This thesis concerns iterative solvers for poromechanics problems. The problems in the studies have involved linear poromechanics, non-linear poromechanics, and poromechanics under large deformation. We included high order discretizations, applied linearization techniques and splitting methods to develop new solvers. We studied the robustness and convergence of these solvers. By studying the fixed stress method as an iterative solver for poromechanics, we developed an optimized version of it. Furthermore, by extending the convergence analysis in the time domain, we developed a new version of the fixed stress method that is partially parallelized. This splitting method was combined with linearization techniques to develop solvers for non-linear poromechanics. By studying the convergence of the linearisation schemes, we developed new solvers and extended the applicability to more complex phenomena, for instance poromechanics with large deformation.

We consider a non-linear extension of Biot's model for poromechanics, wherein both the fluid flow and mechanical deformation are allowed to be non-linear. We perform an implicit discretization in time (backward Euler) and propose two iterative schemes for solving the non-linear problems appearing within each time step: a splitting algorithm extending the undrained split and fixed stress methods to non-linear problems, and a monolithic L-scheme. The convergence of both schemes is shown rigorously. Illustrative numerical examples are presented to confirm the applicability of the schemes and validate the theoretical results.

In this work, we study the parallel-in-time iterative solution of coupled flow and geomechanics in porous media, modelled by a two-field formulation of the Biot's equations. In particular, we propose a new version of the fixed stress splitting method, which has been widely used as solution method of these problems. This new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization. We present a rigorous convergence analysis of the method and a numerical experiment to demonstrate the robust behaviour of the algorithm.Terzaghi [1], which was extended to a more general three-dimensional theory by Biot [2,3]. Biot's model was originally developed to study geophysical applications such as reservoir geomechanics, however, nowadays it is widely used in the modeling of many applications in a great variety of fields, ranging from geomechanics and petroleum engineering, to biomechanics or food processing. There is a vast literature on Biot's equations and their existence, uniqueness, and regularity, see Showalter [4], Phillips and Wheeler [5] and the references therein.Reliable numerical methods for solving poroelastic problems are needed for the accurate solution of multi-physics phenomena appearing in different application areas. In particular, the solution of the large linear systems of equations arising from the discretization of Biot's model is the most consuming part when real simulations are performed. For this reason, a lot of effort has been made in the last years to design efficient solution methods for these problems. Two different approaches can be adopted, the so-called monolithic or fully coupled methods and the iterative coupling methods. The monolithic approach consists of solving the linear system simultaneously for all the unknowns. The challenge here, is the design of efficient preconditioners to accelerate the convergence of Krylov subspace methods and the design of efficient smoothers in a multigrid framework. Recent advances in both directions can be found in [6,7,8,9] and the references therein. These methods usually provide unconditional stability and convergence. Iterative coupling methods, however, solve sequentially the equations for fluid flow and geomechanics, at each time step, until a converged solution within a prescribed tolerance is achieved. They offer several attractive features as their flexibility, for example, since they allow to link two different codes for fluid flow and geomechanics for solving the coupled poroelastic problems. The design of iterative schemes however is an important consideration for an efficient, convergent, and robust algorithm. The most used iterative coupling methods are the drained and undrained splits, which solve the mechanical problem first, and the fixed-strain and fixed-stress splits, which on the contrary solve the flow problem first [10,11].Among iterative coupling schemes, the fixed stress splitting method is the most widely used. This sequential-implicit method bas...

This paper describes the challenge of managing and optimizing the production of a large land based oilfield with hundreds of ESP-boosted wells arranged in widely distributed well clusters which production converges to major trunklines traversing the field. The Rubiales field, located in the eastern plains of Colombia has challenging features, characteristics and layout that demand effective model-based production optimization and control. The field's gathering system feeds the commingled production to two central field processing plants.The flow of the numerous wells and streams of the network are interdependent as there are no gas separation facilities at the clusters or at any other location in the network between the wellhead sources and the entry to the processing plants. This creates an interdependency of well streams. Thus, any production change at a single well affects the pressure and rate of all other wells in the network and consequently the total field production. The water rate from each individual producing well strongly depends on the drawdown and the stage of depletion of that particular well, and how it is controlled by varying the speed of its ESP. High water cuts of most producing wells and the constraints on water treatment and disposal at the field level dictates a need for frequent readjustment of individual well ESP speed.Adjusting ESP speeds to maximize the field oil production, subject to field water production constraints, must also take into account a variety of additional constraints related to system limitations, ESP performance, power consumption, production operations and reservoir recovery strategy. One cannot rely solely on operational intuition and empirical field practice for individual ESP control. Rather, a modelbased optimization system has been implemented, taking into account all field and well constraints. The implemented system is robust, fast and easy to tune. Furthermore, inflow of heavy and viscous Rubiales oil into the horizontal wellbores is driven by a strong and active aquifer in a highly heterogeneous and permeable reservoir. This results in rapid changes of produced water cut in response to small changes in drawdown, demanding effective tuning of a predictable well inflow function for the purpose of optimization.This paper describes the model-based optimization system employed in the Rubiales field. The system is customized to the large scale and special features of Rubiales, such as the demanding production performance of its wells, the constraints of facilities, and the objective to maximize profit given by production revenues less OPEX.

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