Hyper-Reduced-Order Models for Subsurface Flow Simulation

Chapter 9 - Fluid Flow Equations

It is becoming common to use automatic or assisted techniques for history matching of oil and gas reservoirs. These techniques are complicated by the need to update the properties of the model that are linked to the geological representation of the reservoir but also affect the misfit between observations and prediction. In this work we compare updating of regional properties for the former while updating local properties to satisfy the latter. We apply an automatic history matching approach to the Nelson field, a channelized turbidite reservoir, in order to better match the production rates of oil and water by well. We also integrate time lapse seismic data and we compare the result with production history matching output. We control the shale distribution within and between the major channel complexes by updating on the basis of geological environment variables which are regionally distributed. We compare this approach with more local updating by using pilot points with Kriginig. By updating the reservoir properties by environment variables we reduce the total production history match misfit by 22% for six years of data. We also improve the forecast of a further three years by 20%. By updating locally after by environment, this improvement is 60% and 30% in the matching and forecasting periods respectively. By integrating time lapse seismic data in history matching to update by environment, the production misfit is reduced by an equivalent amount while forecasting is improved by 19%. Seismically we reduce the misfit by 7%. Appropriate updating of the reservoir model is very important for reservoir management to reduce risk in development and to select appropriate models for forecasting. By updating the reservoir based on honouring the geological information or by using the geostatistical tools we can obtain a better representative of the reservoir with stronger prediction capability.

Reservoir characterization and flow prediction modeling form the fundamental basis of modern integrated Asset Management. As our oil industry evolves, increasingly sophisticated component technologies are being developed and applied in support of these characterization and modeling objectives. The oil industry is developing and embracing these new component technologies at a stunning pace - sometimes without full knowledge as to how each component might be consolidated into an overall solution to address real business issues. Forging through the maze of available technologies to identify a process which will address a specific business question is a challenge that requires several insights: an understanding of the business objective and risk, an understanding of specific details as to what each technology offers and, an understanding of how each technology can be linked to form an overall solution. This is a complex quest which often results in uncertainty within both company management and employees who are unclear as to how any individual technology will "pay out" or how a particular subprocess will be used as part of a whole to "make a difference". During ongoing efforts to evaluate new reservoir modeling technologies at Caltex, a team of Caltex specialists developed a combination work process/data flow model for reservoir modeling and characterization. The work/data process model definitions are expressed in the form of Reservoir Modeling Process (RMP) maps. These maps can be used to understand and define the overall objectives of an integrated team effort and accordingly allocate resources and project deliverables for Reservoir Characterization and Flow Modeling. The RMP process maps are scaleable in that various degrees of sophistication, and commensurate business risk-reduction, can be implemented. Examples of RMP application and scalability are offered from the historic use within Caltex CPI Indonesia. Current limitations and future evolution of the RMP process maps is also discussed.

A hybrid coupled model of surface and subsurface flow for surface irrigation

Chapter 14 - Reservoir Flow Simulation

Ensemble methods present a practical framework for parameter estimation, performance prediction, and uncertainty quantification in subsurface flow and transport modeling. In particular, the ensemble Kalman filter (EnKF) has received significant attention for its promising performance in calibrating heterogeneous subsurface flow models. Since an ensemble of model realizations is used to compute the statistical moments needed to perform the EnKF updates, large ensemble sizes are needed to provide accurate updates and uncertainty assessment. However, for realistic problems that involve large-scale models with computationally demanding flow simulation runs, the EnKF implementation is limited to small-sized ensembles. As a result, spurious numerical correlations can develop and lead to inaccurate EnKF updates, which tend to underestimate or even eliminate the ensemble spread. Ad hoc practical remedies, such as localization, local analysis, and covariance inflation schemes, have been developed and applied to reduce the effect of sampling errors due to small ensemble sizes. In this paper, a fast linear approximate forecast method is proposed as an alternative approach to enable the use of large ensemble sizes in operational settings to obtain more improved sample statistics and EnKF updates. The proposed method first clusters a large number of initial geologic model realizations into a small number of groups. A representative member from each group is used to run a full forward flow simulation. The flow predictions for the remaining realizations in each group are approximated by a linearization around the full simulation results of the representative model (centroid) of the respective cluster. The linearization can be performed using either adjoint-based or ensemble-based gradients. Results from several numerical experiments with two-phase and three-phase flow systems in this paper suggest that the proposed method can be applied to improve the EnKF performance in large-scale problems where the number of full simulation is constrained.

This report documents a project to improve subsea flow measurements to allowthe flow rates of individual wells to be known more accurately. Becauseproduction revenues are determined from flow rates, improved flow measurementreduces the financial risk to stakeholders, including producers and the U.S.government. Improved measurement also helps to improve reservoir recovery. Inthis project, commercially available flow models were evaluated for theirsuitability as virtual flow meters, to augment or replace hardware devices forthe determination of flow rates in deep water wells. The goal of this projectwas to develop methods for evaluating flow models, and to report on theefficacy of these models to predict flow rates. Several flow models were usedin the study, which also includes a survey of the available models. 1. Introduction 1.1. What is a Virtual Flow Meter? Virtual flow meters (VFM) are a method for determining the flow rate of a wellby means of modeling the flow. A VFM model may consist of a single well to anentire field of co-mingled wells. The models are fed data from sensorsinstalled at various measurement points (nodes). Typically pressure andtemperature sensors are used, which are available downhole (perhaps at multiplepoints), at the wellhead, and at the surface. With the pressure and temperaturedata at each node provided as an input, the VFM model computes the flow rate ofthe well. Typically, the model is tuned periodically using available flow data. Once tuned, the model is capable of determining the well flow rate with onlythe pressure and temperature nodal data. The flow models employed with VFMsystems may be built from first principles, but this is not the only method forVFM modeling. Flow models may also be derived from statistical analysis ofmeasurements acquired at different nodes along the production system. An important observation is that VFM software is not something that must beinstalled at the same time as the physical multiphase flow meters, as long ascare is taken to install proper sensors to be used at the various nodes. Thefault tolerant nature of the VFM system is an advantage, since a failed "hard"sensor will reduce the number of nodes, but will not cause a complete loss ofinformation. While VFM technology appears to be maturing rapidly, it is not yet generallyaccepted an equivalent replacement for good physical metering. VFM models areemployed as a back up to a physical meter to be used in case the primarymetering system is unable to function, or as an augmentation of physicalmetering to reduce measurement uncertainty. Some of the commercial VFMs use the technique of Nodal Analysis for the flowrate prediction. This method utilizes available transmitters (pressure andtemperature) and a mathematical model of the complete flow path from bottomhole to the topside facility. This method requires considerable input data, including well and pipeline profiles, the diameters of all components ofsections, sensor locations, fluid properties, choke geometry, etc.

I. Principles, Basic Equations and Analytical Solutions.- Review of stochastic theory of transport in groundwater flow.- On the identification of the parameters of groundwater mass transport.- Heat and mass transfer in unsaturated porous media with application to thermal energy storage.- Basin-scale transport of dissolved species in groundwater.- Variable density fluid flow in the brackish transition zone between fresh and saline groundwater.- Stationary principles for flow and transport in aquifers.- II. Modelling Flow and Transport in Porous-Like Media.- An overwiew of groundwater modelling.- Incorporating assurance into groundwater quality management models.- The eigenvalues approach for solving linear groundwater flow problems.- New method for diffusive transport.- Comparison of fast equation solvers for groundwater flow problems.- Modelling flow and transport through porous media in vector computers.- Eulerian-Lagrangian method for solving transport in aquifers.- Hydrodynamic dispersion in model porous media.- The random walk method in pollutant transport simulation.- Modeling of solute transport with the random walk method.- III. Modelling Flow and Transport in Fractured Media.- Modelling of flow through fractured rocks: geostatistical generation of fractures networks. Stress-flow relationship in fractures.- Combined seismic and hydraulic method of modeling flow in fractured low permeability rocks.- Stochastic continuum representation of fractured rock permeability as an alternative to the REV and fracture network concepts.- Flow in three-dimensional fracture networks using a discrete approach.- MINC: an approach for analyzing transport in strongly heterogeneous systems.- A stochastic particle transport model based on directional statistics of flow through fracture networks.- IV. Consideration of Fluid-Solid Phase Interactions and Heterogeneities in Modelling.- Advances in modelling water-rock interaction in aquifers.- Parameters for modelling the transport of cadmium as influenced by the chemical properties of groundwater and aquifer material.- Micro-scale modelling in the study of plume evolution in heterogeneous media.- Modelling the increasing dispersivity with FE transport models using the multilayer concept.- Random-walk method to simulate pollutant transport in alluvial aquifers or fractured rocks.- V. Multiphase Flow and Transport Modelling.- Advances in modeling of water in the unsaturated zone.- FLuid-mechanical aspects of the migration of chemicals in fractured media.- About the numerical analysis of dynamics in multi-component-continua.- VI. Aquifer Parameter Identification by Models.- State-of-the-art of the inverse problem applied to the flow and solute transport equations.- Estimation of spatial covariance structures with application to hydrological, hydro-chemical and isotopic data from aquifers: state-of-the-art and adjoint state maximum likelihood cross-validation methods.- Bayesian identification of steady-state (anisotropic) groundwater flow models.- VII. Data Gathering and Utilization of Models.- Present limitations and perspectives on modelling pollution problems in aquifers.- Advances in the assessment of data worth for engineering decision analysis in groundwater contamination problems.- A comparative analysis of mathematical mass transport codes for groundwater pollution studies.- New approaches and applications in subsurface flow modeling: 3-D finite element analysis of dewatering for an electro-nuclear plant.- Large scale 3-D groundwater flow modeling in highly heterogeneous geologic medium.- Gathering of data for modelling.- Present status of coastal aquifer modelling: short review.- Conclusions.- Transport modeling.- Solution methods in groundwater flow and transport.- Transport processes in groundwater: chemical and biological aspects.- Application of models.- Author Index.- List of Participants.

Linearized reduced-order models for subsurface flow simulation

Understanding soil water, groundwater, and shallow surface water dynamics as an integrated hydrological system is critical for understanding the Earth’s critical zone, the thin outer layer at our planet’s surface where vegetation, soil, rock, and gases interact to regulate the environment. Computational tools that take this view of soil moisture and shallow surface flows as a single integrated system are typically referred to as integrated surface/subsurface hydrology models. We extend the open-source, highly parallel, subsurface flow and reactive transport simulator PFLOTRAN to accommodate surface flows. In contrast to most previous implementations, we do not represent a distinct surface system. Instead, the vertical gradient in hydraulic head at the land surface is neglected, which allows the surface flow system to be eliminated and incorporated directly into the subsurface system. This tight coupling approach leads to a robust capability and also greatly simplifies implementation in existing subsurface simulators such as PFLOTRAN. Successful comparisons to independent numerical solutions build confidence in the approximation and implementation. Example simulations of the Walker Branch and East Fork Poplar Creek watersheds near Oak Ridge, Tennessee demonstrate the robustness of the approach in geometrically complex applications. The lack of a robust integrated surface/subsurface hydrology capability had been a barrier to PFLOTRAN’s use in critical zone studies. This work addresses that capability gap, thus enabling PFLOTRAN as a community platform for building integrated models of the critical zone.

In physical modeling of phonation, the pressure drop along the glottal constriction is classically assessed with the glottal geometry and the subglottal pressure as known input parameters. Application of physical modeling to study phonation abnormalities and pathologies requires input parameters related to in vivo measurable quantities commonly corresponding to the physical model output parameters. Therefore, the current research presents the inversion of some popular simplified flow models in order to estimate the subglottal pressure, the glottal constriction area, or the separation coefficient inherent to the simplified flow modeling for steady and unsteady flow conditions. The inverse models are firstly validated against direct simulations and secondly against in vitro measurements performed for different configurations of rigid vocal fold replicas mounted in a suitable experimental setup. The influence of the pressure corrections related to viscosity and flow unsteadiness on the flow modeling is quantified. The inversion of one-dimensional glottal flow models including the major viscous effects can predict the main flow quantities with respect to the in vitro measurements. However, the inverse model accuracy is strongly dependent on the pertinence of the direct flow modeling. The choice of the separation coefficient is preponderant to obtain pressure predictions relevant to the experimental data.

Flow simulation for a horizontal well with slotted screen and ICD completions based on the wellbore–annulus–reservoir model

An increasing number of field development projects include rigorous uncertainty quantification workflows based on parameterized subsurface uncertainties. Reservoir model calibration workflows for reservoir simulation models including historical production data, also called history matching, deliver non-unique solutions and remain technically challenging. In addition, the validation process of the reservoir simulation model often introduces a break of the conceptual connection to the geological model. This raises questions on how to quantify the deviation between the calibrated simulation model and the original geological model. Workflow designs for history matching require scalable and efficient optimization techniques to address project needs. Derivative-free techniques like Markov Chain Monte Carlo (MCMC) are used for optimization and uncertainty quantification. Adjoint techniques derive analytical sensitivities directly from the flow equations. For history matching those sensitivities are efficiently used for property updates on grid block level. Both techniques have different characteristics and support alternative history matching strategies like global vs. local, stochastic vs. deterministic. In this work both techniques will be applied in an integrated workflow design to the Norne field. The Norne field is a North Sea oil-and-gas reservoir with approximately 30 wells, with one third being used for WAG injection for pressure support. Field data was previously released by Statoil and made available for a public benchmark study (NTNU Norway) testing history matching techniques including production and time-lapsed seismic data. We focus on well production data for history matching. MCMC is used for global parameter updates and uncertainty quantification in a Bayesian context. An implementation of an adjoint technique is applied for analytical sensitivity calculations and local parameter adjustments of rock properties. History matching results are presented for field wide and well-by-well production data. Consistency checks between updated and original geological model are presented for rock property distribution maps. Geostatistical measures including spatial correlations are used to quantify deviations between updated and original geological model. In conclusion scalability and performance efficiency of the practical workflow implementation is discussed with a perspective of a consistent feedback loop from history matching to geological modeling.

History matching and uncertainty quantification in subsurface flow settings typically require large numbers of flow simulations. The use of surrogate models in place of high-fidelity simulations can lead to substantial reductions in computational cost. In this chapter, we discuss deep neural network surrogate models in detail. Two categories of deep-learning models, namely physics-informed and data-driven methods, are considered. One particular data-driven approach, the recurrent residual U-Net, is described and applied for an example involving oil-water flow in 3D channelized geomodels. The accuracy of this surrogate model for flow predictions with new realizations is demonstrated, and the method is then used in rejection-sampling-based history matching. The chapter concludes with a discussion of potential directions for future research on deep-learning surrogate models.

Summary In the field of subsurface flow simulation, deep-learning-based surrogate modeling is shown as a promising approach to significantly reduce the computational cost associated with full-physics reservoir simulations. However, the successful construction of highly accurate deep-learning-based models often requires a large number of training simulations, which can be very time-consuming to generate by itself for large-scale systems. The incorporation of physics constraints during the training process was shown to be an effective approach to reduce the training cost and to improve model accuracy. However, it remains challenging to reimplement the physics constraints correctly for modeling training when it comes to reservoir models with complex physics or gridding methods. In this study, we propose a physics-constrained surrogate (PCS) model training approach for production optimization that integrates multifidelity training data, together with a new way of implementing the physics constraints that leverages an existing mature reservoir simulator. The proposed approach starts with model pretraining on a relatively large number of low-cost, low-fidelity training data, followed by model fine-tuning with a smaller number of high-fidelity training samples and the inclusion of physics constraints. The physics constraints are implemented by adding the residuals of discretized governing equations into the loss function. Training with multifidelity data and incorporating physics constraints allows for reduced reliance on high-fidelity data while enhancing physical consistency. Systematic comparison studies were performed for both 2D and 3D cases, and it was shown that the proposed approach can reduce the computational costs associated with training data generation by about 80%, while achieving a similar level of prediction accuracy of the surrogate model. In addition, under a small number of training simulations (i.e., 50 equivalent high-fidelity runs), our proposed PCS model with multifidelity data can reduce the prediction error by 90%, in comparison with the model trained with only high-fidelity data. Finally, the trained surrogate model was applied to a well-control optimization problem. In comparison with the use of full-order simulations, the total computational time can be reduced by 97.7%.