1 Introduction

Multiphase flow in geological porous media is of considerable importance in many practical applications, including production of oil and gas from subsurface formations, geological sequestration of CO2 and contaminants transport and clean-up in soil. The most common description of multiphase flow in porous media at larger scales uses a phenomenological extension of Darcy’s law (Richards 1931; Muskat and Meres 1936). This is the basis for numerical modelling and uncertainty assessment. Phenomenological parameters, such as relative permeability, are typically measured experimentally. These measurements are notoriously time-consuming and costly to perform and experimental data are often scarce. Relative permeability is a manifestation of the underlying pore-scale physics and depends on the pore structure of the rock and the physical characteristics of the solid and the fluids that occupy the pore space (Øren and Bakke 2003). This has spurred an increasing interest in digital rock analysis (DRA) approaches to compute phenomenological parameters that can be used to inform larger-scale continuum models.

In DRA, effective or averaged rock properties are derived from numerical simulations on a digital model of the pore space. The models are typically obtained by high-resolution micro-computed tomography imaging (micro-CT) of sub-samples extracted from the rock in question. Numerical simulations on the model can be performed with a wide range of methods. These can be grouped into direct numerical simulation (DNS), performed directly on a segmented 3D image volume, and pore network models (PNM). In DNS, solvers for mass and momentum balance such as particle-based methods (Liu et al. 2007) or direct Navier–Stokes solvers (Raeini et al. 2014) are coupled with methods to track interfaces, such as the volume-of-fluids method (Raeini et al. 2014) or diffuse interface methods (Anderson et al. 1998; Alpak et al. 2018) such as lattice-Boltzmann (Ramstad et al. 2012; Alpak et al. 2018) or phase-field methods (Koroteev et al. 2014). However, all these methods are extremely computationally expensive and have very limited room for tuning, either through initial or boundary conditions. The computational problem intensifies for complex rocks due to the need to work on images as large as possible in order to capture the representative elementary volume of the rock.

PNM do not work directly on a grid or mesh of the pore space. Instead, the pore space is discretized into a coarse-grained network of pores and throats. Conventionally, pores and throats are represented as abstracted simplified geometrical features that are amenable to analytical or semi-analytical expressions for properties such as capillary pressure, hydraulic conductance and phase saturation (Øren et al. 1998). However, these properties can also come from DNS or even pore-scale flow experiments. Typically, PNMs assume quasi-static pore-scale physics and displacement rules, i.e. the displacement is so slow that viscous forces can be neglected and only capillary and buoyancy forces need to be accounted for. This reduces the computational cost significantly. By relying on capillary equilibrium displacement rules abstracted from decades of observations and experience in multiphase flow experimentation (Lenormand et al. 1983), a balance is found between a very high computational efficiency, interpretability and physical accuracy (Øren et al. 1998; Valvatne and Blunt 2004). This allows extensive variation of modelling parameters (Foroughi et al. 2020) and is perhaps the main reasons why pore network modelling has been one of the most widely used numerical tools for studying capillary dominated two-phase flow at the pore-scale (Fatt 1956a, b, c; Bryant and Blunt 1992; Øren et al. 1998; Patzek 2001; Øren and Bakke 2003; Valvatne and Blunt 2004; Ryazanov et al. 2010).

Due to the abstracted geometry and simplified physics, which are key to computational efficiency (Wang et al. 2021a), the predictive capabilities of pore network modelling have been challenged (Sorbie and Skauge 2012). Historically, predictive capabilities of PNM improved significantly with the ability to reproduce essential geometrical and topological features of the pore space using networks extracted from micro-CT images (Lindquist and Venkatarangan 1999; Lindquist et al. 2000; Silin and Patzek 2006; Dong and Blunt 2009; Ruspini et al. 2017) or process-based reconstructions (Bakke and Øren 1997; Øren and Bakke 2002, 2003). A number of early studies demonstrated that quasi-static PNMs could match experimental relative permeability curves for simple rocks (Øren et al. 1998; Øren and Bakke 2003; Valvatne and Blunt 2004; Blunt et al. 2013). The development of an optimized set of displacement rules (Ruspini et al. 2017) and an improved network extraction algorithm (Øren et al. 2019) have further improved the predictive capabilities of PNM. Fluid distributions predicted by PNM have been validated by comparison with micro-CT imaging on a pore-by-pore basis (Øren et al. 2019). For a water wet sandstone, the pore filling discrepancy was less than 3% for primary drainage and less than 7% for imbibition. Recently, pore level fluid occupancy and filling sequences predicted by PNM have been validated against time-resolved synchrotron beamline data of imbibition in a sandstone and a carbonate (Bultreys et al. 2019).

An often overlooked fact in DRA is that in most practical cases a micro-CT image of the pore space alone is not sufficient to arrive at a predictive numerical model. The pore space geometry typically requires further adjustments because parts of the pore space are not sufficiently resolved. Furthermore, the wettability or contact angle distribution must be tuned to experimental data (Rücker et al. 2019) as this cannot be deduced from micro-CT scans of the pore structure alone. For practical applications, it is thus “best practice” to anchor digital rocks models to available experimental data (Saxena et al. 2019, 2021).

The high computational efficiency and flexible geometrical representation used in PNMs make them excellently suited to represent complex rocks having broad pore size distributions that cannot be captured in a single micro-CT image. However, when a significant fraction of the percolating porosity is below the resolution of the micro-CT image (Leu et al. 2014; Soulaine et al. 2016; Saxena et al. 2017), conventional pore network modelling is no longer valid and a multiscale PNM approach is needed. These typically gather information from images at different scales and fuse them into a single network. Two principal approaches can be followed: (i) adding all the pores explicitly by connecting pore networks at different scales to each other (Mehmani and Prodanovic 2014; Jiang et al. 2013; Prodanovic et al. 2015), or (ii) only explicitly resolving pores at the largest scales and representing the smaller pores in an averaged or continuum sense by adding extra Darcy-type network elements to the network (Bekri et al. 2005; Bauer et al. 2012; Bultreys et al. 2015, 2016a). The first approach is more detailed, but quickly experiences computational difficulties when the scale difference becomes large. The second approach does not have this problem and is therefore well suited to study systems with large differences in pore sizes (Bultreys et al. 2016b; Ruspini et al. 2016).

In this work, we describe a generalized multiscale imaging and pore network modelling approach to analyse complex rocks. A dry/wet micro-CT imaging sequence (Knackstedt et al. 2013) is used to spatially map resolved and unresolved porous regions. The unresolved porosity regions are classified into different rock types (i.e. porosity types). The porosity and rock-type maps are used to extract a multiscale pore network model. The essence of the multiscale PNM is to represent resolved pores as abstracted simplified geometrical features and unresolved porosity regions as a Darcy-type porous medium (Darcy pores). Each Darcy pore is populated with single and multiphase flow properties derived from high-resolution images of unresolved porosity regions. Relative permeability and capillary pressure scanning curves are used to calculate individual waterflooding saturation curves for each Darcy pore based on their local saturation state after primary drainage.

We apply the multiscale PNM workflow to two heterogeneous reservoir rocks, a mixed wet thinly laminated sandstone and an oil wet carbonate. The PNM is anchored to experimental data using the procedure proposed by Masalmeh et al. (2015). One of the advantages of this anchoring procedure is that more easy-to-obtain experimental data such as capillary pressure can be used to predict much more costly to measure parameters such as the imbibition relative permeability curve which is one of the key parameters required for reservoir modelling. We calculate imbibition relative permeabilities for the two samples in a blind test and compare the results with high-quality experimental data.

2 Methods and Materials

2.1 Multiscale Digital Rock Workflow

The novel workflow used in this work to analyse complex rocks integrates multiscale imaging, numerical modelling and the use of available experimental data to constrain modelling assumptions at different scales (Masalmeh et al. 2015). It incorporates heterogeneities at smaller scales and captures their effect on rock properties at larger scales. Figure 1 illustrates the multiscale imaging and modelling workflow. The imaging workflow is a down-scaling process and consists of two main steps;

  1. 1.

    Classification: determination and spatial mapping of rock types at a given scale.

  2. 2.

    Selection: choice of representative samples for imaging at a smaller scale (i.e. higher resolution).

Fig. 1
figure 1

Multiscale imaging (top) and modelling (bottom) workflow. Multiscale imaging begins with a micro-CT scanning of the plug scale with subsequent rock typing. Higher-resolution micro-CT scans of physical sub-samples at sub-plug scale and electron microscopy (SEM and FIB-SEM) are, then, used to resolve details down to the nanometre level. This information is then integrated by computing effective properties from explicitly resolved higher-resolution scales, which are then propagated into a multiscale pore network that contains explicitly resolved pores and “Darcy pores”. From the multiscale PNM upscaled effective properties can be computed

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Imaging commences at the largest scale which in this work is the plug scale. The plug is scanned both in a dry and wet state (fully saturated with contrast X-ray attenuating brine) by micro-CT imaging following Arns et al. (2003), Golab et al. (2010). The two images are registered and the difference image is used to generate a 3D porosity map which spatially map the porosity and the connectivity of resolved and unresolved porous regions, as shown in Fig. 2. The voxel size of the plug scans is typically in the range of 10 µm to 20 µm.

The unresolved porous regions are classified into different rock types, i.e. different porosity classes as a consequence of rock components and/or diagenetic processes. We use the porosity map to define porosity-based rock types and their 3D spatial mapping. The porosity map is a logical starting point for defining rock types since it allows the use of conventional cross-variograms between porosity and permeability and scaling of capillary pressure using the Leverett J-function (Leverett 1941). However, more advanced pore-scale rock typing methods, such as those based on morphological descriptors or Minkowski functionals, can also be used (Ismail et al. 2013; Jiang and Arns 2020).

Based on the spatial mapping of rock types (i.e. rock-type map), representative regions are selected for extraction of sub-plugs to be micro-CT imaged at higher resolution, typically at 1 µm to 5 µm voxel size. If the sub-plug contains significant unresolved porosity, the classification and selection steps described above are repeated at the sub-plug scale. In this case, representative 3D pore space models are derived from scanning electron microscopy (SEM) images using a process-based modelling (PBM) workflow (Bakke and Øren 1997; Øren and Bakke 2003). PBM was initially developed for sandstones but has been extended to carbonates (Roth et al. 2011). In essence, 3D pore space models are reconstructed from 2D sections and validated against SEM images using a range of statistical correlation tests.

Fig. 2
figure 2

Porosity map from micro-CT: dry scan (left), wet scan where the rock sample has been saturated with brine doped with a contrast agent (centre). The porosity map is constructed from the difference (right) between wet and dry scans

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The numerical modelling workflow is illustrated in Fig. 1 (bottom). It is a hierarchical upscaling process. Since in many cases the scales are not clearly separated, computing effective properties of heterogeneous rocks implies an intrinsically coupled combination of pore-scale simulation and upscaling. The workflow consists of three main steps;

  1. 1.

    Evaluation: characterization of average properties at the smaller scale.

  2. 2.

    Population: propagation of properties at the smaller scale into regions at the larger scale.

  3. 3.

    Upscaling: calculation of average or effective properties at the larger scale.

These steps are general and are the same as those used in conventional steady-state upscaling at larger scales (Ekrann and Aasen 2000; Pickup and Hern 2002). Characterization of average properties at the smaller scale can come from flow experiments or from pore-scale modelling (Kløv et al. 2003; Rustad et al. 2008). Simulations at the larger scale are conventionally done using continuum-based flow simulators (Aarnes et al. 2007; Odsæter and Rustad 2015). Here, we use a novel multiscale pore network model to simulate flow at the larger (plug) scale.

2.1.1 Multiscale Pore Network Model

The multiscale pore network model is constructed by combining a 3D porosity map and a 3D rock-type map (labels grid). Individual networks are extracted for the resolved pores and for each of the rock types following the method presented in Øren et al. (2019). Resolved pores are described in a conventional PNM manner. Pores belonging to an unresolved porosity region are treated as a continuum and denoted as Darcy pores. Each of them has a rock-type index and an average porosity value which is measured directly from the porosity map. Geometrical properties that are used to characterize Darcy pores include volume, cross-sectional area, length and radius.

A virtual link is created where there is a contact area between pores from different networks. No volume or length is assigned to virtual links and we only store the cross-sectional area. In this way, independent pore networks are merged into a single multiscale pore network. Figure 3 shows an example of a multiscale pore network of a Savonnieres carbonate rock sample (Bultreys 2016). For visualization purposes, the rock-type map and the pore network are overlaid to demonstrate the equivalence of the geometrical representation of the PNM and the micro-CT image. Clearly, the multiscale pore network honours the 3D spatial distribution of resolved pores and Darcy pores.

Fig. 3
figure 3

Rock-type map and the corresponding multiscale pore network model for a Savonnieres carbonate rock sample (Bultreys 2016). Explicitly resolved pores are shown in red, rock type or Darcy pores are shown in blue and the solid is shown in white

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Properties which are assigned to the resolved pores, such as capillary pressure and hydraulic conductances, are calculated analytically or semi-analytically from the simplified geometrical description (Øren et al. 1998). Darcy pores, on the other hand, must be populated with single- and two-phase continuum rock properties. These properties are calculated from the high-resolution pore space models using a traditional PNM which incorporates optimized displacement rules (Ruspini et al. 2017). The following properties must be provided:

  • Permeability as a function of porosity, \(\mathbf{K} = f(\phi )\)

  • Primary drainage and imbibition capillary pressure curves, \(P_c(S_w)\)

  • Primary drainage and imbibition relative permeability curves, \(k_r(S_w)\)

  • Capillary pressure and relative permeability scanning curves

For numerical implementation, the capillary pressure function for each rock type is made dimensionless by using the Leverett J-function

$$\begin{aligned} J(S_w) = \frac{P_c(S_w)\sqrt{\frac{K}{\phi }}}{\sigma }. \end{aligned}$$
(1)

Permeability versus porosity trends for each rock type are generated from sub-sampling of the high-resolution images. Figure 4 shows an example of the directional permeability data calculated from sub-sampling. To capture the significant scatter in permeability at a given porosity, we use a power law correlation of the form

$$\begin{aligned} K = \left[ (\alpha A_{\min}+(1-\alpha )A_{\max}\right] \phi ^b \end{aligned}$$
(2)

where \(\alpha\) is a normal distributed random number in the range [0–1] and \(A_{\min}\) and \(A_{\max}\) define the lower and upper limits for the permeability at a given porosity, as illustrated in Fig. 4. Figure 5 shows an example of \(k_r(S_w)\) and \(P_c(S_w)\) scanning curves calculated on a high-resolution rock-type model of a water wet sandstone. The saturation increment between the calculated scanning curves is \(\varDelta S_w = 0.1\).

Fig. 4
figure 4

Porosity versus permeability correlation calculated (by PNM) from sub-samples of a high-resolution model. A power law corresponding to Eq. (2) is plotted in grey line and the upper/lower limit boundaries are shown in dotted lines

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Fig. 5
figure 5

Relative permeability and capillary pressure scanning curves calculated on a high-resolution model of a water wet sandstone

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2.1.2 Modelling of Two-Phase Flow

We simulate two-phase (oil–water) immiscible flows in the multiscale PNM. The invading fluid is injected through an external reservoir that is connected to every pore at the inlet side of the network. The displaced fluid escapes through the outlet face on the opposite side. No-flow boundary conditions are imposed along the sides parallel to the main flow direction.

Fluid displacements are modelled using the assumption of quasi-static pore-scale physics: the displacements are so slow that viscous forces can be neglected and only capillary and buoyancy forces need to be accounted for. The fluids are assumed to be in capillary equilibrium at each saturation state. Predicting the evolution of the displacement therefore reduces to calculating the entry capillary pressure associated with available pore elements and invade them following the order of entry pressures. We note that in the limiting case where all the porosity is resolved, the multiscale PNM is identical to a traditional single scale PNM (Øren et al. 2019; Ruspini et al. 2017). In the other limiting case where no porosity is resolved, multiscale PNM simulations are identical to steady-state upscaling in the capillary limit (Ekrann and Aasen 2000; Pickup and Hern 2002; Kløv et al. 2003; Rustad et al. 2008).

Primary drainage: Initially, the PNM is fully saturated with water and is strongly water wet. Then, oil (non-wetting phase, nwp) enters the network representing oil migration into the reservoir. At every stage of the process, nwp invades the pore with the smallest entry capillary pressure. This forms the basis for the invasion percolation algorithm used to model drainage processes (Wilkinson and Willemsen 1983). Entry or threshold capillary pressures for explicitly resolved pores are calculated using the MS-P method (Mayer and Stowe 1965; Princen 1969). Entry or threshold capillary pressure for Darcy pores, \(P_{c}^e\), are scaled by the J-function

$$\begin{aligned} P_{c}^e = J(S_w=1)\sigma \sqrt{\frac{\phi }{K}}. \end{aligned}$$
(3)

This ensures that Darcy pores have different entry pressures and local capillary pressure curves. Darcy pores are considered to be available for invasion if a percolating path from the pore to the inlet exists and if the relative permeability of the oil phase is greater than zero.

Waterflooding: The capillary pressure drops during water injection and water invades pores with the lowest displacement capillary pressure. The displacement pressure for resolved pores is calculated following (Øren et al. 1998; Ruspini et al. 2017). Since water is connected throughout the network, the local water saturation in a Darcy pore at a given global capillary pressure is determined by the local capillary pressure curve assigned to the pore. This depends on the local pore properties (i.e. porosity, permeability and rock-type index) and on the global capillary pressure at the end of primary drainage.

Figure 6 illustrates the method used to determine the local \(J(S_w)\), and thus \(P_c(S_w)\), for a Darcy pore. We calculate the local \(J(S_w)\) function for a given initial \(S_w\) by interpolating between the two closest scanning curves. The same methodology is used to determine the local waterflood relative permeability functions that are assigned to each Darcy pore. The incorporation of scanning curves constitutes a major improvement compared to previous multiscale PNMs that used only a single imbibition capillary pressure curve and a single relative permeability curve for all the Darcy pores (Bultreys et al. 2016a).

Fig. 6
figure 6

Methodology to model scanning curves in waterflooding processes. The dashed line corresponds to the primary drainage \(J(S_w)\) function. The red lines correspond to the set of scanning curves calculated from high-resolution images of the rock type. The local \(J(S_w)\) function assigned to each Darcy pore depends on the initial \(S_w\) and is determined by interpolating between the two closest curves. The blue lines show the \(J(S_w)\) curves used for two different pores

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Effective Properties: Effective or upscaled properties that are calculated from the multiscale PNM include absolute permeability and saturation dependent functions such as relative permeability and capillary pressure curves. At each moment the fluids are assumed to be in capillary equilibrium and \(p_c\) is constant throughout the simulation domain \(\varOmega\). The average water saturation, \(\overline{S}_w\), is determined by pore volume weighted averaging,

$$\begin{aligned} \overline{S}_w = \frac{\int _\varOmega V({\mathbf{x}} ) \phi ({\mathbf{x}} ) S_w({\mathbf{x}} ) {\text{d}} \varOmega }{\int _\varOmega V({\mathbf{x}} ) \phi ({\mathbf{x}} ){\text{d}} \varOmega } \end{aligned}$$
(4)

where V is the volume of the pore located at x. The upscaled capillary pressure curve, \(\overline{P}_c(\overline{S}_w)\), is constructed by calculating the average water saturation at different global \(p_c\) values.

For laminar flow, the flow-rate of fluid k between two connected pores i and j is given by

$$\begin{aligned} q_{k,ij} = \frac{g_{k,ij}}{L_{ij}} \left( P_{k,i} - P_{k,j} \right) \end{aligned}$$
(5)

where \(L_{ij}\) is the spacing between the pores, \(g_{k,ij}\) is the effective hydraulic conductance and \(P_{k,i}\) is the pressure of phase k in pore i. The hydraulic conductances of resolved pores are calculated as described in Øren et al. (1998). The conductances of Darcy pores are calculated using

$$\begin{aligned} g_{k,i} = K_i k_{rk,i}(S_{w,i}) A_i \end{aligned}$$
(6)

where \(A_i\) is the cross-sectional area and \(S_{w,i}\) is the local water saturation. \(K_i\) is the absolute permeability and \(k_{rk,i}\) is the relative permeability function of phase k assigned to pore i. \(k_{rk,i}\) depends on the rock-type index of the pore and the local saturation at the start of the displacement process. Similarly to resolved pores, the effective hydraulic conductance for flow between two connecting Darcy pores is assumed to be the harmonic mean of the two pores and the connecting throat.

The fluids are immiscible and conservation of mass at each pore requires that

$$\begin{aligned} \sum_j{q_{k,ij}} = 0 \end{aligned}$$
(7)

where j runs over all the pores connected to pore i. Equations (5) and (7) gives rise to a set of linear equations for the pore pressures. The effective permeability \(\overline{K}\) of the multiscale PNM is computed by imposing a constant pressure gradient across the model and letting the system relax using a LDL\(^T\) Cholesky decomposition method to solve for the pore pressures (Golub and VanLoan 1996). The total flow-rate is calculated from the pressure distribution and the absolute permeability is determined using Darcy’s law.

Relative permeability is calculated similarly. At a given global saturation \(\overline{S}_w\), we calculate the flow of each phase separately by solving the single-phase pressure equation twice to obtain phase permeability \(\overline{K}_k\) (\(k =\) wetting, non-wetting) and the effective relative permeability \(\overline{k}_{rk} = \overline{K}_k \cdot \overline{K}^{-1}\) at the saturation point \(\overline{S}_w\). The upscaled relative permeability curve, \(\overline{k}_{r},(\overline{S}_w)\) is constructed by calculating the phase permeability at different global saturation.

2.1.3 Anchoring the Multiscale PNM to Experimental Data

We anchor the multiscale PNM to experimental data using the workflow proposed by Masalmeh et al. (2015). The complete workflow requires porosity, absolute permeability, drainage and imbibition capillary pressure curves (after ageing the rock with crude oil to restore its reservoir wettability). The measured porosity is compared to the image-based porosity calculated from the porosity map. The relative difference between the two porosity values should be less than \(\pm {10}\%\). If the porosity mismatch is larger than this, it is a strong suggestion that the image is not representative. This can be due to imaging or registration artefacts or that the two porosity measurements are not from the same rock volume.

The pore size distribution of the multiscale PNM is anchored by comparing simulated and measured drainage mercury–air capillary pressure (MICP) curves. If the curves are different, we tune the pore size distribution of the Darcy pores by adjusting the permeability versus porosity correlation for each rock type. This is done by changing the \(A_{\min}\) and \(A_{\max}\) values in Eq. (2). The size of the resolved pores is not adjusted. The tuning of pore sizes is done automatically by minimizing the difference between the simulated and measured MICP curves.

Figure 7 shows a comparison between measured and simulated (after tuning) MICP curves for a heterogeneous carbonate. A multiscale PNM was extracted from images of the actual sample (Fig. 7 (left)) used in the experiment. PNM capillary pressure curves are typically calculated assuming that only a single face of the model is open for invasion. Figure 7 shows that in this case the simulated capillary pressure curve significantly overestimates the measured MICP curve. By imposing the same boundary conditions in the simulations as in the experiment (i.e. injection from all six faces of the sample), the match with the measured data improves significantly.

Fig. 7
figure 7

Comparison between measured and simulated MICP curves for a heterogeneous carbonate sample (left) after tuning the pore sizes of Darcy pores. The MICP curves are scaled according to the brine-oil interfacial tension. Directional capillary pressure curves (i.e. invasion only from one face of the model) are also shown. A 2D slice through the 3D porosity map of the sample is shown in the left image

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The wettability or contact angle distribution for a given system varies with parameters such as surface roughness, mineralogy, oil composition and brine salinity in ways that are difficult to predict (Rücker et al. 2019). The advancing contact angle \(\theta _a\) to be assigned to different pores must therefore be anchored or calibrated to experimental data (Sun et al. 2020). In this work, we use imbibition capillary pressure (after ageing the rock with crude oil to restore its reservoir wettability) to do this. However, there is no unique way of determining the microscopic (i.e. pore scale) distribution of \(\theta _a\) from macroscopic measurements such as capillary pressure. Different microscopic configurations can produce similar macroscopic behaviour, especially for mixed wet systems where spatial correlations of contact angles exist. For samples with no spontaneous imbibition of water, an initial guess of the advancing contact angle can be derived from the drainage capillary pressure curve using the method described in Masalmeh and Jing (2006).

2.2 Rock Samples

Reservoir samples were carefully selected based on the availability of high-quality relative permeability and imbibition capillary pressure data, in addition to standard porosity and permeability information.

2.2.1 Laminated Sandstone Rock (Sample SS4)

Sample SS4 consists of moderately sorted sandstones with planar and cross laminae with bimodal grain size distribution in alternating coarse-grained and fine-grained laminae. The coarse-grained laminae are well sorted and very porous with well-connected pores and average grain sizes of 0.359mm. The fine-grained laminae are volumetrically minor and moderate to well-sorted, with average grain sizes of 0.119mm. Pore sizes are smaller and volumetrically less relevant in fine-grained laminae, due to the smaller grain sizes and grain-to-grain compaction. Only limited amounts of cements are observed to cause porosity reduction. The cements present in sample SS4 are grain-coating chlorite, and locally, anhydrite, quartz overgrowths and authigenic potassium feldspar.

Sample SS4 shows a very wide pore size distribution ranging from approximately 30 µm to 0.01 µm. Figure 8a shows the central YZ-slice of the 3D overview scan of the full plug. The voxel size is 10.1 µm. The sample is clearly laminated and an-isotropic. The coarse lamina is only 5–7 grains in width. Table 1 summarizes the basic properties of sample SS4.

Table 1 Dimensions and basic properties of the two samples
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Fig. 8
figure 8

a Overview scan of sample SS4. The voxel size is 10.1 µm. b Overview scan of sample Ca1. The voxel size is 16.5 µm

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2.2.2 Dual Porosity Heterogeneous Carbonate (Sample Ca1)

Sample Ca1 is a reservoir carbonate. It consists of a grain-supported fabric with variable grain sizes ranging mostly from 30 to 100 µm (peloids and foraminifera), reaching up to around 1mm or more (composite grains and red algae fragments). Cements around grains and in larger pores reduced significantly the depositional porosity. The mineralogy of grains and cements is calcite. Ca1 shows a very broad and heterogeneous pore size distribution (Masalmeh and Jing 2008) covering more than three orders of magnitude, from 0.02 to 40 µm. Figure 8b shows the central YZ-slice of a dry state scan of the full plug. The voxel size is 16.5 µm. The sample is clearly heterogeneous. The basic properties of sample Ca1 are summarized in Table 1.

2.2.3 Relative Permeability and Capillary Pressure Measurement

The relative permeability data were measured using the steady-state method, following the Shell internal standard protocols (Kokkedee et al. 1996; Rücker et al. 2021), combining in-situ saturation monitoring with accurate pressure measurements and bump floods at the end of the tests (Berg et al. 2021a, b). The imbibition capillary pressure curves were measured on twin samples, using multi-speed centrifuge, on set-ups equipped with automatic data acquisition (Masalmeh et al. 2014). For both sets of tests, numerical simulations [using Shell internal simulator, MoReS (Regtien et al. 1995; Berg et al. 2021a)] were performed on the experimental data to extract relative permeability and capillary pressure curves (Sorop et al. 2015; Berg et al. 2021b).

3 Results and Discussion

We applied the multiscale imaging and modelling workflow to calculate imbibition relative permeability curves for the two reservoir rocks described above. The predictive capability of the multiscale PNM was assessed in a blind test where only drainage and imbibition capillary pressure data were used for anchoring the model. The measured imbibition relative permeability was held back and only used for validation.

3.1 Laminated Sandstone (Sample SS4)

Figure 9a shows a central slice of the 3D plug scale porosity map. The resolved porosity is \(\simeq 7\%\) and the unresolved porosity is \(\simeq 14\%\). The unresolved porosity regions were divided into two rock types (RT1 and RT2) based on the MICP data (see Fig. 9c) and using porosity-based rock typing. A central slice through the 3D rock-type map is displayed in Fig. 9b. A multiscale PNM was extracted from the central region of the sample. The model contained more than 7 million pores.

Fig. 9
figure 9

a 3D porosity map, b rock-type map at the plug scale and c pore throat size distribution derived from MICP data for sample SS4. Resolved pore is black, RT1 is brown, RT2 is grey, and solid is yellow

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A 12mm horizontal sub-plug was extracted from the plug. Figure 10a shows an overview scan of the sub-plug. The voxel size is 9.3 µm. A Region of Interest (ROI) scan of the sub-plug is displayed in Fig. 10b. The resolution of the ROI scan is 1.6 µm. Inter-pore filling clay and clay lining on grain surfaces can clearly be seen in the ROI scans (see Fig. 10c). A single scale PNM was extracted from the ROI scan and used to calculate the input rock curves for rock type RT1.

Fig. 10
figure 10

a Overview scan of the 12mm sub-plug. The voxel size is 9.3 µm. b ROI scan of the sub-plug. The voxel size is 1.6 µm. c close-up of the ROI scan

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Figure 9c shows that rock type RT2 represents pore sizes in the range [1–0.01 µm]. No high-resolution images of the sample in this range were available. Instead, we generated a high-resolution model of RT2 from the MICP data assuming a fractal pore size distribution (i.e. Brooks-Corey \(P_c\) model). A fractal dimension of \(D_f=2.2\) fitted the MICP data well for the pore size range covered by RT2. Rock curves were generated using the fractal model described in Li (2004). Wetting phase relative permeability was derived using the Purcell method (Purcell 1949) while non-wetting phase relative permeability was calculated using the Burdine approach (Burdine 1953).

Figure 11 shows the relative permeability rock curves for the two rock types. There is strong hysteresis between drainage and imbibition relative permeabilities, especially for RT1. This is due to the bimodal grain (and pore) size distribution of the sample and to the different wettability states for primary drainage (strongly water wet) and imbibition (weakly oil wet). In primary drainage, oil is the non-wetting phase and preferentially invades the largest coarse-laminae pores. These pores are well connected and the oil relative permeability increases quickly. In forced imbibition water is the non-wetting phase. It preferentially invades the largest oil filled pores and the oil relative permeability drops sharply. However, the imbibition water relative permeability does not increase as steeply as expected, akin to the oil relative permeability in primary drainage. This is because water layers established during primary drainage make water injection in oil wet media similar to an ordinary percolation process and not an invasion percolation process as primary drainage (Blunt 2017).

Fig. 11
figure 11

Sample SS4; relative permeability rock curves for rock type RT1 (left) and RT2 (right). The receding contact angle for primary drainage is in the range [0°–10°] (i.e. strongly water wet). The advancing contact angle is distributed in the range [110°–130°] (i.e. weakly oil wet)

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Two-phase (oil–brine) flow is simulated in the multiscale PNM. The interfacial tension is 31.9 mN/m and the brine and oil densities are 1106 kg/m3 and 759 kg/m3, respectively. Flow is in the vertical direction (i.e. aligned with the coarse-grained lamina). The multiscale PNM was anchored to the experimental MICP data by tuning the permeability versus porosity correlations for rock types RT1 and RT2. Figure 12a compares the simulated and measured MICP curves. The curves are scaled according to the oil–brine interfacial tension. We terminated the primary drainage simulations at the experimental \(S_{wi} \simeq 0.18\).

The wettability or advancing contact angle assigned to the PNM was anchored to the measured imbibition capillary pressure curve. The experimental curve displayed little or no spontaneous water imbibition and an initial guess of the advancing contact angle was estimated using the method described in Masalmeh and Jing (2006). The final “best fit” to the measured \(P_c(S_w)\) curve was obtained using a Gaussian distribution of contact angles in the range [120°–140°] for the resolved pores and [110°–130°] for RT1 and RT2. Figure 12b compares the simulated and measured imbibition capillary pressure curves.

It was difficult to obtain a “perfect” match to the measured capillary pressure curve by only adjusting the range of the contact angle distribution for the different rock types. This is not surprising since there is no unique way of determining the pore-scale distribution of wettability from macroscopic measurements such as capillary pressure. In addition, the measured capillary pressure stems from a “twin” sample having slightly different porosity and permeability.

Fig. 12
figure 12

a Measured and anchored simulated MICP curves and b measured and anchored simulated imbibition capillary pressure curves

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The imbibition relative permeability “predicted” by the anchored multiscale PNM is compared with experimental data in Fig. 13. The calculated oil relative permeability curve is in close agreement with the measured data across the entire saturation range. For \(S_w <0.3\), the simulated water relative permeability displays clear percolation threshold effects (Wilkinson and Willemsen 1983; Larson et al. 1981; Heiba et al. 1983). Overall, however, we find the agreement between measured and simulated relative permeability quite encouraging, especially considering the very broad pore size distribution of sample SS4.

Fig. 13
figure 13

Plug level comparison between simulated (blue lines) and measured (red squares) imbibition relative permeability curves for sample SS4

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The discrepancy in \(k_{rw}\) at low water saturations is mainly due to different inlet boundary conditions in the simulations and in the experiments. In the simulations, only water is injected at an infinitesimal small rate. At low \(S_w\), water is forced to flow through connected layers established at the end of primary drainage. The water layers are pinned (Øren et al. 1998) and retain approximately the very low conductance from the end of primary drainage and the \(k_{rw}\) is small. Any appreciable increase in \(k_{rw}\) requires a spanning cluster of water-filled pores. In steady-state relative permeability experiments, both oil and water are injected simultaneously. At low \(S_w\) (i.e. small fractional flow of water) in an oil wet media, water is the non-wetting phase and the principal flow of water is through the movement of ganglia (Armstrong et al. 2016; Rücker et al. 2015). This is different from flow through connected pathways of each phase, as assumed in PNM and as implicitly assumed in the use of traditional multiphase Darcy’s law.

3.2 Dual Porosity Heterogeneous Carbonate (Sample Ca1)

Figure 14a shows a central slice of the plug porosity map generated from the difference image (dry-wet images). The resolved porosity is \(\simeq 2\%\) and the unresolved porosity is \(\simeq 28\%\). The unresolved porosity was classified into two porosity classes or rock types (RT1 and RT2). The approximate range of pore throat sizes for the two rock types is indicated in Fig. 14c. A 2D slice through the generated rock-type map which characterizes the spatial distribution of the different rock types is shown in Fig. 14b. A multiscale PNM was extracted from the images. The model contained more than 3.2 million pores.

Fig. 14
figure 14

a 3D porosity map, b rock-type map at the plug scale and c pore throat size distribution derived from MICP data for sample Ca1. Resolved pore is black, RT1 is brown, RT2 is grey, and solid is yellow. The distribution has two distinct peaks, one at 7 µm and at 0.4 µm

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A high-resolution 3D model for RT1 was generated from an 8mm sub-plug extracted from the left bottom corner of the plug. Figure 15 shows an overview scan of the sub-plug. The size of the scan is 1300 × 1300 × 2400 voxels and the voxel size is 3.37 µm. A single scale PNM was extracted from the image and used to calculate the input rock curves for rock type RT1.

Fig. 15
figure 15

Sub-plug micro-CT image for sample Ca1, XZ-plane (left) and XY-plane (right)

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Fig. 16
figure 16

BSE images of sample Ca1

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A high-resolution 3D model for rock type RT2 was generated from SEM images using process-based modelling (Bakke and Øren 1997; Øren and Bakke 2003; Roth et al. 2011). Several SEM images were acquired at different resolutions. The overview image in Fig. 16a shows the approximate locations where the higher-resolution images were acquired from. Figure 16b shows an image with a resolution of \(\sim\)0.02 µm that was used as input to the process-based modelling. Three different models were generated to capture variations of the different properties. A single scale PNM was extracted from each of them and used to generate rock curves for RT2.

Figure 17 shows the relative permeability rock curves for rock types RT1 and RT2. The relative permeability hysteresis is smaller than that for SS4 (see Figure 11), although the contact angles assigned to the rock types are similar. This is of course due to different pore structures, i.e. different topological and geometrical features, pore size distributions and local connectivity.

Fig. 17
figure 17

Sample Ca1; relative permeability rock curves for rock type RT1 (left) and RT2 (right). The receding contact angle for primary drainage is in the range [0°–10°] (i.e. strongly water wet). The advancing contact angle is distributed in the range [105°–115°] (i.e. weakly oil wet)

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Primary drainage and imbibition displacements are simulated in the multiscale PNM. The interfacial tension is 27.0mN/m and the brine and oil densities are 1098 kg/m3 and 782 kg/m3, respectively. Flow is in the vertical direction. The PNM was anchored to the experimental MICP data by tuning the permeability versus porosity correlations for rock types RT1 and RT2. Figure 18a compares the simulated and measured drainage capillary pressure curves. There is a small difference between the two curves at high \(S_w\) values. This is the region characterized by the resolved pores which are not adjusted. The discrepancy is due to the fact that the simulation and the measurements are done on different rock samples. The primary drainage simulation terminated at the experimental \(S_{wi}\simeq 0.09\).

Fig. 18
figure 18

a Measured and simulated mercury–air intrusion capillary pressure curves (MICP) for sample Ca1. b Measured and simulated oil–brine imbibition capillary pressure curves for different contact angle distributions

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We calibrated the wettability or advancing contact angles for the multiscale PNM by comparing simulated and measured imbibition capillary pressure curves. The experimental sample displayed little or no spontaneous water imbibition and an initial guess of the advancing contact angle was estimated using the method described in Masalmeh and Jing (2006). The “best fit” to the measured \(P_c(S_w)\) curve was obtained using a Gaussian distribution of contact angles in the range [120°–130°] for the resolved pores, [105°–115°] for RT1 and [100°–105°] for RT2. Figure 18a compares the simulated and measured imbibition capillary pressure curves. The agreement between the two curves is compelling, especially considering that they come from different samples.

The simulated imbibition relative permeability for the calibrated contact angle case is compared with experimental data in Fig. 19. The simulated oil relative permeability is in close agreement with the measured data across the entire saturation range. As for sample SS4, the simulated water relative permeability curve displays clear percolation threshold effects at low water saturations (\(S_w < 0.3\)). This is mainly due to different inlet boundary conditions in the experiment and in the simulation and to the PNM modelling assumptions that fluids can only flow through connected pathways. The simulated water relative permeability slightly under-predict the experimental data for \(0.3< S_w < 0.5\) but is in close agreement with the data for \(S_w > 0.5\). Overall, we find the agreement between measured and simulated imbibition relative permeability encouraging, especially considering the heterogeneous pore size distribution of the sample.

Fig. 19
figure 19

Plug level comparison between simulated (blue lines) and measured (red squares) imbibition relative permeability for sample Ca1

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The good agreement between numerical results and experimental data for samples SS4 and Ca1 is promising. Although we need to examine a much larger group of petrophysically heterogeneous rock samples, these preliminary results suggest that multiscale DRA may unlock the possibility of augmenting core flood relative permeability experiments. These measurements are the most expensive and lengthiest part of any core analysis program, often resulting in data arriving too late to impact business decisions. Using digital rock methods, combined with a reduced set of core analysis data, may lead to significant cost savings and acceleration of the experimental program as well as a more robust assessment of the uncertainties in critical subsurface parameters.

Further work on pore-scale validation of the simulated fluid distributions with in-situ micro-CT imaging is currently being pursued (Wang et al. 2021b). This approach can provide confirmation that the good match between simulated and measured relative permeabilities is a result of improved microscopic displacement rules. Furthermore, such experiments can be used to validate and improve the rock typing approach presented here, by distinguishing zones with different multiphase flow behaviours in the pore space.

4 Conclusion

In this work, we present a multiscale digital rock workflow to characterize petrophysical and multiphase flow properties for complex rocks. The workflow incorporates pore structures imaged at different scales and merges them into a single multiscale pore network model. A dry/wet micro-CT imaging sequence is used to spatially map the porosity at the larger scale and to classify unresolved porous regions into rock types. The essence of the multiscale PNM is to represent resolved pores at the larger scale in a conventional PNM manner and to represent smaller-scale pore structures in a continuum manner using relative permeability and capillary pressure saturation functions calculated on high-resolution images. The multiscale workflow incorporates experimental data to constrain modelling assumptions at different scales.

The workflow was applied to two complex reservoir rocks, a thinly laminated sandstone and a complex carbonate with dual porosity pore size distribution and spatial heterogeneity. Tests were performed in a blind fashion where only capillary pressure data were used for anchoring and the results of the workflow were validated against independently measured imbibition relative permeability only after the modelling was completed. The numerical predictions were found to be in good agreement with the experimental data.