Report

Finite-Element-Based Characterisation of Porescale Geometry and its Impact on Fluid Flow Lateef Akanji Supervisors Prof. Martin Blunt Prof. Stephan Matthai Outline 1. Research Objectives 2. Development of Single-phase Pore-scale Formulation and Numerical Model 3. Workflow and Model Verification 4. Validation: Application to Porous Media 2 Research Objectives To characterize pore-scale geometries and derive the constitutive relationship governing single and multiphase flow through them To contribute to a better understanding of the physics of fluid flow in porous media based on first principle numerical approach To investigate the dependency of fluid flow on the pore geometry which is usually neglected on the continuum scale To develop a constitutive relationship which allows a more rigorous assessment of fluid flow behavior with implications for the larger scale 3 Outline 1. Research Objectives 2. Development of Single-phase Pore-scale Formulation and Numerical Model 3. Workflow and Model Verification 4. Validation: Application to Porous Media 4 Development of Single-phase Pore-scale Formulation and Numerical Model (1/2) The general p.d.e. governing fluid flow at pore scale is given by the u Navier – Stokes equations as: 2 t u u u P For an incompressible fluid conservation of mass takes the form u 0 For a steady-state system, the substantial time derivative goes to zero i.e. 2 u u u P For slow laminar viscous flow with small Reynold’s number, the advective acceleration term drops out and we have the linear Stokes equations: 2u P p 2 2 h u x y y 2 2 x, y, z x 2 y 2 z 2 5 Development of Single-phase Pore-scale Formulation and Numerical Model (2/2) FEM discretisation and solution sequence Define a function x, y, z that obeys: 1 2 Step 1: We solve Poisson’s equation for x, y, z with homogeneous b.c. Step 2: We compute the pressure field using x, y, z – this ensures that x, y, z P 0 fluid pressure, P tetrahedron Since we define the velocity by: u u 0 P x, y, z μ u Dependent variables are placed at the nodes. 6 Outline 1. Research Objectives 2. Development of Single-phase Pore-scale Formulation and Numerical Model 3. Workflow and Model Verification 4. Validation: Application to Porous Media 7 Workflow and Model Verification Task Tool Model Generation CAD ( Rhino ) Meshing ICEM - CFD Mesher Simulation Visualization (1/7) CSMP++ MayaVi, vtk, Paraview 8 Model Verification, Step1: Porosity (2/7) Porosity Pore Volume / (Grain Volume + Pore Volume) Vp Vb 9 Model Verification, Step2: Pore Radius Computation (3/7) Pore radii rd GRAIN 2 3.35 µm PORES Derivative of f(x,y) 2 0 3.35 µm Pore Radius (μm) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 10 Model Verification, Step3: Pore Velocity (4/7) Placement of 7 FEM Placement of 14 FEM Placement of 21 FEM 11 Model Verification, Step3: Pore Velocity (5/7) Error analysis Case a b c 9860 9860 9860 Channel length (µm) 30 30 30 Number of Elements 7 14 21 22.62 2.54 0.92 22.8 13.64 2.0 Pressure gradient (Pa-m-1) Channel velocity mismatch b/w analytical and numerical (%) Volume flux mismatch b/w analytical and numerical (%) 12 Model Verification, Step3: Pore Velocity (6/7) Velocity (µms-1) 13 Model Verification, Step4: Effective Permeability (7/7) keff q AP 14 Outline 1. Research Objectives 2. Development of Single-phase Pore-scale Formulation and Numerical Model 3. Workflow and Model Verification 4. Validation: Application to Porous Media (Results) 15 (Validation) Porous Media with Cylindrical Posts (1/10) 16 Application to Porous Media (2/10) Sample I: Ottawa sandstone (Talabi et al., SPE 2008) meshing simulation thresholding 4.5mm Velocity (x 10-5 ms-1) Micro-CT scan CAD Hybrid mesh 0 2 4 6 8 10 12 14 Velocity profile 0.0 2.0 4 .0 Velocity (x 10-5 ms-1) 6.0 8.0 10.0 12.0 1714.0 Application to Porous Media (3/10) Pore radius distribution LV60 Sandstone Ottawa Sandstone Sombrero beach carbonate Pore Radius (μm) 0 10 20 30 40 50 60 Pore Radius (μm) 70 80 0 10 20 30 40 50 60 70 80 18 (4/10) Application to Porous Media Computed versus Measured Permeability 3D Lab Expt 2D Num. Simulation Ottawa sand Dimension (mm) Porosity (%) Permeability (D) 4.5 x 4.5 x 4.5 35 45 4.5 x 4.5 39 31 4.1 x 4.1 x 4.1 37 40 4.1 x 4.1 40 29 LV60 sand Dimension (mm) Porosity (%) Permeability (D) Sombrero beach carbonate sand Dimension (mm) Porosity (%) Permeability (D) - 4.5 x 4.5 36 28 19 (5/10) Application 3D Granular Packs Permeability vs. Concentration for Single Sphere Numerical Experiment 6 Permeability (x 10-14 m2) r 0.5 0.4764 r 0.5 r 0.45 0.r6 0.55 0.4764 0.618 0.3284 r r 0.625 0.7 r0.2022 0.15 0.041 0 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 Concentration 20 3D Granular Packs (6/10) Xavier Garcia 21 3D Granular Packs (7/10) CAD geometry Fluid Pressure 22 (8/10) Sample 1 Φ= 32.3 Φ= 33.52 Φ= 35.80 2.4 mm Φ= 37.02 Φ= 38.43 23 (9/10) Sample 2 Φ= 32.43 Φ= 33.52 Φ= 35.57 Does the detail really matter? 2.4 mm Φ= 36.81 Φ= 37.63 24 (10/10) X 10 -5 Permeability versus Porosity 25 Single-phase Advection in Porous Media (1/2) Ottawa 26 Single-phase Advection in Porous Media (2/2) LT-M 27 Conclusions (1/1) I have presented a Finite-Element-Based numerical simulation work flow showing pore scale geometry description and flow dynamics based on first principle This is achieved by carrying out several numerical simulation on micro-CT scan, photomicrograph and synthetic granular pack of pore scale model samples In order to accurately model fluid flow in porous media, the φ, r, pc, k distribution must be adequately captured 28 Future work (1/1) Two-phase flow with interface tracking testing for snap-off and phase trapping using level set method (Masa Prodanovic – University of Texas @ Austin) drainage Investigate dispersion in porous mediaimbibition (Branko Bijeljic) Capturing snap-off during imbibition Courtesy: (Masa Prodanovic – University of Texas @ Austin) Courtesy: (Masa Prodanovic – University of Texas @ Austin) 29 Acknowledgements PTDF Nigeria CSMP++ Group 30 THANK YOU! 31