Upscaled Value Research Articles

A coupled permeability thresholding with effective medium theory upscaling technique for highly heterogeneous reservoirs

Reservoir simulations are performed with fine grid geological models as primary input into the reservoir simulators. These fine grid geological models contain ten million to one hundred million cells while current serial reservoir simulators can only handle one hundred thousand to one million cells. This disparity in scales prohibits direct input of fine grid geological models into serial simulators. However, traditional upscaling techniques are unable to reproduce the fine-scale dynamic flow behavior at the coarse-scale and up-scaled permeability values are dependent on the applied boundary conditions. The problem is exacerbated if isolated flow units are present. These problems motivated the need to couple the permeability thresholding and effective medium theory (EMT) approaches to develop a practical and accurate quasi-global single-phase permeability upscaling technique that uses the reservoir simulator in the upscaling process. The workflow involves identifying and making the no-flow cells slightly permeable. This is followed by defining two models for each region to be scaled up, one of which includes the heterogeneous permeability field in the region and the other the upscaled value. Also, a fine-scale gridded ring is constructed around the target region. Then the upscaled value of the permeability is resolved by minimizing the difference between the well water production rates (WWPR) of the two runs. The technique is shown to mimic the channel fine-scale permeability distribution at the coarse-scale. It is also shown to reproduce the physics of flow for two-phase flow problems as evident in the well oil production rate, injection well pressure and pressure distribution profile. Sensitivity analyses on different scale-up factors, well locations and sequentially active wells show the superiority of the novel technique over the upscaling and downscaling with effective medium theory (UDEMT) and local upscaling techniques - albeit at the cost of more CPU time. The new method takes more CPU time than local upscaling method but the accurate results justifies the extra CPU time. Our workflow should be extended to three dimensions and fractured reservoirs as well as coupled with existing upscaling techniques for near wellbore and relative permeability upscaling.

Low- and High-Resolution X-Ray Tomography Helping on Petrophysics and Flow-Behavior Modeling

Summary Reservoir studies involve spatial scales ranging from pore models to field-simulation grids. X-ray tomography is a nondestructive technique of inspection and characterization of samples that works from microns in the pore scale to centimeters in drill cores, which makes it a useful tool for modeling porous systems in reservoir-rock samples. The generated models serve to simulate physical phenomena helpful to determine both static and dynamic properties. Various techniques of preprocessing, porous-medium modeling, and simulation are applicable depending on the degree of heterogeneity of the sample. This work presents a workflow to make the porosity and permeability models from images acquired with axial computed tomography (CT) at different resolution scales. The workflow was used in sandstone and carbonate samples from Brazilian outcrops. The density and porosity of the sample were determined at the millimeter scale using a medical scanner and the distributions of pore size, grain size, and absolute permeability were determined using a synchrotron beamline, which provided high-resolution data at the micrometer scale. Integrating the data obtained from both scales, the combined description of the drill-core porosity served as a base to build a permeability model for the sample with the dimensions of that one scanned by the medical CT. The approach to generate the model depends on the type of heterogeneities associated with the sample. The homogeneous sandstone sample was modeled as a matrix with undefined porosity at the resolution of the medical tomography. The carbonate sample showed defined porosity at different scales: Moldic and vugular porosity were defined on the medical scale, while interparticle porosity was set on the synchrotron scale. Permeability was simulated in the micrometric model using an approach that integrates Stokes law and Darcy’s law in a percolating porous medium. Subsequently, the permeability was transferred to the millimetric model, creating the 3D distribution of porosity and permeability. Finally, the upscaled value for the samples was determined. This study contributes to the approaches regarding techniques of rock upscaling and digital rock physics and presents a workflow and some discussions of the multiscale approach.

Mini array of quantum Hall devices based on epitaxial graphene

Series connection of four quantum Hall effect (QHE) devices based on epitaxial graphene films was studied for realization of a quantum resistance standard with an up-scaled value. The tested devices showed quantum Hall plateaux RH,2 at a filling factor v = 2 starting from a relatively low magnetic field (between 4 T and 5 T) when the temperature was 1.5 K. The precision measurements of quantized Hall resistance of four QHE devices connected by triple series connections and external bonding wires were done at B = 7 T and T = 1.5 K using a commercial precision resistance bridge with 50 μA current through the QHE device. The results showed that the deviation of the quantized Hall resistance of the series connection of four graphene-based QHE devices from the expected value of 4×RH,2 = 2 h/e2 was smaller than the relative standard uncertainty of the measurement (<1 × 10−7) limited by the used resistance bridge.

Novel permeability upscaling method using Fast Marching Method

Despite computational advances, geo-cellular models are routinely upscaled for reservoir simulation purposes. Geo-cellular models typically incorporate small scale variations obtained from static data; whereas, simulation models capture dynamic data obtained from production of the field. The process of reducing the number of grid blocks in the geological model so that the simulation model can be run efficiently is called the process of upscaling. Upscaling requires combining the small grid blocks from geological model to make a larger grid block, as well as assigning reservoir properties on the coarse scale. Among the physical properties that need to be assigned, the assignment of permeability is the most challenging. Unlike porosity (or saturation), permeability is a dynamic property and capturing the fine scale dynamic displacement requires a non-linear upscaling of fine scale permeability values.The two types of methods used for permeability upscaling can be broadly categorized as static upscaling and dynamic upscaling. Static upscaling methods average fine scale values to calculate an upscaled value. These methods are efficient but may not capture the dynamic behavior. The dynamic upscaling methods are computationally expensive and may be dependent on boundary conditions. In this paper, we propose a new dynamic method for areal upscaling based on the fast marching algorithm. The Fast Marching Method (FMM) accurately captures the propagating pressure front as a function of time. We define the objective function as the difference in the propagation time between fine scale and coarse scale models. By minimizing the objective function, we calculate the permeability of the upscaled model.The Fast Marching Method is an extremely efficient method for calculating the pressure front propagation time based on reservoir properties. This method is based on solving the Eikonal equation using an upwind finite difference approximation. The advantage of this method is that, without running any flow simulation, we can calculate the pressure front location (radius of investigation) as a function of time. In this paper, using 2D and 3D models, we demonstrate that the calculated effective permeability using the new method can reproduce the dynamic behavior of the fine scale model.

New dynamic permeability upscaling method for flow simulation under depletion drive and no-crossflow conditions

The main purpose of upscaling in reservoir simulation is to capture the dynamic behavior of fine scale models at the coarse scale. Traditional static or dynamic methods use assumptions about the boundary conditions to determine the upscaled properties. In this paper, we show that the upscaled properties are strongly dependent on the flow process observed at the fine scale. We use a simple no-crossflow depletion drive process and demonstrate that an upscaled property is not a constant value. Instead, if the goal is to match the performance of the fine scale model, the upscaled permeability changes with time. We provide an analytical solution to determine the upscaled permeability and present the value of upscaled permeability under limiting conditions. Our equation suggests that it is possible that upscaled value can fall outside the range of fine scale values under certain conditions. We show that for pseudo steady state flow, using common averaging methods like arithmetic or even geometric averaging methods can lead to optimistic results. We also show that the no-crossflow solution is significantly different than crossflow solution at late times. We validate our method by comparing the results of the method with flow simulation results in two and multi-layered models.

Representative hydraulic conductivity of hydrogeologic units: Insights from an experimental stratigraphy

A critical issue facing groundwater flow models is the estimation of representative hydraulic conductivity assigned to the model units. In this study, an experiment-based, high-resolution hydraulic conductivity map offers a test case to evaluate this parameter. Various hydrogeological units are distinguished, each is of irregular shape with distinct heterogeneity pattern created by physical sedimentation. Extending a previous study which used numerical upscaling to compute equivalent conductivities for these units (at two upscaling scales) [Zhang, Y., Gable, C.W., Person, M., 2006. Equivalent hydraulic conductivity of an experimental stratigraphy – implications for basin-scale flow simulations. Water Resources Research 42, W05404. doi:10.1029/2005WR004720], this study compares them with local statistics and effective conductivities predicted by a stochastic theory. Results suggest that for a system with moderate ln K variance (4.07) and low topographic slope (∼1°), the arithmetic mean ( K A ) provides a good estimate for the maximum principal component ( K max ) of the equivalent conductivity. The minimum principal component ( K min ) lies between the harmonic and geometric means: its closeness to the geometric mean is affected by heterogeneity pattern and upscaling scale. Using K max (alternatively, the arithmetic mean), geometric mean, and ln( K ) variance, the stochastic theory predicts a K min that is consistent with the up-scaled value. Similarly, knowing K min , K max predicted by theory is also consistent with the up-scaled value. For most deposits (some with variance greater than 1), a low-variance version of the theory is more accurate than a high-variance version. However, the increase of topographic slope (to ∼4°) and total ln K variance (to 16) result in increased deviation of K max from K A . High variance also results in significantly larger anisotropy ratio, possibly due to the dominance of preferential flow. Finally, for select units, equivalent conductivity exhibits scale effect. Field scale representative elementary volume thus does not exist and upscaling the full unit is necessary to obtain the representative conductivity.

Scaling of land surface temperature using satellite data: A case examination on ASTER and MODIS products over a heterogeneous terrain area

Land surface temperature (LST) is a key parameter in numerous environmental studies. Surface heterogeneity induces uncertainty in pixel-wise LST. Spatial scaling may account for the uncertainty, however, different approaches lead to differences in scaled values. Satellite-retrieved LST may be representative of the pixel-wise LST and useful for scaling analysis, but the limited accuracy of retrieved values adds uncertainty into the scaled values. Based on the Stefan–Boltzmann (S–B) law, this study proposed scaling approaches for LST over flat and relief areas to explore the combined uncertainties in scaling using satellite-retrieved data. To take advantage of simultaneous, multi-resolution observations at coincident nadirs by the Advanced Spaceborne Thermal Emission Reflection Radiometer (ASTER) and the MODerate-resolution Imaging Spectroradiometer (MODIS), LST products from these two sensors were examined for part of the Loess Plateau in China. 90-m ASTER LST data were scaled up to 1 km using the proposed approaches, and variation in the LST was generally reduced after scaling. Amongst the sources of uncertainties, surface heterogeneity (emissivity) and different scaling approaches resulted in very minor differences, with a maximum difference of 0.2 K for the upscaled LST. Terrain features, taken as an areal weighting factor, had negligible effects on the upscaled value. Limited accuracy of the retrieved LST was the major uncertainty. The overall LST increased 0.6 K on average with correction for terrain-induced angular effect and 0.4 K for both angular and adjacency effects over the study area. Accounting for terrain correction in scaling is necessary for rugged areas. With terrain correction, the upscaled ASTER LST achieved an agreement of − 0.1 ± 1.87 K and a root mean square error (RMSE) of 1.87 K overall with the 1-km MODIS LST rectified by Wan et al.'s approach [Wan, Z., Zhang, Y., Zhang Q., Li, Z.-L. (2002b), Validation of the land-surface temperature products retrieved from Terra Moderate Resolution Imaging Spectroradiometer data. Remote Sensing of Environment, 83, 163–180]. Refining the rectification approach resulted in a better agreement of − 0.2 ± 1.57 K and a RMSE of 1.58 K.

Modelling heat transfer of carbon nanotubes

Modelling heat transfer of carbon nanotubes is important for the thermal management of nanotube-based composites and nanoelectronic devices. By using a finite element method for three-dimensional anisotropic heat transfer, we have simulated the heat conduction and temperature variations of a single nanotube, a nanotube array and a part of a nanotube-based composite surface with heat generation. The thermal conductivity used is obtained from the upscaled value from the molecular simulations or experiments. Simulations show that nanotube arrays have unique cooling characteristics due to their anisotropic thermal conductivity.

Upscaling and dispersion for transport in heterogeneous media

This paper focuses on upscaling of the transport equation for heterogeneous porous media with random flow. We consider the local flow field being a stationary random field and develop an upscaling by the recently developed coarse graining method which is based on filtering procedures in Fourier space. The coarse graining method is used to obtain an upscaled dispersion tensor which depends on the given length scale of the upscaling. We give explicit results for the scale-dependent dispersion coefficient in lowest-order perturbation theory. For finite length scales the upscaled dispersion models the effect of the unresolved subscale flow fluctuations, and for a global upscaling the upscaled value agrees with the well-known macrodispersion coefficient, which is, however, nearly approached for length scales larger than tenfold of the correlation length.

Upscaling for Reservoir Simulation

Upscaling for Reservoir Simulation