Effects Of Small-scale Heterogeneities Research Articles

Upscaling acoustic wave equation using renormalization group theory

Seismic waves interact with a broad range of heterogeneities as they propagate through the earth. Simulating this full range of scales for wave propagation requires capturing heterogeneities of all scales, which can be computationally unaffordable. In such cases, we have relied on macroscopic representations of media obtained through an upscaling process that preserves the effects of small-scale heterogeneities (in comparison with the wavelengths of interest). Here, we discuss the application of the renormalization group (RG) theory-based upscaling to the 2D acoustic wave equation. RG-based upscaling requires constructing a special Fourier operator and is implemented using a domain decomposition in conjunction with an “expansion and truncation” method to mitigate edge effects. We test this upscaling method on several benchmark models and in the context of reverse time migration. The upscaled models obtained using this method indicate a good consistency for generated waveforms, whereas the runtime for simulations is reduced by at least an order of magnitude.

Characterizing locality- and scale-dependent heterogeneity in conglomerate core and associated fluid flow using X-ray CT imaging

Heterogeneous conglomerates preserving matrix and clasts caused the complex distribution of pore spaces and can eventually influence fluid flow. To evaluate the influence of complex heterogeneous features in conglomerate rock on both pore distribution and fluid flow, two conglomerate cores were analyzed using X-ray computed tomography (X-ray CT). Conglomerate cores (Core I and Core II) characterized by different distributions of matrix and clasts represented highly heterogeneous features; in Core I (clast-supported conglomerate), clasts were evenly distributed through the entire core, but in Core II (matrix-supported conglomerate), large-sized clasts were dominantly positioned at the bottom of core. Such heterogeneous features caused differences in volume of pores (Core I: 7,409.54 mm3 and Core II: 17,525.83 mm3). Based on X-ray CT image analysis, the numerical model (single-phase fluid simulation) was implemented to evaluate the permeability of conglomerate at different scales. Especially, due to inner heterogeneous features (e.g., matrix, clasts, and cracks) in conglomerate cores, permeabilities were changed significantly dependent on the selected location and size of sub-domains representing a few centimeters (kh and kv: 1.18×10-12 to 1.01×10-10m2). This suggests that the effect of small-scale heterogeneity should be evaluated carefully when heterogeneous conglomerate cores were analyzed. Additionally, the measurement scale for X-ray CT analysis should be large enough to capture such heterogeneous features (e.g., the size of clasts).

Robust processing of airborne laser scans to plant area density profiles

Abstract. We present a new algorithm for the estimation of the plant area density (PAD) profiles and plant area index (PAI) for forested areas based on data from airborne lidar. The new element in the algorithm is to scale and average returned lidar intensities for each lidar pulse, whereas other methods do not use the intensity information at all, use only average intensity values, or do not scale the intensity information, which can cause problems for heterogeneous vegetation. We compare the performance of the new algorithm to three previously published algorithms over two contrasting types of forest: a boreal coniferous forest with a relatively open structure and a dense beech forest. For the beech forest site, both summer (full-leaf) and winter (bare-tree) scans are analyzed, thereby testing the algorithm over a wide spectrum of PAIs. Whereas all tested algorithms give qualitatively similar results, absolute differences are large (up to 400 % for the average PAI at one site). A comparison with ground-based estimates shows that the new algorithm performs well for the tested sites. Specific weak points regarding the estimation of the PAD from airborne lidar data are addressed including the influence of ground reflections and the effect of small-scale heterogeneity, and we show how the effect of these points is reduced in the new algorithm, by combining benefits of earlier algorithms. We further show that low-resolution gridding of the PAD will lead to a negative bias in the resulting estimate according to Jensen's inequality for convex functions and that the severity of this bias is method dependent. As a result, the PAI magnitude as well as heterogeneity scales should be carefully considered when setting the resolution for the PAD gridding of airborne lidar scans.

Effect of small-scale heterogeneity on biopolymer performance in carbonates

Polymer flooding is a well-established chemical method for enhancing oil recovery in sandstones; however, it has a limited application in carbonates. This is due to the harsh reservoir conditions in carbonates including high temperature, high salinity, and high heterogeneity with low permeability. This paper numerically investigates the effect of Schizophyllan biopolymer on oil recovery from carbonates. The effect of biopolymer on oil recovery was predicted by running several 1D simulations. Biopolymer flow behavior was modeled based on experimental data. The results showed that the effect of the investigated biopolymer on oil recovery was not much pronounced compared to conventional waterflooding. This is due to small-scale heterogeneity, which increased effective shear rate and hence, decreased in-situ polymer viscosity. Formation permeability, polymer viscosity, and oil saturation maps were consistent in justifying this observation. The findings of this study were supported by fractional flow and mobility ratio analyses. This work highlights the importance of small-scale heterogeneity of the core in modeling polymer flooding, particularly the shear effect on polymer viscosity.

CO2-brine relative permeability and capillary pressure of Tuscaloosa sandstone: Effect of anisotropy

Relative permeability and capillary pressure are known as essential properties that have substantial impacts on the accuracy of reservoir simulations. The effect of small-scale heterogeneity and lamination in the rock structure is often ignored during the measurement of capillary pressure and relative permeability curves in core samples. This study highlights the remarkable impact of anisotropy on the multiphase flow properties of stratified formations. A series of steady-state CO2-brine drainage and imbibition tests are conducted at reservoir conditions in horizontal and vertical core samples of the Tuscaloosa sandstone from the Cranfield CO2 injection site in Mississippi. The relative permeability curves represent an anisotropic behavior influenced by the heterogeneous and laminated structure of realistic rock samples. The CO2 saturation profiles during drainage and imbibition cycles indicate that the phase distribution in the pore space is controlled by core-scale heterogeneity in the porosity distribution among the laminations that causes capillary pressure inhomogeneity. Using the saturation profile during the imbibition cycle, the trapping characteristic of the horizontal and vertical rock samples are compared and we found that the capillary trapping is less likely in the vertical direction. Furthermore, the centrifuge-measured capillary pressure demonstrates distinctive characteristics for horizontal and vertical core samples. Since the flooding experiments are performed under capillary controlled flow, the capillary pressure contrast in the laminated structure of the rock strongly affects the relative permeability. The presented results can potentially improve the accuracy of the large-scale simulations for the CO2 post-injection period, in which the vertical displacement has an important role in the plume migration.

The systematic effect of streambed conductivity heterogeneity on hyporheic flux and residence time

Abstract A systematic understanding of hyporheic flux (HF) and residence times (RT) is important as they are a major control of biogeochemical processing in streambeds. Previous studies addressing the effect of heterogeneity in streambed hydraulic conductivity (K) on HF and RT have come to deviating conclusions depending on the specific study design and the selection of heterogeneity cases being investigated. To more systematically evaluate the effects of conductivity heterogeneity on HF and RT, we simulated hyporheic exchange induced by idealized streambed ripples over a large range of heterogeneities. Conductivity heterogeneity was represented in the simulations in terms of 10,000 different heterogeneity realizations from a geostatistical model based on continuous Gaussian and discrete indicator random fields. We demonstrate that any isotropic homogeneous K-field, as an average of a heterogeneous K-field, can only match RT or HF of the respective heterogeneous K-field, but never both. We found exponential correlations of RT and HF with the variance of heterogeneous conductivity. Based on these correlations, an equivalent anisotropic homogeneous conductivity tensor K can be derived. This equivalent anisotropic K efficiently accounts for the effects of small scale heterogeneity on HF and RT. It can be calculated from the median and variance of the hydraulic conductivity distribution of the targeted heterogeneous sediment, without explicitly characterizing the sediment texture.

Nonlinear effects of locally heterogeneous hydraulic conductivity fields on regional stream–aquifer exchanges

Abstract. Computational experiments are performed to evaluate the effects of locally heterogeneous conductivity fields on regional exchanges of water between stream and aquifer systems in the Middle Heihe River basin (MHRB) of northwestern China. The effects are found to be nonlinear in the sense that simulated discharges from aquifers to streams are systematically lower than discharges produced by a base model parameterized with relatively coarse effective conductivity. A similar, but weaker, effect is observed for stream leakage. The study is organized around three hypotheses: (H1) small-scale spatial variations of conductivity significantly affect regional exchanges of water between streams and aquifers in river basins, (H2) aggregating small-scale heterogeneities into regional effective parameters systematically biases estimates of stream–aquifer exchanges, and (H3) the biases result from slow paths in groundwater flow that emerge due to small-scale heterogeneities. The hypotheses are evaluated by comparing stream–aquifer fluxes produced by the base model to fluxes simulated using realizations of the MHRB characterized by local (grid-scale) heterogeneity. Levels of local heterogeneity are manipulated as control variables by adjusting coefficients of variation. All models are implemented using the MODFLOW (Modular Three-dimensional Finite-difference Groundwater Flow Model) simulation environment, and the PEST (parameter estimation) tool is used to calibrate effective conductivities defined over 16 zones within the MHRB. The effective parameters are also used as expected values to develop lognormally distributed conductivity (K) fields on local grid scales. Stream–aquifer exchanges are simulated with K fields at both scales and then compared. Results show that the effects of small-scale heterogeneities significantly influence exchanges with simulations based on local-scale heterogeneities always producing discharges that are less than those produced by the base model. Although aquifer heterogeneities are uncorrelated at local scales, they appear to induce coherent slow paths in groundwater fluxes that in turn reduce aquifer–stream exchanges. Since surface water–groundwater exchanges are critical hydrologic processes in basin-scale water budgets, these results also have implications for water resources management.

Numerical and analytical study of effects of small scale heterogeneity on CO2/brine multiphase flow system in horizontal corefloods

Here we provide a comprehensive study of the fundamental physics both numerically and analytically, of the combined influence of heterogeneity, viscous forces, gravity, and capillarity on multiphase flow of CO2 and brine. Specifically, steady-state 3D coreflood displacements in heterogeneous cores over a range of relevant conditions are simulated to study the impact of sub-core heterogeneity on CO2/brine displacements over a wide range of flowrates. Various degrees of heterogeneity are generated based on the normal random distribution as well as for real models of cores based on 3D X-Ray tomography. A 2D semi-analytical model considering gravity and permeability heterogeneity is developed for predicting brine displacement efficiency over a wide range of capillary numbers that provides good agreement with the simulated 3D results. The analytical derivation is general and the provided solution can estimate the flow regimes for horizontal core floods efficiently.

Effect of small-scale heterogeneities on interpretation of crustal compositions exemplified by a layered anorthosite

The composition of the lower crust has a significant effect on geodynamic processes because it influences physical rock properties such as densities and seismic velocities. Compositional differences in lower crustal rocks are potentially large and exist on the scales of centimeters up to kilometers resulting in non-unique seismic and gravity data. While larger heterogeneities can be detected as reflections on seismic profiles, irregular small-scale compositional variations are not likely to be discovered, but will influence the averaged seismic velocities and densities of an area. The extent and effects of such small-scale heterogeneities are explored on an exposed high-grade layered anorthositic body by providing a detailed field map, petrological descriptions, pycnometry measurements as well as whole rock and mineral analyses combined with thermodynamic phase equilibria calculations. To evaluate the results of our thermodynamic calculations, densities and mineral modes obtained from the modeled phase equilibria are compared to measured densities and estimated mineral modes from rock samples. The proportion of mafic to ultramafic (plagioclase-poor) rocks in the mapped field area amounts to 10–15% but higher proportions of these rock types in the lower crust are feasible. To further study the effects of compositional variations, we have generated mixtures of mafic to ultramafic and anorthositic/intermediate rocks until the average properties of these mixtures are comparable to those of mafic granulites (3000–3100 kg/m3; 7.1–7.3 km/s). Mixtures of anorthosite with 40–45% and of tonalite with 50–60% high-grade mafic to ultramafic rocks yield average densities and seismic velocities similar to mafic granulites although they still contain 50–60 vol.% plagioclase. Hence small-scale mixing of certain rock types may result in the overestimation of the proportion of mafic (garnet) granulites in the lithologic interpretation of crustal compositions from seismic data. Since the transition to eclogite-facies in plagioclase-rich rocks is shifted to higher pressures and anorthositic/intermediate eclogites yield lower densities, a lower crust with higher modal amounts of plagioclase may not always provide the significant densification needed for certain geodynamic settings (e.g. delamination or subsidence).

Modeling for geophysical monitoring of multiple phase saturation of rocks: Applications to CO2 sequestration

A numerical study of measured petrophysical properties was used to develop and demonstrate a workflow for monitoring [Formula: see text] sequestration in a subsurface reservoir. Amplitude versus offset (AVO) attributes can be sensitive discriminators for [Formula: see text] presence. The sensitivity of AVO attributes to [Formula: see text] saturation increased when an upscaling scheme that propagated the effects of small-scale heterogeneities was used. A global sensitivity analysis was then performed to study the importance of various rock properties that would be used as inputs in reservoir characterization. An extensive rock properties database for a carbonate reservoir containing natural variability was used. The various rock properties were ranked by their importance to the quality of the remote characterization. Defining and using the intrinsic correlations among petrophysical properties were essential to evaluating whether a particular reservoir was amenable (sensitive) to this AVO characterization method.

Analysis of Heavy Oil Recovery by Thermal EOR in a Meander Belt: From Geological to Reservoir Modeling

The objectives of this work is to assess the impact of reservoir heterogeneities on heavy oil recovery of a reservoir analogue of meander belt through the Steam Assisted Gravity Drainage (SAGD) process by using numerical models. These models are obtained with different scales of upscaling of the geological model. Meander belts consist of point bar deposits, characterized by a 3D complex internal architecture, with different scales of heterogeneities, which distribution is associated with the depositional processes. Based on a 3D outcrop description of a meander belt analogue to the Canadian heavy-oil fields, the approach includes three steps: 1) the construction of a reference static reservoir model based on a very fine description of the outcrops in terms of architecture and geological heterogeneities, 2) upscaling of the grid at different scales using different upscaling factors in order to evaluate their impact on the heterogeneity distribution in the reservoir, 3) reservoir SAGD simulations using horizontal well doublet (steam injector and producer) across the meander belt, so as to assess the impact of upscaling of heterogeneities on heavy oil production. The impact of heterogeneities on simulation results are evaluated for several upscaling stages. Results show that heterogeneity distribution has an impact on fluid flow at different stages of production. On the fine gridded model, small scale heterogeneities impact the steam chamber development and fluid flow in the wellbore vicinity at the beginning of the steam injection, whereas large scale heterogeneities strongly influence oil recovery during the whole recovery process and lower the efficiency of the reservoir drainage. On coarser grids, the effect of small-scale heterogeneities can be diminished, depending on the upscaling stage. The geomechanical effect is not taken into account in this work, the objective being to assess the impact of heterogeneities on oil recovery. The performance of SAGD is clearly linked to the steam chamber development, which depends on the degree of heterogeneities present in the reservoir. The simulation workflow and the sensitivity study on the upscaling method contribute to a better restoration of the heterogeneity distribution in the reservoir. The negative effect of these heterogeneities during the oil recovery must thus be quantified in order to monitor the thermal production at crucial periods of the production process.

Upscaling of small-scale heterogeneities to flow units for reservoir modelling

The effect of small-scale heterogeneities on fluid flow through a reservoir is determined using one of so called mathematical upscaling methods, that as such are fairly well established. Less well established, however, are the operations that provide their input, i.e. construction of reservoir model and acquisition of data. We thereto propose systematic description of facies and bed and systematic data sampling. We upscale the ensuing model and data through a two-step flow simulation, i.e. first on the flow cell and then on the flow unit scale. This work provides for a straightforward and easily applicable upscaling method that incorporates the effects of all heterogeneities.

Effects of small scale heterogeneity of vegetation on radiative transfer in forest fire

Wildland fires are driven by vegetation which is a medium that can be very heterogeneous at several scales. These heterogeneities from shoot to stand levels are likely to affect fire behavior. In the context of theoretical fire modelling, we investigated the effects of departure from randomness in radiative transfer. A methodology derived from approach in other turbid media for solar radiation was adapted to our problem. From several Mediterranean fuel type calculations, conditions on vegetation parameters that provide heterogeneity effects were exhibited. Fuel description was done from field measurements and plant architecture modelling. Results show that both a strong spatial variation in fuel distribution and vegetation dense enough were required to provide heterogeneity effects. The combined effect of fuel surface area, cover fraction and heterogeneity size was studied. Radiative transfer in heterogeneous maritime pines and dense shrub stands is shown to be very significantly affected by heterogeneity. On the contrary, lower effects are expected from Aleppo pine and light shrub fuel distributions. Further work will determine the appropriate resolution scale for vegetation description of each species and a way to take into account heterogeneity when no detailed information is available on fuel distribution. © 2006 Published by Elsevier B.V.

Significance of Legumes for the Distribution of Plant Species in Grassland Ecosystems at Different Altitudes in the Alps

The composition of plant species in grassland communities is determined in part by negative and positive interactions among neighbouring plants; such interactions could be dispersal or – in the case of legumes – symbiotic N2 fixation or resource acquisition. We hypothesised that small scale distribution of plant species may be in part affected by (i) the plant's narrow dispersal in the close neighbourhood, (ii) by the N input into the soil through symbiotic N2 fixation by legumes, and (iii) by small scale heterogeneity of N availability in the soil. To address these hypotheses, in four species-rich semi-natural and natural grasslands along an altitudinal gradient from 900 to 2100 m a.s.l., the association of species that grew within a radius of 5 cm of selected plant species (central plant, CP) was studied. The distribution of the plant species within the plant community was not random and the species composition that grew within a radius of 5 cm of a CP varied with altitude. The strength of the association between a CP and its neighbouring species decreased with increasing altitude. The 5 cm radius was most dominated by individuals of the same species as the CP, suggesting that narrow dispersal (either through seeds or through vegetative propagates) may be most important for the species composition within the sward. δ15N values of non leguminous species growing near legumes compared to non leguminous species that grew far from legumes indicate that N from symbiotic N2 fixation may be another factor influencing the association of plant species. Effects of small scale heterogeneity in apparent N availability in the soil were detected to be the least important for the species composition.

The effect of small-scale heterogeneity on the arrival time of waves

SUMMARY Small-scale heterogeneity alters the arrival times of waves in a way that cannot by explained by ray theory. This is because ray theory is a high-frequency approximation that does not take the finite frequency of wavefields into account. We present a theory based on the first-order Rytov approximation that predicts well the arrival times of waves propagating in media with small-scale inhomogeneity with a length scale smaller than the width of Fresnel zones. In the regime for which scattering theory is relevant we find that caustics are easily generated in wavefields, but this does not influence the good prediction of finite frequency arrival times of waves by scattering theory. The regime of scattering theory is relevant when the characteristic length of heterogeneity is smaller than the width of Fresnel zones. The regime of triplications is independent of frequency but it is more significant the greater the magnitude of slowness fluctuations.

EFFECTS OF SPATIAL VARIABILITY ON THE SCALING OF LAND SURFACE PARAMETERIZATIONS

Understanding and modelling physical and dynamical processes over heterogeneous land surfaces have becomea central focus of many recent studies. There is aconsiderable debate, however, over how to representthe effects of spatial heterogeneity in mesoscale andglobal scale models. Here, a computationally efficientanalytical approach is presented to evaluate scalingproperties of land surface representations. It isshown that the effects of spatial variability may not benegligible for commonly encountered land surfaces andassociated parameterizations. Second-order correction termsinvolving variances of the parameters and covariancesof each pair of land surface parameters are developedto account for the effects ofheterogeneity. Using this analytical approach, weshow that the detail of spatial heterogeneity may not beimportant for the infrared radiation and reflectedsolar radiation from the surface, while sensible andlatent heat fluxes are shown to be sensitive toheterogeneity. Assumptions related todifferent parameterizations for the same physicalprocess could potentially lead to different inferencesregarding the influence of spatial heterogeneity. Theproposed approach, however, is capable of identifying therole of different parameterizations in estimating theinfluence of spatial heterogeneity. These analyticalresults are consistent with the results of severalrecent numerical and field experiments that deal withthe effects of small-scale heterogeneity in landsurface characteristics.

Effects of soil moisture aggregation on surface evaporative fluxes

The effects of small-scale heterogeneity in land surface characteristics on the large-scale fluxes of water and energy in the land-atmosphere system has become a central focus of many climatology research experiments. The acquisition of high resolution land surface data through remote sensing and intensive land-climatology field experiments (like HAPEX, FIFE, and BOREAS) has provided data to investigate the interactions between microscale land-atmosphere interactions and macroscale models. To determine the effect of small scale heterogeneities, the spatially averaged evaporative fraction is analytically derived for spatially variable soil moisture and soil-atmospheric controls on evaporation at low soil moisture. This average evaporative fraction is compared with the evaporative fraction determined using the spatially averaged soil moisture, as if from a lumped, or aggregated, land surface model. Results show that the lumped-model based evaporation will over estimate evaporation during periods of low atmospheric demands (early morning/late afternoon, Winter periods, etc.) and under estimate evaporation during periods of high demand (midday Summer periods.) The accuracy of using ‘effective’ parameters in lumped macroscale models depends on the variability of soil moisture and the sensitivity of the soil-vegetation system to low soil moisture.

Scaling, soil moisture and evapotranspiration in runoff models

The effects of small-scale heterogeneity in land surface characteristics on the large-scale fluxes of water and energy in the land-atmosphere system has become a central focus of many of the climatology research experiments. The acquisition of high resolution land surface data through remote sensing and intensive land-climatology field experiments (like HAPEX and FIFE) has provided data to investigate the interactions between microscale land-atmosphere interactions and macroscale models. One essential research question is how to account for the small scale heterogeneities and whether ‘effective’ parameters can be used in the macroscale models. To address this question of scaling, the probability distribution for evaporation is derived which illustrates the conditions for which scaling should work. A correction algorithm that may be appropriate for the land parametrization of a GCM is derived using a second order linearization scheme. The performance of the algorithm is evaluated.

Scaling Water and Energy Fluxes in Climate Systems: Three Land-Atmospheric Modeling Experiments

The effects of small-scale heterogeneity in land-surface characteristics on the large-scale fluxes of water and energy in the land-atmosphere system have become a central focus of many of the climatology research experiments. The acquisition of high-resolution land-surface data through remote sensing and intensive land-climatology field experiments (like HAPEX and EIFE) has provided data to investigate the interactions between microscale land-atmosphere interactions and macroscale models. One essential research question is how to account for the small-scale heterogeneities and whether “effective” parameters can be used in the macroscale models. To address this question of scaling, three modeling experiments were performed and are reviewed in the paper. The first is concerned with the land-surface hydrology during rain events and between rain events. The second experiment applies the Simple Biosphere Model (SiB) to a heterogeneous domain and the spatial and temporal latent beat flux is analyzed. The third experiment uses thermatic mapper (TM) data to look at the scaling of the normalized vegetation index (NDVI), latent heat flux, and sensible heat flux through either scaling of the TM-derived fields using the TM data or the fields derived from aggregated TM data. In all three experiments it was found that the surface fluxes and land characteristics can be sealed, and that macroscale models based on elective parameters are sufficient to account for the small-scale heterogeneities investigated. The paper also suggests that the scale at which a macroscale model becomes valid, the representative elementary scale (REA), is on the order 1.5–3 correlation lengths, which for land processes investigated appears to be about 1000–1500 m. At scales less than the REA scale, exact patterns of subgrid heterogeneities are needed for accurate small-scale modeling.

Significance of small-scale spatio-temporal heterogeneity in phytoplankton abundance for energy flow in Mytilus edulis

The effect of small-scale heterogeneity of phytoplankton abundance on mussel (Mytilus edulis L.) energy flow was studied by comparing filtration rate, consumption and assimilation efficiency in two groups of mussels fed under constant (control) and fluctuating (24 min) seston regimes of the same average. The experiment was carried out twice. Mussels appeared to regulate their filtration rate in response to varying seston concentration in one experiment, but not in the other. This behavioral difference may have been caused by differences in mean food level. Mean consumption was not different in the two feeding regimes, but assimilation efficiency was probably lower under fluctuating seston concentration. This suggests that small-scale heterogeneity in phytoplankton abundance would not affect the total transfer of energy between plankton and benthos. However, since a higher proportion of consumed energy was probably lost as faeces and/or pseudofaeces under the fluctuating seston regime, it may be expected that growth rate of mussels is depressed by small-scale heterogeneity in phytoplankton availability.