Its really dark down there: Uncertainty in groundwater hydrology - - PowerPoint PPT Presentation

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Its really dark down there: Uncertainty in groundwater hydrology - - PowerPoint PPT Presentation

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Its really dark down there: Uncertainty in groundwater hydrology Larry Winter University of Arizona QUIET 2017 SISSA July 21, 2017 NSF: 1707658-001 Spatial scales and typical dynamics Individual pore : 10 m 10 mm radii, 0.1 10

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It’s really dark down there: Uncertainty in groundwater hydrology

Larry Winter University of Arizona

QUIET 2017 SISSA July 21, 2017 NSF: 1707658-001

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Spatial scales and typical dynamics

Individual pore: 10 µm – 10 mm radii, 0.1 – 10 cm length Dynamics: Poiseuille Eqn, Navier-Stokes Eqns Explicit porous microstructures: 1 cm – 1 m sample lengths Dynamics: Navier-Stokes Eqns Laboratory: 1 – 10 m3 blocks Dynamics: Stokes Flows / Darcy’s Law Field: 10m – 1 km Dynamics: Darcy’s Law Local aquifer: 1 – 10 km Dynamics: Diffusion (Darcy’s Law) Basin-scale: 1 – 104 km Dynamics: Diffusion (Darcy’s Law)

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Typical Scales of Measurement/ Observation

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Flow through porous media: alternate representations

µm-mm cm-km

Pore Space Porous Contjnuum

Κ(r x )

Porous microstructure

  • Void and solid phases
  • Navier-Stokes equations
  • Detailed pore geometry

{

χ(x) = 1 if x ∈ pore space 0 otherwise Continuum

  • Darcy’s Law, advection-diffusion
  • Effective parameters
  • Hydraulic conductivity [L/t]
  • Head [L], velocity, concentration
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Highly heterogeneous media

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Model (theory) Domain of Application Scales Assumptions in addition to IBCs & forcing functions NSE Pore-Pore network 10µ - cms (1) Newton's 2nd Law (2) Conservation

  • f mass

Darcy's law Elementary volume of a porous medium cm-m Continuum representation of porous medium. Uniform material Continuity eqn Eqn of state Flow eqn 2D Transmissivity with Sy Unconfjned aquifer km T doesn't vary with head 2D Transmissivity with S Confjned aquifer km (1) Confjning beds are plane and parallel, (2) One principal direction of K perpendicular to confjning beds, (3) head gradient independent of z, (4) ∆h/∆t doesn't depend on z Difgusion eqn cm-km Known K(x)

Models, their applications, scales, and assumptions The … equations for the circulation of a fluid in a porous medium [relating to Darcy’s law] are significant only for [small] volumes of a porous medium

  • - Marsily
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Measurement scales: REV

Measurement scale

1

Porosity REVs

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REV?

Photo 0.15 cm tjp 0.31 cm tjp 0.63cm tjp 1.27 cm tjp Berea sandstone

Pufg permeameter images.

Tidwell et al., 1999

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Pore microstructures

Berea Sandstone (Courtesy Ming Zang)

Simulatjon Berea Sandstone

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Biological Porous Media: Human Pancreas

Murakami et al., “Microcirculatory Patterns in Human Pancreas,” (1994)

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Flow through porous media: Lab scale

Center for experimental study of subsurface environmental processes Colorado School of Mines

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Flow through porous media: Field scale

Capillary rise Infjltratjon Plant uptake (transpiratjon) Evaporatjon usgs Recharge Interfmow

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System of aquifers

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Aquifer systems

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High Plains Aquifer

450,000 km2 Elevation: 2400m – 355m Few streams The Great Plains produce about 25% of US crops and livestock. Great reliance on ground water for agriculture 30% of all ground water pumped for irrigation in the United States.

Courtesy USGS

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Karst systems

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Sources of uncertainty

Winter and Tartakovsky, 2013

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Flow through porous structures: experiments

Moroni et al., 2001

Computational

  • Eulag CFD Simulator (Prussa et al., 2006).
  • Immersed boundary method for pore spaces

(Smolarkiewicz and Winter, 2010)

Physical

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Computational experiments

Hyman et al. (2012)

Particle trajectories – Yellow is fast

Synthetjc Beads Volcanic Tuf Smolarkiewicz and Winter (2010)

Synthetic medium

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Heterogeneous velocities

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Expanding (left) and Contracting Regions (right)

(Hyman and Winter, Phys Rev E, 2013)

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Field scale and larger

https://www.swstechnology.com/ Kansas Geological Survey

USGS Modflow https://water.usgs.gov/ogw/modflow/

Computational Physical

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System dynamics: Continuum representation

Parameters

Conductivity: K(x), [K] = m/s Permeability: k(x) = (µ / g ρ) Κ = m2 Transmissivity: T(x), [T] = m2/s Storativity: S, [S] = 1 Dispersion coefficient: D, [D] = m2/s

State variables

Hydraulic head: h(x, t), [h] = m Darcy flux: q(x, t), [q] = m/s Flow rate: Q(x, t), [Q] = m3/s Concentration: c(x, t), [c] = M/m3

∇•K∇h S∂h ∂t + F = Flow Mass Transport −K(x)∇h q = Darcy’s Law Continuity ∇•q + + F) = 0 (S∂h ∂t

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Groundwater Flow: Some Foundational Problems

Inverse problem. Estimate basic parameters (hydraulic conductivity) at a given scale of analysis (porous microstructures -- aquifers) from data. Most are highly heterogeneous, e.g., K(x) = Ki(x) if x ∈ material i 1st Forward Problem (Heterogeneities). Determine effects of material heterogeneities on flow/transport at a given scale. Scale-up. Scale observations of heterogeneous parameters up to effective parameters at a larger scale. Scale-Down. Scale parameters averaged at a larger scale down to realistic distribution of heterogeneities at a smaller scale. 2nd Forward Problem (Prediction). Quantify uncertainties about system states arising from incomplete knowledge of parameters and model structure for a specific aquifer.

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Scale-up

Effective parameters Statistically uniform

  • Stationary and ergodic. Glimm and Kim,

1998

  • Single hydro-geological material

produced at more or less the same time by more or less the same process.

  • Asymptotic expansions. Gelhar and

Axness, 1983, Winter et al., 1984; Fannjiang and Papanicolaou, 1997 Statistically heterogeneous media

  • Separable scales. Winter and

Tartakovsky, 2001. Clark et al., in prep.

  • Self-similarity. Neuman, 1994. Molz,

2004

Yeh et al, 2009 Neuman, 1994

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Scale-down and Inverse Problem

Realizations of pore spaces with specified correlations (Adler, 1992) or physical properties, e.g., Minkowski functionals of integral geometry (Hyman and Winter, 2014) can be produced by thresholding Gaussian random fields.

Statistical interpolation

  • Spatial covariance, structure

function, Kriging: γ(∆x) = E[ ||K(x + ) – K(x)||2]

  • Monte Carlo simulation

Sequential estimation Thresholded fields

Thresholded surface Simulated pore space

Thresholded Gaussian Fields

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1st Forward problem: Effect of heterogeneities

Zhu et al, 2015 12 realizations

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1st Forward problem: Effect of heterogeneities

Zhu et al, 2015

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2nd Forward Problem: Prediction

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Bayesian Hydrogeology

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Predictions

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Models of reduced complexity

Reduced physics Lattice Boltzmann.

Chen and Doolen,

Continuous time random walk

https://www.weizmann.ac.il/EPS/People/Bria n/CTRW/ Berkowitz, 2006.

State transition diagrams.

Winter and Tartakovsky (2009)

Jump processes Reduced dimensionality Orthogonal polynomials. Xiu and Karniadakis, 2003; Zhang and Lu, 2004;

Xiu and Tartakovsky, 2006

Wavelet transforms. Foufoula-Georgieu

LBM CTRW

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RCM for particcle breakthrough

105 particles Σ, Φ ~ slow and fast states Lx = Ly = 1.28 cm, Lz = 2,56 cm σ, φ ~ residence times per state vΣ, vΦ ~ constant velocities vΦ >> vΣ Vertical velocities

  • C. Clark – UA
  • J. Hyman – LANL
  • A. Guadagnini -- Politecnico
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Transitions and break through

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Results

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Continuous time Markov chain model