Empirical Transfer Function Research Articles

Streamline simulation of counter-current imbibition in naturally fractured reservoirs

Describing fluid transport in naturally fractured reservoirs entails additional challenge because of the complicated physics arising from matrix–fracture interactions. In this paper, the streamline-based simulation is generalized to describe fluid transport in naturally fractured reservoirs through a dual-media approach. The fractures and matrix are treated as separate continua that are connected through a transfer function, as in conventional finite difference simulators for modeling fractured systems. The transfer functions that describe fluid exchange between the fracture and matrix system can be implemented easily within the framework of the current single-porosity streamline models. In particular, the streamline time of flight concept is utilized to develop a general dual-porosity, dual-permeability system of equations for water injection in naturally fractured reservoirs. The saturations equations are solved using an operator splitting approach that involves ‘convection’ along streamline followed ‘matrix–fracture’ exchange calculations on the grid. The proposed formulation reduces to the commonly used dual-porosity model when the flow in the matrix is considered negligible. For modeling matrix–fracture interactions, two different transfer functions are examined: an empirical transfer function (ETF) and a conventional transfer function (CTF). The ETF allows for analytical solution of the saturation equation for dual porosity systems and is used to validate numerical implementation. Results obtained using the CTF are compared with a commercial finite-difference simulator (ECLIPSE) for waterflooding in five-spot and nine-spot patterns. The streamline approach shows close agreement in terms of recovery histories and saturation profiles with a marked reduction in numerical dispersion and grid orientation effects. Finally, an examination of the scaling behavior of the computation time indicates that the streamline approach is likely to result in significant savings for large-scale field applications.

Mean square error of the Empirical Transfer Function Estimator for stochastic input signals

The accuracy of the Empirical Transfer Function Estimator (ETFE) with stochastic input has been described with a deterministic and a stochastic approach. The results for bias and variance are different, but the mean square error is infinite in both approaches. In the deterministic approach where the input is considered as a single realization of a stochastic process, a number of frequencies has no finite ETFE estimate. The average number of problematic frequencies will be constant if the number of observations increases.

Borehole Response Studies at the Garner Valley Downhole Array, Southern California

The Garner Valley Downhole Array (GVDA) consists of a set of seven downhole strong-motion instruments ranging from 0- to 500-m depth. One of the objectives of this experiment is to estimate site response and study wave propagation as the energy travels from the bedrock underneath the site up through the soil column. The GVDA velocity structure is studied by computing synthetic accelerograms for a small event located at an epicentral distance of 10 km. These synthetics simulate well the data recorded at the borehole stations. In addition, theoretical transfer functions are calculated using the obtained velocity model and compare well with the empirical transfer functions from 54 recorded events. It is also observed that the downgoing wave effect is predominant in the first 87 m and is strongly reduced at depth. Using the velocity structure at GVDA and the transfer function results, it has also been possible to develop a simple method to compute the incident wave field, which is needed in nonlinear site response for instance. Recently there have been many comparative studies between horizontal-to-vertical (H/V) spectral ratios and traditional spectral ratios. Although many of these studies show that H/V spectral ratios can reproduce the shape of the site response curve, most show differences in the amplitude level. In the case of Garner Valley, where we have both surface and multiple borehole instruments, we find that this discrepancy in amplitude of the site response estimates is because the vertical component has significant site response associated with it due to S -to- P conversions that begin in the weathered granite boundary at 87-m depth. Manuscript received 27 August 2001.

Application of minimum crest factor multisinusoidal signals for “plant-friendly” identification of nonlinear process systems

Abstract Guidelines for specifying the design parameters of minimum crest factor multisine signals generated per the approach of Guillaume et al. are presented. These guidelines are evaluated in the identification and control of nonlinear process systems. The minimum crest factor multisine signals offer some distinct advantages over both Schroeder phased multisine signals and multi-level Pseudo-Random Sequences (multi-level PRS) with respect to “plant-friendliness” considerations. These signals can be used to reduce the effects of nonlinearity in obtaining an Empirical Transfer Function Estimate (ETFE). As an example, the ETFE of a Rapid Thermal Processing (RTP) reactor simulation is presented. The effectiveness of the minimum crest factor multisine signals is also discussed and illustrated in the identification and control of a simulated continuous stirred tank reactor using “Model-on-Demand” estimation and Model Predictive Control. Since the performance of the “Model-on-Demand” estimator is highly dependent upon the quality of the identification data, the CSTR case study provides a compelling example of the usefulness of the proposed design procedure.

Bias of indirect non-parametric transfer function estimates for plants in closed loop

We consider non-parametric indirect transfer function estimates for plants in closed loop. Such estimates are derived from the corresponding closed loop system transfer function estimate and knowledge of the controller. In particular, we give expressions for the bias of such estimates. We consider the application of the results to the spectral and empirical transfer function estimates under the assumption of Gaussian noise. In these cases, the bias can be expressed in terms of the experiment time, controller gain and the levels of disturbance and noise. We also briefly consider corresponding expressions for the variance.

Waves at Holderness from X-band radar

During the Holderness Experiment of 1994–1995 and 1995–1996, data were collected by a coastal deployment of an X-band radar with simultaneous deployments of various moored and bottom-mounted wave measurement devices. The radar data consist of sequences of images of the sea surface at intervals of 1.5 h over several months. Three-dimensional wave number-frequency spectra were derived from the sequences of captured images using an analysis method developed by Young et al. [J. Geophys. Res. 90(C1) (1985) 1049]. Some aspects of this analysis were refined. An empirical transfer function was derived relating the radar image spectra to the wave spectra measured by other instruments. A new calibration procedure for obtaining wave height from the radar spectra was also derived. The radar spectra were compared with directional spectra collected by a bottom-mounted InterOcean S4DW directional wave-current meter and with Datawell Waverider data. The capabilities of X-band radar in a coastal situation are discussed including the derivation of water depth required to optimise the analysis.

Chironomids, temperature and numerical models: a reply to Seppälä

We reply to comments by Seppälä (2001) concerning the suitability of the surface-water temperature measurements and mean July air temperature estimates used by Olander et al. (1999) to derive quantitative chironomid-temperature transfer functions. We use new data for water temperatures based on two-hourly recordings from thermistors installed in 32 lakes and air-temperature data from an automatic weather station at one lake to refute Seppälä’s accusations that our field data are unsatisfactory. We also respond to Seppälä’s suggestions about factors influencing chironomid distribution and abundance and suggest that there is a serious confusion of temporal and spatial scales between Seppälä’s ideas and the Olander et al. study. We attempt to explain the underlying logic behind empirical transfer functions that form a major part of Holocene quantitative environmental reconstructions and the major assumptions behind such transfer functions as Seppälä appears not to understand the basis or the assumptions of such widely used numerical models in palaeolimnology. We respond to Seppälä’s comments that lakewater pH and distance beyond tree-line have been inadequately measured and answer his erroneous understanding that a helicopter was used to facilitate lake sampling. We briefly consider Seppälä’s suggestion that a helicopter can totally mix the surface water of a lake for many hours.

Identification of empirical transfer function coefficients for a hollow tile based on detailed models of coupled heat transfers

Numerical simulation of transient heat transfer through a hollow tile, two air cells deep, is carried out using the SIMPLE algorithm. The coupling between the heat transfers by conduction, convection and radiation is taken into account. The heat fluxes computed for realistic time-varying excitations are used to derive empirical transfer function coefficients using an identification technique. Dynamic responses of the hollow tile predicted using both the transfer function coefficients and the simulation program are compared for thermal excitations that differ from those used to derive the coefficients. The results show a very good agreement between the two procedures’ predictions.

The Effect of Riverine Nitrate Loads on Winter Concentrations in the Great Belt, Denmark

This paper presents a method that models the fraction of the NOx concentrations (nitrate and nitrite) in a marine recipient resulting from riverine runoff from land. The analysis is performed on the surface waters in the Great Belt, Denmark, during winter seasons. The method is based on simulation of two independent NOx fractions: First, the “background” fraction, which results from the oceanographic mixing in the Belt Sea between Kattegat water and Baltic Sea water, and second, the nutrient load from land. The calculation of the background fraction is based on the finding that the Kattegat water has high salinity and high NOx contents whereas the Baltic Sea water has low salinity and low NOx contents. By means of an empirical relation between salinity and NOx a specific salinity measurement in the mixing zone can be related to a “background” NOx concentration. Measurements in the Great Belt show to be higher than the calculated “background” concentrations, which indicates the presence of a second NOx fraction. The second fraction is defined as the difference between the calculated background concentration and the measured NOx concentration. The hypothesis of the present paper is that this second fraction is dominated by the river runoff. Cross correlation analysis between the time series of the river NOx load and the time series of surplus concentration in the Great Belt reveals a time lag between maximum river load and maximum surplus concentration during winter seasons of approx. 1 month. An empirical transfer function has been developed in order to connect the daily NOx river loads with the surplus concentration in the Great Belt. The development of the NOx-concentration in the Great Belt due to river run off is modelled for 8 specific winter periods of 3 months (December - February) between 1988/1989 and 1995/1996. The model simulates concentrations on a weekly basis and quantifies the NOx-increase during each winter based on the river load data. The analysis indicates that the NOx concentration can be doubled during winters with high runoff, giving rise to an increase of approx. 60 μgN/1 at the end of February. In winters with minimal runoff no significant increase above the background level is found.

On the Statistical Accuracy of the Empirical Transfer Function Estimator

At its introduction, the accuracy of the Empirical Transfer Function Estimator ETFE has been described for periodical input and for a realization of a stationary stochastic process as input. Later, an infinite value for the variance of the ETFE has been derived by treating the input as a stationary stochastic process. The results for the three different types of input are compared. An example clarifies the experimental conditions for each type of input. It shows under which conditions the theory of a single realization of a stochastic process applies and when a full stochastic treatment is necessary. This has consequences for the method of averaging that has to be applied to improve the accuracy.

Design of Minimum Crest Factor Multisinusoidal Signals for "Plant-Friendly" Identification of Nonlinear Process Systems

Guidelines for specifying design parameters for minimum crest factor multisine signals per the approach of Guillaume et al. are presented. These guidelines are evaluated for the identification of nonlinear process systems. The minimum crest factor multisine signals offer some distinct advantages over both Schroeder phased multisine signals and m-level Pseudo-Random Sequence (m-level PRS) signals with respect to "plant-friendliness" considerations. These signals can be used to reduce the effects of nonlinearity in obtaining an empirical transfer function estimate (ETFE). As an example, the ETFE of a Rapid Thermal Processing (RTP) reactor simulation is constructed. "Plant-friendly" issues are also discussed and illustrated in the identification and control of a CSTR simulation via "Model-on-Demand" estimation. This provides a compelling example, since the "Model-on-Demand" estimator is a data-driven nonlinear identification approach.

Experiments and Modelling of Water Injection in Water-wet Fractured Porous Media

Abstract For predicting the performance of water injection in naturally fractured reservoirs, scale-up of the recovery data from immersing an oil-saturated core into water is commonly used. Oil recovery from some of the naturally fractured reservoirs of the North Sea has been better than what was predicted using the immersion laboratory experiments. In the field, the matrix blocks do not become surrounded by water at once; they experience an advancing fracture-water level (FWL). In this paper, the results of experiments of water injection in fractured porous media comprising a number of water-wet matrix blocks are reported for the first time. The blocks experienced an advancing fracture-water level (FWL). Immersion-type experiments were performed for comparison; the dominant recovery mechanism changed from co-current to counter-current imbibitions when the boundary conditions changed from advancing FWL to immersion-type. We performed single block experiments of co-current and counter-current imbibitions and found that co-current imbibitions led to more efficient recovery. Kansas chalk and Berea sandstone were investigated. A column of three blocks of Berea sandstone (Φ = 0.22, k = 0.62 µm2, pore volume (PV) = 8,800 ? 10−6 m3) and a stack of 12 blocks (four rows and three columns) of an outcrop Kansas chalk (Φ = 0.30, k = 0.002 - 0.005 µm2, PV = 13,900 ? 10−6 m3) were used. Breakthrough recoveries were 0.2 - 0.4 for the Berea and 0.2 - 0.6 of PV for the chalk experiments. Corresponding ultimate recoveries were around 0.5 and 0.65 of PV; oil recovery from low permeability chalk was better than that of high permeability Berea. Fracture apertures in all the above experiments were in the range of 150 - 200 µm. An approximate mathematical model was developed for counter-current imbibition. It was found that the late-time matrix-fracture transfer function simplifies to an exponential function. Hence, the physical significance of the empirical transfer function of Aronofsky et al. was demonstrated. The exponential transfer function was incorporated in a model, which was used to match the water injection experiments performed on a stack of very low permeability Austin chalk (Φ = 0.05, k = 0.00001 - 0.00005 µm2, PV = 287 ? 10−6 m3). These experiments were dominated by counter-current imbibition. Introduction Water injection is known as an important method for oil recovery from some fractured reservoirs. In water-wet fractured reservoirs, the capillary pressure contrast between the fracture and the matrix media provides the main driving force for water imbibition which can be an efficient recovery mechanism(1). Field application of water injection in fractured reservoirs has been implemented since the early fifties(2). Many issues, however, remain unresolved in the understanding of this process. Since the early studies, it was understood that recovery behaviour from a block totally covered by water is different than the same block in contact with water from some faces and with oil from other faces(2). However, the majority of studies have centred on immersion-type boundary conditions(3). Intuitively, if a block is surrounded by water, oil is forced to flow in the opposite direction of water flow, hence by counter-current imbibition.

Site effects at Euroseistest—II. Results from 2D numerical modeling and comparison with observations

This paper presents results of numerical modeling of site response for Euroseistest. Ground motion across a very detailed model of the subsoil of this valley has been simulated for vertically incident SH waves. The predominance of locally generated surface waves is very clear in the synthetic seismograms. These results are then compared with published studies of observed site effects at this basin and with a detailed analysis of two events in the time domain. It is discussed in which sense it is possible to obtain a good fit between observations and 1D models, even though the real behavior involves locally generated Love waves. For this reason, it can be misleading to rely on an incomplete observation such as empirical transfer functions. Finally, it is stressed that in order to predict ground motion in alluvial valleys the information contained in the phase cannot be neglected.

A Model to Interpolate Monthly Mean Climatological Data at Bavarian Forest Climate Stations

Six methods were used to interpolate the monthly mean climatological data from German climate stations to three Bavarian forest climate stations. The observed forest climatological data at the Bavarian forest climate stations were used as the reference data to which the interpolated data were compared. The results show that, for monthly mean daily maximum temperature at valley and plain forest climate stations, each of the six interpolation methods can give accurate estimates. For monthly mean daily maximum temperature, minimum temperature, air temperature and water vapor pressure at mountain forest climate stations, topographically aided interpolation can give the most accurate estimates. Barnes interpolation combined with empirical transfer functions can give accurate estimates forall climate variables at the plain and valley forest climate stations, and it can also give accurate estimates for monthly mean wind speed and monthly precipitation at the mountain forest climate station. The empirical transfer functions are very important for estimating the forest climatological data. These transfer functions will be used for reconstruction of long-term forest climatological data in Bavaria.

Empirical transfer function determination by the use of multilayer perceptron

Neural Networks are relevant statistical methods to extract information from data when physical phenomena are very complicated and cannot be described in terms of theoretical analysis. Scatterometers are active microwave radar which accurately measure the power of the backscatter signal versus incident signal in order to calculate the normalized radar cross section ( σ 0) of the ocean surface. We use multilayer perceptrons in order to determine the Geophysical Model Function and to estimate the variability of the signal of ERS-1, ERS-2 and NSCAT scatterometers.

Forest climatology: reconstruction of mean climatological data for Bavaria, Germany

A method is presented that allows the reconstruction of forest climatological data using data available from routine weather stations. Data from 32 routine weather stations were used to estimate the monthly mean values of daily mean air temperature, daily maximum temperature, daily minimum temperature, water vapour pressure, wind speed and precipitation at eight forest climate stations in Bavaria. The data obtained at these stations were used to establish empirical transfer functions to transform data interpolated from the weather stations to values that are valid for the different meteorological conditions in the forests. These empirical transfer functions between observed and interpolated climatological data are derived using a universal regression technique. The results show that using empirical transfer functions reduced the mean absolute errors between observed and estimated monthly mean climatological data significantly as compared to simple interpolation. A 31 year (1965–1995) monthly mean forest climatological data set in Bavaria, reconstructed using Barnes interpolation and the empirical transfer functions, was used to compare the forest microclimate with the surrounding mesoclimate. In addition, the climates of three typical forest regions were compared.

Rotating Stall Control in a High-Speed Stage With Inlet Distortion: Part II—Circumferential Distortion

This paper presents the first attempt to stabilize rotating stall in a single-stage transonic axial flow compressor with inlet distortion using active feedback control. The experiments were conducted at the NASA Lewis Research Center on a single-stage transonic core compressor inlet stage. An array of 12 jet injectors located upstream of the compressor was used for forced response testing and feedback stabilization. Results for a circumferential total pressure distortion of about one dynamic head and a 120 deg extent (DC(60) = 0.61) are reported in this paper. Part I (Spakovszky et al., 1999) reports results for radial distortion. Control laws were designed using empirical transfer function estimates determined from forced response results. Distortion introduces coupling between the harmonics of circumferential pressure perturbations, requiring multivariable identification and control design techniques. The compressor response displayed a strong first spatial harmonic, dominated by the well-known incompressible Moore–Greitzer mode. Steady axisymmetric injection of 4 percent of the compressor mass flow resulted in a 6.2 percent reduction of stalling mass flow. Constant gain feedback, using unsteady asymmetric injection, yielded a further range extension of 9 percent. A more sophisticated robust H∞ controller allowed a reduction in stalling mass flow of 10.2 percent relative to steady injection, yielding a total reduction in stalling mass flow of 16.4 percent.

Rotating Stall Control in a High-Speed Stage With Inlet Distortion: Part I—Radial Distortion

This paper presents the first attempt to stabilize rotating stall in a single-stage transonic axial flow compressor with inlet distortion using active feedback control. The experiments were conducted at the NASA Lewis Research Center on a single-stage transonic core compressor inlet stage. An annular array of 12 jet-injectors located upstream of the rotor tip was used for forced response testing and to extend the compressor stable operating range. Results for radial distortion are reported in this paper. First, the effects of radial distortion on the compressor performance and the dynamic behavior were investigated. Control laws were designed using empirical transfer function estimates determined from forced response results. The transfer functions indicated that the compressor dynamics are decoupled with radial inlet distortion, as they are for the case of undistorted inlet flow. Single-input-single-output (SISO) control strategies were therefore used for the radial distortion controller designs. Steady axisymmetric injection of 4 percent of the compressor mass flow resulted in a reduction in stalling mass flow of 9.7 percent relative to the case with inlet distortion and no injection. Use of a robust H∞ controller with unsteady nonaxisymmetric injection achieved a further reduction in stalling mass flow of 7.5 percent, resulting in a total reduction of 17.2 percent.

Frequency domain identification with generalized orthonormal basis functions

A method is considered for the identification of linear parametric models based on a least squares identification criterion that is formulated in the frequency domain, To this end, use is made of the empirical transfer function estimate (ETFE), identified from time-domain data. As a parametric model structure use is made of a finite expansion sequence in terms of recently introduced generalized basis functions, being generalizations of the classical pulse and Laguerre and Kautz types of bases. An asymptotic analysis of the estimated models is provided and conditions for consistency are formulated. Explicit and transparent bias and variance expressions are established, the latter ones also valid in a situation of undermodeling.

The Simulation of Daily Temperature Time Series from GCM Output. Part I: Comparison of Model Data with Observations

For climate change impact analyses, local scenarios of surface variables at the daily scales are frequently required. Empirical transfer functions are a widely used technique to generate scenarios from GCM data at these scales. For successful downscaling, the impact analyst should take into account certain considerations. First, it must be demonstrated that the GCM simulations of the required variable are unrealistic and therefore that downscaling is required. Second, it must be shown that the GCM simulations of the selected predictor variables are realistic. Where errors occur, attempts must be made to compensate for their effect on the transfer function‐ generated predictions or, where this is not possible, the effect on the transfer function‐generated climate series must be understood. Third, the changes in the predictors between the control and perturbed simulation must be examined in the light of the implications for the change in the predicted variable. Finally, the effect of decisions made during the development of the transfer functions on the final result should be explored. This study, presented in two parts, addresses these considerations with respect to the development of local scenarios for daily maximum (TMAX) and minimum (TMIN) temperature for two sites, one in North America (Eau Claire, Michigan) and one in Europe (Alcantarilla, Spain). Part I confirms for a selected GCM that simulations of daily TMAX and TMIN, whether taken from the nearest land grid point, or obtained by interpolation to the site location, are inadequate. Differences between the GCM 1 3 CO2 and observed temperature series arise because of a 08C threshold in the model data. At both sites, variability is suppressed during periods affected by the threshold. The thresholds persist into the perturbed simulation, affecting not only GCM-predicted 2 3 CO2 temperatures but also, because the duration and timing of the threshold effect changes in the perturbed simulation, the magnitude and seasonal distribution of the 2 3 CO2 ‐1 3CO2 GCM differences. Comparison of modeled and observed 500-hPa geopotential height (Z500) and sea level pressure (SLP) shows that, although systematic errors of the type associated with the 08C threshold in the temperature data are absent, significant errors do occur in certain seasons at both sites. For example, SLP is poorly modeled at Alcantarilla, where the control and observed means differ significantly in every season. The worst results at both sites are in summer. These results will affect the performance of the transfer functions when initialized with model data. Whereas little change is found to occur in SLP at either site between the 1 3 CO2 and 2 3 CO2 simulation, there is a noticeable increase in Z500. Other things being equal, therefore, the temperature changes predicted by the transfer functions are likely to be greatest when Z500 contributes the most to the explained variances. In Part II, a range of transfer functions are developed from the free atmosphere variables and validated, using observations. The performance of these transfer functions when initialized with model data is evaluated in the light of the findings in Part I. The sensitivity of the perturbed climate scenarios to a range of user decisions is explored.