Laboratory and Numerical Study of Sinkhole Collapse Mechanisms Ming Ye (mye@fsu.edu) Department of Scientific Computing Florida State University FSU Karst Workshop 9/25/2015
Research Team (2014 Present) Ming Ye (Scintific Computing) Xiaoming Wang (Math) Kamal Tawfiq Daniel Kuncicky (Civil Engineering) (Experimental Design) Roger Pacheco Castro (GFDI) Boohyun Nam (Univ. Central Florida) Xiaohu Tao (Hohai University) Jian Zhao (Hohai University)
Sinkholes
Conceptual Model of Sinkhole Formation Cover collapse sinkholes Subsidence sinkholes Dissolution sinkholes Ground surface Ground surface Ground surface Ground surface Water table Cohesive soil Piezometric surface elastic Water table Cohesive soil Water table Water table elastic plastic elastic Piezometric surface Cohesive Piezometric surface Cohesive Piezometric surface plastic soil soil Rock Karst cave Rock Karst cave Rock Karst cave Rock Karst cave Ground surface Ground surface Ground surface Ground surface Water table Piezometric surface Water table Water table Water table Non-cohesive soil Non-cohesive soil Piezometric surface Non-cohesive Piezometric surface Non-cohesive Piezometric surface soil soil Rock Karst cave Rock Karst cave Rock Karst cave Rock Karst cave
Geological Patterns of Spatial Distribution It is more important to know when and where sinkholes will happen. http://interactive.orlandosentinel.com/sinkholes/
Scientific/Engineering Questions What are controlling factors of sinkhole collapse? ü Geological (long-term) ü Geotechnical (short-term) ü Hydrological (long/short-term) How to understand the interaction between the factors in the forecast context? Are sinkhole collapse predictable in a deterministic sense? How to make probabilistic predictions? How to characterize the factors for improving our understanding? ü Geological survey ü Geophysical survey ü Laboratory experiments
Research Approaches Laboratory experiment Since it is unknown when and where sinkholes happen in the field, laboratory experiments are indispensable. Field investigation Field investigation is needed to design laboratory experiments, develop conceptual models, and validate analytical and numerical models. Analytical analysis Given the complicated conditions of hydrology, soil mechanics, and geology at the field scale, simplification is necessary for fundamental understanding. Numerical simulation For predictive analysis (deterministic or stochastic), numerical modeling is needed.
Laboratory Experiments A/D Box Computer FSU GFDI Inflow Water tank for unconfined aquifer Ground Surface Rain Deformation Sensor Water tank for unconfined aquifer Inflow Water tank for confined aquifer Unconfined aquifer Sand Intermediate confining unit Karst Conduit Confined aquifer Water tank for confined aquifer Outflow Ou Water Pressure Transduser Laser Distance Measurer Soil Pressure Transduser Optic Fiber Valve Flow Meter PBS/NOVA http://www.pbs.org/wgbh/nova/earth/ sinkholes.html History Channel Fox News Tampa Russia Today International
Experimental Results Field observation Laboratory observation
Laboratory Experiment Hohai University, China
Finding 0-200 -400-600 -800-1000 17:59 18:05 18:11 18:17 18:23 18:29 18:35 18:41 18:47 18:53 18:59 19:05 19:11 19:17 19:23 19:29 19:35 19:41-1200 Hydrogeological responses to sinkhole development is observable by monitoring variation of hydraulic head. Once a cavity reaches unconfined layer (sand), change of hydraulic head can be observed far from the cavity. Once a cavity reaches unconfined layer (sand), sinkhole collapse occurs quickly. It may be too late by monitoring hydraulic head in aquifers only. Monitoring should not be focused on aquifers (confined or unconfined) but on aquitards (confining layer such as clay) before damage is made. Dilemma: ü Head change in aquitard does not propagate far. ü It is unknown where sinkhole occur and thus unknown where to put monitoring wells.
Pros and Cons of Experiments Pros ü Observe sinkhole occurrence directly (we have observed different patterns of cavity formation even for the same type of soil) ü Use the equipment for different soils and hydrogeological conditions (e.g., rising head in unconfined aquifer and dropping head in confined aquifer) ü Test controlling factors and their combinations by running different experiments ü Obtain a large amount of data for model validation Cons ü Scaling issue: it is still challenging to scale the experiment to represent field conditions ü Repeatability: Experiments cannot be repeated exactly, because soil packing also vary between experiments, and the variation could make big different for sinkhole occurence
Field Investigation WKP Virginia
Analytical Analysis: Why Are Sinkholes Round? Geological explanation by Dan Doctor at USGS When a void occurs in sediment that has a certain amount of cohesion ('stickiness' among sediment grains), the most stable configuration of the roof of the void is a dome, like the dome of the U.S. Capitol building. If that dome collapses, the vertical sides may remain upright, and the open hole will be circular. (http://io9.com/why-are-sinkholes-round-1614203510) Our analytical explanation Clay Sand Ds Soil Cavity Clay Sand Dc Water table, Hu Piezometric surface, Hc Ó F = F + G' + S f F G' S total Clay dh = [ γ D (1 n ) ((1 n ) γ ( H D ) + F '] A + γ LH + γ dv τld dz ' s s s s w u c loading c w c Dc Forces: Loading (F) Submerged weight (G ) Seepage force (S) Cohesion force (f)
F Ó Clay Dc F = F + G' + S f total dh = [ γ D (1 n ) ((1 n ) γ ( H D ) + F '] A + γ AD + γ dv τld dz = + + + ' s s s s w u c loading c c w c ' [ γsds(1 ns) ((1 ns) γw( Hu Dc) Floading '] A γc ADc γwi ' ADc τldc G' ' A = L [ γsds(1 ns) ((1 ns) γw( Hu Dc) + Floading ' + γcdc + γwi' Dc] τdc L S F total should be zero if the system is in balance. Note that there is a given maximum shear stress, τ f, a property of clay. τ should not exceed τ f, ; otherwise collapse occurs. These lead to ' [ γsds(1 ns) ((1 ns) γw( Hu Dc) + Floading ' + γcdc + γwi' Dc] = τdc τ fdc Assuming L (perimeter) is a constant, according to the isoperimetric inequality, when the plane is a circle, A is the largest. Thus, the failure surface should be a cylindrical surface A L
Mathematical and Computational Modeling Coupling groundwater flow and soil mechanical modeling Pressure and seepage force Groundwater flow Stress-Strain Deformation, porosity, and geometry Stress-Strain σ ij + fi = 0 x j ( σ ' + pδ ) i j x j ij = ρ g s i
Stress-Strain Modeling Force balance σ σ σ h + + + = x x x x ' ' ' 11 21 31 γ w 1 2 3 1 σ σ σ h + + + = x x x x ' ' ' 12 22 32 γ w 1 2 3 2 σ σ σ h x x x x ' ' ' 13 23 33 ' + + + γw γ = 1 2 3 3 0 0 0 If hydraulic head (h) is known, the above equations have six unknowns. The Cauchy equations for strain are needed, which introduce nine more unknowns consisting of three displacements (u 1, u 2, and u 3 ) and six strains (ε ij ). For elastic materials, the constitutional relations (i.e., Hooke equations) between stress and strain are needed to solve the fifteen unknowns.
( σ ' 11 + pδ11) ( σ ' 21+ pδ21) ( σ ' 31+ pδ31) + + = 0... x1 x2 x3 ( σ ' 12 + pδ12) ( σ ' 22 + pδ22) ( σ ' 32 + pδ32) + + = 0... Navier equation x1 x2 x3 ( σ ' 13+ pδ13) ( σ ' 23+ pδ23) ( σ ' 33+ pδ33) + + = ρsat g3...... x1 x2 x3 u1 ε11 =... x 1 u2 ε 22 =... x 2 u 3 ε 33 =... x3 Cauchy equation 1 u1 u2 ε12 = ( + )... 2 x2 x 1 1 u1 u3 ε13 = ( + )... 2 x3 x 1 1 u2 u3 ε 23 = ( + )... 2 x3 x2 1+ ν ν ε11 = σ 11 ' ( σ11 ' + σ22 ' + σ33 ')... E E 1+ ν ν ε22 = σ 22 ' ( σ11 ' + σ22 ' + σ33 ')... E E 1+ ν ν ε33 = σ 33 ' ( σ11 ' + σ22 ' + σ E E 33 ')... Hookeequation 1+ ν ε12 = σ 12 '... E 1+ ν ε13 = σ 13 '... E 1+ ν ε23 σ 23 '... = E
PFC & FLAC
PFC results for clay. An arch is formed but no collapse. The right figure is for the contact force. PFC results for sand only.
t=1s t=3s t=5s t=8s More advanced computational methods are needed!
Mathematical and Experimental Studies Mechanism of sinkhole formation Physical experiment research Mathematical model research Test system Material mechanics, Elasto-plasticity. Mohr-coulomb, Groundwater flow The water pressure and the soil pressure distribution Cohesive/ Noncohesive soil tests Relationship between the surface subsidence range and the fracture and the depth Data analysis Tests with different condition Relationship between the cave formation and the hydraulic condition Relationship between the soil property and the sinkhole formation The Size of the sinkhole Coupling the seepage and mechanics The characteristic of seepage The formation of cave Mathematical analysis model Mechanism of sinkhole formation Verification with two tests Simulation of sinkhole formation Hydraulic head load Evaluation of sinkhole formation
Conclusion Laboratory experiments are useful to understand the controlling factors of sinkhole collapse. Analytical analysis is useful for gaining fundamental understanding of the mechanisms of sinkhole collapse. Field investigation is indispensable for designing laboratory experiments and developing conceptual models for mathematical and numerical modeling. Numerical modeling is hard! Groundwater modeling and geotechnical modeling have been done, but their coupling for sinkhole modeling has not been attempted.
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