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Understanding the gas migration in highly water saturated sedimentary rock formations is of great importance for safety of radioactive waste repositories which may use these host rocks as barrier. Recent experiments on drainage in argillite samples have demonstrated that they cannot be represented in terms of standard two-phase flow Darcy model. It has been suggested that gas flows along highly localized dilatant pathways. Due to very small pore size and the opacity of the material, it is not possible to observe this two-phase flow directly. In order to better understand the gas transport, a numerical coupled hydraulic-mechanical model at the pore scale is proposed. The model is formulated in terms of Smoothed Particle Hydrodynamics (SPH) and is applied to simulate drainage within a sample reconstructed from the Focused Ion Beam (FIB) images of Callovo-Oxfordian claystone. A damage model is incorporated to take into account the degradation of elastic solid properties due to local conditions, which may lead to formation of new pathways and thus to modifications of fluid transport. The influence of the damage model as well as the possible importance of rigid inclusions is demonstrated and discussed.

The principal objective of this study is to improve the representation and understanding of gas migration in highly water saturated clays. This objective is closely related to several projects that use a deep sedimentary rock formation, such as Callovo-Oxfordian (COx) clay [

In classical approaches, several gas transport mechanisms can be active, separately or simultaneously [

The coupling between the fluid flow hydrodynamics and solid matrix deformation has been tackled recently by several authors. In [

In recent years, due to X-ray computer tomography CT-scan and Focused Ion Beam Scanning Electron Microscope (FIB-SEM) imaging, it become possible to acquire

In order to overcome this difficulty, we propose to implement a model taking into account both two-phase flow inside pore space and its interaction with deformable rock matrix. We aim as well at modelling of the appearance of new pore connections, and for this reason we have given up the Lattice Boltzmann Model, which is solved on underlying regular mesh and thus is not well suited for capturing of fracture initiation. It is important to stress here that these “new” connections are seen not only as true microcracks, but also as a way to take into account the underlying pores connectivity that is not visible on the CT-scans. This will allow us to work with samples that are apparently nonpercolating and to ultimately restore visible connectivity during gas migration.

The model presented in this paper is based on the framework of the Smoothed Particle Hydrodynamics (SPH) which is a Lagrangian mesh-free method [

The article is organized as follows. In Section

Fluid flow inside a porous medium can be described by different equations depending mainly on flow characteristics and the scale chosen for the flow description. At macroscopic scales the Darcy equation and its modifications are widely used [

Momentum conservation equation in Lagrangian frame can be written as

The key problem in the modelling of immiscible multiphase flow is the description of interface dynamics. When both of the fluid phases are described by means of the Navier-Stokes equations (

A widespread approach consists in introducing the continuum surface force [

In the framework of this model, a volumetric surface tension force

Clay material is usually composed of a mixture of various minerals. At the macroscopic scale this mixture can be considered as a continuum medium with homogeneous mechanical properties. Obviously, this is no longer true at the microscopic scale where the clayey matrix and the mineral inclusions of various sizes can be clearly distinguished. Due to their mechanical properties, the mineral inclusions can be considered as incompressible rigid bodies, while the clayey matrix exhibits mainly elastic response to the applied stress [

The deformations inside an elastic material are given by the strain or deformation tensor

If pressure build-up or mechanical stress applied to the clay is high enough, its transport properties may be modified because of the elastic deformation of clayey matrix, as a result of, for example, dilatation of existing pores creating preferential gas paths. However, in the case of nonpercolating pore space the transport of gas phase can only take place while creating new microcracks.

Emergence of such microcracks can be modelled as a modification of mechanical properties of the clayey material under certain conditions. Since the exact models for this kind of features are not available, we propose here to reuse the ideas related to damage and fracture propagation in elastic materials. In the simplest isotropic scalar model, the process of degradation can be quantified by introduction of the damage variable

In order to relate the damage variable to mechanical load inside the clayey matrix, we adopt the criterion proposed by [

To verify if the elastic material fails under the tensile load, the Rankine criterion is applied [

To verify if the elastic phase fails under the shear or compressive load, the Mohr-Coulomb criterion is applied [

An example of constitutive damage law.

It can be assumed that, for a certain load range, the material follows linear elasticity law, and outside this range a nonlinear damage behaviour is observed, which coincides with observations made for clayey materials [

COx clay properties.

Parameter | Value |
---|---|

Young’s modulus | 3 GPa |

Poisson’s ratio | 0.1 |

Tensile strength | 4 MPa |

Uniaxial compressive strength | 20 MPa |

Internal friction angle | |

Strain at elastic limit | 0.0013 |

Ultimate tensile strain | 0.0026 |

Compressive strain at elastic limit | 0.015 |

The total damage

In order to avoid sharp restoration of the damage variable which may cause oscillations of elastic properties, the relaxation is applied to obtain a restored damage value

In their natural state, clayey rocks are highly heterogeneous, especially at the microscale where various rigid inclusions, such as quartz (

Rigid body dynamics is fully described by the Newton-Euler equations,

The Newton-Euler equations as described by (

In order to provide a correct interaction between phases at the solid-fluid interface, the boundary conditions must be defined. In most cases, the velocity of both phases at the contact point

At the interface between the solid phases, the continuity of normal stresses is imposed:

The model presented in previous section must be solved numerically. In this paper, this is done by means of the Smoothed Particle Hydrodynamics (SPH) which is a Lagrangian mesh-free method originally applied for solution of astrophysical problems [

The SPH model follows the idea that, for a given function

The smoothing function

A variety of smoothing functions satisfying these and some additional conditions [

Smoothing functions proposed by [

In SPH model, the medium is approximated by a set of particles which are initially uniformly or randomly distributed in space. Each SPH particle is a Lagrangian particle and is characterized by its position and velocity. It also has certain physical variables such as mass

When

The direct use of (

Another way to calculate the divergence of a scalar function can be resumed as follows [

Using these formulations for scalars and derivative terms, it is possible to discretize various physical equations in order to obtain their SPH analogues which can be solved numerically. For example, the movement of the Lagrangian particles can be described by the following system of equations:

The SPH is usually used to solve systems of partial differential equations which relate the time derivative of variables of interest to spatial derivatives. Since the equations of interest here describing fluid flow, elastic body deformation, and solid body movement satisfy these conditions, let us consider their discretization in detail.

In the framework of SPH, the continuity equation (

The right part of the momentum conservation equation (

In weakly compressible formulation adopted in our model, the incompressibility of the fluid is imposed by the proper choice of the speed of sound

Two principal approaches exist to model the surface tension by using the SPH model. The first one consists in using the pairwise interparticle force which mimics the intermolecular attraction forces [

In our model, the surface tension effects are introduced by discretizing the continuum force model (Section

The most frequently used method to model elastic bodies by means of SPH consists in integration of rate of change of the deviatoric stress tensor [

In order to approximate the force components by SPH, second derivatives of the local displacement can be calculated using the same approach as for the first derivatives. However, this formulation has several disadvantages [

When elastic forces acting on elastic particles are calculated, the particle trajectory can be easily obtained by numerical integration of corresponding equations (

A rigid body can be discretized with a set of Lagrangian points distributed inside the body volume and at the body surface. This spatial description allows a natural coupling of discretized equations of rigid body dynamics with the other equations in the frame of SPH [

In the context of our model, the movement of rigid inclusions results from forces applied on their surfaces by other phases as well as from contact forces between inclusions themselves. In order to decrease the memory use and to save calculation time, one can describe the rigid body by taking into account only the points lying at the body surface and in its close proximity (i.e., the points which may experience the external forces from other phases) without any loss of accuracy since the expected rate of rotation is low, and the final position of the rigid grain is most likely defined by its shape and distribution of external forces at the grain surface rather than by distribution of mass inside this body.

The center of mass of the rigid body represented by a set of

The Newton-Euler equations (

Interaction of the fluid phase described by SPH with solid objects can be obtained in various ways. Boundary forces taking into account viscous stress and fluid pressure are described in [

SPH particles belonging to two different phases or two different objects of the same phase are considered as boundary particles if the distance between them is smaller than smoothing length

In our model, we adopted the simplest way to satisfy the conditions (

Interaction between boundary particles belonging to different solid phases or objects can be modelled by introduction of pairwise interparticle forces based on Lennard-Jones potential [

In our model, we implement the XSPH correction which is a frequently used technique [

The capacity of the model to represent correctly important physical phenomena was tested in various validation tests. A simple test to verify the capacity of the model to simulate fluid flow is the transient Poiseuille-Couette flow between two plane walls

For SPH fluid simulations, the cubic spline smoothing function is selected, which is most often used for fluid flow since it provides most accurate results. Fluid particles are initially placed at regular cubic lattice nodes with

The obtained velocity profiles at different times are compared with analytical solutions in Figure

Transient Poiseuille-Couette flow. Velocity profile evolution with time. Velocity profiles correspond to

The capacity of the SPH to simulate fluid flow in porous media can be tested for a nontrivial geometry represented by periodic array of spheres for which an analytical expression of permeability is available [

The permeability

The ability of SPH to simulate fluid flow for periodic array of spheres with a good accuracy was demonstrated in [

For numerical simulation the unit cell size is

In order to test the sensitivity of the method to various

Coarse and refined discretizations of the sphere for

Numerical errors for the calculated permeability values are summarized in Table

Numerical error for permeability of periodic arrays of spheres for various

| Discretization | ^{2} | | |
---|---|---|---|---|

0.343 | Coarse | | | |

0.45 | Coarse | | | |

0.5236 | Coarse | | | |

| ||||

0.343 | Refined | | | |

0.45 | Refined | | | |

0.5236 | Refined | | | |

It must be noted (see Table

The calculated permeability

Interparticle spacing | ^{2} | err, %. |
---|---|---|

| | |

| | |

| | |

Based on these simulation results, we can expect for our drainage simulations the precision of about 30%, which is satisfactory for simulations performed for coarse geometries with

In order to test the capacity of the model to simulate multiphase flow, let us consider a two-phase (liquid-gas) Poiseuille flow between parallel plane walls located at

The simulation domain is the same as for the considered above transient Poiseuille-Couette flow simulation. The no-slip boundary conditions are applied at the solid phase SPH particles, while spatially periodic boundary conditions are imposed along

The numerical velocity profile is compared to the analytical solution in Figure

Two-phase Poiseuille flow. Analytical solution (black solid line) for velocity profile is compared to the SPH numerical solution (points).

In order to test the continuum force algorithm described in Section

Laplace’s law for spherical bubbles. Pressure step versus

The elastic-fluid coupling can be tested by calculation of the relative volume variation

The volume variations for several pressure steps are compared with analytical solution given by (

Elastic cube volume change due to applied fluid pressure.

In order to test the ability of our code to simulate gas-water flow inside a realistic pore space in repository conditions, we make a very first application to a small sample of Callovo-Oxfordian (COx) clay.

The COx claystone sample EST27405-1 is represented by a series of 103 images of

In order to decrease the computational costs, a coarsening procedure was applied to the original sample consisting in replacing of 8 neighbouring voxels with a single two times larger voxel whose nature (pore space or clayey matrix) was defined based on the majority rule. Two samples of

Largest percolating pore components inside the Callovo-Oxfordian clay sample.

Due to significant computational costs (one simulation for

(a) A disconnected pore component

For the present study, the OpenMP [

Several drainage tests are performed in order to study the model behavior under various conditions. A unit cell for simulations is obtained by cutting a part of the sample containing the selected percolating pore component as well as small surrounding disconnected pores. In order to avoid fluid movement across the cell borders, an elastic layer is added to each lateral side of the sample. Finally, the sample is placed between horizontal solid walls in order to prevent its spatial movement during the simulation as displayed in Figure

Usually, it may be possible to prescribe real values of physical parameters of all the phases considered in the model and to solve numerically the corresponding equations. However, in this case the time step would be prohibitively small (

Two-phase flow in porous media can be characterized by several dimensionless numbers:

Dimensionless numbers characterizing two-phase water-gas flow in clay microfractures.

Number | Re | | Ca | Bo |
---|---|---|---|---|

Expected value | | | | |

Simulation value | | | | |

It can be observed that the Reynolds and capillary numbers observed in our simulations (for mean gas velocity ^{−1}) values; however, they are still sufficiently small and should describe adequately laminar two-phase flow dominated by capillary forces.

As it was previously mentioned, in weakly compressible formulation the speed of sound in the fluid must be high enough to effectively penalize density variations and at the same time small enough to allow reasonably large time steps. The adopted value ^{−1} for both fluid phases satisfies these conditions with relative fluid density variation of order of 0.12% observed in our simulations for the pressure difference

The pressure difference

Since the ratio

Liquid

After the initial compression phase, the saturation evolves almost linearly until the percolation, where the curve changes the slope and the drainage continues with a decreasing velocity due to the fact that water phase is now mainly located near the walls. It can be expected that the sample will be completely drained after a longer time due to viscous drag force. Incorporation of the damage model (Figure

The next series of drainage tests is performed for various confining stresses

Same as for (Figure

(a) Number of damaged elastic SPH particles. (b) The total damage calculated as a sum over all the damaged points.

The percolation times and gas saturation at percolation are summarized in Table

Percolation times

| | |
---|---|---|

10 | | 0.5158 |

8 | | 0.5074 |

6 | | 0.5043 |

4 | | 0.4971 |

| ||

6, no damage | | 0.5039 |

| ||

Incompressible porous matrix | | 0.5159 |

Evolution of the number of damaged points and of the total damage is presented in Figure

A cross-section (SPH points inside the

The seepage velocities of fluid phases during drainage are plotted in Figure

Water (a) and gas (b) seepage velocities during drainage as functions of time for various confining stresses.

(a) Water and gas mean pressures inside the sample as functions of time. (b) Capillary pressure as a function of gas saturation.

The water

It must be noted that the numerically obtained capillary pressure values displayed in Figure

From the presented simulations it can be concluded that the mechanical interaction between the solid matrix and the fluid phases can influence the drainage process in clays in repository conditions. It has been seen as well that the resulting damaged area was relatively wide and did not localize into new fractures. We have tested several approaches in order to better localize damaged particles. We observed that the details of damage model greatly influenced simulation outcome. The results presented in previous chapters were obtained supposing the reversibility of damage variable; that is, elastic properties of damaged particles can be restored if this is indicated by the damage criterion. Also, in order to preserve the volume of the elastic phase, damaged elastic particles continue to interact by pressure gradient term (second term in (

A cross-section (SPH points inside the

When using SPH model for the elastic phase, a fracturing can be observed even without applying a damage model. For the simulations presented in Sections

A still different behaviour can be observed if damaged elastic particles are considered as ghost particles, which no longer interact with any other particles. In this case, the release of the stress accumulated inside the elastic matrix may result in formation of narrow fractures as displayed in Figure

In order to consider the possible influence of the rigid inclusions on the flow conditions, two rigid spheres were incorporated into the elastic matrix as displayed in Figure

A cross-section (a)

The aim of this paper was to propose a modelling approach for the dilatant two-phase flow which is assumed to occur during gas migration within initially water saturated indurated clay rock. This type of flow, unlike classical viscoelastic flows, require an interaction of flow with a deformable solid matrix.

We developed a numerical model which allows simulating a two-phase fluid flow inside a porous medium and taking into account the mechanics of the solid matrix, possibly composed of two different media: a linearly elastic continuous medium and isolated rigid inclusions. Then a damage model is introduced in order to take into account in a controlled manner the degradation of elastic material properties due to local stress-strain state. The model can be applied to the FIB-SEM images of the real clay samples in order not only to study various phenomena at pore scale but also to calculate effective transport properties, that is, permeability for varying pressure gradients and confining stresses taking into account deformation and degradation of the elastic matrix.

The model was solved in the framework of Smoothed Particle Hydrodynamics, which is a Lagrangian mesh-free method. Classical methods to numerically simulate fluid flow are mostly based on Finite Volume Methods (FVM), while in solid mechanics the Finite Element Methods (FEM) are widely applied. These methods allow solving several problems more efficiently than SPH; however, in some complicated situations they may suffer from drawbacks related to their mesh based nature. This includes the cases with deformable and moving interfaces, dealing with large deformations and propagation of fractures. Use of SPH model is very appropriate in this context since it provides the opportunity to describe various types of interactions between different materials within the same framework. The presented SPH model was coded using C++ programming language implementing all the phenomena mentioned above. The SPH is computationally expensive. The main load comes from calculation of neighbouring particles contributions when evaluating densities and forces. Its cost grows significantly with the number of particles located within the smoothing length. Therefore, the original code was parallelized using OpenMP [

For the present study, the OpenMP and CUDA GPU parallelization techniques were applied with a significant gain in numerical performance when run on an ordinary graphical card.

With several elementary tests we have shown that the model is sufficiently stable and robust for use in conditions representative for application to gas-water flow in argillites and that a limited resolution was sufficient to obtain permeability values with a reasonable precision (about 30%).

The model was then applied to simulate drainage in a small

For the particular geometry of our sample containing a percolating pore, the overall drainage process stayed similar, though the softening of clay though damaging resulted in accelerated drainage velocity. Also, the obtained relation for the suction curve of an isolated pore is decreasing with gas saturation, which is the opposite from the experimental relationships obtained for macroscopic samples. One remarkable feature of capillary pressure curves in Figure

The capability of the model to create new pathways due to localized damage was also investigated. We have demonstrated that for a small sample under applied conditions it is difficult to create macroscopic fractures and damage zone stays diffuse. However, taking into account the rigid inclusions resulted, as expected, in creation of new pores between them and the elastic matrix.

We conclude that the developed SPH code is capable of representing all required phenomena for FIB-SEM based geometries in conditions close to the real ones. To be able to obtain more quantitative information about drainage in argillites some further work needs to be conducted. In particular, it is not completely clear whether the macroscopic damage model is appropriate for use if the solid matrix is divided into clayey matrix and rigid inclusions (for COx they constitute about 40% of volume). At the nanometric scale also the mass exchange phenomena can be very important, especially the possibility for gas to dissolve and for water to evaporate, since in tortuous pore space the continuous phase flow may be slowed down or even stopped. Finally, our model should be applied to larger samples in order to minimize the boundary effects and to provide data independent of individual pore space geometry.

A part of this work was presented during JEMP 2016 conference, held on 12–14 October 2016 in Anglet, France.

The authors declare that they have no conflicts of interest.

The financial support from NEEDS MIPOR is gratefully acknowledged. The help of Alain Genty (CEA) who provided a GPU cluster for simulations is greatly appreciated.