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Poroelasticity – Biot Method

Poroelasticity – Biot Method. after Leake and Hsieh (USGS Open File Report 97-47). Poroelasticity. s T Total Stress. In porous media, total stress is distributed between fluid and solid particles s E = s T - s F

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Poroelasticity – Biot Method

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  1. Poroelasticity – Biot Method after Leake and Hsieh (USGS Open File Report 97-47)

  2. Poroelasticity sT Total Stress In porous media, total stress is distributed between fluid and solid particles sE = sT - sF If system initially at equilibrium, subsequent changes relate only to applied stress sF Fluid pressure sE Effective stress on solids

  3. Poroelasticity The land surface in Fresno, CA dropped about 9 m in 20 years due to overpumping • Pumping reduces fluid pressure, which increases effective stress on solid particles, causing deformation. • When the pore space compacts, it “squeezes” fluids, further altering pore fluid pressures. • The change in fluid pressure causes more deformation. And so on…. • The compaction can stress subsurface infrastructure like well casings, collapse open-hole wells, cause the earth to fracture, and more. • Plus the loss in pore space often is unrecoverable. www.fresno.gov

  4. Poroelasticity HORIZONTAL FAULTING Little vertical displacement Big vertical displacements Flow to axis of valley Impermeable basement step

  5. Terzaghi: Flow modeling only -> vertical compaction only Darcy’s Law Biot: 2-way flow & structure modeling -> horizontal and vertical Darcy’s Law plus solid displacement term Plane Strain plus fluid pressure term

  6. Poroelasticity Upper reservoir - effectively limitless Ground surface moves freely Permeable upper reservoir Almost impermeable confining unit Permeable lower reservoir Pump in lower reservoir Head drops in time No lateral movement Impermeable step No flow, No movement Impermeable basement below No Flow, No movement

  7. Darcy’s Law Application Mode Hydraulic head analysis:H = p / ( rfg ) + y Let H equal CHANGE in hydraulic head H = Ht – H0 H = change in hydraulic head p = fluid pressure rf = fluid density g = gravity 1/M = Biot Modulus (i.e., specific storage at constant strain) K = hydraulic conductivity ab = Biot-Willis coefficient (i.e., DVf/DVt at constant head) = time rate change in strain dweak term: -alphab*(px_test*u_time+py_test*v_time)

  8. Darcy’s Law Boundary and Initial Conditions zero change in head H = 0 H = 0 zero change in head H = 0 zero change in head K H = 0 no flow H = H(t) change in head varies with time H0 = 0 at t = 0 K H = 0 no flow K H = 0 no flow

  9. Plane Strain Let u equal CHANGE in displacement u = ut – u0 u = displacement vector E = drained Young’s modulus n = Poisson’s ratio ab = Biot-Willis coefficient (i.e., DVf/DVt at constant p) rf = fluid density gr = gravity H = change in hydraulic head

  10. Plane Strain Boundary and Initial Conditions Free surface u = 0 zero horizontal displacement u = 0 zero horizontal displacement u = v = 0 at t = 0 zero displacement u = v = 0 u = v = 0 zero displacement

  11. Results: Poroelasticity – Biot Method difference in vertical drop produces lateral extension -> fractures displacement contours stack up near impermeable step t = 10 years Surface is displacement Contours are displacement. Arrows are fluid veocities.

  12. Poroelasticity – Biot Method Surface is head; contours are displacement; arrows are fluid velocities. Flow is from right to left.

  13. Displacements: Biot vs. Terzaghi Similar displacements far from impermeable feature (step). Depart near step. Biot displacements are model results. Terzaghi compaction estimates are scripted post processing calculations from conventional flow model. Db = Ssk b ( H0 - H ) Biot displacements - dashed Terzaghi compaction- solid

  14. Poroelasticity – Biot Method horizontal strain ux horizontal displacement u 2-way coupling gives horizontal as well as vertical deformation Shown are results at the ground surface.

  15. Points of interest • Multiphysics: Combination of predefined Fluid Flow and Structural Mechanics application modes. • Linking flow-to-structure and structure-to-flow requires straightforward commands. • Terzaghi approach provides reasonable estimates of vertical compaction over the majority of the fied. • Biot poroelasticity approach yields a better estimate of displacement near impermeable objects because it accounts for both vertical and horizontal displacements.

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