ABSTRACT:

A numerical approach for calculating the 3D virgin stress distribution in a subsurface model is developed. A superposition principle and an inversion technique are used to calculate a consistent stress field that satisfies the available measurements as well as equilibrium and compatibility equations. An elastic unloading response of the rock is assumed and applicability of the method for non-linear rocks is discussed. The solution to the forward problem is obtained using the finite element method. The ill-posed inverse problem is solved by minimizing a least-squares functional and by using regularization. The convergence of the algorithm is demonstrated with a numerical example. The work has application in the field of petroleum geomechanics where large-scale subsurface modeling requires stress initialization. The main advantage of the presented algorithm is that a consistent three-dimensional initial stress field that matches the available scarce stress data is constructed and can be used in the subsequent modeling.

1. INTRODUCTION

There are many subsurface applications that require the knowledge of in-situ stresses. In the petroleum industry, these are well design, evaluation of the cap rock integrity, compaction and subsidence calculations. The virgin in-situ stress can be used directly in a green field development and also as an initial state for calculating the mechanical response to fluids production from or injections into underground formations. To estimate the initial stress field it is often assumed that the stress is uniform in lateral directions and one of the principal stresses is vertical. Then the vertical and horizontal stress profiles are estimated by averaging the available stress indications such as the data obtained from density logs, leak-off tests, borehole breakouts and regional stress maps. There are three major drawbacks of such an approach that need to be addressed. First, the average stress “profile” may not be sufficiently accurate in cases of laterally heterogeneous fields (e.g., due to complex surface topology or presence of salt domes). Second, it does not make full use of the very limited data. And third, it may not be in equilibrium to be used as the initial stress distribution in 3D geomechanical modeling. The modeling tools may adjust the uniform stress profile until it satisfies equilibrium equations. However, the resulting adjusted initial stress field does not necessarily satisfy the original field data such as leak-off pressures. Quantitative estimation of the stress distribution is extremely challenging due to the lack of data. Not only are direct stress measurements scarce, but also there are large uncertainties in the geology (configuration and history of deformation) and material properties. The approach undertaken in this paper is based on the fact that regardless of the deformation history of a subsurface, the stress field still must satisfy the equilibrium equations. With certain assumptions, it may even be argued that the bulk of the current stress is due to present external loading, i.e., distributed load (e.g., gravity and pressure source) and boundary conditions. If this is the case then the current in-situ stress can be estimated by virtually unloading the system.

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