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The focus of this thesis

Dans le document The DART-Europe E-theses Portal (Page 45-51)

Based on table-top experiments and numerical simulations, we study the fast pro-cesses of fluid flow in deforming granular media and evolving fractures. Air is in-jected at constant overpressure into saturated or dry granular media confined inside Hele-Shaw cells. We vary boundary conditions, injection pressure and initial satu-rating fluid of the medium. In addition, we present a sub-project focused on slow processes with experiments on reactive flow in fractured calcite samples. The rest of this section presents a summary of the specific experiments we performed.

We present an exploratory study where we experiment with the combination of two-phase and granular flows by performing air injection into saturated mono-layers of beads. Here, we study the formation of viscous fingering and fracturing patterns that occur when air at constant overpressure invades a radial Hele-Shaw cell containing a liquid-saturated deformable porous medium, i.e. during the flow of two non-miscible fluids in a confined granular medium at high enough rate to

de-form it. The sample is created by preparing monolayers of glass beads in cells with various boundary conditions, ranging from a rigid disordered porous medium to a deformable granular medium with either a semi-permeable or a free outer boundary.

By injecting air at constant overpressure into a deformable saturated monolayer of beads, we have a system where particles may be displaced by a viscous pressure gradient and/or interfacial forces between the two fluids. Since it is a single layer of beads without imposed confining pressure, granular stress is expected to depend on the boundary condition and number of particles in contact ahead of the flow, rather than build-up of normal stress against the plates. As a result of competi-tion between viscous and capillary forces, and build-up/relaxacompeti-tion of friccompeti-tion during flow, we observe transitions between finger opening and pore invasion in the vis-cous fingering regime, for example during initial compaction or outer decompaction during flow in the open cell. The resulting patterns are characterized in terms of growth rate, average finger thickness as function of radius and time, and fractal properties. Based on experiments with various injection pressures, we identify and compare typical pattern characteristics when there is no deformation, compaction, and/or decompaction of the porous medium. We show that the patterns formed have characteristic features depending on the boundary conditions. The study is part of fundamental research on pattern morphology in various deforming systems, which is important for increased understanding of flow in any deformable porous medium.

Next, we present an experimental study on flow regimes and pattern formation during air injection into confined granular media. In this experiment, we inject air at constant overpressure into a dry dense granular medium inside a linear Hele-Shaw cell, where air escapes at the outlet while beads cannot. As opposed to similar experiments with open outer boundary conditions [56,58,60], after the flow compacts the medium there is no decompaction. Due to the confined nature of the experiment, it is thought to be a laboratory analog to pneumatic and hydraulic fracturing of tight rock reservoirs where the free boundary at the surface is very distant from the injection zone, i.e. in a situation where fracture propagation stops long before reaching a free surface, such that the surrounding medium is not decompacted.

We thus observe the material behavior (at high enough overpressure to displace beads) to have a transition from fluid-like to solid-like during experiments, and that eventual invasion patterns initially expand like viscous fingering in the fluid-like regime, crossing over to stick-slip propagation of the tips as the medium becomes more solid-like, until it reaches a final structure when the compacted medium has reached a completely solid-like behavior. What is less obvious, is how the flow patterns in this system change with the injection pressure. By varying the imposed overpressure, we identify and describe the different flow regimes. The motivation of this setup is to study the granular fingering instability in compacting granular media. We characterize typical properties of the emerging structures, such as their fractal dimension, typical thickness and invasion depth, growth dynamics, as well as flow regimes depending on injection pressure.

Then, with the same experimental setup as above, we present a study where we focus on characterizing the deformations and the evolution of the pressure in the medium surrounding the channel growth. Displacement fields are found

experimen-tally with a Digital Image Correlation technique, while we calculate the evolution of the pressure field in the porous medium numerically. Two solutions of the pres-sure field are evaluated; A steady-state configuration of the prespres-sure field, which is estimated by solving the 2D Laplace equation with fixed pressures in the air cluster and at the outlet boundary. The other one is a solution where the imposed pressure initially diffuses into the pore space with a diffusion constant based on the poros-ity of the material and the viscosporos-ity of the fluid. This pressure field is calculated over time by solving the 2D diffusion equation with the Crank-Nicholson method.

The two solutions for the pressure are compared to check if, when and where the diffusing pressure field satisfies the Laplace solution. By investigating the obtained deformation data and pressure gradients across the granular medium, we discuss the deformations, growth dynamics and rheology.

Finally, we present preliminary results in a chapter about a sub-project studying slow transformation processes, where we perform reactive flooding experiments in fractured carbonate reservoir rock. Here, calcite core samples (cylinders of 3.8 cm diameter and 1.7 to 2.6 cm in length) are fractured into two pieces through their principal axis by loading them across their diameter until failure. After being frac-tured, the pieces are put back together and mounted inside a Hassler cell (cylindrical confinement). During the experiments the system is flooded with a reactive fluid at a constant flow rate parallel to the fracture plane. Here, we inject distilled water at a constant flow rate of 0.1 ml/min through the fractured samples in experiments of various durations. A local topographical profile of the fracture surfaces is measured before and after the reactive flooding by using a white light interferometer, which in principle has a height resolution down to a few nanometers. With this data we compare the initial and final profiles to investigate the evolution of surface roughness and fracture aperture for different durations of flooding, and investigate the impact of flow direction on the changes.

In a closely related project, where the work in this thesis also contributes, acous-tic emissions during experiments with air injections into dry granular media are recorded with piezo-electric shock accelerometers. Scientific papers discussing the results are shown in the appendix of this thesis. In Fourier analysis of the obtained signals, the evolution of the emissions is characterized in the frequency domain dur-ing the different stages of deformation and fracturdur-ing, and the processes creatdur-ing the various types of signals are discussed in comparison with the optical data. Here, it is shown that different stages of the invasion process can be identified acous-tically in terms of characteristic frequencies and distinct microseismic events. In addition, microseismic events caused by particle rearrangement in the later stages of the experiments are counted and fitted with characteristics of real world seismic events [75]. In combination with the optical data of deformations, an energy based localization technique for the sources of these microseismic events is developed [76].

The work done in this thesis project is ultimately a part of further understanding the mechanisms of complex processes involved in transforming reservoirs due to fluid flow, and is also partly involved in the development of acoustic localization techniques which may have industrial applications.

Chapter 3

Basic concepts and methods

3.1 Flow in porous media

Porous materials can be divided into two parts, the solid part forming the porous matrix, and the pore-space which is the interstitial space where fluids can be present.

If the pore-space is interconnected, the porous material is permeable, i.e. the pores are connected and fluids are able to flow through the material using these pathways.

In a granular medium, like soil, the space between grain contacts is connecting the whole pore-space making it very permeable. Porosity is the ratio of the volume of the pore-space to the total volume of a porous sample and is given by

φ = Vpores

Vtot

= 1−ρs, (3.1)

where ρs is the solid fraction. The porosity is a measure of how porous a solid is, and is a dimensionless number between 0 and 1. Permeability describes the ability of fluid flow through a porous medium and is often related to the porosity.

In granular media, the average permeability in a sample is given by the Kozeny-Carman relation [59, 62, 75] (in this case for a packing of spherical beads)

κ= a2φ3

180(1−φ)2, (3.2)

where a is the bead diameter.

Fluid flow is driven by pressure, viscous and external forces as described in the Navier-Stokes fluid equations of motion:

ρ ∂~v

∂t +~v· ∇~v

=−∇P +µ∇2~v+F .~ (3.3) Equation (3.3) states that the acceleration of a fluid element with density ρ is driven in the direction of the pressure force −∇P, viscous forceµ∇2~v, and external forces F~ such as gravity. However, the Navier-Stokes equations require sufficient boundary conditions to be solved. In porous media, the boundary conditions are very complex and is practically impossible to define precisely. However, it gives a view on the forces involved. Instead, rather than considering the fluid flow on pore scale, it is common and useful for flow at small enough Reynolds number to average

the flow in porous media on sample scale by using the phenomenological Darcy law.

This law relates the volume flux per unit area, or Darcy velocity, ~v = QA to the permeability, viscosity, pressure gradient and gravity as

~v =−κ

µ(∇P −ρ~g). (3.4)

In cases where gravity can be neglected, as for a horizontal Hele-Shaw cell, the Darcy velocity is found by

v =−κ

µ∇P. (3.5)

In immiscible two-phase flow in porous media, e.g. when gas invades a porous medium saturated with a liquid, capillary and interfacial forces are added to the problem. Capillary forces originate from the adhesive forces between a fluid and a solid surface, and interface tension comes from the cohesive forces between the molecules of a liquid. Wettability is a measure on the tendency of a fluid droplet to smear out on a smooth solid surface in the presence of a second fluid. It is measured in terms of contact angle θ, i.e. the angle between the solid surface inside the droplet and the tangent of the fluid-fluid interface at the triple point between the two fluids and the solid. Normally in a porous medium with two fluids present, one fluid has stronger contact forces with the solid than the other. The fluid with stronger contact forces is said to be wetting, withθ ∈[0,90], while the other is said to be non-wetting, with θ ∈ (90,180]. In two-phase flow where the non-wetting fluid is displacing the wetting one, called drainage, the invading fluid meets capillary thresholds it must overcome to displace the defending fluid out of narrow pore-necks.

The narrower the pore-neck, the higher is the pressure threshold, which is given by the Young-Laplace equation as

Pcap =γ 1

R1

+ 1 R2

cosθ, (3.6)

whereγ is the interfacial tension andR1 andR2 are the vertical and horizontal sizes of a given pore-neck. Since fluid flow tend to follow the path of least resistance, the pore geometry at the fluid-fluid interface influence the flow path taken by the invading fluid.

The flow regime during drainage of a horizontal porous medium depends on the ratio between the driving and stabilizing forces involved in flow and pore-invasion, described by the dimensionless capillary number Ca [40, 41]. The capillary number is the ratio of viscous pressure drop over capillary pressure drop at the characteristic pore scale, and can be found by

Ca= µva2

γκ , (3.7)

wherevis the Darcy velocity. For low capillary numbers (Ca≪1) capillary fingering dominates and for higher capillary numbers (Ca→1) there is a crossover to viscous fingering. In the capillary fingering regime, the invading fluid displaces the defending fluid pore-by-pore with slow build-up and relaxation of the invading fluid pressure

fluctuating around the capillary threshold values. In the viscous fingering regime, a viscous pressure gradient across the defending fluid leads to a flow instability where more advanced parts of the invading cluster grow on expense of less advanced ones in finger like intrusions. The instability is explained by the pressure field in the defending liquid; The defending liquid can be considered incompressible (∂ρ/∂t= 0), and for incompressible flow in porous media we can combine the continuity equation

∂φρ

∂t +∇ ·(ρ~v) = 0 ⇒ ∇ ·~v = 0 (3.8) and the Darcy law in eq. (3.5) to get the Laplace equation∇2P = 0 for the pressure in the defending liquid. Since the pressure can be assumed uniform in the invading gas cluster, the driving force∇P is effectively screened behind the longest fingers [1].

In a horizontal Hele-Shaw cell the pressure field is given by the 2-D Laplace equation

2P

∂x2 +∂2P

∂y2 = 0. (3.9)

A fingering instability similar to viscous fingers occurs during fluid injection at sufficient overpressure into a dense granular medium saturated with the same fluid, i.e. branched channels empty of grains open up even in the absence of surface ten-sion. In this situation, the invading air displaces grains by momentum exchange and open up pathways of increased local permeability. These channels expand quickly in the flow direction, and are screening the pressure gradient from the less advanced ones, inhibiting their growth. For compressible gas flow in a dry dense granular medium, the gas pressureP evolves in the medium according to the equation [59,77]

∂P

∂t =D∇2P −~vg· ∇P − P

φ∇ ·~vg, (3.10)

where ~vg is the local granular velocity and D is a diffusion constant. If the terms involving the granular velocity are negligible, equation (3.10) reduces to the diffusion equation for the pressure between the grains, which in 2-D is given by

∂P

∂t =D∇2P =D ∂2P

∂x2 + ∂2P

∂y2

. (3.11)

The diffusion constantD can be approximated for granular media as [75]

D= κP0

φµ = a2φ2P0

180(1−φ)2µ, (3.12)

assuming an ideal gas, which is valid for conditions without strong density variations, i.e. in a system where the pressure is not varying by orders of magnitude. Here, P0

is the atmospheric pressure, and it is assumed the Carman-Kozeny expression (3.2) is valid for the permeability κ[59, 75]. In a rigid 1-dimensional porous medium, the analytical solution for the 1-D diffusing pressure is

P(x, t) =Pin 1− x

L −

X

n=1

2Pin

πn sinnπx L

en

2π2

L2 Dt, (3.13)

where L is the system length. Equation (3.13) shows that for increasing time t, the diffusing pressure goes towards the steady-state Laplace solution P(x) = Pin 1−Lx

.

Dans le document The DART-Europe E-theses Portal (Page 45-51)