Algebraic Multiscale Framework for Fractured Reservoir Simulation

Despite welcome increases in the adoption of renewable energy sources, oil and natural gas are likely to remain the main ingredient in the global energy diet for the decades to come. Therefore, the efficient exploitation of existing suburface reserves is essential for the well-being of society. This has stimulated recent developments in computer models able to provide critical insight into the evolution of the flow of water, gas and hydrocarbons through rock pores. Any such endeavour, however, has to tackle a number of challenges, including the considerable size of the domain, the highly heterogeneous spatial distribution of geological properties, as well as the intrinsic uncertainty and limitations associated with field data acquisition. In addition, the naturally-formed or artificially induced networks of fractures, present in the rock, require special treatment, due to their complex geometry and crucial impact on fluid flow patterns.From a numerical point of view, a reservoir simulator’s operation entails the solution of a series linear systems, as dictated by the spatial and temporal discretization of the governing equations. The difficulty lies in the properties of these systems, which are large, ill-conditioned and often have an irregular sparsity pattern. Therefore, a brute-force approach, where the solutions are directly computed at the original fine-scale resolution, is often an impractically expensive venture, despite recent advances in parallel computing hardware. On the other hand, switching to a coarser resolution to obtain faster results, runs the risk of omitting important features of the flow, which is especially true in the case of fractured porous media.This thesis describes an algebraic multiscale approach for fractured reservoir simulation. Its purpose is to offer a middle-ground, by delivering results at theoriginal resolution, while solving the equations on the coarse-scale. This is made possible by the so-called basis functions – a set of locally-supported cross-scale interpolators, conforming to the heterogeneities in the domain. The novelty of the work lies in the extension of these methods to capture the effect of fractures. Importantly, this is done in fully algebraic fashion, i.e. without making any assumptions regarding geometry or conductivity properties.In order to elicit the generality of the proposed approach, a series of sensitivity studies are conducted on a proof-of-concept implementation. The results, which include both CPU times and convergence behaviour, are discussed and compared to those obtained using an industrial-grade AMG package. They serve as benchmarks, recommending the inclusion of multiscale methods in next-generation commercial reservoir simulators.Petroleum Engineerin

Ensemble-Based History Matching for Channelized Petroleum Reservoirs

The oil industry is at the backbone of global economy and, as natural resources are becoming scarce, there is a pressing need for efficient extraction strategies. This has led to the development of reservoir models and simulators, able to predict future field behaviour, when paired with accurate geological information. However, the data obtained through in-situ measurements, such as seismic surveys, is insufficient to represent the large number of unknowns (porosity, permeability, pressure and fluid saturation in each grid cell). In response to this issue, the scientific community designed computer-assisted history matching algorithms, which are able to provide estimates for model parameters by conditioning on the log of observed production data. The Ensemble Kalman Filter, in particular, is becoming the industry standard, because of its ease of implementation and natural ability to handle uncertainty. However, as past studies have pointed out, reservoirs with complex structural features, such as curved or branching channels, raise difficulties because of the higher-order dependencies induced between the state variables. Another important drawback is the appearance of ensemble collapse, which leads to poor estimates and causes the filter to diverge. The Subspace EnKF is a recently developed history matching framework, able to address both of these issues, by using parameterizations to constrain the ensemble members to different subregions of the parameter space. The main goal of the Master's project was to adapt this framework to 2D channelized petroleum reservoirs, composed of two types of rocks (permeable sand and background shale). For this purpose, we studied polynomial kernel Principal Components Analysis and proposed a novel analytical solution to the preimage problem. The experiments showed that our method surpasses the fixed-point iterative scheme, suggested in the literature, especially in terms of scalability and computational expense. Next, we paired the resulting KPCA parameterization with the Iterative Ensemble Smoother and the Subspace EnKF. Our comparative history matching experiment revealed that the latter is able to successfully avoid ensemble collapse. Finally, we suggested training set clustering as a means to accommodate the subspace parameterizations to the prior information and conducted a sensitivity study on the Subspace EnKF, which yielded encouraging results.Risk and Environmental ModellingApplied MathematicsElectrical Engineering, Mathematics and Computer Scienc

Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (F-AMS)

This paper introduces an Algebraic MultiScale method for simulation of flow in heteroge-neous porous media with embedded discrete Fractures (F-AMS). First, multiscale coarse grids are independently constructed for both porous matrix and fracture networks. Then, amap between coarse-and fine-scale is obtained by algebraically computing basis functions with local support. In order to extend the localization assumption to the fractured media, four types of basis functions are investigated: (1)Decoupled-AMS, in which the two media are completely decoupled, (2)Frac-AMS and (3)Rock-AMS, which take into account only one-way transmissibilities, and (4)Coupled-AMS, in which the matrix and fracture interpolators are fully coupled. In order to ensure scalability, the F-AMS framework permits full flexibility in terms of the resolution of the fracture coarse grids. Numerical results are presented for two-and three-dimensional heterogeneous test cases. During these experiments, the performance of F-AMS, paired with ILU(0) as second-stage smoother in a convergent iterative procedure, is studied by monitoring CPU times and convergence rates. Finally, in order to investigate the scalability of the method, an extensive benchmark study is conducted, where a commercial algebraic multigrid solver is used as reference. The results show that, given an appropriate coarsening strategy, F-AMS is insensitive to severe fracture and matrix conductivity contrasts, as well as the length of the fracture networks. Its unique feature is that a fine-scale mass conservative flux field can be reconstructed after any iteration, providing efficient approximate solutions in time-dependent simulations.Petroleum Engineerin

Adaptive Algebraic Multiscale Solver for Compressible Flow in Heterogeneous Porous Media

An adaptive Algebraic Multiscale Solver for Compressible (C-AMS) flow in heterogeneous oil reservoirs is developed. Based on the recently developed AMS [Wang et al., 2014] for incompressible linear flows, the C-AMS extends the algebraic formulation of the multiscale methods for compressible (nonlinear) flows. Several types of basis functions (incompressible and compressible with and without accumulation) are considered to construct the prolongation operator. As for the restriction operator, C-AMS allows for both MSFV and MSFE methods. Furthermore, to resolve high-frequency errors, Correction Functions and ILU(0) are considered. The best C-AMS procedure is determined among these various strategies, on the basis of the CPU time for three-dimensional heterogeneous problems. The C-AMS is adaptive in all aspects of prolongation, restriction, and conservative reconstruction operators for time-dependent compressible flow problems. In addition, it is also adaptive in terms of linear-system update. Though the C-AMS is a conservative multiscale solver (i.e., only a few iterations are employed infrequently in order to maintain high-quality results), a benchmark study is performed to investigate its efficiency against an industrial-grade Algebraic Multigrid (AMG) solver, SAMG [Stuben, 2010]. This comparative study illustrates that the C-AMS is quite efficient for compressible flow simulations in large-scale heterogeneous 3D reservoirs.Geoscience & EngineeringCivil Engineering and Geoscience

Projection-based Embedded Discrete Fracture Model (pEDFM)

This work presents a new discrete fracture model, namely the Projection-based Embedded Discrete Fracture Model (pEDFM). Similar to the existing EDFM approach, pEDFM constructs independent grids for the matrix and fracture domains, and delivers strictly conservative velocity fields. However, as a significant step forward, it is able to accurately model the effect of fractures with general conductivity contrasts relative to the matrix, including impermeable flow barriers. This is achieved by automatically adjusting the matrix transmissibility field, in accordance to the conductivity of neighboring fracture networks, alongside the introduction of additional matrix-fracture connections. The performance of pEDFM is investigated extensively for two- and three-dimensional scenarios involving single-phase as well as multiphase flows. These numerical experiments are targeted at determining the sensitivity of the model towards the fracture position within the matrix control volume, grid resolution and the conductivity contrast towards the matrix. The pEDFM significantly outperforms the original EDFM and produces results comparable to those obtained when using DFM on unstructured grids, therefore proving to be a flexible model for field-scale simulation of flow in naturally fractured reservoirs.Petroleum Engineerin

A copula-based sensitivity analysis method and its application to a North Sea sediment transport model

This paper describes a novel sensitivity analysis method, able to handle dependency relationships between model parameters. The starting point is the popular Morris (1991) algorithm, which was initially devised under the assumption of parameter independence. This important limitation is tackled by allowing the user to incorporate dependency information through a copula. The set of model runs obtained using latin hypercube sampling, are then used for deriving appropriate sensitivity measures. Delft3D-WAQ (Deltares, 2010) is a sediment transport model with strong correlations between input parameters. Despite this, the parameter ranking obtained with the newly proposed method is in accordance with the knowledge obtained from expert judgment. However, under the same conditions, the classic Morris method elicits its results from model runs which break the assumptions of the underlying physical processes. This leads to the conclusion that the proposed extension is superior to the classic Morris algorithm and can accommodate a wide range of use cases.Petroleum EngineeringApplied Probabilit