## Numerical Studies for Petrol and Gas Reservoir Problems

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 general, partial differential equations with random input data can be thought of as outer problems, since they entail repeated solution of the deterministic forward problem, namely, the inner problem, for different values of the random parameters. The main challenges in solving such problems are: identification of random variables; discretization of probability, space and time domains. Within the scope of this project, random variables will be constructed primarily using the well-known Karhunen-Love expansion, in which the random variables are expressed as Fourier series expansion. To handle the probability domain, we will consider two different approaches: Stochastic Galerkin method and Multilevel Monte Carlo method. While the multilevel Monte Carlo method is an example of sampling technique, the stochastic Galerkin is based on the orthogonal projection. The multilevel Monte Carlo method is a type of the classical Monte Carlo method but it has a better convergence. On the other hand, an important feature of the stochastic Galerkin approach is the separation of the spatial and stochastic variables. This allows a reuse of established Galerkin techniques. While we prefer to discontinuous Galerkin (DG) methods as a discretization technique for spatial domain, the rational deferred correction method is applied as a post processing technique with the standard time discretization techniques. The reason is to choose discontinuous Galerkin method is that the local mass conservation is a crucial property for the problems in reservoir simulation and fluid dynamics. Compared with the DG methods, finite difference method is not able to handle complex geometric, the finite volume method is not capable of achieving high-order accuracy and the standard continuous finite element lacks the ability of local mass conservation. Moreover, for convection diffusion problems, the DG methods produce stable discretization without the need for stabilization strategies and they allow for different orders of approximation to be used on different elements in a very straightforward manner. Further, with the help of the rational deferred correction method, we obtain better accuracy using comparatively small number of time nodes.

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)

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