Hume, Laurène

Moonen, Peter

Poncet, Philippe

English

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Submitted on 15 Jul 2019
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Permeability estimation using vortex-based method and
synthetic samples from X-ray micro-CT
Laurène Hume, Philippe Poncet, Peter Moonen
To cite this version:
Laurène Hume, Philippe Poncet, Peter Moonen. Permeability estimation using vortex-based method
and synthetic samples from X-ray micro-CT. 14me Journées d’étude des milieu poreux (JEMP), Oct
2018, Nantes, France. ￿hal-02116155￿
Permeability estimation using vortex-based method and
synthetic samples from X-ray micro-CT
L. Humea, P. Ponceta, P. Moonenb,c
aUniv. Pau & Pays de l’Adour, CNRS, LMAP-IPRA, UMR 5142, 64000 Pau, France.
bUniv. Pau & Pays de l’Adour, CNRS, DMEX-IPRA, UMR 3360, 64000 Pau, France.
cUniv. Pau & Pays de l’Adour, CNRS, TOTAL, LFCR-IPRA, UMR 5150, 64000 Pau, France.
Keywords: Digital rock physics, Grid-particle methods, High-performance-computing, Reference data.
1. Introduction
The investigation of fluid flow at the pore scale, based on X-ray scans of a porous medium of
interest, is one of the most challenging problems in CFD. However, the predictive power of many
existing simulators remains to be demonstrated. Model validation is crucial, as direct observation of the
flow field inside real porous media is generally not possible, and since model and experiment generally
exhibit differences in geometry, boundary conditions and/or physics. In this study, we present the cross-
validation of a new flow simulator from a numerical point of view via a permeability estimation. The
pore scale geometry used in the computation is obtained using X-ray tomography.
2. A next-generation hybrid grid-particle solver for pore-scale modeling
Flow simulation at pore scale is difficult because of the inherent complexity of the geometry, which
includes a fluid-solid interface with possible roughness. The high resolution needed to capture relevant
geometrical details has to be handled without using tremendous memory resources, excluding traditional
assembling methods like finite elements or finite volumes. In our study, we use a robust hybrid grid-
particle method [1, 2] to solve the advection-diffusion problem. This complete problem writes in the
domain Ω 
u = ū in solid domain,
−div(2µ(α, u)D(u)) = f −∇P in fluid domain ,
divu = 0 in Ω,
∂tα+ u · ∇α− η∆α = 0 in Ω,
(1)
where u is the fluid velocity, D(u) = (∇u + ∇uT )/2, µ is the fluid viscosity, f is the external force,
P is the pressure, α is the transported quantity and η is its diffusion coefficient. The fluid velocity
field is first computed on a Cartesian grid, then particles manage transport. Interpolation between grid
and (moving) particles needs to conserve mass and positivity for the sake of physical coherence. Such
coupled diffusion-transport model applies to heterogeneous fluids [6] and viscous mixtures [5]. Velocity
field computation relies on the solution of the 3D Stokes equation in its penalized formulation [3]. In our
solver, this is carried out using an adapted velocity-vorticity formulation based on the method described
in [4] for external flows. This allows to efficiently satisfy adherence and slipping conditions at fluid-solid
interface.
3. Simulation setup and results
An important point is the validation of the model by assessing its performances via direct measure-
ments. To assess the capability to numerically predict the permeability of a porous medium, a dedicated
experimental setup has been built. For each studied configuration, a 3D tomographic scan is made from
the dry state. The reconstructed geometry is extracted and the resulting 3D geometrical virtual volume
is used as input for numerical computation.
FICPM - Nantes, France - October 8-10, 2018.
Different configurations are tested: first, a network of smooth micro-spheres, packed in the cylindrical
cavity of a Hassler-like cell, is used for validation. This configuration of interest consists of a pack of
smooth micro-spheres with nominal diameter equal to 553 ± 11 µm. This diameter was selected to have
around 10 spheres across the diameter of the test cell. The region of interest will be around 5003 voxels,
which can be handled by a typical numerical solver, and each sphere is described by around 50 voxels
across its diameter, which is sufficient to have an accurate geometrical representation. The voxel size
is set to 15µm. This sample permits the validation of an accurate 3D model, as it presents a complex
topology. While the micro-spheres uses a synthetic medium, a second test can be set which relies on
a natural geometry. A rock sample, precisely a Bentheimer stone (a well-documented sandstone), is
chosen to conduct this real case test. A rendering of the fluid-solid splitting for this geometry is shown
in figure 1.
Figure 2 depicts a rendering of the resulting velocity field computing over the domain, while figure
3 shows particle advection according to the computed field. Measurements using different fluids, using
multi-modal sphere-packs and at higher spatial resolutions are foreseen as well. In order to validate
the numerical process, the solution fields are processed to get the permeability of the sample via the
averaged Darcy law. The numerical permeability values are finally compared to experimental ones on
the same samples.
Figure 1: Rendering of fluid and solid
phase inside a 2993 Bentheimer sand-
stone
Figure 2: Norm of velocity field inside
beads network (2573 grid)
Figure 3: Rendering of advected parti-
cles from computed velocity field
References
[1] R. Chatelin, P. Poncet, Hybrid grid-particle methods and Penalization: A Sherman-Morrison-Woodbury approach to
compute 3D viscous flows using FFT, J. Comp. Phys., 269, 314-328 (2014).
[2] R. Chatelin, P. Poncet, A hybrid grid-particle method for moving bodies in a 3D Stokes flow with variable viscosity,
SIAM J. Sci. Comp., 35(4), B925-B949 (2013).
[3] R. Chatelin, D. Sanchez and P. Poncet, Analysis of penalized variable viscosity 3D Stokes equations coupled with
transport and diffusion, ESAIM: Math. Model. Numer. Anal., 50:2, 565-591 (2016).
[4] M. El Ossmani and P. Poncet, Efficiency of multiscale hybrid grid-particle vortex methods, SIAM MMS, 8(5), 1671-
1690 (2010).
[5] B. Noetinger, L. Hume, R. Chatelin and P. Poncet, Effective viscosity of a random mixture of fluids, Phys. Rev. Fluids,
3(1), 014103 (2018).
[6] D. Sanchez, L. Hume, R. Chatelin and P. Poncet, Analysis of 3D generalized Stokes problem coupled to transport-
diffusion for miscible shear-thinning heterogeneous microscale flows: Applications to digital rock physics and mucociliary
clearance,Submitted.
FICPM - Nantes, France - October 8-10, 2018.


Permeability estimation using vortex-based method and synthetic samples from X-ray micro-CT

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Permeability estimation using vortex-based method and synthetic samples from X-ray micro-CT

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