PDF are used to locally quantify pore network connectivity. An effective way of characterizing the pore network connectivity globally is to consider the topology of sub networks consisting of nodes and bonds with specific size ranges—the connectivity function [Vogel and Roth, 2001]. The connectivity function is based on the Euler number χ, which for a pore network is simply defined as math formula, where NN and NB are the number of nodes and bonds in the network, respectively. The connectivity function, math formula is defined as

display math(1)

where the number of nodes and bonds with radii equal to or larger than r is denoted by NN(r) and NB(r), respectively. Each bond accounting for NB(r) is only used to connect nodes that are counted in NN(r), and obviously for the smallest radius rmin in the network we have NN = NN(rmin) and NB = NB(rmin).

[18] Also, to facilitate the generation of stochastic networks of arbitrary volume, we introduce node densities math formula, and hence the number of bonds in Step 4 can then be calculated as, [ρN − χV(rmin)]V(Ω) in order to have the same connectivity function in the stochastic network as in the original one.

[19] Statistical information is extracted from the original pore network which was constructed from the 3-D binary image of the rock (of volume V and node density ρN,) and this is then used to generate an equivalent network which can be of arbitrary size. The various steps in this process are listed below [Jiang et al., 2011]:

[20] (1) Choose a domain Ω of arbitrary volume V(Ω) for the stochastic network.

[21] (2) Based on the node density ρN, create ρNV(Ω) nodes, random values are generated for each node parameter (radius, volume, coordination number, shape factor and clay volume) according to the statistical information.

[22] (3) The centers of the nodes are positioned randomly in domain Ω, ensuring that overlap of the nodes is avoided.

[23] (4) Create bonds (by assigning values to radius, volume, shape factor, total length, half-length 1, half-length 2, and clay volume) according to the connectivity function χV(r), including boundary bonds, and then random values are generated for each bond parameter.

[24] (5) While implementing the strongest external correlation between properties of a node and its adjacent bonds, the bonds are positioned between the various pairs of nodes while simultaneously honoring the connectivity function and the bond lengths.

[25] (6) The boundary bonds are then positioned between the faces (inlet or outlet), based on the bond lengths.

2.4.Multiscale Network Integration

[26] In the following, we consider the generation of a two-scale network by integrating a fine-scale network (NetFine), extracted from an image at one length-scale, and a coarse-scale network (NetCoarse), extracted from an image at another, larger, length-scale. For example, this could be the networks of micropores and macropores in Figure 2. The integration method can be summarized as follows:

[27] (1) Determine the domain Ω (nesting domain) in which NetCoarse is located;

[28] (2) Generate a stochastic NetFine in Ω without intersecting NetCoarse;

[29] (3) Integrate NetFine and NetCoarse into a single network by connecting the networks through addition of fine-scale bonds between coarse-scale nodes and nearby fine-scale nodes.

[30] In broad outline, we superimpose NetFine onto NetCoarse and remove any bonds of NetFine that intersect bonds of NetCoarse. The bonds of NetFine intersecting with nodes of NetCoarse are first removed and then reintroduced in step (3) above to create a proper network configuration. This step also gives some significant computational benefits. Note that NetCoarse and NetFine often have a range of network elements (nodes and bonds) of similar sizes, that is when the gap between the length scales of the respective networks is relatively small. This means that nodes from NetFine are not necessarily smaller than the nodes from NetCoarse that they will be connected to.

[31] In the first step, the domain Ω is usually the nesting domain (or a subdomain) of the original coarse-scale network. Commonly, at the coarse scale the pore space (e.g., the image with macropores in Figure 2) is modeled by a pore network with a relatively small number of nodes and bonds (e.g., hundreds of nodes and bonds), which is not sufficient to obtain representative information that can be used to reconstruct a stable stochastic network being equivalent to the original one. In this case, we choose the extracted coarse-scale (sub-) network as NetCoarse and its nesting (sub-) domain in order to capture the coarse-scale structural information as far as possible. However, if the original network is large enough from a statistical point of view, we can use a stochastic coarse-scale network in an arbitrary domain Ω as NetCoarse.

[32] In the second step, we generate a stochastic network NetFine that is equivalent to the original fine-scale network (e.g., the micropore network in Figure 2) in the chosen domain Ω. We choose to locate network elements of NetFine in the remaining space Ω\Ωc of Ω that excludes the regions, denoted by Ωc, that are already occupied by elements of NetCoarse. To avoid fine-scale nodes and bonds from intersecting coarse-scale nodes and bonds, we simply consider nodes as balls and bonds as cylinders without actually altering original pore shapes and pore-wall roughness described by shape factor, radius and volume, as shown in Figure 3, with their respective inscribed radii. We discard the fine-scale balls and cylinders if they intersect any nearby coarse-scale balls or cylinders. Note that here we uniformly distribute fine-scale nodes into the space Ω\Ωc without taking into account any spatial correlations. This will be discussed in detail in a future paper, as in complicated materials like carbonate rocks correlations between microporosity and macroporosity play an important role.

Figure 3.

Connections between a coarse-scale node (red) and a set of fine-scale nodes (small balls) by fine-scale bonds (small cylinders).

[33] Commonly, even if a subdomain is chosen for NetCoarse, the difference between the length scales of NetCoarse and NetFine is such that NetFine will contain a very large number of pore elements, making network flow simulation practically impossible. To address this issue, we reduce the density of the original fine-scale network in Ω\Ωc without significantly jeopardizing the integrated pore structure and petrophysical properties. We introduce the fraction of fine-scale network 0< fF ≤ 1 to determine the actual density of NetFine in Ω\Ωc, hence the number of nodes in NetFine is given by

display math(2)

where ρN,F is the node density of the original fine-scale network, respectively. Similarly, the number of bonds are reduced proportionally and they are created satisfying the reduced connectivity function fFχV,F(r), where χV,F(r) is the connectivity function of the original fine-scale network. The reduction results in a decrease of the average coordination number in NetFine, the impact of which will be discussed later. We emphasize that this selection of a fraction of the smallest pores is simply done for practical reasons; our method can generate cases with all these micropores present (fF = 1.0) but we would practically be unable to perform any calculations on such a “complete” model in the integrated network and so they are not presented here.

[34] Because we uniformly distribute fine-scale nodes into the space Ω\Ωc, reduction of the network density leads to an increase in the average distance between nodes. To deal with this, we implement bonds with lengths (generated from statistical information) that are smaller than the physical distance between the nodes that they connect. However, our network model is able to deal with these slightly unrealistic (in spatial terms) network configurations. The more realistic case, where we include spatially correlated large/small pore system network configurations (e.g., microporosity located just small regions near macropores), will be discussed in the future.

[35] In the last step of linking each coarse-scale node to adjacent fine-scale nodes by fine-scale bonds, we need to address the following issues:

[36] (1) For coarse-scale nodes define adjacent fine-scale nodes;

[37] (2) Determine a cross-scale coordination number for each coarse-scale node, which is the maximum number of fine-scale nodes it could be connected to;

[38] (3) Generate cross-scale connections.

2.4.1. Adjacent Fine-Scale Nodes

[39] For two nodes N1 and N2, again approximated as balls with respective inscribed radii r1 and r2, linked by a bond as shown in Figure 4a, we define the internode distance di(N1, N2) as

display math(3)

where d(N1, N2) is the Euclidean distance between the two node centers.

Figure 4.

(a) The definition of internode distance between nodes N1 and N2; (b) the set Sa of adjacent fine-scale nodes (black) for a coarse-scale node N (blue), consisting of the nodes that intersect with the pink layer of thickness di,max.

[40] For an original fine-scale network (e.g., the extracted network of micropores shown in Figure 2), we calculate the maximum internode distance di,max across all nodes and the distribution of internode distances.

2.4.2. Cross-Scale Coordination Number

[41] The cross-scale coordination number of a coarse-scale node in a two-scale network is defined as the number of fine-scale nodes that it can be connected to by fine-scale bonds.

[42] For an original fine-scale network (Figure 5a) with a set of nodes {N1, N2, …, Nn} and a set of bonds {B1, B2, …, Bm}, we insert s concentric spheres at the center, p, of the network (e.g., s = 9 in Figures 5b and 5c)) of radii (0 < r1 < r2 < … < rs), such that the largest sphere is still contained in the fine-scale network. We then count the numbers NBj, j=1,…,s of bonds that penetrate (Figure 5c) the respective concentric spheres. In other words, if d(p, N) is the Euclidean distance between p and the center of node N, and Nk1 and Nk2 are the two nodes linked by bond Bk, such that d(p, Nk1) < d(p, Nk2), then

display math(4)

where # is the cardinality operation to obtain the number of elements in a set. Furthermore, we define a so-called cross-scale coefficient γCSC as

display math(5)

where aj is the surface area of the jth sphere. This coefficient is considered to be the number of bonds that penetrate a unit area.

Figure 5.

A series of concentric spheres (b, c) inserted in the domain of a (fine) network (a), where (c) shows intersection of bonds with spheres of different radii.

[43] To determine the cross-scale coordination number, we consider next that for a coarse-scale node part of its surface is already occupied by connecting coarse bonds. Then, for each coarse-scale node Nc with radius rc and single-scale coordination number zc, the cross-scale coordination number zcs(Nc) (≥ 0) is computed as

display math(6)

where Ak is surface area of the spherical dome associated with a (coarse) bond with radius of rk, which is Ak = 2πrk(r2rk2)1/2.

2.4.3. Cross-Scale Connections

[44] The final step is to implement the cross-scale connections between NetCoarse and NetFine (see Figure 6). For a given coarse-scale node Nc in NetCoarse, identify the set Sa of all adjacent fine-scale nodes Nf that have internode distance di(Nf, Nc) less than the maximal internode distance for NetFine, di,max, calculated using equation (3). Calculate the cross-scale coordination number zcs(Nc), using equations (4) and (5). Generate zcs(Nc) values from the internode distance distribution and select the fine-scale nodes from Sa for which di(Nf, Nc) match those values most closely. For these fine-scale nodes create bonds according to the statistical information from NetFine about bond radius, volume, length, and shape factor and the correlation between node radius and bond radius [Jiang et al., 2011].

Figure 6.

(a) A coarse-scale network (red) and a fine-scale network (black); (b) the set Sa of fine-scale nodes within the shadow as adjacent candidates of node N1; (c) the resultant cross-scale connections.

[45] After all coarse-scale nodes have been accessed, the final two-scale network results are as shown in Figure 6c.

[46] For generation of a multiscale network involving multiple length scales, we simply repeat the two-scale network generation process between the further scales. For example, if networks from three length-scale images are available, the network from the smallest length scale is first inserted into that of the middle length scale. A stochastic network of this two-scale network is then inserted into the corresponding network for the largest length scale.

3. Application of the Multiscale Method

[47] The multiscale network generation algorithm is now applied to CT images of a vuggy carbonate at three different resolutions [Vik, 2011]. This vuggy carbonate experimental data was kindly supplied to us by Professor A. Skauge at the University of Bergen, where the multiresolution micro-CT scans were performed. It is used in this work simply as an example data set to illustrate the multiscale method proposed here. A cylindrical laboratory core sample was selected (Figure 7) and three CT images were acquired from the whole sample (core A), a subsample (plug B), and a sub-subsample (plug C), respectively. The image of core A is a gray-level image with different resolutions in three directions, 0.305 mm in both the X and Y directions and 1.5 mm in the Z direction. A central cuboid in the Z direction was cut out of this image with area 70 mm ×70 mm, and the image was segmented to identify the pore space (vugs). The resolution in the Z direction was artificially increased to approximate the resolution in the other directions by dividing each voxel in the Z direction into five identical subvoxels. Thus the reconstructed binary image, denoted by IA, has a volume of approximately 70×70×210 mm3. Similar cut-outs and segmentations were applied to plugs B and C, resulting in binary images IB and IC, respectively. The basic features of these three binary images (IA, IB, and IC) are shown in Table 1.

Figure 7.

Three 3-D CT images (IA, IB, and IC) at three length scales for a vuggy carbonate rock with permeability of 30–45 Darcy [Vik, 2011].

Table 1. Basic Features of the Three Binary Images Shown in Figure 7
ImagesVolume in VoxelsResolution (µm)Image Porosity (%)Volume (mm3)

3.1. Networks at Individual Scales

[48] For the three binary images IA, IB, and IC shown in Figure 7, we have extracted networks NetIA, NetIB, and NetIC as described in section 2.2. Table 2 lists the major properties of each network and the single-phase permeabilities calculated by our pore-network flow model. We note that the permeabilities of the vug network (NetIA), although highly variable, are in the range of the measured permeabilities (30–45 Darcy), but they are extremely large compared to those for the macropores network (NetIB) and micropores network (NetIC).

Table 2. Summary of Network Properties From the Three Images (IA, IB, and IC) and the Corresponding Pore Networksa
Extracted Networks
Network PropertiesNetIANetIBNetIC
  1. a

    For the absolute permeability the range of values corresponding to flow in the three main directions is shown.

Number of nodes NN66311325211518
Number of bonds NB97071892717929
Density of nodes ρN (mm−3)0.0062120.6813083.43
Average coordination number math formula2.862.843.08
Network porosity ϕ (%)15.774.5515.47
Absolute permeability k (mD)25327∼4839540.0127.22∼234.55

[49] Figure 1a shows the pore size distribution (PSD) for the three networks, revealing very large pores in NetIA with only a very small range of pore sizes overlapping with those of NetIB. On the other hand, NetIB and NetIC have a wide range of overlapping pore sizes. However, it is clear that the derived pore size distribution strongly depends on the image size and resolution and that none of the three images contains the complete information on the multiscale pore system in this carbonate rock, although we find in this case that the absolute permeability is largely determined by the vugs. The latter indicates that the network of vugs on its own is percolating (i.e., it is joined up and can flow), unlike NetIB (which cannot flow). However, in this case, the smaller pore network NetIC is well connected, yielding a considerable permeability considering the small pore sizes. This example also demonstrates that there is a trade-off between voxel size and physical sample size in the CT-scanned technology; this is evident in the corresponding connectivity functions, for instance, which are shown in Figure 8.

Figure 8.

The connectivity functions for the three networks shown in Table 2.

3.2. Two-Scale Networks—Single Phase Properties

[50] In our two-scale network generation, we select the whole or subnetwork (rather than use the equivalent stochastic network) of the extracted network of large-scale pores as coarse network and the stochastic network of smaller-scale pores as fine network. For integration of the two networks NetIA and NetIB, we select a subnetwork of NetIA with volume 24.8×24.8×74.7mm3 as NetCoarse and its domain as the nesting domain Ω for the target two-scale network, while we take an equivalent stochastic network of NetIB as NetFine with a network fraction fF = 0.1. The basic features, including the absolute permeabilities, of NetCoarse, NetFine and the integrated network are shown in Table 3. For the well-connected large pores (vugs), integration with the macropore network only leads to a very small increase in permeability. Moreover, the large difference in pore sizes between NetIA and NetIB makes it unlikely that integration has a big impact on the properties of the coarse-scale network.

Table 3. Basic Features of Two Two-Scale Integrated Networks for Vugs and Macropores (Left), as Well as for Macropores and Micropores (Right)a
 Vuggy (NetIA) + Macro (NetIB) poresMacro (NetIB) + Micro (NetIC) pores
 Nesting domain ΩNesting domain Ω
 Volume V(Ω) = 24.8×24.8×74.7 mm3Volume V(Ω) = 4×4×3.43 mm3
 NetCoarse – a sub extracted networkNetCoarse – a sub extracted network
Two-scale networksNetFine – a stochastic network of macroporesNetFine – a stochastic network of micropores
  1. a

    NetFine is the network with reduced density on its own, that is without the cross-scale connections.

  fF = 0.1  fF = 0.1 
math formula2.862.482152.496162.853.1563.212
ϕ (%)9.130.48229.609774.251.495.79
SymbolsNN – Number of nodes.
 NB – Number of bonds.
  math formula – Average coordination number.
 ϕ – Network porosity.
 K – Absolute permeability.
 fF – Fraction of fine-scale network.
  NetIB => NetCoarse

[51] For integration of the two networks NetIB and NetIC, we choose a subnetwork of NetIB as NetCoarse and its volume (4×4×3.43 mm3, see Table 3) as the nesting domain Ω of the target two-scale network, while we take an equivalent stochastic network of NetIC as NetFine with a network fraction fF = 0.1. Observe from Table 3 that the integration of the micropore network of reduced density NetFine, which on its own is well connected, with the nonpercolating macropore network, leads to a permeability (302 mD) that is significantly larger than that of both NetCoarse and NetFine. Notice that in this case the range of pore sizes of NetCoarse and NetFine significantly overlap and that NetFine has also a considerable impact on the porosity of the integrated network.

[52] It is clear from the above example that if a coarse-scale network is already fully connected, as is the case NetIA, then integration with a fine-scale network does not enhance its permeability very much. On the other hand, if the coarse-scale network is poorly connected, as is the case for NetIB, then integration with a fine-scale network may significantly enhance the combined network flow properties. In this case, the density of the integrated fine-scale network, given by the fraction fF, will also have an impact. As mentioned in section 2.4, fF has been introduced to keep flow calculations feasible.

[53] Figure 9 gives some idea of the impact of fF (up to a value of fF ∼0.45) on the porosity and permeability for the two-scale network of macropores and micropores (NetIB and NetIC). As expected, the porosity increases linearly with fF. As NetIB on its own is not percolating, even a small fraction fF of NetIC, shows a sharp increase of the permeability. For larger fF the permeability tends to become almost constant, in this case approximately for fF > 0.2 (see Figure 9).

Figure 9.

Impact of fine-scale network density on porosity and permeability for integration of networks NetIB and NetIC.

3.3. Two-Scale Networks—Two Phase Properties

[54] In this section, we present example two phase relative permeability and capillary pressure network calculations carried out on the two-scale network constructed above. These results are presented here to illustrate differences in two-phase flow properties which can arise between single networks and two-scale integrated networks and to show the effects of the micropores. It is not possible to make a direct comparison with experiment, but work on this is currently in progress. The central contribution of this paper is the multiscale integration method. However, some observations specific to the actual integrated networks presented here are discussed.

[55] Primary drainage (PD) has been simulated in the two-scale integrated network (shown in Table 3) between micropores and macropores described above where oil (nonwetting phase) displaces water from an originally water-wet fully saturated rock. The resulting PD relative permeability curves and drainage capillary pressures are shown in Figure 10 for two-scale networks having a range of values of fF from fF = 0.01 to 0.4. In the PD results in Figure 10a, the water relative permeability krw curve and oil relative permeability kro curve move in different directions as the density of the micropores increases, with the oil curve showing rather more variability. The effect of the presence of the micropores is very significant on the relative permeability curves, but their precise influence is not very clear looking directly at these calculated curves in Figure 10. The situation is clarified by plotting the corresponding (water phase) fractional flow curves, fw(Sw) (where fw =1/(1+(krow)/(krwo))), with the 2 phase viscosities equal in this case, μw = μo. Figure 11 shows the fractional flow curves for selected cases in Figure 10a with fF values 0.01, 0.03, 0.05, 0.10, 0.20 and 0.40; these examples are chosen to span either side of the region where the single phase permeability becomes constant (fF ∼ 0.1). Figure 11 shows that the fractional flow curves at very low values (0.01, 0.03, 0.05) show some variability but they settle to a fairly constant curve at fF ≥ 0.1. For the lowest value of fF = 0.01 (i.e., 1% of the micropore network present) along with the (nonpercolating) medium size pores, the fractional flow curve in Figure 11 indicates that the water → oil displacement would be very unfavorable (assuming no hysteresis in the relative permeabilities). This is probably due to the fact that the oil in the larger pores is isolated and easily bypassed; however, this is the case for this particular network and may not be general.

Figure 10.

Impact of fine-scale network density on relative permeability curve (a) and capillary pressure curve (b) for fF from 0.01 up to 0.4.

Figure 11.

Fraction flow curves for water (fw(Sw)) for selected relative permeability curves in Figure 10(a) for fF values 0.01, 0.03, 0.05, 0.10, 0.20, and 0.40.

3.4. The Three-Scale Network

[56] The full method described above has been applied to the integration of all three-scale networks from NetIA, NetIB, and NetIC and the resulting network is shown (in part) in Figure 12. The color coding in this network shows the various type of pore/bond in the final network and the large vug in this figure is particularly clear.

Figure 12.

Integrated three-scale network from NetIA, NetIB, and NetIC. Only one tenth of the network is displayed to illustrate the multiscale network. The large gray nods are vugs from NetIA, the brown bonds and yellow nodes are macropores from NetIB, while the purple nodes and bonds are micropores from NetIC.

[57] To investigate the multiphase flow properties of the three-scale pore structure of the carbonate rock shown Figure 12, we chose the full size physical domain (4 mm ×4 mm × 6.86 mm) of image IB as the nesting domain of the two-scale network between micropores and macropores. The resulting network contains approximately 3 million network elements. If we then used the domain (210 mm ×70 mm × 70 mm) of IA for the integrated three-scale network of micropores, macropores and vugs, we would obtain an extremely large resultant network with about 35,700 million network elements. In order to keep the pore network to a manageable size for any desk computer available currently, we choose a two-scale network of micropores and macropores in the physical size domain (4 mm ×4 mm × 6.86 mm) of IB but of network size fraction fF = 0.2 when it is integrated into the three-scale network. In the generation of the three-scale network, we choose a small block of image IA (232×232×232 voxels) as the nesting domain Ω and the corresponding network (ϕ = 13.48%, k = 40.679 Darcy) as NetCoarse, and then we generate a three-scale network with network size coefficient of 0.01 of the two-scale network (See Figure 12). This three-scale network model is very heterogeneous in all of its properties, pore size distribution, network topology and its single-phase and two-phase flow functions as discussed below.

[58] The PSD of the original three networks of images IA, IB, and IC are shown in Figure 13 along with that of the full three-level network. The heterogeneity of the pore system is obviously quite extreme but is quite common in carbonate rocks. There are many small pores in this model although they are not apparent in the PSD in Figure 13 since they are not volumetrically very significant in this case; the porosity of the micropore system is 0.05% compared with a total porosity of the system of 13.53%.

Figure 13.

Pore size distribution (volumetric fraction) from three original networks of images IA, IB, and IC and a three-scale integrated network.

[59] To compare the connectivity of the three samples in Figure 7, we calculate the specific Euler number for individual pore networks for micropores, macropores and vugs and the three-scale integrated pore system. The specific Euler number vs. pore size for the three-level network is shown in Figure 14 along with the same quantities for the three sub networks; the total function is quite complex for the full network and it covers 4 orders of magnitude in pore size. Figure 14 shows that the pore connectivity increases dramatically from −0.00155 mm−3 up to −7.0582 mm−3 of the specific Euler number, in other words, almost all the isolated vugs in block B8 are connected by newly added macropores and micropores. As a result of the integration, the predicted permeability is calculated to be k = 42.3 D which is just ∼1.66 D higher than the corresponding two-scale network permeability which is k = 40.679 D.

Figure 14.

Absolute connectivity function of micropore, macropore, vuggy, and integrated pore networks.

[60] Figure 15 shows the calculated oil/water drainage relative permeabilities of the networks for the vugs, the micropores and the full three-level network (the medium pore system does not percolate and so no relative permeability can be calculated for this). As discussed above, this result is presented simply to show that the presence of the smallest pores does make a difference to the two-phase relative permeability of the system.

Figure 15.

Relative permeabilities of microporosity, vugs, and three-scale networks.

4. Conclusions

[61] Building on the method for generating stochastic networks previously published by us, a novel methodology is described in this paper to combine information from pore networks extracted from CT images or numerical models, at widely different length scales to construct a single network [Jiang et al., 2011; Jiang, 2008]. Using the micro-CT measurements at various scales of resolution (2.86, 14.29, and 205 μm) from a vuggy carbonate data set supplied by Vik and Skauge of CIPR at the University of Bergen (Norway), we have used this novel method to construct both two-level and three-level pore networks. These generated multiscale networks contain information from all resolution scales and contain pore elements (pores and bonds) with over 4 orders of magnitude of length scale from micropores to vugs; the example we use is a complex vuggy carbonate available to us.

[62] Having generated the two-scale and three-scale integrated networks from this data, we have carried out statistics and measures as well as single-phase and two-phase flow simulations. As a practical issue in the integrated networks, we could only incorporate a certain fraction, fF, of the very smallest micropores (fF ∼ 0.2 in the three-scale networks). However, with increased computer power this would not be a limitation of our method. The absolute specific Euler function vs. pore radius for the three-level network shows the complexity of the final network and how the sub networks combine to give the total function for the final model. In this case, the two-scale medium/small pore network permeability is actually controlled by the micropores since the medium pore network does not percolate. The absolute permeability of the total three-scale network is dominated by the largest scale vug network and is only slightly enhanced by the addition of the medium sized and smaller pores since the vug network actually percolates. However, this situation depends on the actual rock sample and the results here are quite as expected given the percolation properties of the various scale networks.

[63] Using both the two-scale and three-scale networks, we have carried out some limited two phase network modeling simulations of both oil/water drainage, assessing capillary pressure and relative permeability. These calculations are intended to be illustrative rather than predictive since it is the multiscale construction method that is the novel contribution of this paper. However, the two-scale relative permeability simulations showed that wide variability in this function can be seen in networks having micropores present (at a range of values of fF) because of the changes in connectivity introduced by this smaller pore system. However, plotting the fractional flows (fw versus Sw) clarified the situation and shows that for very low fF the networks give different relative permeabilities but as fF ≥ 0.1, the fractional flows start to overlap and the two phase flow is dominated by the combined micropore/macropore systems. Likewise, some relative permeability calculations have been carried out on the two-level and three-level networks, but these are purely for illustration and no general points can be drawn from these.

[64] Although we believe that the method described in this paper is a significant development, there is space for future improvement, as follows: (i) the spatial correlation between small-scale and large-scale porosities can be incorporated more rigorously, and (ii) three-scale or even higher-scale integrated networks are still too large to handle in our current network models. There are two approaches to the second problem. On the first issue, we will take into account the spatial correlation when integrating network by introducing a method to simulate this relationship and also find approaches to shrink the resultant network size as much as possible without destroying the natural structure of multiscale pore systems. Second, we could try to port our codes onto larger and more powerful machines, but the value of the method is highest if it can be implemented on PCs. However, we will work on parallelization of our two-phase flow simulation code so that it can run the full physical domain of coarse-scale network to capture the maximum heterogeneity of rocks and so the most accurate prediction of petrophysical properties. These are exciting times for studies of the digital petrophysics of complex multiscale rocks—the challenges are very significant but the prize is enormous.


[65] We would like thank Arne Skauge and Bartek Vik of the Centre for Integrated Petroleum Studies (CIPR) at the University of Bergen, Norway, for generously allowing us to use their multiscale data on a vuggy carbonate to illustrate the methodology reported in this paper. The authors would also like to thank their Heriot-Watt University colleagues Sebastian Geiger and Jingsheng Ma for their helpful discussion and comments on this work.