In this dissertation we discuss some upscaling methods for flow and transport in heterogeneous reservoirs. We studied realization-based multi-phase flow and transport upscaling and ensemble-level flow upscaling. Multi-phase upscaling is more accurate than single-phase upscaling and is often required for high level of coarsening. In multi-phase upscaling, the upscaled transport parameters are time-dependent functions and are challenging to compute. Due to the hyperbolic feature of the saturation equation, the nonlocal effects evolve in both space and time. Standard local two-phase upscaling gives significantly biased results with reference to fine-scale solutions. In this work, we proposed two types of multi-phase upscaling methods, TOF (time-offlight)- based two-phase upscaling and local-global two-phase upscaling. These two methods incorporate global flow information into local two-phase upscaling calculations. A linear function of time and time-of-flight and a global coarse-scale two-phase solution (time-dependent) are used respectively in these two approaches. The local boundary condition therefore captures the global flow effects both spatially and temporally. These two methods are applied to permeability distributions with various correlation lengths. Numerical results show that they consistently improve existing two-phase upscaling methods and provide accurate coarse-scale solutions for both flow and transport. We also studied ensemble level flow upscaling. Ensemble level upscaling is up scaling for multiple geological realizations and often required for uncertainty quantification. Solving the flow problem for all the realizations is time-consuming. In recent years, some stochastic procedures are combined with upscaling methods to efficiently compute the upscaled coefficients for a large set of realization. We proposed a fast perturbation approach in the ensemble level upscaling. By Karhunen-Lo`eve expansion (KLE), we proposed a correction scheme to fast compute the upscaled permeability for each realization. Then the sparse grid collocation and adaptive clustering are coupled with the correction scheme. When we solve the local problem, the solution can be represented by a product of Green's function and source term. Using collocation and clusering technique, one can avoid the computation of Green's function for all the realizations. We compute Green's function at the interpolation nodes, then for any realization, the Green's function can be obtained by interpolation. The above techniques allow us to compute the upscaled permeability rapidly for all realizations in stochastic space.

In this dissertation, we focus on the uncertainty quantification problems where the goal is to sample the porous media properties given integrated responses. We first introduce a reduced order model using the level set method to characterize the channelized features of permeability fields. The sampling process is completed under Bayesian framework. We hence study the regularity of posterior distributions with respect to the prior measures. The stochastic flow equations that contain both spatial and random components must be resolved in order to sample the porous media properties. Some type of upscaling or multiscale technique is needed when solving the flow and transport through heterogeneous porous media. We propose ensemble-level multiscale finite element method and ensemble-level preconditioner technique for solving the stochastic flow equations, when the permeability fields have certain topology features. These methods can be used to accelerate the forward computations in the sampling processes. Additionally, we develop analysis-of-variance-based mixed multiscale finite element method as well as a novel adaptive version. These methods are used to study the forward uncertainty propagation of input random fields. The computational cost is saved since the high dimensional problem is decomposed into lower dimensional problems. We also work on developing efficient advanced Markov Chain Monte Carlo methods. Algorithms are proposed based on the multi-stage Markov Chain Monte Carlo and Stochastic Approximation Monte Carlo methods. The new methods have the ability to search the whole sample space for optimizations. Analysis and detailed numerical results are presented for applications of all the above methods.