Adaptive Multiscale Streamline Simulation and Inversion for High-Resolution Geomodels

A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids

Conventional finite-element method for solving Helmholtz equation on highly heterogeneous media usually require very finely discretized mesh to accurately represent the medium property variations, and therefore is computationally expensive. We present a continuous Galerkin generalized multiscale finite element method for solving Helmholtz equation in media with variable velocity and mass density on coarse mesh. Instead of using polynomial basis functions, our multiscale method use multiscale functions obtained by multiplying the eigenfunctions from local spectral problems with an appropriately selected multiscale partition of unity to form the approximation space on the coarse mesh. These multiscale basis can efficiently capture the information of fine scale features of the medium without solving the problem on the fine mesh. We show by a numerical example that our multiscale method can greatly reduce the dimension of the coefficient matrix of Helmholtz equation system, as well as the computational time, with very small error.

Streamline-based assisted and automatic history matching techniques have shown great potential in reconciling high resolution geologic models to production data. However, a major drawback of these approaches has been incompressibility or slight compressibility assumptions that have limited applications to two-phase water-oil displacements only. We propose an approach to history matching three-phase flow using a novel compressible streamline formulation and streamline-derived analytic sensitivities. First, we utilize a generalized streamline model to account for compressible flow by introducing an 'effective density' of total fluids along streamlines. Second, we analytically compute parameter sensitivities that define the relationship between the reservoir properties and the production response, viz. water-cut and gas/oil ratio (GOR). These sensitivities are an integral part of history matching, and streamline models permit efficient computation of these sensitivities through a single flow simulation. We calibrate geologic models to production data by matching the water-cut and gas/oil ratio using our previously proposed generalized travel time inversion (GTTI) technique. For field applications, however, the highly non-monotonic profile of the gas/oil ratio data often presents a challenge to this technique. In this work we present a transformation of the field production data that makes it more amenable to GTTI. Further, we generalize the approach to incorporate bottom-hole flowing pressure during three-phase history matching. We examine the practical feasibility of the method using a field-scale synthetic example (SPE-9 comparative study) and a field application. Recently Ensemble Kalman Filtering (EnKF) has gained increased attention for history matching and continuous reservoir model updating using data from permanent downhole sensors. It is a sequential Monte-Carlo approach that works with an ensemble of reservoir models. Specifically, the method utilizes cross-covariances between measurements and model parameters estimated from the ensemble. For practical field applications, the ensemble size needs to be kept small for computational efficiency. However, this leads to poor approximations of the cross-covariance matrix, resulting in loss of geologic realism. Specifically, the updated parameter field tends to become scattered with a loss of connectivities of extreme values such as high permeability channels and low permeability barriers, which are of special significance during reservoir characterization. We propose a novel approach to overcome this limitation of the EnKF through a 'covariance localization' method that utilizes sensitivities that quantify the influence of model parameters on the observed data. These sensitivities are used in the EnKF to modify the cross-covariance matrix in order to reduce unwanted influences of distant observation points on model parameter updates. The key to the success of the sensitivity-based covariance-localization is its close link to the underlying physics of flow compared to a simple distance-dependent covariance function as used in the past. This flow-relevant conditioning leads to an efficient and robust approach for history matching and continuous reservoir model updating, avoiding much of the problems in traditional EnKF associated with instabilities, parameter overshoots and loss of geologic continuity. We illustrate the power and utility of our approach using both synthetic and field applications.

As the role of reservoir flow simulation increasingly impacts existing operations and field development decisions, it follows that rigor, fitness and consistency should be imposed on the calibration of reservoir flow models to dynamic data through history matching. Although a wealth of history matching techniques exist in the petroleum literature that propose novel algorithms or share case studies, seldom does the content guide the modeler in fit-for-purpose reservoir model calibration for an operating asset. To evaluate the applicability of these diverse techniques against standards required for reservoir management, an internal study was performed to benchmark four assisted history matching (AHM) techniques commonly promoted in the oil and gas industry. The techniques were vetted against a comprehensive suite of modeling requirements for multiple asset classes, integrating a variety of historical dynamic data types through the calibration of reservoir properties that control flow behavior from the field to inter-well scale. The methods benchmarked were: (1) Design of Experiments (DoE)-based, (2) Ensemble Kalman Filter and Ensemble Smoother, (3) Genetic Algorithm and (4) Generalized Travel Time Inversion. This manuscript focuses solely on the DoE-based technique.In order to consistently benchmark the techniques, a set of standards was defined against which each was evaluated to determine its suitability for widespread history matching applications. The standards involve: the capacity to parameterize (and therefore calibrate) a diversity of reservoir flow model attributes, the capacity to integrate different types of dynamic data, the level of independence from the flow simulator and the capability to provide probabilistic outcomes for predictive uncertainty assessment. Of the four techniques, the DoE-based approach uniquely satisfied all requirements. Its history matching workflow has the flexibility to incorporate any form of reservoir model parameter and to assimilate a history matching error metric for any individual or group of historical data types; therefore, benchmarking established DoE-based techniques as unambiguously the most compliant with generic asset modeling requirements. The approach was also identified as the most straightforward, both theoretically and in practical computation, and therefore applicable to the broadest range of practitioners. Perhaps most importantly, the approach demonstrated the capacity for accurate quantification of uncertainty (or non-uniqueness) in reservoir quality resulting from an exhaustive, although approximate, exploration of model parameter space and the associated history matching error metric(s).This manuscript compiles the results and insights gained from benchmarking of the DoE-based techniques through the proposal of a comprehensive assisted history matching workflow. The workflow is designed for generality while providing best practices that guide the modeler in fit-for-purpose application. Limitations of the workflow are also recognized. Key components include: selection and screening of calibration parameters based on statistical significance, development of surrogate models to characterize the relationship between parameters and the simulated historical data being integrated into the reservoir model, use of the surrogate models for exhaustive yet efficient exploration of parameter space to identify (non-unique) history matched models, and (deterministic or probabilistic) discrete reservoir model selection for use in forecast-based decision making. Each step of the AHM workflow is presented from a conceptual and applied perspective, and field applications are provided to demonstrate key concepts. Although the applications presented include two deepwater Gulf of Mexico assets, the workflow and insights provided are developed from benchmarking across a diverse suite of asset types.

Summary Multiscale methods have been developed as a robust alternative to upscaling and to accelerate reservoir simulation. In their basic setup, multiscale methods use both a restriction operator to construct a reduced system of flow equations that can be solved on a coarser grid and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. When combined with a local smoother, this gives an iterative solver that can efficiently compute approximate pressures to within a prescribed accuracy and still provide mass-conservative fluxes. We present an adaptive and flexible framework for combining multiple sets of such multiscale approximations. Each multiscale approximation can target a certain scale; geological features such as faults, fractures, facies, or other geobodies; or a particular computational challenge such as propagating displacement and chemical fronts, wells being turned on or off, and others. Multiscale methods that fit the framework are characterized by three features. First, the prolongation and restriction operators are constructed by use of a nonoverlapping partition of the fine grid. Second, the prolongation operator is composed of a set of basis functions, each of which has compact support within a support region that contains a coarse gridblock. Finally, the basis functions form a partition of unity. These assumptions are quite general and encompass almost all existing multiscale (finite-volume) methods that rely on localized basis functions. The novelty of our framework is that it enables multiple pairs of prolongation and restriction operators—computed on different coarse grids and possibly also by different basis-function formulations—to be combined into one iterative procedure. Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators that have different spatial resolution and that each individual operator can be designed to target, among others, challenging grids, including faults, pinchouts, and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.

We explore the use of efficient streamline-based simulation approaches for modeling partitioning interwell tracer tests in hydrocarbon reservoirs. Specifically, we utilize the unique features of streamline models to develop an efficient approach for interpretation and history matching of field tracer response. A critical aspect here is the underdetermined and highly ill-posed nature of the associated inverse problems. We have investigated the relative merits of the traditional history matching ('amplitude inversion') and a novel travel time inversion in terms of robustness of the method and convergence behavior of the solution. We show that the traditional amplitude inversion is orders of magnitude more non-linear and the solution here is likely to get trapped in local minimum, leading to inadequate history match. The proposed travel time inversion is shown to be extremely efficient and robust for practical field applications. The streamline approach is generalized to model water injection in naturally fractured reservoirs through the use of a dual media approach. The fractures and matrix are treated as separate continua that are connected through a transfer function, as in conventional finite difference simulators for modeling fractured systems. A detailed comparison with a commercial finite difference simulator shows very good agreement. Furthermore, an examination of the scaling behavior of the computation time indicates that the streamline approach is likely to result in significant savings for large-scale field applications. We also propose a novel approach to history matching finite-difference models that combines the advantage of the streamline models with the versatility of finite-difference simulation. In our approach, we utilize the streamline-derived sensitivities to facilitate history matching during finite-difference simulation. The use of finite-difference model allows us to account for detailed process physics and compressibility effects. The approach is very fast and avoids much of the subjective judgments and time-consuming trial-and-errors associated with manual history matching. We demonstrate the power and utility of our approach using a synthetic example and two field examples. We have also explored the use of a finite difference reservoir simulator, UTCHEM, for field-scale design and optimization of partitioning interwell tracer tests. The finite-difference model allows us to include detailed physics associated with reactive tracer transport, particularly those related with transverse and cross-streamline mechanisms. We have investigated the potential use of downhole tracer samplers and also the use of natural tracers for the design of partitioning tracer tests. Finally, we discuss several alternative ways of using partitioning interwell tracer tests (PITTs) in oil fields for the calculation of oil saturation, swept pore volume and sweep efficiency, and assess the accuracy of such tests under a variety of reservoir conditions.

An iteratively adaptive multi-scale finite element method for elliptic PDEs with rough coefficients

A multi-scale finite-volume (MSFV) method for solving multiphase flow problem in highly heterogeneous media was recently developed. In contrast with classical upscaling techniques, the goal of multiscale methods is not simply to capture the large-scale effects of the fine-scale heterogeneity, but to provide an efficient tool for solving large flow problems with fine-scale resolution. The MSFV is based on a fractional flow formulation of the problem: first an equation for the total velocity is solved, then a fine-scale velocity field is reconstructed, finally the phase-saturation distribution is obtained by solving the nonlinear transport equations. In addition to the original fine grid the MSFV method employs an imposed coarse grid and a dual coarse grid. The first step is to compute the effective parameters that have to be used for solving the global flow problem on a coarse grid. This is done by means of a set of basis functions, which are numerical solutions computed on the cells of the dual grid. From these basis functions, the fluxes across the coarse-block boundaries are computed and the transmissibilities are extracted. Then the conservative fine-scale total-velocity field is reconstructed by solving a local flow problem in each coarse cell. In this paper we extend the MSFV method to account for gravity effects, which enables our method to work for fairly realistic problems, e.g. dead-oil problems or black-oil problems. To correctly account for gravity effects an additional local problem has to be solved on each block of the dual grid. This problem provides an additional basis function (gravity basis function) that represents a local correction to the fluxes computed in the absence of gravity.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

Flow simulation of multimillion grid cell models with hundreds of wells and decades long production history can be extremely time-consuming. This often limits the applicability of assisted history matching techniques. A pragmatic solution to this problem is grid coarsening which is now embedded in many commercial reservoir simulators. Instead of up-scaling geological models in external packages, grid cells are automatically amalgamated within the simulator while preserving flux distribution and reducing the total number of active cells. The resulting speedup can be significant, often only at small loss of accuracy. Both characteristics are essential elements of any inversion technique in a multimillion grid cell environment. For water-flood history matching, we have utilized a commercial finite-volume simulator and the streamline-based generalized travel time inversion whereby water-cut behavior is matched by adjusting inter-well permeabilities. To apply the assisted history matching technique on high resolution models, a flux reconstruction method is devised which makes full use of the benefits of automatic grid coarsening. It approximates fluxes in the original geological grid by recalculating transmissibility at the fine scale and redistributing coarse scale fluxes accordingly. The inversion is thus conducted on the fine scale grid, while the forward simulation model uses the coarse grid. While solving for an inexpensive coarse model, the calibration takes into account fine grid resolution. Upscaling is not required and the degree of coarsening can be adjusted during the matching process. The proposed method was successfully tested on a supergiant carbonate oilfield with about hundred wells and large-scale water injection. The history match improved dramatically at relatively low numerical cost, which also allowed for investigating multiple sensitivities. The results were verified against non-coarsened model. The significant increase in efficiency makes this a potential method of choice for cases, where previously assisted history matching techniques could not be deployed due to excessive run times.

SummaryWe present the development and field application of a workflow for multiscale reservoir-model calibration that seamlessly integrates production data into the reservoir description from the facies to the grid-cell scale. To start with, the permeability field is parameterized using a novel grid-connectivity-based transformation basis that can be applied with any model geometry, including unstructured and corner-point grids. The parameterization basis functions emerge from spectral decomposition of the grid-connectivity Laplacian and are related to the structural harmonics of the grid. To reconcile data with model resolution during history matching, we first use the coarsest-scale basis functions to identify the large-scale variability. Additional smaller-scale basis elements are then adaptively incorporated to successively refine the model to a level supported by data resolution. During refinement, the inclusion of more detailed basis functions into the parameterization is determined by generic modal frequency when the prior model is unavailable or by using prior information when available. In the final step of the workflow, a streamline-based inversion is performed to locally adjust the reservoir model at grid-cell resolution along preferential-flow paths defined during the coarser-scale parameterization.We demonstrate the suitability and effectiveness of the developed workflow through application to an offshore turbidite reservoir with frequent well intervention, including shut-ins and recompletions. The static model has over 300,000 cells, a complex channelized interpretation with faults, four injector/producer pairs with deviated wells, and over eight years of production history, including water cut and pressure data. The grid-connectivity-based parameterization effectively updates the prior regional permeability at scales and in locations warranted by the data, while preserving the geologic continuity and avoiding ad hoc redefinition of regions given the sparse well pattern. The multiscale calibrated-permeability field indicates flow communication previously unrecognized in static geologic interpretation or manual history matching.

Summary Numerical modeling of the in-situ conversion process (ICP) is a challenging endeavor involving thermal multiphase flow, compositional pressure/volume/temperature (PVT) behavior, and chemical reactions that convert solid kerogen into light hydrocarbons, which are tightly coupled to temperature propagation. Our investigations of grid-resolution effects on the accuracy and performance of ICP simulations have demonstrated that ICP-simulation outcomes—specifically, chemical-reaction rates, kerogen-accumulation profiles, and oil-/gas-production rates, may exhibit relatively large errors on coarse grids. Coarse grids are attractive because they deliver favorable computational performance. We have developed a novel multiscale modeling method for simulating ICP that reduces numerical-modeling errors and reproduces fine-scale-simulation results on relatively coarse grids. The method uses a two-scale solution method, in which the reaction kinetics of the solids is solved locally on a fine-scale grid, with interpolated temperatures obtained from coarse-grid simulations of thermal flow and fluid transport. We demonstrate the accuracy and efficiency of our multiscale method with representative 1D models. It is shown that the method delivers accurate solutions for key ICP performance indicators with very little computational overhead compared with corresponding coarse-scale models. The robustness of the multiscale method has been verified over a number of physical-parameter ranges with a limited-scope sensitivity study. Numerical results show that the multiscale method consistently improves the simulation results and matches the fine-scale reference results closely.

An adaptive local–global multiscale finite volume element method for two-phase flow simulations

Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity

Model calibration or history matching has commonly been conducted on a single deterministic model by a "manual" trial-and-error approach. With recent advances in the application of design of experiments for using numerical models in probabilistic forecasting, the feasibility of the manual history match process used for a single deterministic model has become questionable. In manual history matching, a structured approach is used in which the sequence of adjustments has been from global, then to flow units, followed by local changes in model properties. However, the applicability of this approach to probabilistic history matching has not been demonstrated. Moreover, the industry's lack of experience in using assisted history-matching tools and the careless application without using a structured logic that is based on engineering judgment can lead the users to several potential pitfalls resulting in unrealistic solutions. This work shows that in using evolutionary algorithms, a structured (staged) assisted history-matching methodology can be applied by considering probabilistic ranges of relevant input parameters and tailored objective functions for each stage of the process. The key "heavy-hitter" parameters with the highest impact on the history-match process are identified and introduced through conducting sensitivity runs. A physically-sound and proper set of parameters with realistic ranges are introduced at each stage of the history-match process in a logical order. This paper shows that a workflow can be designed so that the ranges of selected parameters that are used to attain the best solutions at each stage can be carried over to the next stage to continue the history-match process. Since this workflow is conducted in a probabilistic semi-automatic manner, a diverse set of solutions can be obtained with the flexibility to guide the process as is traditionally done in manual history matching. The workflow is demonstrated through its application to history match production data from the Tengiz super-giant carbonate oil field located on the shores of the Caspian Sea in the Republic of Kazakhstan.