© Copyright, 2010 , Saudi Aramco, All Rights Reserved

J A Vargas-Guzman, PhD

Darwin Australia

August, 2014

Structural Uncertainty in

Unconventional Reservoirs

2014

Contents

Motivation

Insights

Tight Gas Resource Modelling

Technological Innovations for Clastic Reservoir Modelling

Motivation

Unbiased estimation of basin centered gas resources

Modeling of stochastic fields with extremely skewed probability distributions

Explain abnormal pressure decline, and Sw

Predict sweet spots to sustain productivity

Provide directions to stimulation, and improve well-productivity

Motivation

Motivation

Vargas-Guzman. Unbiased resource evaluations with kriging and stochastic models of heterogeneous

rock properties. Natural Resources Research, 2008

Delimit and evaluate resources

Predict structural controls

Integrate seismic and well data

Predict sweet spots, volumes and locations

Provide conditions for geomechanical and flow modelling

Predict subseismic faults

Model compartments

Help predict outcomes from stimulation

Improve production history match

Motivation

Tight Gas Resource Modeling Challenges

Modeling Geological Structural Uncertainty

Predict probability of sweet spot locations

Predict rock bodies (i.e., geometries and numerical sequence stratigraphy boundaries)

Reconstruct deformation processes

(i.e., structural geology)

Model probability fields of fractures

(i.e., natural, hydraulic fractures)

Tight Gas Resource Modeling Challenges

Structural Modeling Challenges Gigantic Grids

Model

300 million cells

89 faults

Seismic conditioning

FAULTS

Khan and Vargas-Guzman. Modeling nonlinear beta probability fields. Geostatistics

Oslo 2012 Quantitative Geology and Geostatistics

Structural Modeling Challenges Complex Grids

Sub-Seismic Faults

Erosional Unconformity

Compartmentalization

Vargas-Guzman and Liu ,Enhanced compartmentalization of a complex reservoir with sub-seismic faults

from geological inversion. Journal of Petroleum Science and Engineering, 2008

Structural Modeling Challenges Resolution

Structural Modeling Challenges Multiple Resolution

Forsyth and Vargas-Guzman. Innovative petrophysics to understand the spatial gas distribution in a

conventional gas reservoir with unexpected unconventional characteristics, SPWLA-2013

Hydrothermal mineralization and oxidation of pyrite controlled by a vertical fracture

Structural Modeling Challenges Structural Controls of Diagenesis

Vargas-Guzman, atal., Identification of high permeability zones from dynamic data using

streamline simulation and inverse modeling of geology.Journal of Petroleum Science and

Engineering, 2009

Geometry and Analogs

Structural Modeling Challenges Rock Bodies

Vargas-Guzman, et al., A High-Resolution Reservoir simulation study for a giant offshore field

using a model constrained to complex clastic rock-bodies, SPE JOT (July 2012)

Rock Bodies Modeling Challenges Object Modeling

10

km

Rock Bodies Modeling Challenges MPS Modeling of Geobodies

TRAINING IMAGE

SIS JFb3_1

Template MPS Model

MPS+SIS Rock Types

Vargas-Guzman, Effect of multipoint heterogeneity on nonlinear

transformations for geological modeling: porosity-permeability

relations revisited. Journal of China University of Geosciences,

2008

Finite Elements

Stochastic Geometry

Heterogeneous Rock Properties

Physical Constraints

Structural Modeling Challenges Rock Bodies

Vargas-Guzman and Qassab,. Spatial conditional simulation of facies objects for modelling

complex clastic reservoirs. Journal of Petroleum Science and Engineering, 2006

Technological Innovations for Clastic Reservoir Modeling

Technological Innovations Downscaling Seismic

Seismic Data Integration

Technological Innovations Downscaling Seismic

Vargas-Guzman etal., Integration of 3D seismic impedance into high resolution

geocellular models using non-collocated downscaling SPE 2011

Cocumulant Interpretation & Modeling Beyond Linear Correlations

Technological Innovations Non-Linear Tools

Vargas-Guzman. The Kappa model of probability and higher-order rock sequences. Computational

Geoscience, 2011

0.00 0.50 1.00

PROPORTIONS

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

CU

MU

LAN

TS

32

3 )(3 fffff

432

4 6127 ffff 5432

538815715 fffff

FLUVIAL

DOMINATED

Mahakam

RhoneFly

IrrawaddyCopper

Nile

Mississippi

Higher Order Cumulants

Technological Innovations Non-Linear Tools

Vargas-Guzman. The Kappa model of probability and higher-order rock sequences. Computational

Geoscience, 2011

0.00 2.00 4.00 6.00

HO Random Variable

0.00

0.40

0.80

1.20

Pro

ba

bility D

en

sity F

un

ctio

n

-5.0 -2.5 0.0 2.5 5.0Logarithm of HO Random Variable

0.00

0.05

0.10

0.15

0.20

0.25

Pro

ba

bility D

en

sity F

un

ctio

n

Technological Innovations Non-Gaussian Tools

3!3

3

1

2

13

1

233

exp

3

133

1

)3

()!3(2

1)(

31

vvv

absvf

PDF’s with Cumulant Parameters

Technological Innovations Non-Gaussian Tools

Vargas-Guzman. Heavy tailed probability distributions for non-Gaussian simulations with higher-order

cumulant parameters predicted from sample data SERRA, 2012

Itô’s Stochastic Integration

𝑦2 𝑥 = 𝑦1 𝑥 ∙ 𝜕𝑦1 𝑥𝑥

0

𝑦2 𝑥 = 12𝑦12(𝑥) + 𝟏

𝟐 𝝈𝟐 ∙ 𝒙

Nonlinear term

Correction

Technological Innovations Nonstationary SPDE Modeling Tools

The non-linear Itô’s component behaves as a

stationary nonlinear model.

Itô’s Stochastic Integration

Extensions

𝑦3(𝑥) =1

2∙ 3𝑦13(𝑥) + 3 𝒚𝟏 𝒙 𝝈𝟐 ∙ 𝒙 + 𝟏

𝟐 𝝈𝟐 ∙ 𝒙

𝑦3 𝑥 = 𝑦1 𝑥 ∙ 𝜕𝑦1 𝑥𝑦1 0

∙ 𝜕𝑦1 𝑥𝑦1 0

= 𝑦2 𝑥 ∙ 𝜕𝑦1 𝑥𝑦1 0

𝑦2 𝑥 = 12𝑦12(𝑥) + 𝟏

𝟐 𝝈𝟐 ∙ 𝒙

Technological Innovations

Vargas-Guzman. Unified principles for nonlinear nonstationary random fields in stochastic

geosciences, In: Mathematics of Planet Earth, Springer, 2013

Technological Innovations Nonstationary SPDE Modeling Tools

Vargas-Guzman. Unified principles for nonlinear nonstationary random fields in stochastic

geosciences, In: Mathematics of Planet Earth, Springer, 2013

Insights

Risk analysis of development projects needs to be based on full geo-cellular models

Cumulant parameter pdfs enable nonlinear models from non-Gaussian inputs at multiple resolutions

The existing probabilistic modelling technology has limitations, and process driven approaches must be included

Structural uncertainty has to be embraced in every reservoir development project

Stochastic Partial Differential Equations SPDE can be integrated with data and geostatistical technology

Insights

Thank you

BACKUP SLIDES

Residual Random Variables of Order G

0

]1[ vu

11122]2[ vvu

31111 3233]3[ vvvu

][3 46 4

1

2

2

4

1

3

1

2

11

234

4]4[ vvvvu

3210

5

1

4

15

3

1

210

2

1

310

1

45

5]5[ vvvvvu

][3 4

1

2

2

4

1

]4[ vu

][10 32

5

1

]5[ vu

Nonlinear Stationary Models

3!3

3

1

2

13

1

233

exp

3

133

1

)3

()!3(2

1)(

31

vvv

absvf

dv

vvvabs

vvv

3

31

211

23

31

331

31111

23

3!3

33exp

3

13 )()!3(2

33

PDF’s with Cumulant Parameters

Nonlinear Stationary Models

Itô’s Stochastic

Integration

-0.00060

-0.00050

-0.00040

-0.00030

-0.00020

-0.00010

0.00000

0.00010

0.00020

0.00030

0.00040

0 200 400 600 800 1000

x 1

00

00

0

WIENER PROCESS

𝑦1(𝑥) = 𝜕𝑊(𝑥)𝑥

0

𝑦1 𝑥 ∙ 𝜕𝑦1 𝑥

𝜕𝑦1 𝑥

𝑦1 𝑥

Itô’s Stochastic Integral 𝑦2 𝑥 = 𝑦1 𝑥 ∙ 𝜕𝑦1 𝑥 𝑦1 0

The non-linear Itô’s component behaves as a

transformation of the collocated stationary input

Itô’s Stochastic

Integration

𝑦2 𝑥 = 12𝑦12(𝑥) + 𝟏

𝟐 𝝈𝟐 ∙ 𝒙

Nonlinear term from

Classic Integration

Parametric Correction

Itô’s Stochastic

Integration

Nonlinear term

𝑦3(𝑥) =12∙ 3 𝑦1

3(𝑥) + 3 𝑦1 𝑥 𝜎2 ∙ 𝑥

Parametric Correction

𝑦3(𝑥)

= 12 𝑦

2(𝑥) 𝜕𝑦 𝑥𝑦1 0

Second Power Input RV

Novel Extensions

-8

-6

-4

-2

0

2

4

6

8

0 100 200 300 400

2nd

3rd

5th

10th

Stochastic Integration of

Non-Gaussian Case

-5.0 -2.5 0.0 2.5 5.0Logarithm of HO Random Variable

0.00

0.05

0.10

0.15

0.20

0.25

Pro

babi

lity

Den

sity

Fun

ctio

n

0.001

0.01

0.1

1

10

100

1000

10000

0 100 200 300 400

2nd

3rd

5th

10th

0.00 2.00 4.00 6.00

HO Random Variable

0.00

0.40

0.80

1.20P

roba

bilit

y D

ensi

ty F

unct

ion

Stochastic Integration of

Non-Gaussian

Insights

Expected values of Ito’s models could help to avoid Monte

Carlo by direct estimation of output pdf parameters

Cumulant parameter pdfs enable nonlinear models from

non-Gaussian inputs

Extensions of Ito’s integration enable non-stationary models

with drifts and nonlinear (Newton-Leibniz) terms

Physical models are non-stationary due to integration, and

stationary differentials may be enabled from spde and

boundary conditions

Expected values on stochastic integrals lead to a unification of

nonlinear and non-stationary models with Gaussian and non-

Gaussian stationary input differences

Thank You

Rice et al. Economic Geol. 2005

Paleozoic

Tertiary volcanic rocks

Dome

POTOSI - CERRO RICO MOUNTAIN OF SILVER

Gigantic Models

High resolution

Gigantic hydrocarbon reservoirs

with non-stationary model properties

Haradh

Hawiyah

Uthmaniyah

Shedgum

Ain-Dar

www.aramco.expats.com

Motivation