Isometric Unfolding of Stratigraphic Grid Units for Accurate Property Populating – Mathematical Concepts

Abstract

In traditional methods used to populate stratigraphic units, the distortions can be very important and affect the setting up of the static and dynamic parameters necessary to the reservoir simulation, hence the simulation results. These distortions result from the mapping between the original curvilinear stratigraphic grid and the intermediate cartesian grid in which the property populating is processed.To minimize the deformation and improve the populating process, we propose a new original isometric unfolding process based on the minimization of the elastic tensor deformation. This method could be applied for every type of deposit: horizontal, parallel to top, parallel to bottom, proportional.Starting from a structural model defined into a coordinate line grid, the user chooses a reference iso-chronological level represented by a triangulated surface. This level can be the top, the bottom or any other characteristic surface of a litho stratigraphic unit. The contacts between this surface and fault surfaces are explicitly extracted as coincident edges. These coincident edges are used to constraint an unfolding process which minimizes the elastic deformation tensor on the whole surface and then, respecting the above constraints, join the fault lips opened by geological tectonic events.To exploit this first 2D Unfolding for volumetric purposes, we use the coordinate line grids as a 3D support and define a complete mapping between the original curvilinear structural grid and its image in the deposition (unfolded) domain. This mapping which expresses the geometrical transformation between the today's world and the depositional period could now be used to transport today's well trajectory stations in the unfolded volume for geostatistical computation and, on the way back, define the geometrical relations needed for upscaling.In this paper we will focus on mathematical concepts of the algorithms used in the whole unfolding process and present some results throw actual case studies. Main Objectives:Present the mathematical concept and methodology for a new unfolding method based on the minimization of the elastic tensor deformation. With this method, we will be able to associate to a stratigraphic grid unit in structural position to its cell to cell image in the depositional space. This mapping could be used to transport well trajectory stations in the unfolded volume for geostatistical computation. New aspects covered :We propose a new original isometric unfolding process based on the minimization of the elastic tensor deformation. We apply this algorithm to unfold iso-chronological surfaces and to unfold volumes in several depositional modes: Top and bottom conformable, proportional and parallel to any surface.

Uso da função distância na modelagem geométrica de corpos de minério e proposta da distância estratigráfica aplicada ao cálculo de variogramas experimentais

“…Superfícies dobradas b-) Superficies desdobradas. Fonte: Horna et al (2010) Neste processo é necessário que para cada ponto de uma superfície da camada a ser desdobrada, deve haver um ponto correspondente na superfície de referência (representados pelos pontos azuis na Figura 4.9). Os contatos das falhas/horizontes devem coincidir corretamente e, ao remover alguma aresta ou simplificar a malha, não deve haver a remoção de informação necessária.…”

“…Figura 4.6: Ilustração do processo de extração da superfície de referência. Fonte: O Autor, baseado emHorna et al (2010) …”