Abstract: Efficient conditioning of a 3-D fine-scale reservoir model to multiphase production data using streamline-based coarse-scale model inversion and geostatistical downscaling

AAPG Bulletin ◽  
2000 ◽  
Vol 84 (2000) ◽  
Author(s):  
Thomas Thai-Binh Tran1
Keyword(s):  
Scale Model ◽  
Production Data ◽  
Fine Scale ◽  
Reservoir Model ◽  
Coarse Scale ◽  
Model Inversion

Multiscale Data Integration Using Markov Random Fields

2002 ◽  
Vol 5 (01) ◽  
pp. 68-78 ◽  
Author(s):  
Sang Heon Lee ◽  
Adel Malallah ◽  
Akhil Datta-Gupta ◽  
David Higdon
Keyword(s):  
Data Integration ◽  
Posterior Distribution ◽  
Random Fields ◽  
Spatial Modeling ◽  
Data Sources ◽  
Production Data ◽  
Fine Scale ◽  
Reservoir Model ◽  
Coarse Scale ◽  
Well Data

Summary We propose a hierarchical approach to spatial modeling based on Markov Random Fields (MRF) and multiresolution algorithms in image analysis. Unlike their geostatistical counterparts, which simultaneously specify distributions across the entire field, MRFs are based on a collection of full conditional distributions that rely on the local neighborhoods of each element. This critical focus on local specification provides several advantages:MRFs are computationally tractable and are ideally suited to simulation based computation, such as Markov Chain Monte Carlo (MCMC) methods, andmodel extensions to account for nonstationarity, discontinuity, and varying spatial properties at various scales of resolution are easily accessible in the MRF framework. Our proposed method is computationally efficient and well suited to reconstruct fine-scale spatial fields from coarser, multiscale samples (based on seismic and production data) and sparse fine-scale conditioning data (e.g., well data). It is easy to implement, and it can account for the complex, nonlinear interactions between different scales, as well as the precision of the data at various scales, in a consistent fashion. We illustrate our method with a variety of examples that demonstrate the power and versatility of the proposed approach. Finally, a comparison with Sequential Gaussian Simulation with Block Kriging (SGSBK) indicates similar performance with less restrictive assumptions. Introduction A persistent problem in petroleum reservoir characterization is to build a model for flow simulations based on incomplete information. Because of the limited spatial information, any conceptual reservoir model used to describe heterogeneities will, necessarily, have large uncertainty. Such uncertainties can be significantly reduced by integrating multiple data sources into the reservoir model.1 In general, we have hard data, such as well logs and cores, and soft data, such as seismic traces, production history, conceptual depositional models, and regional geological analyses. Integrating information from this wide variety of sources into the reservoir model is not a trivial task. This is because different data sources scan different length scales of heterogeneity and can have different degrees of precision.2 Reconciling multiscale data for spatial modeling of reservoir properties is important because different data types provide different information about the reservoir architecture and heterogeneity. It is essential that reservoir models preserve small-scale property variations observed in well logs and core measurements and capture the large-scale structure and continuity observed in global measures such as seismic and production data. A hierarchical model is particularly well suited to address the multiscaled nature of spatial fields, match available data at various levels of resolution, and account for uncertainties inherent in the information.1–3 Several methods to combine multiscale data have been introduced in the literature, with a primary focus on integrating seismic and well data.3–9 These include conventional techniques such as cokriging and its variations,3–6 SGSBK,7 and Bayesian updating of point kriging.8,9 Most kriging-based methods are restricted to multi-Gaussian and stationary random fields.3–9 Therefore, they require data transformation and variogram construction. In practice, variogram modeling with a limited data set can be difficult and strongly user-dependent. Improper variograms can lead to errors and inaccuracies in the estimation. Thus, one might also need to consider the uncertainty in variogram models during estimation. 10 However, conventional geostatistical methods do not provide an effective framework to account for the uncertainty of the variogram. Furthermore, most of the multiscale integration algorithms assume a linear relationship between the scales. The objective of this paper is to introduce a novel multiscale data-integration technique that provides a flexible and sound mathematical framework to overcome some of the limitations of conventional geostatistical techniques. Our approach is based on multiscale MRFs11–14 that can effectively integrate multiple data sources into high-resolution reservoir models for reliable reservoir forecasting. This proposed approach is also ideally suited to simulation- based computations, such as MCMC.15,16 Methodology Our problem of interest is to generate fine-scale random fields based on sparse fine-scale samples and coarse-scale data. Such situations arise when we have limited point measurements, such as well data, and coarse-scale information based on seismic and/or production data. Our proposed method is a Bayesian approach to spatial modeling based on MRF and multiresolution algorithms in image analysis. Broadly, the method consists of two major parts:construction of a posterior distribution for multiscale data integration using a hierarchical model andimplementing MCMC to explore the posterior distribution. Construction of a Posterior Distribution for Multiscale Data Integration. A multiresolution MRF provides an efficient framework to integrate different scales of data hierarchically, provided that the coarse-scale resolution is dependent on the next finescale resolution.11 In general, a hierarchical conditional model over scales 1,. . ., N (from fine to coarse) can be expressed in terms of the product of conditional distributions,Equation 1 where p(xn), n=1, . . ., N, are MRF models at each scale, and the terms p(xn|xn-1) express the statistical interactions between different scales. This approach links the various scales stochastically in a direct Bayesian hierarchical modeling framework (Fig. 1). Knowing the fine-scale field xn does not completely determine the field at a coarser scale xn+1, but depending on the extent of the dependence structure modeled and estimated, it influences the distribution at the coarser scales to a greater or lesser extent. This enables us to address multiscale problems accounting for the scale and precision of the data at various levels. For clarity of exposition, a hierarchical model for reconciling two different scales of data will be considered below.Equation 2 From this equation, the posterior distribution of the fine-scale random field indexed by 1 given a coarse-scale random field indexed by 2 can be derived as follows.

Analytical Upgridding Method To Preserve Dynamic Flow Behavior

2010 ◽  
Vol 13 (03) ◽  
pp. 473-484 ◽  
Author(s):  
Seyyed Abolfazl Hosseini ◽  
Mohan Kelkar
Keyword(s):  
Single Phase ◽  
Flow Behavior ◽  
Pressure Profile ◽  
Scale Model ◽  
Optimum Number ◽  
Fine Scale ◽  
Coarse Scale ◽  
Flow Process ◽  
Number Of Layers ◽  
Pressure Profiles

Summary A geocellular model contains millions of gridblocks and needs to be upscaled 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 that preserves the pressure profile at the upscaled level. It is well established that the more complex the flow process, the more detailed the level of heterogeneity needed in the simulation model. In general, ideal upscaling is the process that 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 obtain a better match between fine scale and coarse scale, we also need to use appropriate upscaling of the reservoir properties. The new method 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 method differs fundamentally from the other methods that try to preserve heterogeneities. In those methods, gridblocks are combined that have similar velocities (or other properties) by assuming constant pressure drop across the blocks. Instead, we combine the gridblocks that have similar pressure profiles, although to release some of our assumptions such as having constant velocities in gridblocks, we balance our equation with the K2 term. The procedure is analytical and, hence, very efficient, but preserves the pressure profile in the reservoir. The gridblocks (or layers) are combined in a way so that the difference between fine- and coarse-scale pressure profiles is minimized. In addition, we also propose two new criteria that allow us to choose the optimum number of layers more accurately so that a critical level of heterogeneity is preserved. These criteria provide insight into the overall level of heterogeneity in the reservoir and the effectiveness of the layering design. We compare the results of our method with proportional layering and the King et al. method (King et al. 2006) and show that, for the same number of layers, the proposed method captures the results of the fine-scale model better. We show that the layer merging not only depends on the variation in the permeability between the gridblocks (K2 term), but also on the relative magnitude of the permeability values by combining 1/K2 and K2 terms.

Two-Stage Upscaling of Two-Phase Flow: From Core to Simulation Scale

SPE Journal ◽  
2006 ◽  
Vol 11 (03) ◽  
pp. 304-316 ◽  
Author(s):  
Arild Lohne ◽  
George A. Virnovsky ◽  
Louis J. Durlofsky
Keyword(s):  
Capillary Pressure ◽  
Effective Properties ◽  
Absolute Permeability ◽  
Scale Model ◽  
Phase Flow ◽  
Fine Scale ◽  
Coarse Scale ◽  
Two Stage ◽  
Two Phase ◽  
Rate Dependent

Summary In the coarse-scale simulation of heterogeneous reservoirs, effective or upscaled flow functions (e.g., oil and water relative permeability and capillary pressure) can be used to represent heterogeneities at subgrid scales. The effective relative permeability is typically upscaled along with absolute permeability from a geocellular model. However, if no subgeocellular-scale information is included, the potentially important effects of smaller-scale heterogeneities (on the centimeter to meter scale) in both capillarity and absolute permeability will not be captured by this approach. In this paper, we present a two-stage upscaling procedure for two-phase flow. In the first stage, we upscale from the core (fine) scale to the geocellular (intermediate) scale, while in the second stage we upscale from the geocellular scale to the simulation (coarse) scale. The computational procedure includes numerical solution of the finite-difference equations describing steady-state flow over the local region to be upscaled, using either constant pressure or periodic boundary conditions. In contrast to most of the earlier investigations in this area, we first apply an iterative rate-dependent upscaling (iteration ensures that the properties are computed at the appropriate pressure gradient) rather than assume viscous or capillary dominance and, second, assess the accuracy of the two-stage upscaling procedure through comparison of flow results for the coarsened models against those of the finest-scale model. The two-stage method is applied to synthetic 2D reservoir models with strong variation in capillarity on the fine scale. Accurate reproduction of the fine-grid solutions (simulated on 500'500 grids) is achieved on coarse grids of 10'10 for different flow scenarios. It is shown that, although capillary forces are important on the fine scale, the assumption of capillary dominance in the first stage of upscaling is not always appropriate, and that the computation of rate-dependent effective properties in the upscaling can significantly improve the accuracy of the coarse-scale model. The assumption of viscous dominance in the second upscaling stage is found to be appropriate in all of the cases considered. Introduction Because of computational costs, field-simulation models may have very coarse cells with sizes up to 100 to 200 m in horizontal directions. The cells are typically populated with effective properties (porosity, absolute permeability, relative permeabilities, and capillary pressure) upscaled from a geocellular (or geostatistical) model. In this way, the effects of heterogeneity on the geocellular scale will be included in the large-scale flow calculations. The cell sizes in geocellular models may be on the order of 20 to 50 m in horizontal directions. However, heterogeneities on much smaller scales (cm- to m- scale) may have a significant influence on the reservoir flow (Coll et al. 2001; Honarpour et al. 1994), and this potential effect cannot be captured if the upscaling starts at the geocellular scale.

scholarly journals Integrating coarse-scale uncertain soil moisture data into a fine-scale hydrological modelling scenario

2011 ◽  
Vol 8 (3) ◽  
pp. 6031-6067
Author(s):  
H. Vernieuwe ◽  
B. De Baets ◽  
J. Minet ◽  
V. R. N. Pauwels ◽  
S. Lambot ◽  
...  
Keyword(s):  
Soil Moisture ◽  
Moisture Content ◽  
Soil Moisture Content ◽  
Epistemic Uncertainty ◽  
Hydrological Modelling ◽  
Scale Model ◽  
Fine Scale ◽  
Coarse Scale ◽  
Soil Moisture Data ◽  
External Data

Abstract. In a hydrological modelling scenario, often the modeller is confronted with external data, such as remotely-sensed soil moisture observations, that become available to update the model output. However, the scale triplet (spacing, extent and support) of these data is often inconsistent with that of the model. Furthermore, the external data can be cursed with epistemic uncertainty. Hence, a method is needed that not only integrates the external data into the model, but that also takes into account the difference in scale and the uncertainty of the observations. In this paper, a synthetic hydrological modelling scenario is set up in which a high-resolution distributed hydrological model is run over an agricultural field. At regular time steps, coarse-scale field-averaged soil moisture data, described by means of possibility distributions (epistemic uncertainty), are retrieved by synthetic aperture radar and assimilated into the model. A method is presented that allows to integrate the coarse-scale possibility distribution of soil moisture content data with the fine-scale model-based soil moisture data. To this end, a scaling relationship between field-averaged soil moisture content data and its corresponding standard deviation is employed.

Substituting of a Thermodynamic Simulation with a Metamodel in the Scope of Multiscale Modelling

2016 ◽  
Vol 879 ◽  
pp. 1207-1212 ◽  
Author(s):  
Piotr Macioł ◽  
Danuta Szeliga ◽  
Łukasz Sztangret
Keyword(s):  
Thermodynamic Simulation ◽  
Multiscale Simulation ◽  
Scale Model ◽  
Precipitation Kinetic ◽  
Fine Scale ◽  
Coarse Scale ◽  
Computational Point ◽  
Computing Power ◽  
Thermodynamic Computations ◽  
Scale Models

A typical multiscale simulation consists of numerous fine scale models, usually one for each computational point of a coarse scale model. One of possible ways of limiting computing power requirements is replacing fine scale models with some simplified and speeded up ersatz ones. In this paper, the authors attempt to develop a metamodel, replacing direct thermodynamic computations of precipitation kinetic with an advanced approximating model. MatCalc simulator has been used for thermodynamic modelling of precipitation kinetic. Typical heat treatment of P91 steel grade was examined. Selected variables were chosen to be modelled with approximating models. Several attempts with various approximation variants (interpolation algorithms and Artificial Neural Networks) have been investigated and its comparison is included in the paper.

Adaptive Local-Global Multiscale Simulation of the In-situ Conversion Process

SPE Journal ◽  
2016 ◽  
Vol 21 (06) ◽  
pp. 2112-2127 ◽  
Author(s):  
Faruk O. Alpak ◽  
Jeroen C. Vink
Keyword(s):  
Injection Rate ◽  
Multiscale Method ◽  
Scale Model ◽  
Fine Scale ◽  
Coarse Scale ◽  
Scale Models ◽  
Pattern Scale ◽  
Heat Injection ◽  
Coarse Grids

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 and are tightly coupled to temperature propagation. Our investigations of grid-resolution effects on the accuracy and performance of ICP simulations demonstrated that ICP-simulation outcomes (e.g., oil/gas production rates and cumulative volumes) may exhibit relatively large errors on coarse grids, where “coarse” means a gridblock size of more than 3 to 5 m. On the other hand, coarse-scale models are attractive because they deliver favorable computational performance, especially for optimization and uncertainty quantification workflows that demand a large number of simulations. Furthermore, field-scale models become unmanageably large if gridblock sizes of 3 to 5 m or less have to be used. Therefore, there is a clear business need to accelerate the ICP simulations with minimal compromise of accuracy. We developed a novel multiscale-modeling method for ICP that reduces numerical-modeling errors and approximates the fine-scale simulation results on relatively coarse grids. The method uses a two-scale adaptive local-global solution technique. One global coarse-scale and multiple local fine-scale near-heater models are timestepped in a sequentially coupled fashion. At a given global timestep, the global-model solution provides accurate boundary conditions to the local near-heater models. These boundary conditions account for the global characteristics of the thermal-reactive flow and transport phenomena. In turn, fine-scale information about heater responses is upscaled from the local models, and used in the global coarse-scale model. These flow-based effective properties correct the thermal-reactive flow and transport in the global model either explicitly, by updating relevant coarse-grid properties for the next timestep, or implicitly, by repeatedly updating the properties through a convergent iterative scheme. Upon convergence, global coarse-scale and local fine-scale solutions are compatible with each other. We demonstrate the much-improved accuracy and efficiency delivered by the multiscale method by use of a 2D cross-section pattern-scale ICP simulation problem. The following conclusions are reached through numerical testing: (1) The multiscale method significantly improves the accuracy of the simulation results over conventionally upscaled models. The method is particularly effective in correcting the global coarse-scale model through the use of the fine-scale information about heater temperatures to regulate the heat-injection rate into the formation more accurately. The effective coarse-grid properties computed by the multiscale method at every timestep also enhance the accuracy of the ICP simulations, as demonstrated in a dedicated test case, in which a constant heat-injection rate is enforced across models of all investigated resolutions. (2) Multiscale ICP models result in accelerated simulations with a speed-up of four to 16 times with respect to fine-scale models “out of the box” without any special optimization effort. (3) Our multiscale method delivers high-resolution solutions in the vicinity of the heaters at a reduced computational cost. These fine-scale solutions can be used to better understand the evolution of the fluids and solids (e.g., kerogen conversion and coke deposition) in the vicinity of the heaters (several-feet-long spatial scale). Simultaneously, with the fine-scale near-heater solutions, the local-global coupled multiscale model provides key commercial ICP performance indicators at the pattern scale (several-hundred-feet-long spatial scale) such as production functions.

scholarly journals A Near–Far-Field Model for Bubbles Influenced by External Electrical Fields

2019 ◽  
Vol 9 (21) ◽  
pp. 4722
Author(s):  
Juergen Geiser ◽  
Paul Mertin
Keyword(s):  
Field Model ◽  
Transport Model ◽  
Near Field ◽  
Bubble Formation ◽  
Dielectric Liquid ◽  
Scale Model ◽  
Far Field ◽  
Fine Scale ◽  
Coarse Scale ◽  
Carrier Fluid

In this paper, we present a model that is based on near–far-field charged bubble formation and transportation in an underlying dielectric liquid. The bubbles are controlled by the dielectric liquid, which is influenced by an external electrical field. This allows us to control the shape and volume of the bubbles in the dielectric liquid, such as water. These simulations are important to close the gap between the formation of charged bubbles, which is a fine-scale model and their transport in the underlying liquid, which is a coarse-scale model. In the fine-scale model, the formation of the bubbles and their influence of the electric-stress is approached by a near-field model, which is done by the Young–Laplace equation plus additional force-terms. In the coarse-scale model, the transport of the bubbles is approached by a far-field model, which is done with a convection-diffusion equation. The models are coupled with a bubble in cell scheme, which interpolates between the fine and coarse scales of the different models. Such a scale-dependent approach allows us to apply optimal numerical solvers for the different fine and coarse time and space scales and help to foresee the fluctuations of the charged bubbles in the E-field. We discuss the modeling approaches, numerical solver methods and we present the numerical results for the near–far-field bubble formation and transport model in a dielectric carrier fluid.

Equation-Free/Galerkin-Free Reduced-Order Modeling of the Shallow Water Equations Based on Proper Orthogonal Decomposition

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
Vahid Esfahanian ◽  
Khosro Ashrafi
Keyword(s):  
Shallow Water ◽  
Basis Functions ◽  
Galerkin Projection ◽  
Scale Model ◽  
Fine Scale ◽  
Coarse Scale ◽  
Reduced Order ◽  
Shallow Water Flows ◽  
The One ◽  
Proper Orthogonal

In this paper, two categories of reduced-order modeling (ROM) of the shallow water equations (SWEs) based on the proper orthogonal decomposition (POD) are presented. First, the traditional Galerkin-projection POD/ROM is applied to the one-dimensional (1D) SWEs. The result indicates that although the Galerkin-projection POD/ROM is suitable for describing the physical properties of flows (during the POD basis functions’ construction time), it cannot predict that the dynamics of the shallow water flows properly as it was expected, especially with complex initial conditions. Then, the study is extended to applying the equation-free/Galerkin-free POD/ROM to both 1D and 2D SWEs. In the equation-free/Galerkin-free framework, the numerical simulation switches between a fine-scale model, which provides data for construction of the POD basis functions, and a coarse-scale model, which is designed for the coarse-grained computational study of complex, multiscale problems like SWEs. In the present work, the Beam & Warming and semi-implicit time integration schemes are applied to the 1D and 2D SWEs, respectively, as fine-scale models and the coefficients of a few POD basis functions (reduced-order model) are considered as a coarse-scale model. Projective integration is applied to the coarse-scale model in an equation-free framework with a time step grater than the one used for a fine-scale model. It is demonstrated that equation-free/Galerkin-free POD/ROM can resolve the dynamics of the complex shallow water flows. Moreover, the computational cost of the approach is less than the one for a fine-scale model.

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