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Summer School In Geophysical Porous Media July 17-28, 2006 Purdue University, West Lafayette, Indiana

MODELING FLOW IN VUGGY MEDIA. Advisor Todd Arbogast - Mathematics, University of Texas at Austin. Diana Battefeld - Physics, Brown University Cheng Chen - Engineering, Northwestern University Abdullah Cihan - Engineering, University of Tennessee, Knoxville

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Summer School In Geophysical Porous Media July 17-28, 2006 Purdue University, West Lafayette, Indiana

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  1. MODELING FLOW IN VUGGY MEDIA Advisor Todd Arbogast - Mathematics, University of Texas at Austin Diana Battefeld- Physics, Brown University Cheng Chen - Engineering, Northwestern University Abdullah Cihan - Engineering, University of Tennessee, Knoxville Daniel Pringle - Geophysics, University of Alaska, Fairbanks Masa Prodanovic – Comp. Applied Math, University of Texas at Austin Danail Vassilev - Computational Mathematics, University of Pittsburgh Hua Zhang- Agronomy, Louisiana State University Summer School In Geophysical Porous Media July 17-28, 2006 Purdue University, West Lafayette, Indiana

  2. Motivation Environmental: Understanding movement of contaminants in vuggy subsurface systems, pumping and protection of water aquifiers. Oil Industry: Oil recovery - Help oil industries to manage oil reserves more efficiently. Geological: System unlike those studied normally by geologists. Mathematical: Model fluid flow and transport in vuggy (dual porosity) media.

  3. What is a Vug? Vug: an orifice in a porous medium relatively large compared to pore size Internal structure of Vug system: X-ray Computed Tomography (CT) Scans of Pipe Creek Reef sample

  4. Where? This sample: Pipe Creek, Texas River basin Area Limestone matrix with caprinid fossils 4-6 cm diameter Cretaceous caprinid fossil

  5. Goal: To characterize the vuggy media’s pore space and to understand flow and tracer transport. We use CT scans, permeability measurements, and experimental tracer curves to achieve this goal. We will proceed as follows: I. CT Image analysisused to characterize pore space & connectivity II. Permeability computationfrom simplified Darcy flow models III. Tracer analysis what we can infer from the experiments

  6. ICT IMAGE ANALYSIS

  7. 3DMA-Rock software http://www.ams.sunysb.edu/~lindquis/3dma/3dma_rock/3dma_rock.html Medial Axis (MA) extraction (Lee-Kashyap-Chu ’94) + some trimming Throat finding + pore partitioning Segmentation (Indicator Kriging, Oh and Lindquist ’99) Pore/throat characterization (distributions) Interphase areas, fluid blob characterization, pore scale saturation…

  8. Medial Axis • Centrally located skeleton • Preserves topology and geometry of a 3D object • a powerful search tool • Obtained by morphological thinning algorithms (3DMA-Rock implements Lee-Kashyap-Chu '94. algorithm) • Branch cluster (node) – digitized version of a point (graph vertex) where medial axis paths meet • Medial axis path (link) – digitized version of a curve (graph link) • First applied to (simulated) porous media by Thovert/Sales/Adler ’93. and to sandstones by Spanne/Thovert/Jacquin/Lindquist/Jones/Adler ’94. Terminology : Medial axis = skeleton= percolating backbone = deformation retract MA computation = morphological skeletonization = morphological thinning/erosion Grass-fire algorithm = burn algorithm = erosion algorithm = distance labeling

  9. Objective: Pore-throat network Most throat finding algorithms minimize area A (and not hydraulic radius R)! R = A/P = (aL2)/(bL)=(a/b)L = cL (a,b,c constants reflective of shape) Throat: R1 > R2 < R3 R1 = c1L1 & R2 = c2L2, assume c1=c2 for close cross-sections, Then R1 < R2 iff L1 < L2 iff c1L12 < c2L22 iff A1 < A2 • Hydraulic radius of fluid pathway cross-section: R = Area/Perimeter • Throat: a pathway cross section of minimal R • Pore: pore space opening enclosed by grain & throat surfaces

  10. Throat Finding Approaches • Delaunay tesselation of sphere packs • “too many” pores obtained; merging close clusters needed • Finney '70, Mason '71, Bryant ’93., Willson ’00. • Morphological thinning based pore partitioning • Baldwin/Sederman/Mantle/Alexander/Gladden '96. • widely used in soil sci.; pore space partitioned, no throats identified • Maximal ball algorithm • Silin/Patzek 2003., results in “ball-n-stick diagram” • Throat finding algorithms • Multi-orientation scanning (Zhao/MacDonald/Kwiecien '94.) • Dijkstra based throats (Venkatarangan/Lindquist '99.) and extensions (Prodanovic/Lindquist/Seright ‘05) MA based • Planar throats (Liang/Ioannidis/Chatzis '99.) MA based • Wedge based throats (Shin/Lindquist '02.) MA based

  11. Sample Analysis

  12. Does upscaling by coarsening work? • cut central 300x300x100 subset of the segmented image • original z-direction slices multiplied 3 times to match resolution in x- & y- directions  3003 • the image coarsened by using majority rule in the boxes of appropriate size 3003 Porosity 17.8% Connectivity 83.3% 1003 Porosity 17.2% Connectivity 83.0% 503 Porosity 16.9% Connectivity 84.7% Connectivity = percentage of pore space in the largest connected component

  13. Vug Space Medial Axis (MA) • Small (MA) structures are destroyed by upscaling (above). Main fluid pathways are preserved: the shortest z-direction pathways are shown below. 503 opens up one more path! 3003 1003 503

  14. Medial Axis Path Statistics

  15. Sample Heterogeneity • Shortest paths across x-, y- and z- directions for 1003 sample • Average geometric tortuosity (ratio of actual length and side-side distance) for the shortest paths is ~1.8 for all directions, and ~2.0 for all possible paths across the sample

  16. Vug-throat network • 3DMA-Rock throat finding algorithms used to identify individual vugs vug surface MA path Branch cluster Relative positions of 10 largest vugs from 3003 sample A vug from 3003 sample, 9983 mm3

  17. Vug-throat Network Statistics Small structures disappear in the upscaled image • If small structures are important for the flow, the simulation in upscaled images will not produce good results • Dead-end vugs (not included the above volume stats) can be identified and occupy 4% of the connected pore space component

  18. IIPERMEABILITY COMPUTATION

  19. Flow and Transport Modeling Provide link between structure and experiment. Can system be understood in terms of an effective permeability? What controls tracer transport? Matrix (no vugs) k = 10 mDarcy Sub-sample (~10 cm)3 k = 100 Darcy Darcy Flow Advection Diffusion Equation

  20. Flow and Transport: Approaches Simple pipe flow Simple pipe networks Medial-axis pipe networks 1. Medial axis and statistics 2. CT output structures ‘Darcy code’: modeling 2D Lattice Boltzmann 3. Dual porosity model fitting

  21. Insights from Poiseuille Flow 1 Darcy (D) ~ 10-12 m2 Parallel Large k dominate Series Small k dominate

  22. Insights from Poiseuille Flow 1 Darcy (D) ~ 10-12 m2 d = 10 cm kmx =10 mD, kpipe = 45 D n=2: keff = 90 Darcy r = 1 mm Small constrictions can strongly control keff

  23. Pipe Network Model Dead end v=0 P=1 P=0 INPUT: Medial axis path network NB: more usual to use pore/throat network, but we had medial axis data available first!

  24. q4 P4 P6 q2 q6 q1 q8 P7 P8=0 P1=1 P2 P3 P5 q3 q7 q5 Pipe Network Formulation Poiseuille flow Mass balance Conjugate gradient method on normal equations: ATAx=ATb

  25. Kzu Kxu Kzd Simulated Permeability a Image resolutions. Medial axis flow paths are used for simulation. b Flow directions.

  26. Kzu Kxu Kzd Simulated Permeability a Image resolutions. Medial axis flow paths are used for simulation. b Flow directions. 2-scale problem ! Voxel-sized restrictions to flow Vug scale ~ sample size Upscaling dependence and Variability

  27. Darcy Code: Parssim IN: 3D geometry + physical parameters OUT:(steady-state) velocity field, concentration field Parallel subsurface simulator www.ices.utexas.edu/~arbogast/parssim/ 48 x 48 x 48 voxels (i) Full sample, Coarse resolution ( L= 16.4 cm) k = 416 Darcy (ii) Sub-samples, max resolution ( L = 2.6 cm) k(s1) = 1282 Darcy k(s2) = 170 Darcy k(vug) = 107 D, k(mx) = 0.01D, ΔP = 104 Pa (0.1 atm),

  28. 2-D Lattice Boltzmann Modeling Flow and passive tracer transport Limited input file size: 100 x 100 voxels Q: How to project ‘very’ 3D structure to 2D ? A: Examine 1-2 ‘connecting’ paths instead. Main flow Dead ends 2D toy model made by tracing and projecting 2 paths

  29. Flow Field

  30. t = 0 (sec)

  31. t = 85.3 (sec)

  32. t = 341.2 (sec)

  33. t = 1108.9 (sec)

  34. t = 1706 (sec)

  35. t = 3412 (sec)

  36. t = 6397.5 (sec)

  37. LBM Breakthrough Curve Reynolds number, Re = 0.10 Permeability, k = 1.1 x 106 Darcy Inlet Outlet Breakthrough Curves

  38. Numerical Permeability Results Within 1-2 orders of magnitude of experiments: this was our target! 2 scale problem: really need Lsample > vugs and voxel < mm To finish: 3D Lattice Boltzmann, pore-throat network system Darcy code: systematic examination of upscaling and k values. New numerical scheme in Arbogast and Brunson, submitted 2006.

  39. IIITRACER TRANSPORT

  40. WHAT WE HAVE SEEN SO FAR: VUGS ARE INTERCONNECTED WE CAN EXTRACT AN EFFECTIVE PERMEABILITY APPROXIMATION: DARCY FLOW (only valid at larger scales) BETTER: POISEUILLE FLOW NOW LET’S SEE WHAT WE EXPECT FROM EXPERIMENTS

  41. TOY MODEL:3D cellular vug-network (periodic)

  42. HEURISTIC ARGUMENTS USING STOKES FLOW: causes for marker behaviour Early Breakthrough:Tracer finds a pathway through the network Multiple Plateaus:Tracer finds other (longer) ways through the network – result: new plateaus whenever a new way opens. Abrupt Drop:Time drop correlated to end of injection. A time delay is due to cells on the main paths, acting as tracer reservoirs for a small amount of time. Single Plateau:Weakly interconnected vugs are drained slowly. The large main volumes of vugs act as the reservoir for a longer time, leading to a plateau of constant height until they are drained. Long Tail:Remaining tracer in thin arms is slowly washed out after main volumes are drained – diffusive process (exponential decay). Implement model on computer: test arguments!

  43. TEST HYPOTHESIS (Experimentally/Numerically, 3D) Multiple plateaus:Close one of the main pathways and see if the early plateaus change. Abrupt drop: Increase injection time – drop should be shifted by same time (the delay does not change) Single Plateau: Inject the fluid for smaller time interval – no reservoir build up: Plateau should decrease in height. Long tail: increase the diameter of the smallest arms in the toy model: exp. decay should be faster.

  44. TEST HYPOTHESIS (Experimentally or Numerically, 3D) From CT scan we see multiple pathways Multiple plateaus: experimentally- close one of the main pathways and see if the early plateaus change. In Poiseuille flow model, plateaus are non-existent (no reservoirs).

  45. Macroscopic Models Dual Porosity Model Mobile Immobile Dispersion Advection Diffusion between mobile and immobile region

  46. Dual Porosity Model CXTFIT Diffusion between mobile and immobile regions important in transport. ( i.e. from vugs in/out of porous matrix ). Tracer data CXTFIT free fit

  47. Stream Tube Model L1 • Li effective flow length • ri effective radius • Correlated random variables with known distributions r1 r4 r2 r3 L3 Solute transport in each tube follows ADE Hydrodynamic dispersion Di vi - Poiseuille flow Lo: Length of the column Tortuosity ti=(L0/Li)2 Longitudinal dispersivity a=0.01 m Flux averaged effluent concentration

  48. Parallel 4-Tube Model Tracer data 4 tube model Generally good fit! Fit could be optimized.. No connectivity in model!

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