Geostatistical representation of multiscale heterogeneity of porous media through a Generalized Sub-Gaussian model

Alberto Guadagnini(1), Monica Riva(1), Shlomo P. Neuman(2), Martina Siena(1)

(1) Dipartimento di Ingegneria Civile e Ambientale (DICA), Politecnico di Milano, Milano, Italy(2) Department of Hydrology and Atmospheric Sciences, University of Arizona, Tucson, AZ, 85721, USA Sharing Geoscience Online

Key ObservationsMany earth, environmental, physical, bio / ecological, financial, & other variables show

1. Power-law scaling in midrange of separation scales (lags). Breakdown in power-law scaling at small and large lags. Power-Law scaling extended to virtually all lags (ESS).

2. Heavy-tailed frequency distributions of incrementsA variable Y can be Gaussian or non-GaussianPdfs of spatial increments ∆Y scale with lag Sharp peaks and heavy tails of pdf decaying with lags

( ) ( ) ( )Y s Y x s Y x∆ = + −

Observations

Sharp Peaks and Heavy Tails of Increment Frequency Distributions

Decaying with LagNeural network hydraulic property estimates of

borehole samples at UA Maricopa Agricultural Station

Log K

Porosity

van Genuchten

A. Guadagnini – October 2018

Consider

where is iid subordinator

Model parameters and observation scale

N = 3567 Neutron Porosity data along a deep borehole

Mean= 14%; SD = 6.4%Sampling interval: 0.15 m

data courtesy of Prof. Sahimi

( ) 0U x >

( ) ( ) ( ); , ; ,u l ulY' x U x G xλ λ λ λ=

Several models for U (lognormal, Exponential, Gamma, ...)

Subordinator U renders Y′ a scale mixture of Gaussian RFs

The scale mixture is non-Gaussian (distribution depends on U)

tfBmY′(x) = Y(x) − 〈Y〉

GSG-Generalized Sub-Gaussian Model

GSG ModelConsider where is iid subordinator.It renders Y Subordinated to G Scale mixture of G with random variances

To start, lognormal (LN) subordinator

It renders Y’ normal-lognormal (NLN)

VU e= ( ){ }20, 2 2V α α= −

( ) ( )( )

22

2 21 1 'ln2 2

' 20

1 1'2 2

G

u yu

Yf y e duuσα

π α

∞ − +

− =− ∫

( ) ( ) ( ); , ; ,u l ulY' x U x G xλ λ λ λ= Y′(x) = Y(x) − 〈Y〉

( )U x∝

Y increments are not NLN:

( )

( )( )22 21 2

2 2 21 2 1 2

1 1 ln ln2 22

2 123/2 2 20 0 2 1 1 2 1 2

1( )2 2 2 2

G GG

yu uu u u u

Y

G

ef y du duu u u u u u

α ρσ σ

α ρπ

∆ − + + + − −∞ ∞

∆ ∆ =− + −

∫ ∫

( )G sρ

( ) ( )( )

22

2 21 1 ln2 2

20

1 12 2

G

u yu

Yf y e duuσα

π α

∞ − +

− =− ∫

Scales with lag through (coefficient of correlation of G)

GSG Model

Variogram of Y

Observed nugget effects, attributed in the literature to variability of Y at scales smaller than the sampling interval and/or to measurement errors, may in fact be (at least in part) a symptom of non-Gaussianity.

Integral scale of Y (subordination dampens and does not destroy covariance structure)

( ) ( )( ) ( )2 2 22 2 22 1Y G Ge e eα α αγ σ γ− − −= − +

( )22Y GI e I

α− −=

GSG Model

Deep Borehole Neutron Porosity dataSample pdf of increments of neutron porosity dataSample pdf of neutron porosity data Y′

Excess kurtosis of mean-removed porosity data (continuous curve) and of porosity increments

(symbols) versus lag

NP

(%)

0

10

20

30

40

50

2800 3000 3200 3400

Well 1 Southwest Iran3,567 dataMinimum lag: 15 cmTotal depth: 543 m

1 14%YM =

Sample standard deviation = 6.4%Dashtian et al. [2011]

aα 1.78

ˆaα 1.73 0.05±

Mean of bα

CV of bα

1.75

0.05

aσ 6.1%

ˆaσ 5.98 0.19±

Mean of bσ

CV of bσ

6.15

0.07

Deep Borehole Neutron Porosity dataRiva et al. (2015) WRR

1.78

Mean of CV of

1.75

0.05

6.1%

Mean of

CV of

6.15

0.07

b

a

%

a

s

%

ˆ

a

s

5.980.19

±

b

s

%

b

s

%

a

a

%

ˆ

a

a

1.730.05

±

a

a

%

1.78

ˆ

a

a

1.730.05

±

Mean of

b

a

%

CV of

b

a

%

1.75

0.05

a

s

%

6.1%

ˆ

a

s

5.980.19

±

Mean of

b

s

%

CV of

b

s

%

6

.1

5

0.07

a

1.78

ˆ

a

1.730.05

Mean of

b

CV of

b

1.75

0.05

a

6.1%

ˆ

a

5.980.19

Mean of

b

CV of

b

6.15

0.07

Drilling operation, using a drill press: multiple subsamples are obtainedalong three lines parallel to the core axis (left, central and right).

222 samples in total

One-dimensional variability

96

96.2

96.4

96.6

96.8

97

97.2

97.4

97.6

97.8

98

0 1E-13 2E-13 3E-13 4E-13 5E-13

x [m]

k gas Scheidegger TRUE [m2]

Left Central

Right

N. of observations Mean Standard deviation Skewness Excess kurtosis220 -13.03 0.228 -0.208 -0.161

Dissolved Area

Unaltered Area

• Topography images of calcite crystalsSurface Roughness

Application to geochemical data – assessment of calcite dissolutionJoint work with Philippe Ackerer and Damien Daval (Université de Strasbourg-CNRS ENGEES/EOST)

Images from L. Stigliano (Msc Thesis, 2020)

1. Data processing and analysis

2. Distributions of P’ and ∆P

4. GSG (with given U)

3. Scaling?

Yes

No Classical

geostatisticalmodels

5. Fitting GSG model to statistical moments and/or frequency distributions of P’ and ∆P

6. Parameter estimation

7. GSG model verification: Do GSG

distributions and parameters scale

correctly?

Yes 8. Generation

of GSG functions

No

Vary U

P’

ProcessSimulation

9. State variables

WORKFLOW FOR ANALYSIS

8 ×106 cells

3D-flow/transport

problems

Many variables with scalable statistics exhibit Heavy-tailed increment frequency distributions with sharp peaks at small lags Peak and tail decay with increasing lag

These aspects are captured by a GSG model based on sub-Gaussian subordination Simple ways to estimate parameters

We developed a methodology to generate unconditional and conditional realizations of such random fields Ready to be employed for Monte Carlo flow and transport simulations in randomly heterogeneous systems (see

Libera, A., F. de Barros, M. Riva, and A. Guadagnini, JCH, 2018).

Lead-order analytical solutions for a simple non-Gaussian case of flow and transport in an unbounded unimodal GSG log permeability field Y under mean uniform flow in two spatial dimensions

What we have seen so far!

Summary GSG model can reconcile several aspects

exhibited by many (environmental and other) variablesPower-law scaling of structure functions in midrange of lagsBreakdown in such scaling at small / large lagsHeavy-tailed increment frequency distributions with sharp peaks at small lagsPeak and tail decay with increasing lagExtended power-law scaling or ESSLinear / nonlinear scaling of power exponent

Simple ways to estimate model parameters Lead-order analytical solutions for simple

flow/transport settingsNumerical flow/transport solutions Conditional GSG

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