Vertical Borehole Research Articles

A spectral method aimed at explaining the role of the heat transfer rate when the Infinite Line Source model is applied to Thermal Response Test analyses

The present paper introduces new concepts related to the modeling of vertical Borehole Heat Exchangers for Ground Coupled Heat Pump applications. A sensitivity analysis on how specific parameters affect the ground thermal conductivity kgr estimation when the Infinite Line Source model is used to interpret a Thermal Response Test has been performed. The study has been conducted considering shallow and deep BHEs, with and without geothermal gradient, and for homogeneous and stratified ground thermal conductivities. The qratio parameter scales the external heat rate to a natural heat rate associated with the geothermal gradient. The effect of qratio on the TRT analysis has been related to a specific dimensionless g-transfer function called g0, which incorporates the geothermal gradient. Three in-house built Fortran90 codes implementing the finite-difference models related to coaxial, single and double U-BHE geometries are exploited to evaluate the dimensionless g-transfer functions related to each fluid volume. A spectral method aimed to reconstruct the fluid temperature profiles by superposing two separated convolutions in the time domain exploiting the Fast Fourier Transform leads to considering qratio as the dominant parameter when the ILS model is used to estimate kgr. In the case of a single-layered subsurface, qratio >>1 guarantees the correct ILS-based kgr estimation for any BHE geometry. In the coaxial center-pipe inlet case with a single-layered subsurface and qratio<1, the ILS-based kgr estimation when the g0-function is taken into account can differ by -14% from the correct ILS-based kgr estimation without taking into account the g0-function. In the case of a multilayered subsurface, the qratio parameter indicates when the effective kgr estimated by the ILS model departs from the weighted-thickness average. A departure of 10% occurs for qratio between 2 and 2.5 for the coaxial center-pipe inlet cases considered and the departure increases with decreasing qratio.

A new semi-analytical solution addressing varying heat transfer rates for U-shaped vertical borehole heat exchangers in multilayered ground

Existing analytical solutions for vertical borehole heat exchangers (BHEs) in a multilayered ground normally require a prescribed heat transfer rate at the borehole wall, such as a constant heat transfer rate. However, in reality, the heat transfer rate between a BHE and its surrounding ground varies along the length of the borehole. To account for this variation, this study proposed a new semi-analytical solution named the “MVQ solution”—Multilayered ground with Varying heat transfer rates (Q). The MVQ solution was first verified with an equivalent three-dimensional (3-D) finite-element (FE) model. The results showed that the MVQ solution had a mean absolute percent error (MAPE) of less than 2.50% for the average borehole wall temperature. In comparison, the conventional constant heat transfer rate solution was found to potentially overpredict the temperature by up to 50.82%. Lastly, a parameter study revealed that the inhomogeneity of ground thermal properties strengthened the heat transfer rate variation along the borehole length. Thus, the MVQ solution is indispensable for thermal performance predictions of BHEs in the multilayered ground, and it can improve the design of BHE systems in shallow geothermal applications.

Comprehensive thermal performance analysis and optimization study on U-type deep borehole ground source heat pump systems based on a new analytical model

U-type deep borehole heat exchangers (UDBHEs) can provide low-carbon heating for buildings utilizing medium-deep geothermal energy. This study presents a new analytical model based on the Segmented Finite Line Source (SFLS) method to investigate the heat transfer process of a UDBHE. The developed model is verified by comparing with experimental results. Subsequently, the soil and fluid temperature are analyzed, and the influences of design and operation parameters on UDBHE performance are investigated. The results indicate that the heat extraction capacity increases as the higher thermal conductivity of backfill material, pipe diameter and flow rate. We identify a linear correlation between the horizontal or vertical borehole length and the heat extraction rate. In addition, an integrated model for the optimization study on the U-type deep borehole ground source heat pump (UDBGSHP) system is presented, which comprehensively considers the dynamic heating loads of the buildings and the dynamic heat transfer process in the UDBHEs. With the control method, the electricity consumption of the heat pump decreases from 5110 kWh to 4812 kWh within a 72-h operation. This work can offer guidance for the design and optimization of UDBGSHP systems.

A novel evaluation scheme of resistivity anisotropy in near-tight sandstones using conventional geophysical logs: A case study of the Triassic Chang 8 oil layer, Zhenjing area, Ordos Basin

The resistivity anisotropy characteristic of near-tight sandstone reservoirs is important for reservoir evaluation, but it is a challenge to obtain it by using conventional logging curves in vertical boreholes. Thus, a novel evaluation scheme is established. Firstly, in the Chang 8 Formation of Zhenjing area, Ordos Basin, China, horizontal rock samples (parallel to bedding) and vertical rock samples (perpendicular to bedding) are collected for designed rock electricity experiments and analyzed to obtain the corresponding horizontal and vertical Archie's parameters. Secondly, based on the detection characteristics of acoustic logging instrument and the transformation form of the Archie's equation, an estimation method of vertical resistivity is established. Combined with the horizontal resistivity obtained from conventional deep resistivity logs in vertical boreholes, an estimation method of resistivity anisotropy coefficient is proposed. Thirdly, the relationships between resistivity anisotropy coefficients and water saturations and porosities are analyzed using experiment results, and four adjacent vertical wells in the study area are processed using the estimated method. The processed results demonstrate the feasible of the estimation method. In addition, the cross well profile of resistivity anisotropy coefficient in the target formation is characterized, which shows the target formation is slightly resistivity anisotropic and slightly heterogeneous. The relationships between the resistivity anisotropy coefficient and commonly used logs are discussed, which contribute to rapid analyze the resistivity anisotropy of the target formation. Based on the above methods and processes, the novel evaluation scheme of resistivity anisotropy is established, which contributes to quantify and qualitatively analyze the resistivity anisotropy characteristics of formations in and between vertical boreholes. The application results show that it provides an alternative scheme for continuously evaluating the resistivity anisotropy in boreholes and cross well profiles by using conventional logs and experiments.

Numerical investigation of vertical borehole heat exchanger heat transfer under coupled conditions of groundwater seepage and layered subsurface

Vertical borehole heat exchangers (VBHEs) have been considered to be one of the best sustainable geotechnical technologies for building heating or cooling. In most of the current numerical models for the analysis of VBHE heat transfer, the assumption of subsurface homogeneity is considered. However, the actual subsurface presents a layered structure, and some aquifer groups have groundwater seepage. An improved numerical model is proposed that takes into account not only the coupled condition of groundwater seepage and the layered subsurface but also the heat transfer of VBHEs at different locations, including inside the borehole, borehole wall and outside the borehole. The heat transfer characteristic at these different locations under groundwater seepage and layered subsurface coupled condition is investigated. Moreover, the model is applied to investigate the thermal behaviors of VBHE arrays in two typical arrangements (matrix and staggered arrangement). Furthermore, the influence of groundwater seepage in different layers on the heat transfer ability of the VBHE is discussed. The results demonstrate that the seepage can effectively improve the average heat transfer performance of VBHEs array, and the relative improvement rate is about 62.0%, Compared with the matrix arrangement, the average heat flow of staggered arrangement is about 0.09 kW higher, and the relative increase rate is about 3.3%.

A 3D geological model of El Hierro volcanic island reflecting intraplate volcanism cycles

Fully 3D and multidimensional geological models are key for energy, mining and water resources assessment and management, which has a great economic impact. 3D geological models provide critical information regarding geometric properties of the geological bodies of interest, such as ore deposits or reservoirs, allowing inference of volumes and availability to be exploited. However, defining a useful methodology to create a fully digital 3D geological model of volcanic islands is still very challenging due to the intrinsic complexity observed in the structure of volcanic systems. Two-dimensional geological maps, associated vertical geological cross-sectional diagrams and borehole and water-mining tunnels (galleries) logs were examined on El Hierro Island. A geological modelling procedure using the software GeoModeller obtained the first 3D geological model of El Hierro volcanic island, as a basis for volcanic formation definitions and geometric analysis. Surface, volume and center of mass estimations of the differentiated sub-units allowed a better understanding of the spatial distribution and evolution of the volcanic cycles of El Hierro Island over time. The 3D model provided geometry information correlatable with volcanic activity cycles and lava/pyroclast deposition mechanisms that ultimately defines the geometry of the volcanic units. This information provides new variables valuable for the investigation of the existing relationships between rift zones and giant lateral collapses. In addition, the obtained 3D geological model of a volcanic island will support the building of the first hydrogeological and geothermal model of El Hierro Island, allowing advancement of sustainability of isolated systems.

Interpreting Concentrations Sampled in Long-Screened Wells with Borehole Flow: An Inverse Modeling Approach.

New approaches are needed to assess contaminant mass based on samples from long-screened wells and open boreholes (LSW&OB). The interpretation of concentration samples collected in LSW&OB is complicated in the presence of vertical flow within the well. In the absence of pumping (i.e., ambient conditions), the well provides a conduit for flow to occur between aquifer layers or fractures as a result of head differences. Under pumping conditions, vertical borehole flow may vary with depth depending on far-field heads and hydraulic conductivity; furthermore, if pumping fails to overcome ambient gradients, outflow from the well to the aquifer may occur. Concentration samples thus represent flow-weighted averages of formation concentrations, but the averaging process is commonly unknown or difficult to identify. Recognition of the importance of borehole flow has motivated the use of multi-level wells, packers, and well liners; however, LSW&OB remain common for numerous reasons, including cost, multi-purpose design requirements (e.g., pump-and-treat, water supply), logging, and installation of instrumentation. Here, we present a simple analytical model for flow and transport within a well and interaction with the surrounding aquifer. We formulate an inverse problem to estimate formation concentration based on sampled concentrations and data from flowmeter logs. The approach is demonstrated using synthetic examples. Our results (1) underscore the importance of interpreting sampled concentrations within the context of hydraulic conditions and aquifer/well exchange; (2) demonstrate the value of flowmeter measurements for this purpose; and (3) point to the potential of the new inverse approach to better interpret results from samples collected in LSW&OB.

Numerical modeling of the effects of the radial and axial location of added micro-encapsulated phase change materials in vertical borehole heat exchangers

A numerical model of a vertical borehole heat exchanger (BHE) with micro-encapsulated phase-change material (PCM) was used to examine the thermal effect of the radial and axial location of the added PCM within the borehole. In the radial direction, PCM could be placed at the center of the borehole, at the borehole periphery, or interspersed within the grout. In the axial direction, PCM could be concentrated either in the top, bottom, or mid portions of the borehole grout, or again interspersed evenly. PCM could also be added to the borehole heat transfer fluid. Each location has its own thermal, and therefore economic, advantages and disadvantages, and thus this present work investigated these effects toward decreasing borehole length. Analysis of results showed that the best radial location for the PCM is intermixed within the grout, even though that configuration has a disadvantage of increasing the overall borehole thermal resistance. In the axial direction, it was found that the thermal behavior of the PCM varied over the borehole length, and thus adding the PCM-grout mixture to only the bottom half-depth of the borehole reduced the thermal benefit by only about 19%. Thus, it is concluded that strategic placement of PCM within borehole grout can significantly improve the economics of its use in vertical GCHP systems, but at current costs of PCM, its use in borehole grouts remains an economic challenge.

New Sensor Technology for Determining Rock Pre-destructive States

Based on the experimentally discovered regularity – emission of micro-sized particles during monoaxial deformation of non-homogeneous structures, such as rocks and concretes, a sensor technology to identify their pre-destruction processes is proposed. At nearing the stressed state until the disintegration of the massifs, exponential increase of particles generation appears. The innovation is applicable to risky rock formations where vertical measurement boreholes are made, by which kernels are obtained. Using the specimens formed by the kernels, through monoaxial load by press in laboratory conditions and corresponding particle emissions, the reference pressure values for different stages of the pre-destructive state development are determined. The reference data from the pressure are compared with the results for the deformation loads in the rock structure, corresponding to the generated particles in a borehole model. The values of the stress determine the occurrence of critical states in the structure. This information is an integrated indicator for the timely identification of pre-destructive processes in rocks. The described system increases the accuracy of determining the appearance of failure conditions. One of the strategic applications of the new technology is in seismically active regions.

Analysis on influencing factors of heat transfer characteristics of vertical borehole heat exchanger under stratified soil conditions

In recent years, factors influencing the heat transfer characteristics of the vertical borehole heat exchanger (BHE) is a hot topic in the research of the ground source heat pump (GSHP) systems. Given the complexities involved in environmental situations with respect to groundwater seepage in saturated soil and moisture migration in unsaturated soil, this paper examined the effects of various influencing factors on the heat transfer characteristics of the vertical BHE under stratified soil conditions, using theoretical research, experimental research, and numerical simulations. Subsequently, through altering five influencing factors—groundwater seepage velocity, porosity, saturation, thermal conductivity, and initial soil temperature—the working conditions of the vertical BHE under stratified soil conditions were established, and the heat transfer rate was calculated. The results showed that the weight of the influencing factors on the heat transfer characteristics of the vertical BHE had a descending order: thermal conductivity, groundwater seepage velocity, porosity, saturation, and initial soil temperature. These results can provide an important reference for further optimizing future BHE designs.

Enhancement of shallow ground heat exchanger with phase change material

Heat pumps perform better when coupled with ground as thermal source than with air. In literature, several studies and applications suggest and analyse the use of phase change materials (PCMs) coupled with single or double U-tube vertical borehole heat exchangers (BHEs). Usually, PCMs are mixed with the grouting material during the installation. An alternative solution to vertical BHEs is the use of horizontal ground heat exchangers (HGHEs). The present work investigates the possibility of coupling PCMs with a flat-panel HGHE installed inside a trench 2 m under the ground surface. The study analyses the case in which PCMs are adjacent to the HGHE, taking a cue from alternative coupling technologies which have PCMs added to the backfilling material of the trench where the HGHE is installed. The analysis has been conducted with COMSOL software tool. A simulation model of the system was developed to carry out a parametric analysis. The objective of the simulations is the investigation of the thermal behaviour of the HGHE patent pending coupled with PCMs under cycles of operation which represent how the heat pump could work in GSHP system. The results show the meaningful difference of using the PCM in direct contact with the HGHE.

Influence of Leachate Recirculation on Landfill Degradation and Biogas Production

Population growth combined with the rising standard of living of people around the world is the reason for the ever-increasing production of waste which management is costing states a lot of money for its disposal. Among available waste treatment techniques, landfill is one of the most promoted waste management techniques with the emergence of the bioreactor concept. However, the control of biodegradation parameters in order to accelerate waste stabilization is an important issue. For environmental and economic reasons, the technique of leachate recirculation by injection into the waste is increasingly used to improve the degradation of landfilled waste. The injection of leachate is possible using vertical boreholes, horizontal pipes, infiltration ponds or a combination of these. Indeed, moisture is the main factor in waste degradation and biogas production. The migration of leachate to the bottom of the landfill creates low moisture in the upper areas of the landfill reducing the growth of microbial populations. This results in low or no biogas production. The main objective of the present work is to develop a numerical model of leachate recirculation by injection into the waste to rewet the waste and restart biological activity. The analysis of the results shows that the diffusion of the wet front increases with time and depth. The lateral widening of the wet front is slow in relation to the progression of the wet front towards the bottom of the waste cell. This indicates the predominance of gravity effects over diffusion phenomena. The results reveal that the distributed re-injection is the best mode of leachate recirculation because the moisture distribution on the whole waste mass is totally satisfactory and the biogas generation is more important. Leachate recirculation campaigns should be done periodically to rewet the waste, boost microbial activity and hope for a quicker stabilization of the landfill.

Influence of Different Heat Loads and Durations on the Field Thermal Response Test

Geothermal energy exhibits considerable development potential in space heating. Shallow geothermal energy stored in the soil in the form of low-grade energy is mainly extracted via the ground source heat pump (GSHP) system. GSHP systems use the subsoil as a heat source, typically involving a vertical borehole heat exchanger (BHE) to extract heat from the formation. Accurate measurement of the thermal properties of the formation is very important for the design of BHEs. At present, the most common and effective method to measure the thermal conductivity of the formation in the field is the thermal response test (TRT). However, the test conditions (heat load, test time) during the thermal response test can impact the test results. Therefore, in this study, a borehole with a depth of 130 m was evaluated in the field. The TRT module and the distributed thermal response test (DTRT) module based on distributed optical fiber temperature sensor (DOFTS) technology were used to monitor the test with different working conditions in real-time. In the field tests, geothermal conditions and the evolution of the formation temperature with time and depth were determined. Based on the test results under different heat loads and test times, the influence of the test conditions on the thermal conductivity results was analyzed and described. A constant temperature zone was located at a depth from 25 m to 50 m, and an increasing temperature zone was located at a depth from 50 m to 130 m, with a geothermal gradient of 3 °C/100 m. The results showed that the heat load slightly influenced the thermal conductivity test results. At the initial stage of the test, the temperature significantly increased from 0 to 12 h. After reaching the quasi-stable state, the test time slightly influenced the thermal conductivity test results. The characteristics of the formation thermal recovery stage after the test stage were studied. The heat load decreased, which could shorten the time for the formation to recover the initial temperature. The results could provide a basis for the optimization of thermal response test conditions.

Dynamic modelling of a ground-coupled solar ejector cooling system

ABSTRACT This work presents a solar-driven air conditioning system based on ejector technology with closed-loop vertical geothermal boreholes. Modeling and simulation are carried out using Dymola software for dynamic transient analysis, with the refrigerant R134a. Condenser temperature regulation produces a 267% improvement of performance compared to a solar-only configuration. Solar fraction increases with increasing solar collector area, which benefits the system up to the limit of 27 m2 for a cooling load of 9 kW. The reservoir volume ensuring high solar fraction is 14 m3. Optimal generator saturation temperature ensures a seasonal coefficient of performance value of 0.772 and a maximum overall efficiency of 39%. When the system encounters deviations from the optimum set point, the overall efficiency becomes 37.4%.

Study of the Influence of Thermal Convection on Temperature Measurement in Thermal Control Boreholes during Artificial Ground Freezing

This paper considers the problem of thermal convection of a calcium chloride solution in a vertical borehole. A non-uniform temperature distribution with a given vertical gradient is set at the walls of the borehole. The non-stationary temperature distribution along the borehole axis was analyzed, and its deviations from the temperature at the walls were investigated. From a practical point of view, this problem is important for estimating the error in distributed temperature measurements over the depth of thermal control boreholes during artificial ground freezing. In this study, an area near the bottom of the borehole was identified where the fluid temperature at the borehole axis deviates significantly from the temperature at the wall. The maximum deviations of the fluid temperature from the temperature at the walls, as well as the length of the temperature deviation sections, were determined.

E&amp;P Notes (September 2022)

E&amp;P Notes (September 2022)

A new integrated approach to resolve the saturation profile using high-resolution facies in heterogenous reservoirs

The saturation calculation in complex reservoirs remains a major challenge to the oil and gas industry. In simple formations, a tendency towards simple saturation models such as Archie or Simandoux for clean and shaly reservoirs respectively is always preferable. These models were found to be working effectively in homogeneous formations within which the porosity and permeability are linked in the light of a simple facies scheme. Where the rocks show some degrees of heterogeneity, the well-logs are usually affected by different factors. This adversely results in a compromised or averaged log profiles that may affect the saturation calculations. Four wells drilled across a shaly sand of high heterogeneity have been studied in the Perth Basin, Western Australia. The aim is to resolve the hydrocarbon saturation and explain the high productivity results, despite the high water saturation, obtained through a conducted formation well test across the interested reservoir zones. A new integration technique between a suite of conventional and advanced logging tools together with the capillary pressure measurements has been carried out to generate a high-resolution reservoir saturation profile, that is lithofacies dependent. Three different independent methods were used in the studied wells to calculate the saturation and to reduce the uncertainty of the final estimated profiles. The methods are the resistivity-based saturation, the NMR-based irreducible saturation, and a new application through saturation height modeling. Furthermore, through the workflow, an effective calibration for the magnetic resonance T2 cutoff has been applied that is supported by the excellent reservoir production behavior from such complex reservoir. The methodology will help resolve the saturation calculation as one of the most challenging reservoir parameters, particularly where the resistivity logs are affected in complicated shaly sand environments. The effectiveness of the workflow shines the possibility to predict high resolution facies and saturation profiles in the lack of resistivity logs. A further possibility can complete the analysis on real time basis, which can certainly provide facies and saturation profiles extended to the uncored wells. Application of this methodology in the uncored wells has shown very encouraging results in various well trajectories, either vertical, deviated or horizontal long boreholes.

Geology and Correlative Analysis of Borehole Logs with Geo-electric Sections of Some Parts of Ibadan, Southwestern Nigeria

The property of rocks varies from one geological location to another or even disappears laterally, and these properties are important factors in determining groundwater quantity. This makes this research work focused on the correlative analysis of lithologies and borehole logs with geo-electric sections in the basement terrain of Ibadan, southwestern Nigeria, using Borehole logging and geophysical electrical method. Twenty-two vertical electrical sounding stations and borehole Logsare acquired within the study area with a maximum Schlumberger electrode separation of 100m. The data are interpreted qualitatively and quantitatively by partial curve matching and computer iteration to obtain the first-order geo-electric model parameters. The results of the geo-electric investigation reveal some lithological Layers such as topsoil, clay, weathered laterite basement, and fresh basement. At the same time, those from boreholes include topsoil, loamy soil, friable brown soil, clayey-sand, sandy-clay, laterite weathered basement/saprolite, and Fresh Basement. Electrical resistivity and borehole logs correlation revealed: poor, poor-moderate and moderate-perfect correlations based on a comparison between the number of lithological layers encountered from both vertical electrical sounding derived lithology and those from borehole and depth of occurrence. The depth of water struck grouped into three 0 - 45 m, 45 - 70 m, and 50 - 100 m to bring out the most favourable depth of aquifer potential while three yields categories 0.5-2 lt/s; 2 - 4lt/s and 4 -12 lt/s are observed with the above depth classification respectively. Good yielding fractures increase with depth in the quartzite regions but decrease with depth in both gneisses and the schist rock bodies. The correlations between geo-electric sections and borehole logs have shown that the geoelectric section obtained from the vertical electric section cannot totally be substituted for borehole logging but can serve as alternative means of classifying the sub-surface lithologies in the absence of borehole logging.

Geothermal deep closed-loop heat exchangers: A novel technical potential evaluation to answer the power and heat demands

This paper investigates and optimises the thermal performance of deep closed-loop heat exchanger (DCHE) systems by applying a computational numerical approach. The investigated DCHE configuration accounts for two deep vertical boreholes, an injection and a production well, connected by a horizontal borehole at depth and an insulated pipeline at the surface, establishing an effective closed-loop system. First, a parametric sensitivity study explores the effects of the environmental, design and operating variables on the production temperature. The simulation uses realistic geological and geothermal conditions, depths, circulation rates and injection temperatures. Two complex numerical models are then solved for site-specific DCHEs in different geological scenarios: a foreland basin and a convergent margin hosting low-to-intermediate and high-temperature geothermal resources, respectively. Production temperatures beyond 40–60 °C and 100 °C, sustainable for both heat and electric power generation, are obtained, depending on the geothermal conditions and closed-loop dimensions. Furthermore, circulation rates of 0.02–0.04 m3 s−1 are cost-effective, and the system's efficiency and sustainability increase when a fluctuating and periodic heat extraction strategy is employed. When efficiently operated, DCHEs are a viable solution for renewable energy production and should be integrated into the local heat market and distribution network infrastructure. Field of researchEarth and Planetary Sciences; Energy; Environmental Sciences

Can modeling soil heterogeneity in 2D site response analyses improve predictions at vertical array sites?

This study uses a database of 21 vertical borehole arrays in California to examine whether a two-dimensional (2D) site response analysis framework that accounts for soil heterogeneity via spatially correlated random fields can explain misfits observed from prior one-dimensional (1D) ground response modeling. The main hypothesis is that the overprediction of ground motion at site modal frequencies, consistently observed in many site response validation studies, is caused by soil heterogeneity and 2D/three-dimensional (3D) wave propagation effects that cannot be captured by 1D analyses. We apply classical “within” boundary conditions for borehole input motion along with equivalent incident wave motions derived using a framework developed here to help elucidate the effects of the down-going wave on observed fundamental mode resonances. Results from 2D and 1D analyses are compared to observations using a transfer-function-based taxonomy and residuals of other intensity measures (IMs) including response spectra. The uncertainty in predicted IMs is estimated from the many realizations of 2D models. This 2D approach was found capable of scattering seismic waves and producing transfer function variability resembling the observed event-to-event variability in empirical transfer functions (ETFs). For several sites that exhibit less down-going wave effects in ETFs (i.e. flatter peaks) and/or higher variability in ETFs, median transfer functions from 2D analyses provide a significantly better estimate of the median ETF than conventional 1D deterministic analyses, especially at fundamental modes. In contrast, for some of the sites that are well represented by 1D methods (e.g. Wildlife Liquefaction Array and Treasure Island), 2D methods with generic levels of spatial variability may over-represent the heterogeneity and consequently underpredict amplifications at higher mode frequencies.