# EET 3

## Numerical Earth Models

Instructor |
Prof. Jean-Laurent Mallet |

Duration |
1 day |

Disciplines |
Geophysics |

Level |
Intermediate |

Language |
English |

Book |
Available in EAGE Bookshop |

CPD Points |
5 |

### Course description

- How much can we expect to earn?
- How much could we lose?

Answering these two questions correctly is crucial to making the right decision, which arises immediately after having spent tens of millions of dollars to acquire data on a lease and rights before again spending hundreds of millions of dollars to develop the oil field. Such a decision has to be taken on the basis of a numerical model of the subsurface, integrating all the data and knowledge that have been collected and interpreted by a series of geoscientists, from the geophysical interpretation to the flow simulator. This course presents several types of approaches that can be used to build such models, while pointing out their pros and cons.

### Course outline

The course addresses the following issues:

- What methods are used in classical Computer Aided Design (CAD) and why are they not used to model the subsurface?
- What discrete modelling methods are used in geology to model the geometry, topology and physical properties of the subsurface?
- How does one extract information regarding the geometry and properties of the subsurface from a seismic cube?
- What is a Shared Earth Model (SEM) and what are its intrinsic limitations?
- How does the concept of Unified Earth Model (UEM) allow the limitations of SEMs to be removed?
- Why must flow simulation grids, called FlowGrids, used by flow simulators be built independently of the property model?
- How can FlowGrids be designed optimally to ensure the stability of the numerical algorithms used by flow simulators?
- Why must FlowGrids not be considered as Earth Models?

**Chapter 1 – Classical Computer Aided Design: An Overview**

In the late 1960s and early 1970s, numerical modelling methods were developed for the needs of the car and aircraft industries. These methods, known today under the generic name of Computer Aided Design methods (CAD), are now widely used in all branches of the industry. One can wonder why these methods are not applied as is to the modelling of the subsurface: an answer to this important question is proposed. Limitations of the automatic mapping methods used for many years by the oil and gas industry are also briefly presented. In this chapter we introduce these methods and point out their inadequacy to model the subsurface in an efficient and practical way. In addition, an introduction of the notion of Topological Models is introduced as a mandatory tool to describe the connections between the geological surfaces, such as horizons and faults.

**Chapter 2 – Discrete Modelling**

As pointed out in chapter 1, the classical CAD methods used to model the geometry of manufactured objects are inappropriate to model geological objects. In this chapter, discrete methods specifically designed to model the geometry and properties of the subsurface are proposed. To complete this chapter, some methods used to build the meshes corresponding to discrete models are also presented.

**Chapter 3 – Seismic Interpretation**

In the oil and gas industry, seismic data are the most important source of information related to the geometry and properties of the geological structures. When observing a seismic signal on the Earth’s surface, it appears as a sinusoidal function of time. In practice, this signal is sampled with a constant time step whose magnitude is equal to a few milliseconds. Due to the discrete nature of the data so obtained, discrete Fourier analysis methods are often used to model these seismic signals. In this chapter, we present a slightly different approach, which consists in approximating, locally and continuously, the seismic signal using a centred trigonometric polynomial. In this chapter, several applications of trigonometric polynomials to seismic interpretation are proposed, including auto-picking of seismic horizons, detection of faults and computation of seismic attributes.

**Chapter 4 – Shared Earth Model (SEM)**

Using the tools presented in the previous chapters, it is now possible to envision the construction of a numerical model of the subsurface. Ideally, such a model should play the role of repository of the knowledge related to the structures and properties of the geological domain of interest. As a consequence, this model should be built in such a way that all the geoscientists involved in the modelling process can share and enrich it. In that perspective, the concept of Shared-Earth-Model (SEM) presented in this chapter was introduced in the 1990s. We shall see however, that in the presence of complex fault networks, the Stratigraphic Grids based on pillars and used by the property model of SEMs suffers from dramatic limitations. In the next chapter, we will show that a new unified approach is now proposed to overcome these limitations.

**Chapter 5 – Unified Earth Model (UEM)**

So far, in the oil and gas industry, Stratigraphic-Grids have been used by two populations of geoscientists: geostatisticians and reservoir engineers. However, a close look at the respective problems of each of these geoscientists reveals that they do not have the same needs. For geostatisticians, the (i,j,k)-indexing of the nodes of the cells is used as a discretization of an implicit curvilinear coordinate system. For reservoir engineers, the grid is used to approximate numerically differential equations and the (i,j,k)-indexing of the nodes is only used to retrieve the adjacent cells of each cell. It is clear that we do not need (i,j,k)-indexing to retrieve the neighbouring cells of a cell: this is in particular the case when an unstructured grid is used, where (i,j,k)-indexing is then not possible. In practice, building (i,j,k)-indexed grids is often not possible in presence of a complex Fault-Network and it is then necessary to dramatically simplify this Fault-Network. There are currently two (r)evolutions that should change such an unfortunate current practice dramatically: on the one hand, it is now possible for geostatisticians to parameterize the subsurface without using any (i,j,k) grid and, on the other, the new generation of flow simulators does not care about (i,j,k)-indexing of cells. Based on these considerations, in this chapter we show how modelling the subsurface should be addressed in a new, unified way.

**Chapter 6 – FlowGrids**

Assuming that reservoir engineers are now completely freed from the (i,j,k)-indexing of grids imposed so far by geostatisticians, we propose to reformulate the problems of designing flow simulation grids (FlowGrids) in a new way that does not require simplifying the Fault-Network. For that purpose the real needs of Flow Simulators with regard to the design of these FlowGrids are analysed and we show that these grids can be built without using the traditional technique of extrusion along pillars. In the quest for an optimal design of FlowGrids, the pros and cons of several geometries for their cells (FlowCells) are presented. We show that Stair-Step approximation of faults is certainly an excellent compromise to ensure the stability of numerical approximation used by Flow Simulators.

### Participants' profile

The course is aimed at geoscientists involved in exploration and production projects where seismics play a role and who wish to:

- Learn more about seismic imaging concepts and the terminology used by seismic processors;
- Improve their critical view on the benefits and limitations of the seismic data sets they are using in their projects;
- Have a well-argued selection of the imaging method to apply to the seismic data shot for their projects;
- Have a better appreciation of what they can expect from reprocessing vintage data sets with modern tools.

The course will also benefit students who want to have a first acquaintance to reflection seismics in general and seismic imaging in particular.

### Prerequisites

The course can be understood by geoscientists with a moderate mathematical background. Physical concepts are presented without equations but with a maximum of simple schemes and graphical illustrations. Some basic knowledge of wave propagation theory may help however. A comprehensive list of references is given in the book for those who are interested in more rigorous and mathematical approaches.

### About the instructor

**Jean-Laurent Mallet**graduated from the Ecole-Nationale-Supérieure-de-Géologie (Nancy/France) in 1968. He next diversified his interests into applied mathematics, finishing his research education with a PhD in applied mathematics and computer sciences at the Institut-National-Polytechnique-de-Lorraine (INPL). In his early years, he worked with the CNRS in the field of numerical data analysis and automated mapping. In 1981, he was nominated Professor at the Ecole-Nationale-Supérieure-de-Géologie (ENSG), from which he directed teaching of and research into applications of computer sciences related to geology. Professor Mallet’s research is at the frontier of mathematics, geosciences and the computer sciences. He has contributed in many ways to the progress of sciences and technology in the field of Oil and Gas. He headed the Gocad Research Consortium from 1989 to 2006. He was awarded several international prizes, such as the Italgaz Prize (1997–Italy), the Antony-Lucas Gold Medal (2000–SPE), the Grand Prix Dolomieu (2003–French Academy of Sciences) and the Prix Roubault (2005–Union of French Geologists). Professor Mallet retired from the university in 2006 and he is now Emeritus Professor at the Ecole-Nationale-Supérieure-de-Géologie and Research Fellow at Paradigm-Geophysical.

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