Simulated fluid flow test

The following Chapters (4 and 5) present simulated fluid flow tests and aim to investigate the micro-mechanical behaviour of a porous rock under single phase fluid flow. In this Chapter the theory and the fundamental equations for the setup of the aforementioned tests, as well as the interpretation of the results are discussed. The aim of the simulated fluid flow test is to provide a

good indication of the material’s hydraulic conductivity and the behaviour of

the sample under high pressure. The flow rate, in m3_{/sec, for the liquid flow}

through the porous media is given by

𝑄 = 𝑞Á (2.19)

where Á is the cross sectional area perpendicular to the direction of flow, and 𝑞 is the velocity of the liquid given by Darcy’s Law (Dullien, 1979; Nield and

Bejan, 2006):

𝑞 = − 𝑘

𝜇_𝐷(∇𝑃𝑓− 𝜌𝑔∇𝑧) (2.20)

where 𝑘 is the absolute permeability of the sample, 𝜇_𝐷 is the fluid’s dynamic viscosity, 𝑃_𝑓 is the fluid pressure, 𝑔 is the magnitude

of the gravitational acceleration, 𝜌 is the density of the fluid, 𝑧 is the elevation

in the direction of the flow (which in this case is set to zero as the fluid moves horizontally). In steady-state, the velocity 𝑞 in Eq.(2.19) becomes the interstitial velocity 𝑢0 of the fluid. This can be derived from the combination of

the well-known Navier-Stokes (Itasca Consulting Group, 2008d) and Erqun’s

relations (Ergun., 1952; Jia et al., 2009), Eq.(2.21) and Eq.(2.22), respectively, for fluid flow through beds of granular solids, which for the case of a fixed homogeneous porous material takes the form:

𝜌𝜕𝜀𝑢⃗ 𝜕𝑡 + 𝜌𝑢⃗⃗⃗⃗ ∙ 𝛻(𝜀𝑢⃗ ) = −𝜖𝛻𝑝 + 𝜇𝛻0 2(𝜀𝑢⃗ ) + 𝑓𝛽⃗⃗ (2.21) ∆_𝑝 𝐿 = 150𝜇(1 − 𝜀)2_𝑢 0 𝜀3_𝑑 𝑝 ̅̅̅2 + 1.75(1 − 𝜀)𝜌𝑢02 𝜀3_𝑑 𝑝 ̅̅̅ (2.22)

where, 𝜀 is the porosity of the granular solid, 𝑓_𝛽⃗⃗ is the body force per unit

volume, the interstitial fluid velocity is denoted as 𝑢0, 𝐿 is the height of the

granular solid bed, ∆𝑝 is the pressure difference, 𝑑̅̅̅ is the mean particle 𝑝

diameter, and 150 and 1.75 are constants obtained by experimentation.

During the typical generation process the sample is packed with particles of uniform size (described in detail in Chapter 3 in the uniaxial test procedure). At this stage the assembly is reaching equilibrium with the use of some stabilizing strategies (i.e., target isotropic stress) thus all body forces acting on the particles prior to fluid movement are being eliminated. In the fluid- scheme of the PFC3D, driving forces from the fluid flow are applied to the particles as body forces. The body force is the drag force applied by the particles to the fluid element and is given by:

𝑓_𝛽⃗⃗ = 𝛽(𝑢⃗ ̅ − 𝑢⃗⃗⃗⃗ ) 0 (2.23)

where 𝑢⃗ ̅ is the average velocity of all the particles in a fluid element and 𝛽 is

a coefficient, that for samples of low porosity, is given by:

𝛽 ={150𝜇(1 − 𝜀) + 1.75𝑑̅̅̅(𝑢⃗ ̅ − 𝑢𝑝 ⃗⃗⃗⃗ )𝜌}(1 − 𝜀)0 𝜀2_𝑑

𝑝

̅̅̅2 (2.24)

where 𝜌 is the density of the fluid.

Next a drag force equal in magnitude and opposite in direction is applied to the particles in each given fluid element, given by (Itasca Consulting Group, 2008a): 𝑓_{𝑑𝑟𝑎𝑔} ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ =4 3𝜋𝑟3 𝑓_𝛽⃗⃗ 1 − 𝜀 (2.25)

The total force applied by the fluid to the particles is the sum of the drag force and the force due to buoyancy, given by:

𝑓_{𝑓𝑙𝑢𝑖𝑑}

⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ +_{𝑑𝑟𝑎𝑔} 4

3𝜋𝑟3𝜌𝑔 (2.26)

Furthermore, local non-viscous damping is provided by PFC3D meaning that body forces approach zero for steady motion. If we assume that the assembly of particles is similar to a packed bed, then when there is no flow through the packed bed the net gravitational force (including buoyancy) acts downward. When the flow starts moving, friction forces act upward and counterbalance the net gravitational force. For high enough fluid velocities, the friction force is large enough to lift the particles (Finlayson, 1990; Itasca Consulting Group, 2008d).

Generally, two different formulations can be encountered for the fluid velocity in porous flow: one is the aforementioned interstitial velocity 𝑢₀, and the other is the macroscopic or Darcy velocity 𝜀𝑢⃗ . The interstitial velocity is the actual

velocity of a fluid parcel flowing through the pore space. The macroscopic velocity is the volumetric flow rate per unit cross-sectional area. This is a non- physical quantity calculated on the basis that the flow occurs across the entire cross-sectional area, although in reality the flow only occurs in-between the pore space.

In the case of steady uniform flow, the macroscopic velocity is assumed to be constant and thus the terms on the left-hand side of Eq. (2.21) become zero. On the right-hand side, the term −𝜖𝛻𝑝 is the applied pressure gradient, 𝜇𝛻2_{(𝜀𝑢⃗ ) denotes the momentum loss due to viscosity, and 𝑓}

𝛽

⃗⃗ corresponds to

is negligible for large pressure gradients which in turn induce significant body forces, and therefore Eq.(2.21) can be reduced to:

𝜖𝛻𝑝 = 𝑓_𝛽⃗⃗ (2.27)

Combining Eq.(2.21) and Eq.(2.22) the second order Eq.(2.28) gives the solution of 𝑢0 = √(1 − 𝜀)4𝑑𝑝 3 𝜀3_∆ 𝑝𝜌150 + (𝜀 − 1)4𝜇21502− 150𝜇(𝜀 − 1)2 2𝑑̅̅̅(1 − 𝜀)𝜌1.75_𝑝 (2.28)

Eq.(2.29) was used during the simulations in order to provide the volumetric flow rate results of the discharging liquid through the virtual assembly.

𝑞_𝑣𝑜𝑙 = 𝑢₀ × 𝐴 (2.29)

The macroscopic properties of a real rock cannot be directly described in a DEM model due to the fact that the particles size distribution of the virtual model does not have to copy the actual rock’s grain size distribution (explained in detail in Chapter 3.6). This results in a mismatch between the hydraulics of the real rock and the virtual model in terms of pressure drop and fluid relative velocity. Furthermore, it is actually advantageous to decouple the micro-properties of the DEM specimen from those of the actual rock. This is because attempts to match the porosity of the actual rock would lead to a broader particle size distribution, which in turn lowers the time-step, in order to achieve better accuracy, resulting in impractical simulation time. For these reasons it was considered best to use calibration factors that will alter the fluid flow parameters of the virtual model. The simulated fluid flow tests are presented in Chapters 4 &5.

According to Ergun’s relation in Eq.(2.22) ∆𝑝 𝐿 = 𝐶1𝜇𝑢0+ 𝐶2𝜌𝑢02 (2.30) where 𝐶1 = 150(1 − 𝜀)2 𝜀3_𝑑 𝑝 ̅̅̅2 𝐶₂ =1.75(1 − 𝜀) 𝜀3_𝑑 𝑝 ̅̅̅ (2.31)

In order to match the pressure drop of the DEM specimen with that of an actual rock the terms of Eq.(2.30) on the right hand side should be scaled. The following process results in the scaling factors of viscosity and density used in the virtual model.

Combining 𝐶₁ from Eq.(2.31) with the Kozeny-Carman equation (Carman, 1997), then the permeability of a real rock is given by

𝑘 = 1 180

𝜀3_𝐷2

(1 − 𝜀)2 (2.32)

where 𝐷 is the grain diameter of the real rock. It is concluded that 𝐶1

corresponds to the inverse of the permeability for the DEM specimen and it is given by 𝐶₁ =150(1 − 𝜀)2 𝜀3_𝑑 𝑝 ̅̅̅2 = 150 180 1 𝑘 (2.33)

It is clear that the permeability depends on the specimen’s microparameters thus a calibration factor 𝑎_𝜇 was multiplied with the above equation in order to

match the specimen’s parameters with that of the actual rock with the use of

[𝐶1]𝑃𝐹𝐶× 𝑎𝜇 = [

150 180

1

𝑘]𝑅 (2.34)

where the terms 𝑃𝐹𝐶 and 𝑅 mean that the equations inside the brackets refer

to the PFC model and the real rock respectively. According to the literature the permeability for a limestone rock lies within the range of 2×10-11_{- 4.5×10}- 10_cm2 _{(Nield and Bejan, 2006). Choosing a mean value for the permeability}

the calibration factor is calculated as follows and it refers to the viscosity term of Eq.(2.30) 𝑎𝜇 = [ 𝜀3_𝑑 𝑝 ̅̅̅2 150(1 − 𝜀)2] 𝑃𝐹𝐶 150 180 1 𝑘 (2.35)

The same process was followed regarding the calibration factor 𝑎_𝜌 referring

to the density parameter of Eq.(2.30) with the use of the following relation

[𝐶₂]_𝑃𝐹𝐶× 𝑎_𝜌 = [1.75(1 − 𝜀) 𝜀3_𝑑 ]

𝑅 (2.36)

Using the Kozeny-Carman Eq.(2.32) to calculate the diameter 𝑑 of the real

rock and install it into Eq.(2.36), the calibration factor for the density is given by 𝑎𝜌 = [𝜀3_𝑑 𝑝 ̅̅̅2 ] 𝑃𝐹𝐶 [𝜀3⁄2_√180𝑘] 𝑅 (2.37)

In terms of the coding these factors are used by multiplying the viscosity term with 𝑎𝜇 and the density term with 𝑎𝜌.

In conclusion, in this Chapter the concept of the PFC calculation cycle, as well as the fundamental equations of Lame’s theory, the Navier-Stokes formulation, Ergun’s relation and the process of decoupling the fluid flow

parameters of the virtual model, have been presented. In the next Chapter the theory and equations, which were previously discussed, have been employed in order to describe the fluid flow test performed on a thick-wall cylinder rock sample.

Chapter 3 Calibration of microscopic parameters