Novel permeability upscaling method using Fast Marching Method

Integration of dynamic data typically requires the solution of an inverse problem that can be computationally intensive and practically infeasible for fine scale reservoir models. In this paper we present a new methodology to directly update fine scale geostatistically-based reservoir models by combining gradual deformation parameterization for the fine scale geostatistical model and an upscaling technique for the coarse scale flow simulation model. The proposed methodology includes: Perturbation of the fine scale geostatistical model using the gradual deformation parameterization. Gradual deformation ensures the preservation of the overall geostatistical properties of the fine model. Generation of the coarse scale flow simulation model by upscaling the fine scale geostatistical model. Sensitivity computation of the flow simulation results with respect to the fine scale parameterization. This sensitivity computation is analytical and takes into account the upscaling process. Direct updating of the fine scale geostatistical model using classical optimization process. Direct updating ensures consistency between the fine and coarse scale models. The accuracy of the proposed methodology was improved by calibrating the flow simulation model. The objective of this calibration is to reduce the error introduced by the upscaling step during the flow simulation. We applied successfully our methodology for fine scale reservoir description by integrating permanent down-hole gauge measurements directly into a three-dimensional geostatistical model containing about two million grid blocks. This test is designed to highlight several key issues of the proposed methodology: Efficiency of the upscaling step coupled with gradient-based optimization to speed up the history matching process. Usefulness of the calibration step for a correct integration of upscaling techniques in history matching. Capability of the methodology for maintaining consistency and coherency between fine scale and coarse scale models. Improvement of the reservoir characterization by integrating dynamic data at the fine geostatistical scale. We conclude that the proposed methodology can be used effectively and efficiently for reservoir characterization purposes.

Characterization of naturally fractured reservoirs is complicated. The reservoir model of such reservoirs must represent both fracture and matrix systems and interaction between the two. It is important to choose correct dual continuum model parameters to build the most representative of a fractured reservoir. In this study, we try to improve our understanding of interaction between matrix and fracture and how this can be captured by dual porosity/permeability model. We look at a homogeneous, a simple heterogeneous and a detailed pore-type based heterogeneous model. Fracture act as boundary conditions. First, interaction of forces during various depletion scenarios is studied. Then, oil recovery by gas/water injection is simulated in the fine scale. Dual continium models corresponding to the fine scale models have been built and it has been tried to achieve a good match between fine scale and dual continuum model, by adjusting dual continium model parameters. Observations from this study highlight the importance of capturing the fine scale heterogeneity in fractured reservoir modeling. Furthermore, the ability of reproducing fine scale results, by final simulation model will depend upon selected upscaling/coarsening methodology and how parameters of the coarse scale model are generated/tuned.

Geo-cellular model contains millions of grid blocks and needs to be up-scaled before the model can be used as an input for flow simulation. Available techniques for upgridding vary from simple methods such as proportional fractioning to more complicated methods such as maintaining heterogeneities through variance calculations. All these methods are independent of the flow process for which simulation is going to be used, and are independent of well configuration. We propose a new upgridding method which preserves the pressure profile at the upscaled level. It is well established that more complex the flow process, more the detailed level of heterogeneity is needed in simulation model. In general, ideal upscaling is the process which preserves the "pressure profile" from the fine scale model under the applicable flow process. In our method we upgrid the geological model using simple flow equations in porous media. However, it should be remembered that to get a better match between fine scale and coarse scale, we also need to use appropriate upscaling of the reservoir properties. The new methodology is currently developed for single phase flow; however, we used it for both single phase and two phase flows for 2D and 3D cases. The methodology fundamentally differs from the other methods which try to preserve heterogeneities. In those methods, grid blocks are combined which have similar velocities (or other properties) by assuming constant pressure drop across the blocks. Instead, we combine the grid blocks which have similar pressure profiles. The procedure is analytical and hence very efficient, but preserves the pressure profile in the reservoir. The grid blocks (or layers) are combined in a way so that the difference between fine scale and coarse scale pressure profiles is minimized. In addition, we also propose two new criteria that allow us to choose the optimum number of the layers more accurately so that critical level of heterogeneity is preserved. These criteria provide insight into the overall level of heterogeneity in the reservoir as well as effectiveness of the layering design. We compare the results of our method with proportional layering and King et al.'s method (King 2005) and show that, for the same number of layers, the proposed method better captures the results of the fine scale model. We show that the layer merging not only depends on the variation in the permeability between the grid blocks, but also on relative magnitude of the permeability values. We also show that new method can account for additional variables such as grid block thicknesses and the size.

Geostatistical models of reservoir properties can be hundreds of millions of cells; it is impractical to use them directly in flow simulation due to computational cost. Upscaling techniques are applied to average fine scale permeability values onto coarser flow simulation blocks. In cases where unstructured grids are used or the geology inside the grid block is not aligned with the block geometry, full permeability tensors arise instead of a diagonal tensor. The focus of this work is on development of a method to characterize the full permeability tensor for an unstructured grid block using fine scale heterogeneity information. A single phase flow-based upscaling is performed and a prototype program called ptensor is developed based on the random boundary conditions and optimization technique. Full, symmetric and diagonal permeability tensors are calculated for 2-D and 3-D blocks and sensitivity analysis is performed. Introduction Geostatistical modeling of petrophysical properties can generate fine scale models with hundreds of millions of cells. Using those fine scale models directly in flow simulation is computationally inefficient. Upscaling techniques scale the fine scale models to coarser scale models while preserving the fine scale heterogeneity. A simple averaging is sufficient and reasonable for variables that average linearly; however, in the case of permeability which does not average linearly, a simple arithmetic averaging is inadequate. For complex cases with heterogeneity, flow-based upscaling techniques yield more accurate results (1). In this type of upscaling the flow equation is solved for pressure and the results are used to calculate the block permeability. Commonly unstructured grids are used in order to better capture the flow response near complex reservoir features such as faults and wells. Usually cases that involve the use of irregular block or a heterogeneous permeability field at fine scale require calculation of the full permeability tensor. White and Horne(2) and Gomez-Hernandez(3) proposed different methods to calculate permeability tensor for regular coarse blocks. In recent years, some approaches are presented by Durlofsky(4), Prevost(5) and He(6) to calculate the full permeability tensor for irregular shape grid blocks. This paper introduces a simple, fast and accurate method to calculate full, symmetric or diagonal permeability tensor for any corner point geometry grids. The unstructured grid is surrounded by a bounding box and the geometry is simplified with the fine resolution grid. The steady state flow equation is solved, via finite difference, for the input fine grid cells within a bounding box. The results are used to calculate the permeability tensor of corresponding coarse regular or irregular blocks. Randomly assigned boundary conditions are used and the results are optimized to get the desired full, symmetric or diagonal tensor. Methodology Flow based upscaling is used to calculate effective permeability of coarse block. Consider a single rectangle (2-D) or a cube (3-D) imposed on a fine scale model. The idea here is to calculate the pressure at fine scale with specific boundary conditions applied at the boundary of the coarse block and then use the solution to calculate the full permeability tensor for that coarse block.

History matching of 4D seismic and well production data has been developed recently for reservoir characterization. This is a data driven optimization process to derive reservoir model parameters and constitutes a joint inverse problem. Considering the inherent non‐uniqueness in the inverse problem and the unique feature of Bayesian inference in data integration and uncertainty analysis, this joint inverse problem is formulated in a Bayesian framework and solved stochastically by reconstructing the posterior probability density (PPD) surface using a new multi‐scale Markov Chain Monte Carlo (MCMC) algorithm. In this new MCMC method, a technique of multi‐scaling is used to take advantage of the benefits from both the fine scale model and the coarse scale model. Although the coarse scale does not provide reliable information about the model, it helps speed up the convergence of the fine scale model to a good estimation, and, by exchanging information between the fine and coarse scales, works like a regularization operator to smooth the fine scale model and make it more realistic. The resulting PPD samples can also be used to quantify corresponding uncertainties in order to facilitate risk assessment associated with reservoir decision making and management. We use a numerical example to demonstrate how we derive reservoir's static and dynamic properties as well as quantify uncertainties in a Bayesian framework using the multi‐scale MCMC algorithm.

In order to run reservoir simulation efficiently, a coarse scale (CS) dynamic model is created by upscaling of a fine scale (FS) static model. All history match (HM) changes usually done in the CS dynamic model need to be downscaled to FS for geological justifications and consistency maintenance between the FS static and CS dynamic models. This paper proposes a robust downscaling method for integration of FS static and CS dynamic models. The proposed method downscales a HMDM (dynamic model) to HMSM (static) in multiple steps. Scale-up the ISM (initial) to CS to create an IDM. Identify the cell changes between HMDM and IDM, and transfer the changes to FS to create a MSM (modified). Scale-up the MSM to CS to create to a MDM and calculate the ratios between HMDM and MDM for all cell properties. Transfer the ratios to FS to create a HMSM. Scale-up the HMSM to CS to confirm its identity to the HMDM. Selection of sampling and zone mapping methods is critical in all steps. The proposed method has been successfully applied in a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within/between matrix and fracture/karst. The FS static model contains a 236m × 236m horizontal grid with 593 layers while the CS dynamic model has the horizontal cell sizes in a range of 236m to 944m with 73 layers. Rock regions, permeability, and reservoir connectivity in the CS dynamic model were calibrated using the field historical production data (e.g., static pressure, PLT, interference test, and GOR/water-cut data) to create a HMDM. Since the HM process was performed only in the CS dynamic model, the FS static model and HMDM became inconsistent. Appling the proposed downscaling method has helped the HM team to resolve this issue and resulted in a seamless link between the FS static and CS dynamic models for current and future HM and model updates.

This paper presents the implementation of a novel approach for efficient well placement design based on the dynamic characteristic of the reservoir without the need to run flow simulation. The approach uses State-of-the-Art Technology called the Fast Marching Method (FMM) coupled with Geometric Pressure Approximation to define a Dynamic Reservoir Quality Map. This map is also referred to as the Depletion Capacity (DC) Map. It provides the mapping of future potential locations to further deplete the reservoir. The DC Map is generated by calculating 4 important factors, namely Pore Volume, Mobility of the Hydrocarbon, Reservoir Energy, and the Undrained Volume of the reservoir. The major breakthrough in using this technology is the ability to estimate the pressure distribution (i.e., reservoir energy) and drainage volume (i.e., to estimate the undrained volume) efficiently. These are the 2 factors that are difficult to obtain without conducting traditional flow simulation. The two aforementioned factors were obtained by calculating the FMM diffusive time of flight which can be related to the pressure drop by the Geometric Pressure Approximation theory. Thus, in effect, this calculation represents a pseudo-simulation, which is orders of magnitude faster than conventional simulation. The technique is applicable for both fine scale and coarse scale models with large number of realizations representing geological uncertainties. This approach works well in capturing the primary depletion phenomenon. In this paper, we demonstrate the evaluation of existing well placement of an actual developed field, with 16 wells, located in Tarim Basin, West China, by comparing it to a new design. Additionally, the method is also used to propose future locations for infill drilling. The study is motivated by the big challenges faced when drilling a well at a depth between 6800 m–8000 m. Optimum well placement has the potential to drill optimum number of wells to produce the same reserves. The new design was created with a scenario where 5 exploration wells that been put into production for a couple of years to represent the early depleted condition. The results show the optimum design can be achieved with only 12 wells. A saving of 4 wells compared to the existing well pattern. This is a significant saving considering the drilling challenges. For the infill drilling, the study shows that for this reservoir, adding more wells may not be beneficial from the ultimate recovery point of view but production can be accelerated by drilling up to 2 more wells at the best potential locations as suggested by the DC Map. Best of all, the proposed method optimizes the locations very quickly.

With the advance of CPU power, numerical reservoir models have become an essential part of most reservoir engineering applications. These models are used for predicting future performances or determining optimal locations of infill wells. Hence in order to accurately predict, these reservoir models must be conditioned to all available data. The challenge in data integration for numerical reservoir models lies in the fact that each data has its own resolution and area of coverage. The most common data for reservoir characterization are; well-log/core data, seismic data and production data. Most current approaches to data integration are hierarchical. Fine scale models are used for integrating well-log/core and seismic data while coarse models are used to integrate mostly production data. The drawback of such a hierarchical approach is such that once the scale is changed, data conditioning, maintained in the previous scale, is lost. In this paper, we review a general algorithm as a solution to the multi-scale data integration. Instead of proceeding in a hierarchical fashion, a fine model and a coarse model is kept in parallel throughout the entire characterization process. The link between the fine scale and the coarse scale is provided by non-uniform upscaling. An optimization procedure determines the optimal gridding parameters that provide the smallest possible mismatch between fine and coarse scale reservoir models. A synthetic example application is given and demonstration of the methodology. The upgridding is accomplish by a static gridding algorithm, 3DDEGA. This algorithm aims at preserving geology by minimizing heterogeneity within a coarse grid block. The coarse grids are provided in a corner-point geometry fashion, hence this allows for accurate description of the reservoir with fewer number of grid blocks.

Heterogeneity of the fine-scale models will result in full tensor effects for the upscaled models, especially in reservoirs with channels or fractures. A flow-based upscaling approach integrating multiple flow scenarios is developed to effectively capture the full-tensor effects when upscaling from fine-scale discrete fracture models. The new approach is proposed to deliver computationally affordable simulation models under precision comparable to the high-resolution models. The approach starts with several sets of flow-based single scenario upscaling procedures. A fine-scale discrete fracture model is built up as the reference model for all the successive procedures and comparisons. Then several flow cases (referred as scenarios) are determined according to the target simulation conditions. A global upscaling technique under multipoint flux approximation scheme is introduced to each flow scenario. An integrated output least squares method, which aims to minimize the total bias of simulation results of all the flow scenarios, is adopted to obtain the optimal transmissibility connection list of the coarse-scale models. We design and implement several examples including two synthetic conceptual cases and a real field case. In each case, comparisons are provided for the simulation results of the proposed approach and previous upscaling approaches based on two point flux approximation schemes. The coarse-scale models upscaled by different methods are based on the same fine-scale (reference) model in each case. The results of the upscaled models are also compared to the reference model. The numerical results show that the new approach generates coarse-scale models which are closer to the fine-scale model. Although under certain conditions, traditional upscaling methods can achieve equivalent results, it is proven that the new approach is more robust when applied to more general flow scenarios. It can be noted that it may take a bit more time for the numerical simulation of coarse-scale models upscaled by the new method, due to the introduction of the non-neighbor connections. When compared to the fine- scale model, however, the improvement of computational efficiency is still pretty significant. The last case is a real field case with about 500,000 fine-scale grids and 10,000 coarse-scale grids, which demonstrates the ability of the new approach to be applied in the industry. The novelty of the proposed approach is the optimization technique integrating multiple scenarios to generate a high precision coarse-scale model under multipoint flux approximation scheme. The new method effectively capture the anisotropic features of the coarse-scale models, which is a challenge for flow-based upscaling procedures because a single flow simulation is usually inadequate to obtain full tensor information of the upscaled models.

SummaryOne of the great challenges in reservoir modeling is to understand and quantify the dynamic uncertainties in geocellular models. Uncertainties in static parameters are easy to identify in geocellular models. Unfortunately, those models contain at least one to two orders of magnitude more gridblocks than typical simulation models. This means that, without significant upscaling, the dynamic uncertainties in these models cannot easily be assessed. Further, if we would like to select only a few geological models that can be carried forward for future performance predictions, we do not have an objective method of selecting the models that can properly capture the dynamic-uncertainty range.One possible solution is to use a faster simulation technique, such as streamline simulation. However, even streamline simulation requires solving a pressure equation at least once. For highly heterogeneous reservoir models with multimillion cells and in the presence of capillary effects or an expansion-dominated process, this can pose a challenge. If we use static permeability thresholds to determine the connected volume, it would not account for how tortuous the connection is between the connected gridblock and the well location.In this paper, we use the fast-marching method (FMM) as a computationally efficient method for calculating the pressure/front propagation time on the basis of reservoir properties. This method is based on solving the Eikonal equation by use of upwind finite-difference approximation. In this method, pressure/front location (radius of investigation) can be calculated as a function of time without running any flow simulation. We demonstrate that dynamically connected volume based on pressure-propagation time is a very good proxy for ultimate recovery from a well in the primary-depletion process. With a predetermined threshold propagation time, a large number of geocellular models can be ranked. FMM can be scaled almost linearly with the number of gridblocks in the model. Two main advantages of this ranking method compared with other methods are that this method determines dynamic connectivity in the reservoir and that it is computationally much more efficient. We demonstrate the validity of the method by comparing ranking of multiple geocellular realizations (on Cartesian grid with heterogeneous and anisotropic permeability) by use of FMM with ranking from flow simulation. This method will allow us to select the geological models that can truly capture the range of dynamic uncertainty very efficiently.

Numerical reservoir simulation often requires upscaling of fine-scale detailed models and coarse-scale models are necessary to reduce computational time for dynamic evaluations. However, these simplifications may degenerate results due to loss of resolution of the small-scale phenomena, averaging of sub-grid heterogeneity and numerical dispersion, especially in oil fields where miscible gas is injected. Most of the existing upscaling techniques focus on reproducing the results of a specific geological realization, in a deterministic approach. Nowadays, however, reservoir simulation studies commonly include uncertainty quantifications, which is performed by simulating multiple geological realizations. For that, the use of fine-scale models can be computationally prohibitive and this requires a proper procedure to upscale the coarse-scale simulation models in multiple realizations environment. In this work, we propose and test an ensemble-level upscaling technique for compositional systems with miscible gas injection. The new approach considers the classical Koval factor, calculated for the fine-scale models, as a guide for selecting representative fine-scale models to train pseudo-functions for the coarse-models. Only a few fine models are simulated (about 1%), and the uncertainty quantification process with coarse-scale models can be significantly improved. The proposed workflow is guided by ranking the fine-scale models in increasing order of their Koval Factor. We selected representative models and applied a two-step methodology to improve upscaled coarse-scale results for these models. We then propose a consistent procedure to expand the fitted pseudo-functions to all the coarse models, providing an effective ensemble-level upscaling. The correlation between Koval factor and oil recovery is a useful guide to extrapolate the pseudo-functions obtained for each selected representative model, enabling better coarse-scale simulation results when multiple realizations are considered. This procedure can be applied for continuous miscible gas injection and can be adapted for WAG scheme. This work was motivated by the lack of practical procedures to improve coarse-scale results at the ensemble-level. With our approach, we can better represent uncertainty quantification using coarse-scale models with reduced computational cost and requiring only a few fine-scale simulation runs.

Given a set of inelastic material models, a microstructure, a macroscopic structural geometry, and a set of boundary conditions, one can in principle always solve the governing equations to determine the system’s mechanical response. However, for large systems this procedure can quickly become computationally overwhelming, especially in three-dimensions when the microstructure is locally complex. In such settings multi-scale modeling offers a route to a more efficient model by holding out the promise of a framework with fewer degrees of freedom, which at the same time faithfully represents, up to a certain scale, the behavior of the system. In this paper, we present a methodology that produces such models for inelastic systems upon the basis of a variational scheme. The essence of the scheme is the construction of a variational statement for the free energy as well as the dissipation potential for a coarse scale model in terms of the free energy and dissipation functions of the fine scale model. From the coarse scale energy and dissipation we can then generate coarse scale material models that are computationally far more efficient than either directly solving the fine scale model or by resorting to FE2 type modeling. Moreover, the coarse scale model preserves the essential mathematical structure of the fine scale model. An essential feature for such schemes is the proper definition of the coarse scale inelastic variables. By way of concrete examples, we illustrate the needed steps to generate successful models via application to problems in classical plasticity, included are comparisons to direct numerical simulations of the microstructure to illustrate the accuracy of the proposed methodology.

New upgridding method to capture the dynamic performance of the fine scale heterogeneous reservoir

Reservoir heterogeneity can be detrimental to the success of chemical enhanced oil recovery (EOR) processes. Therefore, it is important to evaluate the effect of uncertainty in reservoir heterogeneity on the performance of chemical EOR. Usually, a Monte Carlo sampling approach is followed were a number of stochastic reservoir model realizations are generated and then numerical simulation is performed to obtain a certain objective function, such as the recovery factor; however, Monte Carlo simulation (MCS) has a slow convergence and requires a large number of samples to produce accurate results. This is computationally expensive when using reservoir simulators. This study applies an extension to MCS using a multi-scale approach. The applied method is known as the multilevel Monte Carlo (MLMC) method and has been only recently applied to problems of flow in porous media. This method is based on running a small number of expensive simulations on the finer scale model and a large number of less expensive simulations on coarser scale models — these are upscaled models of the fine scale model — and then combining the results to produce the quantities of interest. The purpose of this method is to reduce computational cost while maintaining the accuracy of the finer scale model. The results of this approach are compared with reference MCS, assuming a large number of simulations on the fine scale model. This study used MLMC to efficiently quantify the effect of uncertainty in heterogeneity on the recovery factor of different chemical EOR processes. The permeability field was assumed to be the random input. This approach was implemented by writing a MATLAB code to generate the stochastic realizations for the permeability field and also performing the coarsening processes. The code is then coupled with ECLIPSE, which was used as the numerical simulator for the chemical EOR processes to obtain the recovery factor. The code then combines the results obtained from the different scale models to produce the statistical moments for the recovery factor, such as the mean and variance. This method was applied for two-dimensional (2D) and three-dimensional (3D) stylized reservoir models using Gaussian randomly generated permeability fields. Different coarsening algorithms were used and compared, such as the renormalization and pressure solver methods, and polymer and surfactant-polymer (SP) flooding processes where the chemical EOR processes were considered. The results were compared with running Monte Carlo for the fine scale model while equating the computational cost for the multilevel Monte Carlo method. Both of these results were then compared with the reference case, which uses a large number of runs of the fine scale model. The results show that it is possible to robustly quantify spatial uncertainty for chemical EOR processes while greatly reducing the computational requirement, up to two orders of magnitude compared to traditional Monte Carlo. The method can be easily extendable to other EOR processes to quantify spatial uncertainty such as carbon dioxide (CO2) EOR.

Grid coarsening schemes based on the quantitative use of fine scale two-phase flow information are presented and assessed. The basic approach is motivated from a volume average analysis of the fine scale saturation equation including gravitational effects. Extensive results for layered systems are presented. It is shown that coarse grid simulation error correlates closely with specific sub-grid quantities involving higher moments of fine grid variables, which can be computed from the fine scale simulations. By forming a coarse grid that minimises the appropriate sub-grid quantity, optimal coarse scale descriptions can be generated. The overall approach is shown to be applicable to coarse scale descriptions using either rock or pseudo relative permeability curves. The accuracy of the coarse grid calculations is, however, significantly better when pseudo functions are used. The method is applied to determine the optimal number and configuration of coarse grid layers in more general cases and it is shown that coarse grid results do not always improve as the number of coarse layers is increased. Introduction In modern reservoir characterisation, the spatial resolution that may be incorporated into geological models often exceeds the computational capabilities of fluid flow simulators by a significant margin. Therefore, some level of upscaling must be applied to the fine scale geological models before they can be used for practical flow calculations. This upscaling may be a simple block averaging of the single-phase permeability or it may involve the application of a complex upscaling procedure. When the degree of upscaling is very large, the use of a dynamic technique, which may involve the generation of upscaled or pseudo relative permeabilities, is generally required. Several such upscaling methods have been developed and described in the literature; e.g. the Kyte and Berry1, Stone2, Vertical Equilibrium (VE)3 and TW4,a methods. Hewett and coworkers have also suggested approaches based on streamline methods5,6. In general, all of these dynamic methods (except VE) involve some procedure for using the fine grid flows to generate modified pseudo relative permeability and capillary pressure curves at the coarse block scale. When successfully applied, these pseudo functions will accurately incorporate the interaction between small-scale multi-phase fluid flow and heterogeneity, as well as correcting for the numerical dispersion in the coarse grid models. The principal metric is that the upscaling method provides a coarse-scale flow model that accurately reproduces the results (recovery profiles, breakthrough times, etc.) computed using the fine grid model. Pseudo functions do have costs and limitations associated with them, however, and these must be considered when such an approach is applied. Pseudo relative permeabilities may be subject to so-called process dependence, meaning the coarse scale pseudo functions vary with varying global boundary conditions. This in turn can result in a lack of robustness in the coarse scale model. In addition, when pseudo functions are generated through the simulation of a global fine scale flow problem, the computational requirements can be excessive in some cases.