In this case study, we present the implementation of a finite element method (FEM)-based
numerical pore-scale model that is able to track and quantify the
propagating fluid–fluid interfacial area on highly complex micro-computed tomography (

In combination with specialized software packages, a complex high-resolution
modelling domain can be obtained. A numerical workflow based on representative
elementary volume (REV)-scale
pore-size distributions is introduced. This workflow aims at the successive
modification of model and model set-up for simulating, such as a type of
2-phase problem on asymmetric

Understanding the evolution of the fluid–fluid interfaces in 2-phase flow
in porous-media systems is relevant for a series of engineering and
technological applications (e.g. carbon capture and storage (CCS), nuclear waste repository, oil
recovery; Hassanizadeh
and Gray, 1990; Joekar-Niasar et al., 2008; Niessner and Hassanizadeh, 2008;
Reeves and Celia, 1996). The quantity of the (typically unknown parameter)
interfacial area between two phases restricts kinetic interphase mass and
energy transfer (Ahrenholz et al., 2011). In numerical modelling with
classical macro-scale flow models, these processes are not properly taken
into account (Muccino et al, 1998; Ahrenholz et al., 2011). Here, averaged
quantities neglecting the interfacial area are often used (Tatomir et al.,
2015), which are described to lead to negligence of kinetics of mass
transfer, assuming local equilibrium (Niessner and Hassanizadeh, 2009).
In order to resolve this, the capillary pressure(

Based on an extended form of Darcy's law (considering fluid–fluid friction
force and interfacial forces), the mathematical fundamentals of pore-scale
2-phase flow regarding the

Several experimental and numerical approaches were developed to measure the
interfacial area between the two fluid phases. Current research in the field
of CCS focuses on the design and development of field investigation
techniques allowing short response times in on-site plume monitoring and
detection. Detection and quantification of the CO

The modelling approaches by Tatomir et al. (2013, 2014) and Tatomir et
al. (2015) require the a priori knowledge of the fluid–fluid-solid system-specific

The present article presents the corresponding model developed for deriving
this specific

Besides using the resulting

The sandstone sample on which this case study is based originates from
Heletz, Israel, which is a scientifically motivated deep saline CO

Top left: photograph of a well core sample. Top right: thin section image of the sample confirming the well sorted and rounded character. Labels mark the main mineral components quartz, calcite, clay and the pore space. Bottom left: SEM image showing a plane view on a mineral conglomerate of quartz grains interconnected by clayey and carbonatic cement. Bottom right: SEM image highlighting the partially significant amount of cementation (modified from Tatomir et al., 2016).

All modelling in this study was carried out using COMSOL Multiphysics (Version
4.3b). Fluid flow was solved numerically using the incompressible
Navier–Stokes equations (NSEs) as governing equations. Defining a Newtonian
fluid of constant density and laminar flow enables us to solve the governing
equations in the form of momentum balance (Eq. 1) and continuity (Eq. 2) as
follows (Cengel and Cimbala, 2013):

Sharp fluid–fluid interface tracking was realized by implementing the phase-field method as described amongst others in Kim (2012). Here, interfaces are
defined as regions of rapid change in an auxiliary parameter

The change of phase component concentration over time can be described as the
product of diffusion coefficient and chemical potential known as the
Cahn–Hilliard equation (Cahn and Hilliard, 1958). In the present model, this
equation is used in the form of Eq. (3) using the terms mobility

The model described in this study was validated by reproduction of three analytical solutions. First validation refers to single-phase flow and second and third validation at 2-phase flow and fluid–fluid interface tracking.

The NSE model is validated by simulating plane Poiseuille flow. This type of
problem describes pressure-driven steady, incompressible laminar viscous flow
through a two-dimensional channel (Rogers, 1992). Within the channel, a parabolic
velocity profile forms and the maximum velocity (

Model set-up for the model validation using plane Poiseuille flow. Coloured lines indicate isolines of velocity, whereas blue and red colours define lower and higher velocities respectively. The subgeometry used for the analysis is marked by the black box.

For the validation of the phase-field method regarding the emulation of the
sharp fluid–fluid interface, the Rayleigh-Taylor instability has been numerically
designed and modelled. In principle, the Rayleigh-Taylor instability is described
as the gravity-driven unstable displacement of a light fluid f2 with a
heavy fluid f1 (Lord Rayleigh and Strutt, 1883; Taylor, 1950; Mikaelian, 2014).
Initial separation of both fluids is given by a cosine-shaped interface of
the amplitude

Schematic model set-up and phase propagation over time
of the model validation using Rayleigh-Taylor instabilities. All
Rayleigh-Taylor instability stages in the form of linear, nonlinear and
turbulent regime are visualized. Blue and red colours represent heavier
(

Regarding the results,

Interface amplitude change

Further model validation with regard to NSEs and phase-field method in the form
of a two-dimensional capillary rise within a tube was carried out. Here, fluid rises within
a capillary, contracting air due to imbalanced intermolecular adhesive forces
between liquid and solid as well as intermolecular cohesive forces within the
liquid. The corresponding analytical solution derived for the fluid column
height

Model set-up for the model validation via
Young-Laplace capillary rise at pressure equilibrium. Blue and red colours
define water and air respectively. Height of water column is

Comparing the model results in the form of height of water column in the
capillary with the corresponding solution of Eq. (9) gave a deviation of

The numerical workflow steps (e.g. generation of the

In order to test and modify the FEM-based model described in Sect. 3 on the
sample-specific pore geometries (e.g. transitions from big pores to small
bottlenecks), statistical parameters of the

Schematic overview of the workflow for deriving the
final 2-phase flow model and model set-up for complex

The three-dimensional imaging of the sandstone has been performed with a nanotom
S 180

The interfacial area between the two fluid phases for the macro-scale models is represented as volume-averaged. Therefore, the representative elementary volume (REV) with regard to effective interconnected porosity was identified and extracted from digital sample approximations. Generally, a REV-analysis identifies the minimum porous-media volume with similar mechanical and hydraulic properties of the total porous-media volume (Singhal and Gupta, 2010). Further information about the REV can be found in Bear (1988). A detailed description of the workflow and results specifically for Heletz sandstone cores is given in Tatomir et al. (2016).

Results indicate a REV range of a subvolume edge length bandwidth ranging from approx. 120 to 280 voxel. Within this range, fluctuations of porosity are minimal. Subvolumes bigger than 280 voxel edge length show an increase in porosity and can be identified as domain where macroscopic effects are dominant. Subvolumes smaller than 120 voxel can be further distinguished in a transition phase ranging from 50 to 120 voxel, where fluctuations in porosity are relatively low and in domains smaller than 50 voxel, where microscopic effects are dominant and fluctuations are high.

Based on the

Result of the pore-size distribution for an REV-scale subvolume
arbitrarily placed in the sample approximation as number of pores vs. pore
diameter, whereas pore diameters were unified in 30

It can be observed that the majority of pores are represented by diameters up
to 90

Model domains were called idealized pore, idealized pore network,

The idealized pore measures the mean pore diameter

Heletz sandstone digital sample approximation based on idealized and

The idealized pore network domain led from a pore with a diameter

The

The fluid properties for supercritical CO

Fluid properties representing supercritical CO

The temperature is set to 333.15 K, which is the measured temperature in
the Heletz reservoir (Niemi et al., 2016). Analogous to typical on-site
injection, a one-axial pressure drop was chosen to induce fluid flow, whereas
the overall pressure gradient was defined as capillary pressure
(

Outlet pressure

For the model based on idealized geometries, an appropriate time-step size was
found to be

Implementing the initial model and model set-up on the

Results of the pore-size distribution highlighting values for number of pores as well as mean, minimum and maximum pore diameter.

Image series showing the propagation of supercritical CO

Image series showing the propagation of supercritical CO

Since the pressure drop within the domain was kept constant for each
simulation, the calculation of

An exemplary time series of phase and interface propagation within the

Figure 11 visualizes the change of

Plot of

It can be clearly observed that an increased capillary pressure results in a
significant decrease of wetting phase saturation within the domain. However,
at pressures

Figure 12 illustrates the

Given

A case study on Heletz sandstone was carried out to present and test a
workflow for determining the constitutive capillary
pressure–saturation interfacial area relationship based on

3-D bar plots of the

The design of the idealized and extracted

The scanned sandstone sub-core samples do not represent in situ reservoir
conditions, such as the pore space compaction occurring at reservoir depth
(below 1600 m), which implies reduced rock permeability and porosity. The
digital sample approximations used for later modelling represent ex situ rock
characteristics. Comparing permeability results from unpressurized and
pressurized core samples, Tatomir et al. (2016) shows an approximate
overestimation of permeability by a factor of 1.5 (on the same investigated
core samples). However, when compared with the permeability results of
Hingerl et al. (2016), who used a
9 MPa pressurization, the overestimation reaches a factor of 4. Furthermore,
the comparison of capillary pressure–saturation relationships obtained
from

Furthermore, the accuracy of mesh generation via

The composed numerical model could be successfully validated. NSEs were
simplified by defining flow incompressibility. Pressure-driven flow at steep
pressure gradients in highly irregular geometries lead to the formation of
an extremely heterogeneous flow velocity field. Narrow geometric features
such as bottlenecks result in extremely steep pressure gradients and high-flow velocities.
Velocities were often found to exceed the fluid-specific
speed of sound

In this study we have developed a numerical workflow for the determination of
capillary pressure–saturation fluid–fluid interfacial area on highly complex

The

We have not conducted any comparison of with the real REV geometry because the geometry was too complex to resolve with the 2-phase NSE model. However, by preserving the average properties of the medium and the average pore-throat diameters, we attempted to obtain the same kind of accuracy. Future work will concentrate in simulating larger domains on bigger clusters and determine the accuracy of the technique. Also, no comparison with a pore network model was conducted, but it is known that the idealizations imposed by such pore network models also result in incorrect estimation of the pc-S (e.g. Tatomir et al., 2016; Leu et al., 2014; Wildenschild et al., 2005; Wildenschild and Sheppard, 2013).

These results contribute to improving the characterization of the Heletz
reservoir where CO

CCS – carbon capture and storage,
FEM – finite-element method,
GMRES – generalized minimal residual method,
KIS – kinetic interface sensitive (tracers),
MUMPS – MUltifrontal Massively Parallel sparse direct Solver,

We want to show our gratitude to the three reviewers for improving the quality of this manuscript. We want to thank Juliane Herrmann (LIAG) for figure composition and visualization, Stephan Kaufhold (Federal Institute for Geosciences and Natural Resources, BGR) for the SEM analysis as well as Jannes Kordilla and Insa Neuweiler for helpful insights. Edited by: H. Steeb