History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations

Time-lapse seismic data has been widely used for the detailed reservoir characterization, and data assimilation algorithms are commonly used for petroleum reservoir history matching of production data. However, hardly any seismic data has been integrated into the reservoir inverse modeling workflow, due to the large data size, finer gridding, and especially the scarcity in time of the seismic data. Popular ensemble-based reservoir inverse modeling methods such as ensemble Kalman filter (EnKF) also face the problems of high computational cost due to the storage of the intermediate variables, restarting the reservoir simulating process and the inconsistency of the full-step and step-wise simulations. The intrinsic sequential data assimilations characteristics of the EnKF set obstacles to assimilate 4D seismic data. The alternative method can be the ensemble smoother (ES), which is an ensemble based method for data assimilation. The ES is based on a Bayesian updating scheme of the reservoir model to match the production history and improve the production forecast. To improve the algorithm convergence, a multiple data assimilation (MDA) method was proposed by Emerick and Reynolds (2012a). The algorithm we used is called ensemble smoother with multiple data assimilation (ESMDA), we modified the ESMDA to integrate the geophysical data, and created the new workflow to history match both the production and geophysical data. The available production data is well measurements, including oil production rate, well water cut and bottom-hole pressure, as well as time-lapse geophysical data, including P-wave impedance. In this paper, we first formulated the mathematical descriptions of ESMDA, we then down-scaled the seismic data, and modified the data assimilation algorithms, illustrated the history matching workflow for both production and geophysical data, finally, we showed how it works with a case study on a water flooding operation in a synthetic reservoir. In comparison, we also showed the history matching only on the production data, which yields an inferior results than the one matches both production and seismic data.

Sequential assimilation of 4D seismic data for reservoir description using the ensemble Kalman filter

In previous works in this series study, an ensemble Kalman filter (EnKF) was demonstrated to be promising for mesoscale and regional-scale data assimilation in increasingly realistic environments. Parts I and II examined the performance of the EnKF by assimilating simulated observations under both perfect- and imperfect-model assumptions. Part III explored the application of the EnKF to a real-data case study in comparison to a three-dimensional variational data assimilation (3DVAR) method in the Weather Research and Forecasting (WRF) model. The current study extends the single-case real-data experiments over a period of 1 month to examine the long-term performance and comparison of both methods at the regional scales. It is found that the EnKF systematically outperforms 3DVAR for the 1-month period of interest in which both methods assimilate the same standard rawinsonde observations every 12 h over the central United States. Consistent with results from the real-data case study of Part III, the EnKF can benefit from using a multischeme ensemble that partially accounts for model errors in physical parameterizations. The benefit of using a multischeme ensemble (over a single-scheme ensemble) is more pronounced in the thermodynamic variables (including temperature and moisture) than in the wind fields. On average, the EnKF analyses lead to more accurate forecasts than the 3DVAR analyses when they are used to initialize 60 consecutive, deterministic 60-h forecast experiments for the month. Results also show that deterministic forecasts of up to 60 h initiated from the EnKF analyses consistently outperform the WRF forecasts initiated from the National Centers for Environmental Prediction final analysis field of the Global Forecast System.

The ensemble Kalman filter (EnKF) has recently been proposed as a promising parameter estimation approach for constraining the description of rock flow properties, such as permeability and porosity, to reproduce flow measurements that are modeled as nonlinear functions of these properties. One of the key factors that strongly affect the performance of the EnKF is the quality or representativeness of the prior ensemble of property fields used to initialize the EnKF assimilation procedure. The initial ensemble is commonly constructed by assuming a known geological continuity model such as a variogram. However, geologic continuity models are derived from incomplete information and imperfect modeling assumptions, which can introduce a significant level of uncertainty into the produced models. Neglecting this important source of uncertainty can lead to systematic errors and questionable estimation results. In this paper, we investigate the performance of the EnKF under varying levels of uncertainty in the variogram model parameters. We first attempt to directly estimate variogram model parameters from flow data and show that the complex and nonunique relation they have with the flow data provides little sensitivity for an effective inversion with the EnKF. We then assess the performance of the EnKF for estimation of permeability values under uncertain and incorrect initial variogram parameters and show that any bias in specifying variogram parameters tends to persist throughout the EnKF analysis even though locally reasonable permeability updates may be obtained near observation points. More importantly, we show that when variogram parameters are specified probabilistically to account for the full range of structural variability in the initial permeability ensemble, the EnKF update results are quite promising. The results suggest that under uncertain geologic continuity, the EnKF tends to perform better if a very diverse set of property fields is used to form the initial ensemble than when a deterministic and potentially erroneous variogram model is used. Therefore, in applying the EnKF to model calibration problems, it is preferable to overestimate the uncertainty in geologic continuity and to initialize the EnKF procedure with a wide range of variability in property description than to overlook the variogram uncertainty at the risk of introducing systematic bias that cannot be corrected by the EnKF updates. The practical implications of the results in this paper are significant for designing the EnKF for realistic ensemble model calibration problems where the level of uncertainty in the initial ensemble is usually not known a priori.

Recently, the Ensemble Kalman Filter (EnKF) has emerged as an effective tool for performing continuous updating of petroleum reservoir simulation models. The method is firmly grounded on the theory of Kalman filters and sequential Monte Carlo techniques. The ability of the method to sequentially update the spatial properties in petroleum reservoir models, such as permeability and porosity, by integrating the dynamic production data makes it a very attractive approach. Moreover, the method takes into account the production uncertainty in the reservoir models by using error covariance matrices for the measurement vector (Production and injection rates, Gas-Oil ratio, Steam-Oil ratio, etc.) and the state vector (pressure, saturation, permeability, porosity). Similar to the traditional Kalman filter, the covariance matrices have to be tuned to reflect the uncertainty in the model and the measurements. We consider two unconventional oil reservoir models: 1) highly heterogeneous black-oil reservoir model, and 2) heterogeneous SAGD reservoir model. The results will demonstrate the advantage of using the localized EnKF for effective history matching using ensemble sizes relatively lower than what otherwise would be required with the ordinary EnKF. The results will also show the advantages of using prior knowledge available from the wells (permeability and porosity measurements) to generate initial realizations. One of the main practical advantages of history matching using the EnKF over traditional optimization based approaches is its low computational effort. The computational cost is dominated by Monte Carlo simulation of the ensemble of models only. Thus, significant computational time saving is possible by running each of the ensemble simulations on independent processors in a parallel mode. Moreover, the method can be easily integrated with any commercial reservoir simulation software.

Kinematical analyses of mobile radar observations are critical to advancing the understanding of supercell thunderstorms. Maximizing the accuracy of these and subsequent dynamical analyses, and appropriately characterizing the uncertainty in ensuing conclusions about storm structure and processes, requires thorough knowledge of the typical errors obtained using different retrieval techniques. This study adopts an observing system simulation experiment (OSSE) framework to explore the errors obtained from ensemble Kalman filter (EnKF) assimilation versus dual-Doppler analysis (DDA) of storm-scale mobile radar data. The radar characteristics and EnKF model errors are varied to explore a range of plausible scenarios.When dual-radar data are assimilated, the EnKF produces substantially better wind retrievals at higher altitudes, where DDAs are more sensitive to unaccounted flow evolution, and in data-sparse regions such as the storm inflow sector. Near the ground, however, the EnKF analyses are comparable to the DDAs when the radar cross-beam angles (CBAs) are poor, and slightly worse than the DDAs when the CBAs are optimal. In the single-radar case, the wind analyses benefit substantially from using finer grid spacing than in the dual-radar case for the objective analysis of radar observations. The analyses generally degrade when only single-radar data are assimilated, particularly when microphysical parameterization or low-level environmental wind errors are introduced. In some instances, this leads to large errors in low-level vorticity stretching and Lagrangian circulation calculations. Nevertheless, the results show that while multiradar observations of supercells are always preferable, judicious use of single-radar EnKF assimilation can yield useful analyses.

Bayesian estimation theory provides a general approach for the state estimate of linear or nonlinear and Gaussian or non-Gaussian systems. In this study, we first explore two Bayesian-based methods: ensemble adjustment Kalman filter (EAKF) and sequential importance resampling particle filter (SIR-PF), using a well-known nonlinear and non-Gaussian model (Lorenz '63 model). The EAKF, which is a deterministic scheme of the ensemble Kalman filter (EnKF), performs better than the classical (stochastic) EnKF in a general framework. Comparison between the SIR-PF and the EAKF reveals that the former outperforms the latter if ensemble size is so large that can avoid the filter degeneracy, and vice versa. The impact of the probability density functions and effective ensemble sizes on assimilation performances are also explored. On the basis of comparisons between the SIR-PF and the EAKF, a mixture filter, called ensemble adjustment Kalman particle filter (EAKPF), is proposed to combine their both merits. Similar to the ensemble Kalman particle filter, which combines the stochastic EnKF and SIR-PF analysis schemes with a tuning parameter, the new mixture filter essentially provides a continuous interpolation between the EAKF and SIR-PF. The same Lorenz '63 model is used as a testbed, showing that the EAKPF is able to overcome filter degeneracy while maintaining the non-Gaussian nature, and performs better than the EAKF given limited ensemble size.

The genesis of Hurricane Karl (2010) is explored using analyses and forecasts from a cycling ensemble Kalman filter (EnKF) that assimilates routinely collected observations as well as dropsonde measurements that were taken during the Pre-Depression Investigation of Cloud Systems in the Tropics (PREDICT) field campaign. A total of 127 dropsonde observations were collected from six PREDICT flight missions over a 5-day period before and during Karl’s genesis. EnKF analyses that take into account the PREDICT dropsondes provide a detailed four-dimensional overview of the evolving kinematic and thermodynamic structure within the pregenesis disturbance. In particular, the additional field observations are found to increase the low- and midlevel circulation and column moisture in the EnKF analyses and reduce the position error of the low-level vortex. Deterministic forecasts from these analyses show a 24-h improvement in predicting genesis over a control experiment that omits the dropsonde observations. In ensemble forecasts for this event, the more accurate analyses translate into a higher confidence in predicting the intensification of Karl; that is, data assimilation experiments also suggest that initial condition errors at the mesoscale pose large challenges for predicting genesis, thus highlighting the need for improved observation networks and more advanced data assimilation methods.

Multiple assisted history matching (AHM) techniques commonly applied in the oil and gas industry were benchmarked for their applicability to field studies. The benchmarking effort was aimed at exploration of the theoretical as well as practical benefits, challenges and tradeoffs of the AHM techniques as they are utilized for asset development and reservoir management. This paper specifically discusses benchmarking of ensemble-based methods as applied for history matching of a key deepwater asset for the corporation. Two variations of ensemble-based methods are examined: the ensemble Kalman filter (EnKF) (Evensen, 2009) and the ensemble smoother with multiple data assimilation (ES-MDA) (Emerick and Reynolds, 2013). In both techniques, the ensemble of parameter estimates evolves as field data are assimilated into the reservoir characterization, either sequentially using the EnKF or iteratively using the ES-MDA. Both of these variations have been deployed with adaptive bootstrap-based screening (Zhang and Oliver, 2011) to reduce the sampling error of ensemble estimates. One significant strength of ensemble-based methods is their ability to estimate high-resolution model parameters. In this benchmark study, however, the ensemble-based methods were applied for estimation of a group of low-resolution model parameters including regional property multipliers and pilot points. The insights gained from the inverse problem of constraining reservoir simulation models to historical production data can be particularly valuable in guiding the practitioner in re-definition of the prior model uncertainty and towards the model solution. The study shows that a major challenge in the application of ensemble-based methods is parameter over-adjustment resulting from the improper addition of observation errors associated with the actual observations. In practice, multiple experiments are required for the user to iteratively improve the optimal level of the added errors. As the number of observations grows large, the increase in computational cost and decrease in ensemble variability are additional challenges in the application of ensemble-based methods. While assimilating different types of data, a balanced proportion of these data types should be maintained. These points are illustrated and discussed in the field application section of the paper.

There are two broad approaches to quantifying landscape C dynamics – by measuring changes in C stocks over time, or by measuring fluxes of C directly. However, these data may be patchy, and have gaps or biases. An alternative approach to generating C budgets has been to use process‐based models, constructed to simulate the key processes involved in C exchange. However, the process of model building is arguably subjective, and parameters may be poorly defined. This paper demonstrates why data assimilation (DA) techniques – which combine stock and flux observations with a dynamic model – improve estimates of, and provide insights into, ecosystem carbon (C) exchanges. We use an ensemble Kalman filter (EnKF) to link a series of measurements with a simple box model of C transformations. Measurements were collected at a young ponderosa pine stand in central Oregon over a 3‐year period, and include eddy flux and soil CO2 efflux data, litterfall collections, stem surveys, root and soil cores, and leaf area index data. The simple C model is a mass balance model with nine unknown parameters, tracking changes in C storage among five pools; foliar, wood and fine root pools in vegetation, and also fresh litter and soil organic matter (SOM) plus coarse woody debris pools. We nested the EnKF within an optimization routine to generate estimates from the data of the unknown parameters and the five initial conditions for the pools. The efficacy of the DA process can be judged by comparing the probability distributions of estimates produced with the EnKF analysis vs. those produced with reduced data or model alone. Using the model alone, estimated net ecosystem exchange of C (NEE)=−251±197 g C m−2 over the 3 years, compared with an estimate of −419±29 g C m−2 when all observations were assimilated into the model. The uncertainty on daily measurements of NEE via eddy fluxes was estimated at 0.5 g C m−2 day−1, but the uncertainty on assimilated estimates averaged 0.47 g C m−2 day−1, and only exceeded 0.5 g C m−2 day−1 on days where neither eddy flux nor soil efflux data were available. In generating C budgets, the assimilation process reduced the uncertainties associated with using data or model alone and the forecasts of NEE were statistically unbiased estimates. The results of the analysis emphasize the importance of time series as constraints. Occasional, rare measurements of stocks have limited use in constraining the estimates of other components of the C cycle. Long time series are particularly crucial for improving the analysis of pools with long time constants, such as SOM, woody biomass, and woody debris. Long‐running forest stem surveys, and tree ring data, offer a rich resource that could be assimilated to provide an important constraint on C cycling of slow pools. For extending estimates of NEE across regions, DA can play a further important role, by assimilating remote‐sensing data into the analysis of C cycles. We show, via sensitivity analysis, how assimilating an estimate of photosynthesis – which might be provided indirectly by remotely sensed data – improves the analysis of NEE.

This paper develops a parallel implementation of the ensemble Kalman filter (EnKF) based on shrinkage covariancematrix estimation. The EnKF is a sequential Monte Carlo method for parameter and state estimation of highly nonlinearmodels. In the context of ensemble based methods, background error correlations are estimated via anensemble of model realizations. In practice, the model dimension (number of model components) is several timeslarger than the ensemble size owing to the computational effort involved in one single model propagation. Thisconstraint yields to spurious correlations (correlation between distant model components in space) in the backgrounderror estimations. Commonly, localization methods are utilized in order to reduce the impact of spuriouscorrelations. For instance, correlations of model components are reduced based on some function of their physicaldistances. Some EnKF implementations are formulated in such manner that, localization methods are implicitlyinvolved in the assimilation of observations. In this context, one of the best ensemble Kalman filter implementationsis the local ensemble transform Kalman filter (LETKF). In the LETKF, each model component is surrounded by alocal box of size r and then, the information contained within the local domain (i.e., observed components) is used toperform the assimilation. However, for sparse observational networks, r can become sufficiently large with respectto the ensemble size and therefore, the local analysis corrections can be impacted by spurious correlations. Wepropose an ensemble Kalman filter implementation in which the background error covariance matrix is estimated viathe Rao-Blackwell Ledoit and Wolf (RBLW) estimator. This estimator has proven to be better conditioned than thetrue (background) error covariance matrix. In addition, the explicit computation of the RBLW estimator is notrequired in our formulation. This implies enormous savings in terms of memory considering the typical order ofbackground error covariance matrices O(10e8) Furthermore, the proposed EnKF implementation can be efficientlyperformed in parallel: each model component is surrounded by a local box, a local background error covariancematrix is estimated based on the RBLW estimator and then, the ensemble Kalman filter equations are locally used inorder to compute the local analysis corrections. After the local assimilation, each model component is mapped backonto the full model domain from where the global analysis solution is obtained. In operational data assimilation, EnKF implementations are attractive owing to their potential for parallelization. Experiments are performed making use of the Atmospheric General Circulation Model SPEEDY. The proposed method is coded in FORTRAN and MPI is used for the local assimilation of observations. The predicted variables are the zonal and meridional wind components, the temperature and the specific humidity. For each model variable, 8 layers with the T-63 resolution (192 x 96 grid points) are considered which provide a total number of 589,824 model components. The ensemble size is 94. The number of processors utilized are 96, 256, 512, 768, 1024 and 1536. The results reveal that, the use of the RBLW estimator can mitigate the impact of spurious correlations under sparse observational networks and even more, for different number of processors, the computational effort of the proposed EnKF implementation is comparable to that of the LETKF where no covariance estimation is performed. In terms of accuracy, the proposed implementation performs better than the LETKF formulation in the Root Mean Square Error sense.

Impacts of using an ensemble Kalman filter on air quality simulations along the California–Mexico border region during Cal–Mex 2010 Field Campaign

Assimilating time-lapse seismic data in the presence of significant spatially correlated model errors

Even though the ensemble Kalman filter (EnKF) is widely used, history matching reservoirs with facies description has proven to be a major challenge. A preferred technique for estimating large-scale facies fields within the petroleum industry is still missing. In this paper we present a new approach to this problem. Instead of applying the EnKF directly on facies realizations, the approach applies a transformation of facies fields to a specific level-set function, representing distances between facies types. This ensures better agreement with the EnKF Gaussianity assumptions, and the method always returns facies realizations with geological authenticity. The method also offers large flexibility in generating the initial ensemble, which can be done using any geostatistical tool. Further, no modifications of the standard EnKF equations are needed. The methodology is evaluated on two synthetic examples with increasing complexity. In both examples we consider reservoirs with channel structure. The results presented show that the updated models give large improvements in matching the measurements, and the uncertainty of the models is decreased. Further, recovery of the true petrophysical parameters is highly dependent on sufficient information in the measurements, but in one of the examples considered we are able to completely recover the true channel structure. Additional improvements in the quality of the updated facies fields are obtained by proper handling of the distances close to the reservoir boundaries, and conditioning on specific statistical measures to better preserve prior information about channel properties. Introduction The ensemble Kalman filter (EnKF) (Evensen [1], Aanonsen et al. [2]) is one of the most promising tools for assisted history matching of reservoir models. The filter has in particular proven to work well on cases where petrophysical properties are modeled using Gaussian random fields with variogram models. However, many operative oil and gas reservoirs have challenging complex geological structures - channelized reservoirs being one example. These structures typically have rapid spatial variations in the petrophysical properties (e.g., permeability and porosity). The effect of applying the EnKF directly on such reservoirs is that the characteristics of the fields are lost, i.e., sharp gradients are smeared out (dissipated). Several approaches for using EnKF on complex reservoirs have been proposed. The goal is to ensure that the updated fields are facies realizations, or at least approximations of such, and that good history match is obtained. The methods involve use of truncated pluri-Gaussian methods (Liu and Oliver [3, 4]), Gaussian mixture models (Dovera and Della Rossa [5], Sun et al. [6]), discrete cosine transformations (Jafarpour and McLaughlin [7]), kernel methods (Sarma and Chen [8]), and level-set methods (Moreno et al. [9]). Many of the above mentioned methods give promising results, but there are still drawbacks related to e.g., flexibility in how the initial ensemble may be generated, or the method involves complex and large modifications of the standard EnKF implementation. Our goal is to develop a methodology which is both simple, flexible and able to preserve facies structure for the updated fields. To achieve this, we propose a method motivated by the work of Moreno et al. [9], in the sense that level-set functions are used to represent the facies fields. However, in Moreno et al. [9], only perturbations of a fixed "base case" are updated, and it is questionable whether the real uncertainty is properly described by the prior ensemble. The method we propose is not limited to updating perturbations, and the initial ensemble can be computed using any geostatistical tool. This means that using our methodology, the initial ensemble can be conditioned on the correct facies types at the well locations. The method relies on a transformation of the facies field to a specific level-set function - a signed distance function (representing distances between opposite facies types). The distances are then updated using the EnKF, and converted to petrophysical properties when the reservoir simulator is run to the next assimilation time. The advantage with this transformation is that the ensemble of parameters to be estimated is closer to a Gaussian distribution, which is in better agreement with the EnKF Gaussianity assumptions. In addition, we are guaranteed that the updated fields are facies realizations.

In recent years, more traditional history matching methods have been increasingly challenged by sequential data assimilation techniques such as the ensemble Kalman filter (EnKF). There are strong similarities between EnKF and the non-sequential method, randomized maximum likelihood (RML). For a linear forward model the two methods are equal, for a nonlinear forward model there arises some differences (in addition to sequential/batch data assimilation): RML can be iterative, while EnKF is not; RML uses realization-specific gradients/sensitivities to change a model realization while EnKF uses the same covariance for all realizations. We assess the sampling capabilities of RML and EnKF for a weakly nonlinear forward model. Results are compared to a Markov chain Monte Carlo (McMC) method, which samples correctly from the posterior. Our aim is to clarify which of the above mentioned differences between RML and EnKF has the biggest impact on the sampling capabilities. We apply the methods to a two-phase reservoir models small enough to be suitable for McMC. The assessment of RML and EnKF is performed by comparing history matching capabilities, and properties of their posterior distributions to those of the posterior distributions obtained with McMC.