Elements of Mathematical Sedimentary Geology: the GeoChron Model

image of Elements of Mathematical Sedimentary Geology: the GeoChron Model
  • By Jean-Laurent Mallet
  • Format: EPUB
  • Publication Year: 2014
  • Number of Pages: 388
  • Language: English
  • Ebook ISBN: 9789462820081

The purpose of this book is to offer a mathematically well-defined unified framework to model sedimentary terrains. The core of this framework, named ‘GeoChron’, consists of a curvilinear coordinate system (u,v,t) which superimposes on the rectilinear coordinate system (x,y,z) commonly used to locate particles of sediment in the subsurface. For each particle of sediment observed today at location (x,y,z), coordinate (t) represents the geologicaltime (‘Chron’ in GeoChron) at which this particle was deposited whilst (u,v) represent the paleo-geographical coordinates (‘Geol’ in GeoChron) of this particle when it was deposited.

The u (x,y,z), v (x,y,z) and t (x,y,z) functions link the geological space (x,y,z), as it is observed today, to the depositional-space (u,v,t) and the ‘uvt-transform®’ thus defined plays a central role in using GeoChron to model the subsurface. In the (u,v,t) space, the image of each horizon is a horizontal plane where seismic attributes are easier to interpret and where reservoir properties are more accurately modeled.

The first part of this book is dedicated to a theoretical presentation of the GeoChron model. The second part presents applications of this model such as seismic interpretation, fault property modeling, fractured reservoirs characterization, geometrical uncertainties, property modeling, paleo-bathymetry estimation and permeability upscaling. To make this book ‘self sufficient’, the third part consists of annexes presenting the notions of tensors, differential geometry and Discrete Smooth Interpolation (DSI).

Table of Contents

I Theoretical presentation of the GeoChron Model

1 Introducing the GeoChron Model

1.1 Need for a new paradigm
1.2 GeoChron model
1.3 Parametric representation of the G-space
1.4 Swapping coordinates within the G-space
1.5 Notion of uvt-transform
1.6 Counterintuitive properties of the uvt-transform
1.7 Sedimentation rates
1.8 Changing the parameterization
1.9 Building a GeoChron model
1.10 Implicit versus explicit modeling
1.11 Conclusions
2 Deformations of Sedimentary Terrains
2.1 Assessing deformations in the G-space
2.2 Invariants characterizing the deformations
2.3 Tectonic styles
2.4 Non-equivalent parameterizations
2.5 Coherent perturbation of a GeoChron model
2.6 GeoChron versus unfolding methods
2.7 Conclusions
3 Curvature of Sedimentary Terrains
3.1 Notion of curvature
3.2 Gauss’s “Theorema Egregium”
3.3 Curvatures of a GeoChron model
3.4 Curvatures and geology
3.5 Structural axes of sedimentary terrains
3.6 Conclusions
4 Stratigraphic Distance & Riemannian G-space
4.1 Stratigraphic distance
4.2 Introducing the Riemannian G-space
4.3 Geodesics in the G-space
4.4 Sedimentary attributes
4.5 Conclusions

II Applications of the GeoChron Model

5 Property modeling
5.1 Gridding
5.2 Meshes and discrete functions
5.3 Building a numerical GeoChron property model
5.4 Why use the GeoChron model ?
5.5 Conclusions & comments
6 Upscaling permeability
6.1 Notion of upscaled permeability tensor
6.2 Twin boxes Bc and Band associated grids Gc and Gc
6.3 Upscaling techniques
6.4 Upscaling of other properties
6.5 Conclusions
7 Modeling fault properties
7.1 Fault-blocks displacements
7.2 Faults smearing
7.3 Characterizing “leak-points” on faults
7.4 A property model for faults
7.5 Conclusions
8 Fractured reservoirs characterization
8.1 Geo-mechanical model for fractures
8.2 Probability of possible fracturing
8.3 Fitting fractures directions
8.4 On the impact of non-linear geomechanical behavior
8.5 Application to reservoir characterization
8.6 Conclusions & remarks
9 Geometrical uncertainties
9.1 Introducing geometrical uncertainties
9.2 Perturbing the Preferred-Geometrical-Model
9.3 Modeling uncertainties induced by well markers locations
9.4 Modeling uncertainties induced by seismic velocity
9.5 Modeling uncertainties induced by faults positioning
9.6 Modeling uncertainty induced by horizons positioning
9.7 Proposal for a geometrical uncertainties simulator
9.8 Earth models
9.9 Conclusions & remarks
10 Seismic interpretation
10.1 Coupling Seismic interpretation with GeoChron model building
10.2 Acoustic impedance inversion
10.3 Conclusions & perspectives
11 Estimating the paleo-bathymetry
11.1 Introduction
11.2 Pre-processing well data
11.3 Modeling the paleo-bathymetry Z(r)
11.4 Modeling the facies
11.5 Conclusions & perspectives

III Annexes

12 Tensors
12.1 Primal and dual frames
12.2 Notations
12.3 Linear transformation of a frame
12.4 Tensors
12.5 Order two tensors
12.6 Cross-product of vectors in ℝ3
12.7 Notations (continued)
12.8 On the origin of tensors
13 Differential Geometry
13.1 Notion of manifold
13.2 Differential operators
13.3 Euclidean and Riemannian manifolds
13.4 Covariant derivative
13.5 Curvatures
14 Discrete Smooth Interpolation
14.1 Introducing DSI
14.2 Review of some important DSI constraints
14.3 Implementing DSI: case without hard constraints
14.4 Implementing DSI: case with hard constraints
14.5 Reduced DSI equation in the presence of control-nodes
14.6 Conclusions


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