The Fragmentation in a Protoplanetary Disk and the Properties of the Resultant Self-Gravitating Clumps

We study the fragmentation properties in the protoplanetary disk and properties of the resultant self-gravitating clumps using our newly constructed disk model. Our disk model includes the mass inflall term from a molecular cloud core and the photoevaporation winds effect. We adopt the conventional fragmentation criterion to judge whether a protoplanetary disk can fragment. In this work, we follow our previous work to investigate the properties of the resultant self-gravitating clumps. In our calculation, the initial masses of the resultant self-gravitating clumps lie in the range of tens of M_J to more than one hundred of M_J, where M_J is the Jupiter mass. These initial masses can seemingly account for the masses of extrasolar planets in magnitude. We also calculate the subsequent gas accretion of clumps in 1.27 × 10⁴ yr after the formation of self-gravitating clumps. We find that the subsequent gas accretion of self-gravitating clumps is very efficient, and the clump masses grow to hundreds of M_J and the physical radii R_c of clumps increase to about 10 AU. Additionally, we also calculate the orbital migration of clumps. We find that most clumps have short migration timescale to be accreted onto the protostar, and only a small fraction of clumps have long migration timescale (>10⁶ yr) to successfully become gas giant planets. These results are consistent with previous studies.     

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.     

High-dimensional geostatistical history matching

Hydrocarbon reservoir modelling and characterisation is a challenging subject within the oil and gas industry due to the lack of well data and the natural heterogeneities of the Earth’s subsurface. Integrating historical production data into the geo-modelling workflow, commonly designated by history matching, allows better reservoir characterisation and the possibility of predicting the reservoir behaviour. We present herein a geostatistical-based multi-objective history matching methodology. It starts with the generation of an initial ensemble of the subsurface petrophysical property of interest through stochastic sequential simulation. Each model is then ranked according the match between its dynamic response, after fluid flow simulation, and the observed available historical production data. This enables building regionalised Pareto fronts and the definition of a large ensemble of optimal subsurface Earth models that fit all the observed production data without compromising the exploration of the uncertainty space. The proposed geostatistical multi-objective history matching technique is successfully implemented in a benchmark synthetic reservoir dataset, the PUNQ-S3, where 12 objectives are targeted.     

Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain, taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast dependence in the convergence. The method’s convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.     

Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cell-centred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundary conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required.     

Ensemble clustering for efficient robust optimization of naturally fractured reservoirs

In the development of naturally fractured reservoirs (NFRs), the existence of natural fractures induces severe fingering and breakthrough. To manage the flooding process and improve the ultimate recovery, we propose a numerical workflow to generate optimal production schedules for smart wells, in which the inflow control valve (ICV) settings can be controlled individually. To properly consider the uncertainty introduced by randomly distributed natural fractures, the robust optimization would require a large ensemble size and it would be computationally demanding. In this work, a hierarchical clustering method is proposed to select representative models for the robust optimization in order to avoid redundant simulation runs and improve the efficiency of the robust optimization. By reducing the full ensemble of models into a small subset ensemble, the efficiency of the robust optimization algorithm is significantly improved. The robust optimization is performed using the StoSAG scheme to find the optimal well controls that maximize the net-present-value (NPV) of the NFR’s development. Due to the discrete property of a natural fracture field, traditional feature extraction methods such as model-parameter-based clustering may not be directly applicable. Therefore, two different kinds of clustering-based optimization methods, a state-based (e.g., s _w profiles) clustering and a response-based (e.g., production rates) clustering, are proposed and compared. The computational results show that the robust clustering optimization could increase the computational efficiency significantly without sacrificing much expected NPV of the robust optimization. Moreover, the performance of different clustering algorithms varies widely in correspondence to different selections of clustering features. By properly extracting model features, the clustered subset could adequately represent the uncertainty of the full ensemble.     

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

In this paper, we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper, we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regard to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular, we look at how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.     

A numerical method for the solution of the nonlinear turbulent one-dimensional free surface flow equations

Free surface flow of an incompressible fluid over a shallow plane/undulating horizontal bed is characteristically turbulent due to disturbances generated by the bed resistance and other causes. The governing equations of such flows in one dimension, for finite amplitude of surface elevation over the bed, are the Continuity Equation and a highly nonlinear Momentum Equation of order three. The method developed in this paper introduces the “discharge” variable q = η U, where η = elevation of the free surface above the bed level, and U = average stream-wise forward velocity. By this substitution, the continuity equation becomes a linear first-order PDE and the momentum equation is transformed after introduction of a small approximation in the fifth term. Next, it is shown by an invertibility argument that q can be a function of η: q = F(η), rendering the momentum equation as a first order, second degree ODE for F(η), that can be be integrated by the Runge-Kutta method. The continuity equation then takes the form of a first order evolutionary PDE that can be integrated by a Lax-Wendroff type of scheme for the temporal evolution of the surface elevation η. The method is implemented for two particular cases: when the initial elevation is triangular with vertical angle of 120 ^° and when it has a sinusoidal form. The computations exhibit the physically interesting feature that the frontal portion of the propagating wave undergoes a sharp jump followed by tumbling over as a breaker. Compared to other discretization methods, the application of the Runge-Kutta and an extended version of the Lax-Wendroff scheme is much easier.     

Identifying influence areas with connectivity analysis – application to the local perturbation of heterogeneity distribution for history matching

Numerical representations of a target reservoir can help to assess the potential of different development plans. To be as predictive as possible, these representations or models must reproduce the data (static, dynamic) collected on the field. However, constraining reservoir models to dynamic data – the history-matching process – can be very time consuming. Many uncertain parameters need to be taken into account, such as the spatial distribution of petrophysical properties. This distribution is mostly unknown and usually represented by millions of values populating the reservoir grid. Dedicated parameterization techniques make it possible to investigate many spatial distributions from a small number of parameters. The efficiency of the matching process can be improved from the perturbation of specific regions of the reservoir. Distinct approaches can be considered to define such regions. For instance, one can refer to streamlines. The leading idea is to identify areas that influence the production behavior where the data are poorly reproduced. Here, we propose alternative methods based on connectivity analysis to easily provide approximate influence areas for any fluid-flow simulation. The reservoir is viewed as a set of nodes connected by weighted links that characterize the distance between two nodes. The path between nodes (or grid blocks) with the lowest cumulative weight yields an approximate flow path used to define influence areas. The potential of the approach is demonstrated on the basis of 2D synthetic cases for the joint integration of production and 4D saturation data, considering several formulations for the weights attributed to the links.