Many physical phenomena occurring in different areas of science such as fluid dynamics, heat transfer, chemically reacting systems, oil field research, radiation transport, climate science, and structural mechanics, are mathematically modelled by partial differential equations (PDEs) together with appropriate initial and boundary conditions. To simulate complex behaviors of physical systems, engineers and experimental scientists make predictions and hypotheses about certain outputs of interest with the help of simulation of mathematical models. However, due to the lack of knowledge or inherent variability in the model parameters, such real-problems formulated by computational and mathematical forward models generally come with uncertainty concerning computed quantities. Therefore, the idea of uncertainty quantification, i.e., quantifying the effects of uncertainty on the result of a computation, has become a powerful tool for modeling physical phenomena in the last few years.

Mainly, the focus of this project will be mathematical models to describe petroleum or gas reservoir simulations. For such a kind of problems, permeability is desperately needed in the oil industry, however, it is hard to accurately measure the permeability field in the earth, due to the large area of oil reservoir and complicated earth structure. Therefore, identification of permeability parameter is crucial for the efficiency of the mathematical models. Since the classical (deterministic) partial differential equations cannot express the behavior of the physical problem completely, the permeability parameter is represented as a random field.  The mathematical modelling of such a kind of problems corresponds to basically unsteady convection diffusion equations. Therefore, our first interest will be numerical investigation of the time-dependent convection diffusion equation with random input data. Then, we extend the concept to the stochastic optimal control problems. Lastly, by constructing the random data parameters as permeability with the help of the original data, the effect of the mathematical simulations can be observed.

In summary, the work presented in this project lies in the intersection of many mathematical fields such as numerical discretization, probability theory, approximation theory, numerical optimization, and theory of the PDEs. The basic goal of this project is to perform mathematical analysis of the problems, development of numerical solution methods and solution of problems arising in applications of the petrol and gas reservoir problems, modeled with PDEs involving uncertainties due to lack of information or variability of model parameters in physical modeling and numerical simulations. 

Collaborators

• Hamdullah Yücel, Institute of Applied Mathematics, METU (Director)
• Pelin Çiloğlu, Institute of Applied Mathematics, METU
• Sıtkı Can Toraman, Institute of Applied Mathematics, METU
• M. Alp Üreten, Institute of Applied Mathematics, METU

Funded by TUBITAK 1001: 119F022, August 01 2019 - August 01 2021.