Anisotropic Dispersion and Upscaling for Miscible Displacement

SPE Journal ◽  
2015 ◽  
Vol 20 (03) ◽  
pp. 421-432 ◽  
Author(s):  
Olaoluwa O. Adepoju ◽  
Larry W. Lake ◽  
Russell T. Johns
Keyword(s):  
Finite Thickness ◽  
Miscible Displacement ◽  
Cell Permeability ◽  
Numerical Dispersion ◽  
Fine Scale ◽  
Direction Parallel ◽  
Miscible Displacements ◽  
Reservoir Models ◽  
Transverse Dispersion ◽  
Scale Models

Summary Dispersion is the irreversible mixing that occurs during miscible displacements. Dispersion can reduce local displacement efficiency by lessening solvent peak concentration or increase volumetric sweep efficiency by spreading of the injected solvent to more of the reservoir. Dispersion is therefore an important parameter in predicting and simulating miscible displacements. The difficulty of simulating miscible displacement and understanding the effect of dispersion is also compounded by numerical dispersion, which increases the apparent dispersion in finite-difference simulation models. This paper presents an approach to estimate the total longitudinal and transverse dispersion in large-scale media by use of continuous solvent injection in a medium of finite thickness. The simulations are based on the experimental arrangement of Blackwell (1962) to estimate transverse dispersion, with experiments consisting of coinjecting two miscible fluids into different sections of the medium at similar rates. This model arrangement, coupled with analytical solutions for the 2D convection/dispersion equation for a continuously injected solvent, allows us to determine longitudinal and transverse dispersivity simultaneously for the flow medium. In this manner, we investigate the effects of stochastic permeability distributions and other scaling groups affecting first-contact-miscible simulations on dispersion. Sensitivity analysis of dispersion in stochastic permeability fields confirms that both longitudinal and transverse dispersion are scale dependent. Results also show that the effect of increasing autocorrelation of cell permeability in the longitudinal direction (parallel to flow) is to increase longitudinal dispersion, as solvent travels through more continuous layers, while reducing transverse dispersion. Increasing autocorrelation in the transverse direction reduces dispersion in both directions. This reduction is caused by equilibration of solvent concentrations in continuous sections of the reservoir, resulting in reduced solute fingering and channeling. Finally, we developed a simple procedure to use the estimated dispersivities to determine a priori the maximum gridblock size that will maintain an equivalent level of dispersion between fine-scale models and upscaled coarse models. Large gridblock sizes can be used for highly heterogeneous and layered reservoir models. Nonuniform coarsening (upscaling) methods were also recommended and validated for reservoir models with sets of sequential but different permeability distributions. The procedure was extended to multicontact miscible simulations. The sweep and recovery from upscaled multicontact miscible simulations were comparable with those of fine-scale models.

Improved Treatment of Dispersion in Numerical Calculation of Multidimensional Miscible Displacement

1966 ◽  
Vol 6 (03) ◽  
pp. 213-216 ◽  
Author(s):  
D.W. Peaceman
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Miscible Displacement ◽  
Difference Approximation ◽  
Direction Parallel ◽  
Miscible Displacements ◽  
Simplifying Assumptions ◽  
Flow Vectors ◽  
The Stability ◽  
Dispersion Term

Abstract In previous papers by this author on numerical calculations of multidimensional miscible displacement, some simplifying assumptions were made in writing the dispersion term of the differential equation. It was assumed that the flow vector were essentially parallel to the x-axis. However, in most multidimensional miscible displacements, the flow vectors are not so simply oriented with respect to the coordinate axes. This paper derives the correct dispersion term for the more general case; gives a difference approximation for the dispersion term; and derives the stability criterion for the corresponding explicit difference equation. Introduction The differential equation for solvent concentration in miscible displacement is: .........(1) where v is the bulk flow velocity, C is concentration and D is the dispersion coefficient. For simple isotropic dispersion, D is a scalar quantity. however, dispersion in porous media is not generally isotropic since it is usually greater in the direction parallel to flow than in the direction transverse to flow. Hence, D must be treated as a tensor. Scheidegger has shown for an isotropic porous medium that, for the dispersion tensor to be invariant under coordinate transformations, there can be no more than two independent dispersivity factors; these are the longitudinal dispersivity D1, which acts in the direction of flow, and the transverse dispersivity Dt, which acts in the direction perpendicular to flow, In general, both D1 and Dt are functions of the magnitude of the flow velocity. In previous papers, the author made some simplifying assumptions in writing the dispersion term of the differential equation. It was assumed that the flow vectors were essentially parallel to the x-axis and, therefore, that the dispersion term. D C could be replaced by the sum: ...... (2) However, in most multidimensional miscible displacements, the flow vectors are not so simply oriented with respect to the coordinate axes. The purpose of this paper is to derive, without using tensor notation, the correct dispersion term for the more general case, to give a difference approximation for the dispersion term, and to derive the stability criterion for the corresponding explicit difference equation. DERIVATION OF DISPERSION TERM Let x, y be a fixed coordinate system and at any point let v be the velocity vector with magnitude v and angle, measured counter-clockwise from the x-axis. Let q be the vector which describes the rate and direction of flow of solvent due to dispersion. Consider a rotated coordinate system r, s where r is in the direction parallel to v, and s is in the perpendicular direction. Then, for an isotropic medium: SPEJ P. 213ˆ

Upscaling of outcrop information for improved reservoir modelling – exemplified by a case study on chalk

2021 ◽  
pp. petgeo2020-126
Author(s):  
Dongfang Qu ◽  
Peter Frykman ◽  
Lars Stemmerik ◽  
Klaus Mosegaard ◽  
Lars Nielsen
Keyword(s):  
Spatial Heterogeneity ◽  
Spatial Correlation ◽  
Petrophysical Properties ◽  
Fine Scale ◽  
Reservoir Modelling ◽  
Reservoir Models ◽  
Scale Models ◽  
Correlation Information ◽  
Scale Difference ◽  
Subsurface Reservoir

Outcrops are valuable for analogous subsurface reservoirs in supplying knowledge of fine-scale spatial heterogeneity pattern and stratification types, which are difficult to obtain from subsurface reservoir cores, well logs or seismic data. For petrophysical properties in a domain where the variations are relatively continuous and not dominated by abrupt contrasts, the spatial heterogeneity pattern can be characterized by a semivariogram model. The outcrop information therefore has the potential to constrain the semivariogram for subsurface reservoir modelling, even though it represents different locations and depths, and the petrophysical properties may differ in magnitude or variance. However, the use of outcrop derived spatial correlation information for petrophysical property modelling in practice has been challenged by the scale difference between the small support volume of the property measurements from outcrops and the typically much larger grid cells used in reservoir models. With an example of modelling the porosity of an outcrop chalk unit in eastern Denmark, this paper illustrates how the fine-scale spatial correlation information obtained from sampling of outcrops can be transferred to coarser scale models of analogue rocks. The workflow can be applied to subsurface reservoirs and ultimately improves the representation of geological patterns in reservoir models.

Numerical effects on the simulation of surfactant flooding for enhanced oil recovery

2021 ◽  
Author(s):  
Olaitan Akinyele ◽  
Karl D. Stephen
Keyword(s):  
Interfacial Tension ◽  
Cell Size ◽  
Oil Recovery ◽  
Flow In Porous Media ◽  
Numerical Dispersion ◽  
Sharp Change ◽  
Surfactant Flooding ◽  
Fractional Flow ◽  
Time Step ◽  
Scale Models

Abstract Numerical simulation of surfactant flooding using conventional reservoir simulation models can lead to unreliable forecasts and bad decisions due to the appearance of numerical effects. The simulations give approximate solutions to systems of nonlinear partial differential equations describing the physical behavior of surfactant flooding by combining multiphase flow in porous media with surfactant transport. The approximations are made by discretization of time and space which can lead to spurious pulses or deviations in the model outcome. In this work, the black oil model was simulated using the decoupled implicit method for various conditions of reservoir scale models to investigate behaviour in comparison with the analytical solution obtained from fractional flow theory. We investigated changes to cell size and time step as well as the properties of the surfactant and how it affects miscibility and flow. The main aim of this study was to understand pulse like behavior that has been observed in the water bank to identify cause and associated conditions. We report for the first time that the pulses occur in association with the simulated surfactant water flood front and are induced by a sharp change in relative permeability as the interfacial tension changes. Pulses are diminished when the adsorption rate was within the value of 0.0002kg/kg to 0.0005kg/kg. The pulses are absent for high resolution model of 5000 cells in x direction with a typical cell size as used in well-scale models. The growth or damping of these pulses may vary from case to case but in this instance was a result of the combined impact of relative mobility, numerical dispersion, interfacial tension and miscibility. Oil recovery under the numerical problems reduced the performance of the flood, due to large amounts of pulses produced. Thus, it is important to improve existing models and use appropriate guidelines to stop oscillations and remove errors.

Fine-Scale Fire Spread in Pine Straw

Fire ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 69
Author(s):  
Daryn Sagel ◽  
Kevin Speer ◽  
Scott Pokswinski ◽  
Bryan Quaife
Keyword(s):  
Fire Spread ◽  
Small Scale ◽  
Natural Setting ◽  
Fine Scale ◽  
Prescribed Burn ◽  
Fire Dynamics ◽  
Rate Of Spread ◽  
Scale Models ◽  
Fire Front ◽  
Visible Images

Most wildland and prescribed fire spread occurs through ground fuels, and the rate of spread (RoS) in such environments is often summarized with empirical models that assume uniform environmental conditions and produce a unique RoS. On the other hand, representing the effects of local, small-scale variations of fuel and wind experienced in the field is challenging and, for landscape-scale models, impractical. Moreover, the level of uncertainty associated with characterizing RoS and flame dynamics in the presence of turbulent flow demonstrates the need for further understanding of fire dynamics at small scales in realistic settings. This work describes adapted computer vision techniques used to form fine-scale measurements of the spatially and temporally varying RoS in a natural setting. These algorithms are applied to infrared and visible images of a small-scale prescribed burn of a quasi-homogeneous pine needle bed under stationary wind conditions. A large number of distinct fire front displacements are then used statistically to analyze the fire spread. We find that the fine-scale forward RoS is characterized by an exponential distribution, suggesting a model for fire spread as a random process at this scale.

Survival analysis at multiple scales for the modeling of track geometry deterioration

2017 ◽  
Vol 232 (3) ◽  
pp. 842-850 ◽  
Author(s):  
Negin Alemazkoor ◽  
Conrad J Ruppert ◽  
Hadi Meidani
Keyword(s):  
Survival Analysis ◽  
Multiple Scales ◽  
Fine Scale ◽  
Coarse Scale ◽  
Track Geometry ◽  
Time Period ◽  
Scale Models ◽  
Optimal Maintenance ◽  
Maintenance Decisions

Defects in track geometry have a notable impact on the safety of rail transportation. In order to make the optimal maintenance decisions to ensure the safety and efficiency of railroads, it is necessary to analyze the track geometry defects and develop reliable defect deterioration models. In general, standard deterioration models are typically developed for a segment of track. As a result, these coarse-scale deterioration models may fail to predict whether the isolated defects in a segment will exceed the safety limits after a given time period or not. In this paper, survival analysis is used to model the probability of exceeding the safety limits of the isolated defects. These fine-scale models are then used to calculate the probability of whether each segment of the track will require maintenance after a given time period. The model validation results show that the prediction quality of the coarse-scale segment-based models can be improved by exploiting information from the fine-scale defect-based deterioration models.

Modelling and Upscaling Unstable Miscible Displacement Processes: Characterisation of Physical Instabilities

2021 ◽  
Author(s):  
Precious Ogbeiwi ◽  
Karl Stephen
Keyword(s):  
Relative Permeability ◽  
Transport Coefficients ◽  
Miscible Displacement ◽  
Co2 Injection ◽  
Co2 Flooding ◽  
Fine Grid ◽  
Flow Models ◽  
Reservoir Models ◽  
Flood Simulations ◽  
Physical Instabilities

Abstract The compositional simulations are required to model CO2 flooding are computationally expensive particularly for fine-gridded models that have high resolutions, and many components. Upscaling procedures can be used in the subsurface flow models to reduce the high computation requirements of the fine grid simulations and accurately model miscible CO2 flooding. However, the effects of physical instabilities are often not well represented and captured by the upscaling procedures. This paper presents an approach for upscaling of miscible displacements is presented which adequately represents physical instabilities such as viscous and heterogeneity induced fingering on coarser grids using pseudoisation techniques. The approach was applied to compositional numerical simulations of two-dimensional reservoir models with a focus on CO2 injection. Our approach is based on the pseudoisation of relative permeability and the application of transport coefficients to upscale viscous fingering and heterogeneity-induced channelling in a multi-contact miscible CO2 injection. Pseudo-relative permeability curves were computed using a pseudoisation technique and applied in combination with transport coefficients to upscale the behaviour of fine-scale miscible CO2 flood simulations to coarser scales. The accuracy of the results of the pseudoisation procedures were assessed by applying statistical analysis to compare them to the results of the fine grid simulations. It is observed from the results that the coarse models provide accurate predictions of the miscible displacement process and that the fingering regimes are adequately captured in the coarse models. The study presents a framework that can be employed to represent the dynamics of physical instabilities associated with miscible CO2 displacements in upscaled coarser grid reservoir models.

scholarly journals Spatiotemporal Variations in Mercury Bioaccumulation at Fine and Broad Scales for Two Freshwater Sport Fishes

Water ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 1625 ◽  
Author(s):  
Shyam Thomas ◽  
Stephanie Melles ◽  
Satyendra Bhavsar
Keyword(s):  
Climate Change ◽  
Environmental Factors ◽  
Northern Pike ◽  
Space Time ◽  
Fine Scale ◽  
Spatiotemporal Variations ◽  
Scale Models ◽  
Change Context ◽  
Mercury Bioaccumulation ◽  
Broad Scale

Bioaccumulation of mercury in sport fish is a complex process that varies in space and time. Both large-scale climatic as well as fine-scale environmental factors are drivers of these space-time variations. In this study, we avail a long-running monitoring program from Ontario, Canada to better understand spatiotemporal variations in fish mercury bioaccumulation at two distinct scales. Focusing on two common large-bodied sport fishes (Walleye and Northern Pike), the data were analyzed at fine- and broad-scales, where fine-scale implies variations in bioaccumulation at waterbody- and year-level and broad-scale captures variations across 3 latitudinal zones (~5° each) and eight time periods (~5-year each). A series of linear mixed-effects models (LMEMs) were employed to capture the spatial, temporal and spatiotemporal variations in mercury bioaccumulation. Fine-scale models were overall better fit than broad-scale models suggesting environmental factors operating at the waterbody-level and annual climatic conditions matter most. Moreover, for both scales, the space time interaction explained most of the variation. The random slopes from the best-fitting broad-scale model were used to define a bioaccumulation index that captures trends within a climate change context. The broad-scale trends suggests of multiple and potentially conflicting climate-driven mechanisms. Interestingly, broad-scale temporal trends showed contrasting bioaccumulation patterns—increasing in Northern Pike and decreasing in Walleye, thus suggesting species-specific ecological differences also matter. Overall, by taking a scale-specific approach, the study highlights the overwhelming influence of fine-scale variations and their interactions on mercury bioaccumulation; while at broad-scale the mercury bioaccumulation trends are summarized within a climate change context.

Testing plausible upper-mantle compositions using fine-scale models of the 410-km discontinuity

1999 ◽  
Vol 26 (11) ◽  
pp. 1641-1644 ◽  
Author(s):  
James B. Gaherty ◽  
Yanbin Wang ◽  
Thomas H. Jordan ◽  
Donald J. Weidner
Keyword(s):  
Upper Mantle ◽  
Fine Scale ◽  
Scale Models

A Method for Predicting the Performance of Unstable Miscible Displacement in Heterogeneous Media

1963 ◽  
Vol 3 (02) ◽  
pp. 145-154 ◽  
Author(s):  
E.J. Koval
Keyword(s):  
Porous Medium ◽  
Heterogeneous Media ◽  
Research Effort ◽  
Heterogeneous Systems ◽  
Extended Period ◽  
Miscible Displacement ◽  
Longitudinal Dispersion ◽  
Miscible Displacements ◽  
Geometrical Effects ◽  
Displacement Processes

KOVAL, E.J., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF. Abstract Practical miscible displacement processes will be characterized by fingering of the solvent into the oil. The fingering process is brought on by viscosity differences, and can be accentuated by channeling and longitudinal dispersion. The effects of these factors on the efficiency of unstable completely miscible displacements are combined in what is called the K-factor method. This method, analogous to the Buckley-Leverett method, predicts recovery and solvent cut as a function of pore volumes of solvent injected. Experimental data are included and show excellent agreement with theory for a wide variety of sandstone cores and viscosity ratios. Introduction Theoretical considerations, laboratory experiments, and pilot tests lead to the conclusion that miscible displacements in the field will be unstable. In an unstable miscible displacement, the solvent fingers through the oil. This fingering leads to early breakthrough of the solvent and an extended period during which both oil and solvent are produced.For such a system, there appear to be four principal factors which bring about or accentuate the effects of instability: longitudinal dispersion (including geometrical effects), channeling, viscosity differences and gravity differences. Other factors, such as diffusion and flooding rate, can also influence the effects of instability, but at the flooding rates considered in this report, circa 15 ft/D, they are unimportant. Longitudinal dispersion can be thought of as a spreading of the solvent front caused by the presence of microscopic inhomogeneities. Channeling of the solvent occurs when a porous medium has macroscopic inhomogeneities; i.e., gross permeability variations. Viscosity differences lead to fingering of the less viscous solvent. This difference in viscosity accelerates the growth of fingers along paths previously developed because of permeability variations. Gravity differences lead to overriding of the usually less dense solvent. Although gravity effects are generally small at a flooding rate of 15 ft/D, they would, nevertheless, unnecessarily complicate the interpretation of flooding experiments. For this reason, all the experiments reported herein were done with matched density fluids.Fingering and the resultant poor areal sweep were recognized early as the dominant influences on the efficiency and the economics of miscible displacement processes. Much research effort has been spent on ways to minimize fingering and increase areal sweep such as the use of graded viscosity slugs or water slugs. Some researchers attempted to work out ways to prevent fingering completely; i.e., to achieve a stable displacement through gravity or latitudinal dispersion stabilization. Others did not attempt to control fingering but obtained an economic process by merely recycling the solvent and sweeping pattern by pattern.During this period, all aspects of fingering came under close scrutiny. Some researchers reported on how fingering looks and how it is affected by viscosity ratio, geometry, and slug size. Peaceman and Rachford suggested a mathematical approach to the prediction of unstable miscible displacements in relatively homogeneous sand packs, but their work cannot be extended conveniently to heterogeneous systems. Hence for a heterogeneous system, no method is presently available for predicting solvent cut and recovery as functions of pore volumes of solvent injected.The purpose of this investigation was to attempt to fill in the gap in our knowledge concerning the prediction of performance of unstable miscible displacements. Necessarily, the system selected for study was a relatively simple one. The restrictions placed on the system were:The system was linear;The solvent was miscible in all proportions with the oil in place;The solvent was continuously injected into the porous medium;Gravitational effects were eliminated by matching densities andAll the flood rates were high and constant at 15 ft/D to avoid any small rate effect and to minimize any diffusion effects. To simplify and to indicate that both longitudinal dispersion and channeling arise from permeability variations, the effects which they cause or influence have been termed heterogeneity effects. SPEJ P. 145^

Development and Application of Variational Methods for Simulation of Miscible Displacement in Porous Media

1977 ◽  
Vol 17 (03) ◽  
pp. 228-246 ◽  
Author(s):  
A. Settari ◽  
H.S. Price ◽  
T. Dupont
Keyword(s):  
Finite Difference ◽  
Miscible Displacement ◽  
High Order ◽  
Numerical Dispersion ◽  
Convection Diffusion ◽  
Finite Difference Approximations ◽  
Variational Approximations ◽  
Difference Approximations ◽  
Low Dispersion ◽  
Finite Difference Techniques

Abstract Many reservoir engineering problems involve solving fluid flow equations whose solutions are characterized by sharp fronts and low dispersion levels. This is particularly important in tracking small slugs that are characteristic of chemical floods, polymer floods, first- and multiple-contact hydrocarbon miscible polymer floods, first- and multiple-contact hydrocarbon miscible displacements, and most thermal processes. The use of finite-difference approximations to solve these problems when low dispersion levels and small slugs need to be modeled accurately may be prohibitively expensive. This paper shows that the use of high-order variational approximations is a very effective means for economically solving these problems. This paper presents some numerical results that demonstrate that high-order variational methods can be used to solve two-dimensional reservoir engineering problems where finite-difference approximations would require 104 problems where finite-difference approximations would require 104 to 105 blocks. The variational solutions are shown to be essentially insensitive to grid orientation for unfavorable mobility ratios up to M = 100. Introduction The equations describing miscible displacement in a porous medium (convection-diffusion equations) are among the more difficult to solve by numerical means. The character of the concentration equation ranges from parabolic to almost hyperbolic depending on the ratio of convection to diffusion (Peclet number). Consequently, the finite-difference techniques developed for solving the convection diffusion problem can be divided into two categories: those solving the problem as parabolic and those treating the problem as hyperbolic. The parabolic techniques are unsatisfactory when the diffusion becomes small compared with the convection. The methods using central difference approximations for the convection terms oscillate. Price et al. have shown that these oscillations can be eliminated only by using small spatial increments. Methods using upstream difference approximations do not oscillate, but they introduce large truncation errors that have the character of a large diffusion term. Lantz has shown that for many practical problems, reducing the magnitude of numerical dispersion problems, reducing the magnitude of numerical dispersion sufficiently so that it does not mask the physical dispersion will force an impractically fine grid. Several improvements have been suggested, such as transfer of overshoots truncation-error cancellation, and two-point upstream approximations; but none of these is quite satisfactory in the general case. The hyperbolic methods (method of characteristics, point tracking, etc.) also pose many practical problems. These include the complex treatment required for sources and sinks, the need to redistribute points continually when modeling converging and diverging flow, the problem of maintaining a material balance, problems created by complex geometries, and the practical limitation problems created by complex geometries, and the practical limitation of the time-step size. Moreover, these schemes cannot be shown to converge, thereby making the choice of grid size and point distribution fairly arbitrary. Finally, many nonlinearities, such as reactions and adsorption, need to be treated point-by-point, requiring large amounts of computer time and storage. Because the major difficulty in solving the miscible displacement problem is the determination of an accurate approximation to a very sharp concentration front, one of the most promising alternatives to the schemes mentioned above is the use of promising alternatives to the schemes mentioned above is the use of high-order variational approximations, such as those proposed by Ciarlet et al. These methods (which include Galerkin and finite-element methods) are potentially far more accurate for a given amount of computation than the standard finite-difference techniques and, therefore, more able to solve problems that would otherwise be impractical. SPEJ P. 228

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