Fast flow computation methods on unstructured tetrahedral meshes for rapid reservoir modelling

SUMMARY We present a new numerical method to solve the heterogeneous anelastic seismic wave equations with arbitrary high order of accuracy in space and time on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. Using the velocity–stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method relies on the finite volume (FV) approach where cell-averaged quantities are evolved in time by computing numerical fluxes at the element interfaces. The basic ingredient of the numerical flux function is the solution of Generalized Riemann Problems at the element interfaces according to the arbitrary high-order derivatives (ADER) approach of Toro et al., where the initial data is piecewise polynomial instead of piecewise constant as it was in the original first-order FV scheme developed by Godunov. The ADER approach automatically produces a scheme of uniformly high order of accuracy in space and time. The high-order polynomials in space, needed as input for the numerical flux function, are obtained using a reconstruction operator acting on the cell averages. This reconstruction operator uses some techniques originally developed in the Discontinuous Galerkin (DG) Finite Element framework, namely hierarchical orthogonal basis functions in a reference element. In particular, in this article we pay special attention to underline the differences as well as the points in common with the ADER-DG schemes previously developed by the authors, especially concerning the MPI parallelization of both methods. The numerical convergence analysis demonstrates that the proposed FV schemes provide very high order of accuracy even on unstructured tetrahedral meshes while computational cost for a desired accuracy can be reduced when applying higher order reconstructions. Applications to a series of well-acknowledged elastic and anelastic test cases and comparisons with analytic and numerical reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-FV approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

This reference is for an abstract only. A full paper was not submitted for this conference. Abstract The Khuff Formation is a major reservoir in Qatar and the rest of the Middle East Gulf region, and contains some of the world's biggest gas reserves. Geologically the carbonate platform system is complex with fine-scaleheterogeneities which impact the static and dynamic reservoir characteristics. These heterogeneities are depositional, stratigraphic and diagenetic in nature and therefore reservoir characterization needs to be an integrated multidisciplinary and multiscale approach. In order to address these issues a large multidisciplinary and multiscale subsurface study has been launched on a large database including fields from across the region. The aim is to better understand the facies distributions, sequence stratigraphic architecture and the regional reservoir development, in order to better characterize dynamically reservoir behavior. Geologically-based fine-scale reservoir characterization is needed in all field optimization projects' and this starts at the appraisal phase of field evaluations, and not just in mature field scenarios. This necessitates an "anticipation" of the kind of field optimization issues that will be eventually needed. It is also important to think of geological reservoir characterization in dynamic terms at the earliest stage possible, but this must always be underpinned by a strong geological framework. This is achieved by calibration of various dynamic data (well tests, production profiles and well interference tests) with the geological understanding, particularly the sedimentological and diagenetic reservoir characterization. The static reservoir model in Khuff reservoirs is highly dependent on the precise type of sedimentary facies, the stratigraphic context and the type of diagenetic overprint, since these control the fine-scale heterogeneity. However, the dynamic reservoir heterogeneity (characterized by the dynamic data available) is particularly sensitive to the diagenetic overprint (i.e. the development of dolomite in grain-supported facies), and it is seen that the fine-scale permeability modeling and 3D permeability architecture is highly diagenetically influenced and fundamental to understanding the dynamic behavior of the field. Moreover the understanding of the 3D permeability architecture(particularly the high permeability zones around the dolomite-limestone transitions) is important for the application of stimulation techniques, such as acidification. The integration of the fine-scale reservoir characterization into the reservoir model is a key step, but governed by the precise objectives of the reservoir model. In particular the resolution of model must be appropriate to the reservoir model objectives (large scale pore volume calculations vs. fine-scale dynamic sensitivity to geological heterogeneity, and adapted if necessary). In-house reservoir-scale modeling tools have been used to model the fine-scale reservoir heterogeneity, and create multiple realizations of the static reservoir properties. Finally these are modeled dynamically to assess the impact of the geological parameters on the production profiles.

Using commercial numerical reservoir simulators to build a full field reservoir model and simultaneously history match multiple dynamic variables for a highly complex, offshore mature field in Malaysia, had proven to be challenging, manpower intensive, highly expensive, and not very successful. This field includes almost two hundred wells that have been completed in more than 60 different, non-continuous reservoir layers. The field has been producing oil, gas and water for decades. The objective of this article is to demonstrate how Artificial Intelligence (AI) and Machine Learning is used to build a purely data-driven reservoir simulation model that successfully history match all the dynamic variables for all the wells in this field and subsequently used for production forecast. The model has been validated in space and time. The AI and Machine Learning technology that was used to build the dynamic reservoir simulation and modeling is called spatio-temporal learning. Spatio-temporal learning is a machine-learning algorithm specifically developed for data-driven modeling of the physics of fluid flow through porous media. Spatio-temporal learning is used in the context of Deconvolutional Neural Networks. In this article Spatio-temporal Learning and Deconvolutional Neural Networks will be explained. This new technology is the result of more than 20 years of research and development in the application of AI and Machine Learning in reservoir modeling. This technology develops a coupled reservoir and wellbore model that for this particular oil & gas field in Malaysia uses choke setting, well-head pressure and well-head temperature as input and simultaneously history matches Oil production, GOR, WC, reservoir pressure, and water saturation for more than a hundred wells through a completely automated process. Once the data-driven reservoir model is developed and history matched, it is blind validated in space and time in order to establish a reliable and robust reservoir model to be used for decision making purposes and opportunity generation to maximise the field value. The concepts and the methodology of history match of multiple wells, individual offshore platforms, and the entire field will be presented in this article along with the results of blind validation and production forecasting. Results of using this model to perform uncertainty quantification will also be presented. A case study of a highly complex mature field with large number of wells and years of production has been used to be studied and simulated by this data-driven approach. Time, efforts, and resources required for the development of the dynamic reservoir simulation models using AI and Machine Learning is considerably less than time and resources required using the commercial numerical simulators. It is validated that the TDM developed model can make very reasonable prediction of field behavior and production from the existing wells based on modification of operational constraints and can be a prudent complementary tool to conventional numerical simulators for such complex assets.

A new modelling framework is presented for application to a range of three‐dimensional (3D) multi‐scale oceanographic problems. The approach is based upon a finite element discretization on an unstructured tetrahedral mesh which is optimized to represent highly complex geometries. Throughout a simulation the mesh is dynamically adapted in 3D to optimize the representation of evolving solution structures. The adaptive algorithm makes use of anisotropic measures of solution complexity and a load‐balanced parallel mesh optimization algorithm to vary resolution and allow long, thin elements to align with features such as boundary layers. The modelling framework presented is quite different from the majority of ocean models in use today, which are typically based on static‐structured grids. Finite element (and volume) methods on unstructured meshes are, however, gaining popularity in the oceanographic community. The model presented here is novel in its use of unstructured meshes and anisotropic adaptivity in 3D, its ability to represent a range of coupled multi‐scale solution structures and to simulate non‐hydrostatic dynamics. Copyright © 2007 John Wiley & Sons, Ltd.

This paper expounds the value of integrated decision based planning in delivering field development plan (FDP) in a LEAN way. The basic philosophy of lean is waste minimization or elimination of non-value adding activities to improve efficiency, quality and lead-time. Integrated decision based planning is considered as the most pragmatic and efficient approach in integrated reservoir modelling process. Dynamic modeling is the most preferred tool assisting subsurface decisions making. Nevertheless, the data centric approach with support of scaled reservoir model has the advantage over the conventional full field physics-based models specially, in case of a complex reservoir. Embracing the basic lean principles with focused decision based reservoir modeling strategy can establish a new level of performance within the organization in delivering FDPs. The Saih Rawl oil field (SRS) in North Oman is a thin Lower Cretaceous carbonate reservoir with post diagenetic imprint. Post oil fill structural change has resulted in re-saturation and oil trapping due to local capillary imbibitions. The complexities resulting from the tilted water contact, hysteresis in oil mobility and Sor variation with depth, pose a huge challenge in dynamic simulation. In addition, drilling feasibility for horizontal infill wells was quite challenging due to subsurface collision issues and rig footprint interference with existing surface facilities. Integrated decision based planning, linked to the subsurface and surface decisions was adopted for framing the integrated reservoir modeling (IRM) strategy. The IRM strategy with Decision Based Models (DBM), including analytical and sector simulation models, were used to understand the sweep pattern, locate the remaining oil and rank the various water-flood patterns. Data analysis including normalized decline-curve-analysis (DCA) based conduit models and comprehensive field performance analysis using Spotfire (an integrated data visualization and analysis tool by TIBCO), was used to understand the key reservoir management risks and infill potential. Throughout the process, the basic philosophy of lean was adopted embracing several lean tools to improve productivity, quality and lead-time. Out of 12 subsurface feasible options studied, the proposed optimum option envisages an increase in the oil recovery factor by 9% by drilling an additional 92 infill wells in 22 patterns. The successful completion of frontend loading phase of SRS project has achieved reducing in the FDP study time to 19 months compared to an average of 36 months in the past and project implementation 4 years ahead of the original plan. Fast tracking of the project implementation was possible due to standardization of the equipment, maximum utilization of the existing infrastructure and constructive collaboration with the stakeholders. The key enablers for the successful delivering of the SRS FDP study were mainly the integrated decision based planning with data centric approach in reservoir modeling workflow and adoption of basic lean principles This approach emphasizes the importance of adopting lean tools in frontend delivery process. The decision based planning with reservoir models linked to the project decision can significantly improve the efficiency and quality of the FDP. The stakeholder alignment and strong collaboration with key stakeholders of the project can further reduce the lead-time of project execution. The decision based IRM planning used for this study sets a benchmark for future FDP studies. The Urban Plan study approach for this project has also become the standard for other LEAN FDPs.

This paper presents a parallelized version of the edge-based finite element method with a novel post-processing approach for numerical modeling of an electromagnetic field in complex media. The method uses an unstructured tetrahedral mesh which can reduce the number of degrees of freedom significantly. The linear system of finite element equations is solved using parallel direct solvers which are robust for ill-conditioned systems and efficient for multiple source electromagnetic (EM) modeling. We also introduce a novel approach to compute the scalar components of the electric field from the tangential components along each edge based on field redatuming. The method can produce a more accurate result as compared to conventional approach. We have applied the developed algorithm to compute the EM response for a typical 3D anisotropic geoelectrical model of the off-shore HC reservoir with complex seafloor bathymetry. The numerical study demonstrates that the modeling algorithm is capable of simulating the complex topography and bathymetry that is commonly encountered in controlled source electromagnetic problems.

The volume of fluid (VoF) method is widely used in multi-phase flow simulations to track and locate the interface between two immiscible fluids. A major bottleneck of the VoF method is the interface reconstruction step due to its high computational cost and low accuracy on unstructured grids. We propose a machine learning enhanced VoF method based on Graph Neural Networks (GNN) to accelerate the interface reconstruction on general unstructured meshes. We first develop a methodology to generate a synthetic dataset based on paraboloid surfaces discretized on unstructured meshes. We then train a GNN based model and perform generalization tests. Our results demonstrate the efficiency of a GNN based approach for interface reconstruction in multi-phase flow simulations in the industrial context.

In controlled-source electromagnetic (CSEM) inversion with conventional regularization, the reconstructed conductivity image is usually blurry and only has limited resolution. To effectively obtain more compact conductivity models, we apply the concept of multinary transformation to CSEM inversion based on the finite-element method with an unstructured tetrahedral mesh. Within the framework of multinary inversion, the model conductivities are allowed to be clustered only within the designed values, which are usually obtained from other a priori information or from conventional inversion. Synthetic studies show that multinary inversion produces conductivity images with clearer model boundaries compared to the maximum smoothness inversion and the focusing inversion for realistic geoelectric models. We further apply the developed method to a land CSEM survey for mineral exploration. The multinary inversion results are closer to the ground truth compared to the conventional maximum smoothness inversion and the focusing inversion. The developed method and the numerical algorithm provide a new approach and workflow for CSEM inversion when the models need to have clear boundaries and clustering model values. Such geoelectric models could be very useful for geologic interpretation in oil and mineral exploration when a priori information (such as the estimated conductivity values) of the exploration targets is known.

Multi-objective unstructured triangular mesh generation for use in hydrological and land surface models

Detailed geologic studies have become increasingly important especially for secondary- and tertiary-recovery projects. Success or failure of such projects is strongly dependent on a proper understanding of reservoir heterogeneity. In virtually every case, the reservoir has been found to be more complex than originally envisaged. In some reservoirs, complexity has cast serious doubts on viability of the project as initially formulated. With development of modern computers capable of handling much larger numbers of grid blocks, different scales of heterogeneity can be simulated realistically. It is possible to simulate complex fluvial channel architecture over an interval of several hundred feet, as well as the effect on reservoir performance of crossbedding of meter scale or less. Today's reservoir geologist is faced with the challenge of quantifying his conceptual sedimentologic models more precisely. At higher levels of reservoir heterogeneity, probabilistic techniques to predict reservoir configuration between available wells appear promising. Reservoir modeling of individual sandstone bodies is dependent on modern analogs or outcrop-derived prototypes. Textural criteria should be taken into account in the selection of prototypes, as grain size ultimately controls shale barrier distribution and sedimentary structures. Cementation is an important form of reservoir heterogeneity commonly neglected in reservoir modeling. In the case of small-scale reservoir heterogeneity as manifested in cores, carefully planned core analysis programs are required to measure th potential anisotropy caused by sedimentary structures and to determine accurately in-situ properties resulting from clay-mineral diagenesis or overburden stress. End_of_Article - Last_Page 251------------

Parallel adaptive mesh-refining scheme on a three-dimensional unstructured tetrahedral mesh and its applications

Solving the Poisson equation on small aspect ratio domains using unstructured meshes

Computer numerical simulation is a beneficial tool for studying various domains of knowledge. Among the steps in the whole process of numerical simulation is the generation of unstructured meshes. Since the unstructured meshes are usually generated using automatic software, the fundamental knowledge of the unstructured meshes is often neglected. This paper highlighted some useful insights into the unstructured meshes in numerical simulation for several application domains, such as the radiative heat transfer problem, ocean modelling and biomedical engineering. It also reviewed some fundamental concepts and frameworks for element generation in producing unstructured meshes, particularly the Delaunay triangulation and advancing front techniques.

An algebraic turbulence model based on the Baldwin-Lomax model, has been implemented for use on unstructured grids. The implementation is based on the use of local background structured turbulence meshes. At each time-step, flow variables are interpolated from the unstructured mesh onto the background structured meshes, the turbulence model is executed on these meshes, and the resulting eddy viscosity values are interpolated back to the unstructured mesh. Modifications to the algebraic model were required to enable the treatment of more complicated flows, such as confluent boundary layers and wakes. The model is used in conjuction with an efficient unstructured multigrid finite-element Navier-Stokes solver in order to compute compressible turbulent flows on fully unstructured meshes. Solutions about single and multiple element airfoils are obtained and compared with experimental data.

This paper reports on the Beryl petroleum formation, the primary reservoir in the Beryl field, which has a complicated distribution of pressures and fluids controlled by horizontal and vertical permeability restrictions. Horizontal permeability restrictions, the result of extensive faulting, divide the reservoir into eight interrelated areas as determined by careful analyses of pressure and production histories and fluid monitoring. Vertical permeability restrictions, which are the consequence of thin lithologic breaks within the massive sandstone sequence, further divide the eight reservoir areas into seven layers, as evidenced by careful review of repeat formation tester (RFT) data, pressure-transient tests, cased-hole logs, and reservoir performance data. A detailed reservoir model identified areas and intervals of unswept hydrocarbons, resulting in an aggressive workover and drilling program. The model was an essential component in the success of the subsequent reservoir simulation.