Calibrating Multi-Point Geostatistical Models Using Pressure Transient Data

Building of models in the Earth Sciences often requires the solution of an inverse problem: some unknown model parameters need to be calibrated with actual measurements. In most cases, the set of measurements cannot completely and uniquely determine the model parameters; hence multiple models can describe the same data set. Bayesian inverse theory provides a framework for solving this problem. Bayesian methods rely on the fact that the conditional probability of the model parameters given the data (the posterior) is proportional to the likelihood of observing the data and a prior belief expressed as a prior distribution of the model parameters. In case the prior distribution is not Gaussian and the relation between data and parameters (forward model) is strongly non-linear, one has to resort to iterative samplers, often Markov chain Monte Carlo methods, for generating samples that fit the data likelihood and reflect the prior model statistics. While theoretically sound, such methods can be slow to converge, and are often impractical when the forward model is CPU demanding. In this paper, we propose a new sampling method that allows to sample from a variety of priors and condition model parameters to a variety of data types. The method does not rely on the traditional Bayesian decomposition of posterior into likelihood and prior, instead it uses so-called pre-posterior distributions, i.e. the probability of the model parameters given some subset of the data. The use of pre-posterior allows to decompose the data into so-called, “easy data” (or linear data) and “difficult data” (or nonlinear data). The method relies on fast non-iterative sequential simulation to generate model realizations. The difficult data is matched by perturbing an initial realization using a perturbation mechanism termed “probability perturbation.” The probability perturbation method moves the initial guess closer to matching the difficult data, while maintaining the prior model statistics and the conditioning to the linear data. Several examples are used to illustrate the properties of this method.

Significant uncertainty exists in the detailed 3-D distribution of lithofacies, porosity, and permeability in every reservoir. Understanding and modeling the heterogeneous 3-D distribution of these rock properties is critical for improved oil recovery and reservoir management. Geostatistical techniques are being increasingly used to generate alternative heterogeneous 3-D reservoir models that are consistent with the available data. Although a large number of stochastic reservoir models or realizations may be available, a small number of realizations are considered in practice. Due to computer limitations, it is only possible to visualize and perform fine-scale full-field flow simulation on a limited number of realizations. Techniques are reviewed in this paper for ranking a suite of geostatistical realizations so that low-side, expected, and high-side realizations may be reliably chosen. Detailed analysis/flow simulation may then be performed on these realizations that somehow bound the uncertainty in the reservoir. Reservoir management is improved when expected and bounding cases are considered rather than using a limited number of "random" realizations. This paper reviews a number of methods for ranking geostatistical reservoir models. These methods may be classified into three categories. The first category includes statistical methods such as simple statistics, 3-D measures of connectivity, and connectivity to specific well locations. Methods in the second class are based on approximations to flow simulation, e.g., random walk-type results. The third category is for flow-simulation based methods for a simpler process than that being considered for improved oil recovery, e.g., tracer simulation and flow simulation with coarsened models. The applicability of a number of ranking methods is illustrated with a small example. There is no unique ranking index when there are multiple flow response variables and no ranking measure is perfect. Nevertheless, the value of ranking realizations will be quantified by examining the expected loss knowing an economic loss function and the true distribution of uncertainty. Introduction The primary objective of the application of geostatistical tools for reservoir modeling is to create realistic numerical geological models or realizations of the 3-D spatial distribution of lithofacies, porosity. and permeability. An often disturbing fact of geological reservoir modeling is that there are alternative realizations that honor all of the available data equally well and yet yield different reservoir performance predictions. In fact, it is often advertised that "a different realization can be obtained by simply changing a random number seed". A fundamental principle of geostatistics is that of data integration, i.e., all known data should be honored "by construction" and not left to chance. Of course, this is not always possible; it is difficult to constrain detailed 3-D geological realizations to seismic and historical production data. We are restricting ourselves to plausible realizations. There are times when certain realizations would be rejected on the basis of familiarity with the reservoir or data not used in the geostatistical modeling such as production-related historical observations. We are not considering the rejects; only those realizations that meet all basic requirements of reasonableness are considered. Exact prediction / decision making would require exhaustive knowledge of the spatial distribution of porosity and permeability. At any specific instant in time, there is such a single true distribution of petrophysical properties. This true distribution was created by the complex interaction of many different chemical, physical. and biological processes over geological time and would be accessible only through exhaustive sampling. In all practical situations, the unique true distribution will remain unknown. This paper addresses the uncertainty due to incomplete information. We do not consider uncertainty due tothe volume support or scale difference between core measurements and geological modeling cells,the limited flexibility of our geostatistical modeling techniques to reproduce complex nonlinear features, ornumerical approximations in the subsequent flow simulator. P. 105

Ensemble Kalman filter for automatic history matching of geologic facies

The present paper describes the use of pressure derivative and second derivative of integral of pressure in a fractal reservoir with matrix participation with phase redistribution in a geological environment that are not possible by conventional techniques. The analysis of this type of data in reservoir characterization is known as “inverse problem” and one can obtain information about interwell and vertical permeability distribution in a reservoir. The fractal geometry in a dynamic pressure transient tests data plays a very vital role for heterogeneity characterization. The pressure transient response is analyzed for flow in a connected fracture network and fracture with matrix participation. The computer aided matching technique for both pressure and its derivative by nonlinear regression techniques are used in estimating the reservoir properties from measured drawdown/buildup and falloff pressure data of heterogeneous reservoir. In the present paper the fractional calculus approach has been utilized to solve the diffusivity equation with phase redistribution in fractal reservoir. The pressure solution of the diffusivity is in terms of Laplace space and its analytical inversion is not possible. We have obtained numerically inversion of the problem and the pressure, pressure derivative, integral of pressure and its first and second derivative has been calculated. The permeability estimated from pressure transient test data of a well are in good agreement with the identified the geological model.

Reservoir characterization is a crucial aspect of the oil and gas industry, providing insights into the properties and potential productivity of hydrocarbon reservoirs. Advanced geological modeling techniques have significantly enhanced the accuracy and efficiency of reservoir characterization. These techniques integrate diverse datasets and sophisticated algorithms to create detailed subsurface models that inform exploration, development, and production strategies. One of the primary techniques in advanced geological modeling is seismic interpretation, which involves the analysis of seismic data to map subsurface structures. High-resolution 3D seismic surveys provide detailed images of the subsurface, allowing geologists to identify faults, fractures, and stratigraphic features with greater precision. The integration of seismic inversion techniques converts seismic reflection data into quantitative rock property estimates, such as porosity and fluid saturation, further refining the reservoir model. Geostatistical modeling is another critical technique that improves reservoir characterization. By utilizing statistical methods to analyze spatial variability, geostatistics generates multiple realizations of reservoir properties, capturing the inherent uncertainty and heterogeneity of the subsurface. Techniques such as kriging, co-kriging, and sequential Gaussian simulation are commonly used to interpolate and extrapolate data from well logs and seismic surveys, creating probabilistic models that guide decision-making under uncertainty. Petrophysical analysis plays a significant role in enhancing geological models. Well logs, core samples, and laboratory measurements provide direct information about rock and fluid properties. Petrophysical models are developed to correlate these properties with depth and other geological features. Advances in machine learning and data analytics have further refined petrophysical interpretation by enabling the integration of large datasets and the identification of complex patterns that traditional methods might miss. Another advanced technique is the use of reservoir simulation, which models the flow of fluids within the reservoir over time. Reservoir simulators incorporate geological models, petrophysical data, and production history to predict future reservoir performance under different production scenarios. This dynamic modeling is essential for optimizing recovery strategies, enhancing field development plans, and managing reservoir performance. Multi-scale modeling approaches have also become increasingly important. These approaches integrate data and models across different spatial and temporal scales, from pore-scale petrophysical properties to basin-scale geological structures. By capturing the interactions between processes occurring at different scales, multi-scale models provide a more comprehensive understanding of reservoir behavior and improve the accuracy of predictions. The integration of machine learning and artificial intelligence (AI) has revolutionized reservoir characterization. Machine learning algorithms can process vast amounts of data from various sources, such as seismic surveys, well logs, production data, and geological maps, to identify patterns and relationships that enhance the understanding of reservoir properties. AI-driven techniques, including neural networks and support vector machines, are increasingly used for tasks such as facies classification, porosity prediction, and fracture detection, leading to more accurate and efficient reservoir models. The application of big data analytics in geological modeling has enabled the handling and analysis of massive datasets that were previously unmanageable. Advanced data analytics tools facilitate the integration, processing, and visualization of complex datasets, enhancing the ability to derive actionable insights and improve decision-making in reservoir management. Keywords: Techniques, Reservoir Characterization, Advanced, Geological Modeling, Oil, Gas, Industries

Maximum-likelihood (ML), also given its connection to least-squares (LS), is widely adopted in parameter estimation of physiological system models, i.e., assigning numerical values to the unknown model parameters from the experimental data. A more sophisticated but less used approach is maximum a posteriori (MAP) estimation. Conceptually, while ML adopts a Fisherian approach, i.e., only experimental measurements are supplied to the estimator, MAP estimation is a Bayesian approach, i.e., a priori available statistical information on the unknown parameters is also exploited for their estimation. In this paper, after a brief review of the theory behind ML and MAP estimators, we compare their performance in the solution of a case study concerning the determination of the parameters of a sum of exponential model which describes the impulse response of C-peptide (CP), a key substance for reconstructing insulin secretion. The results show that MAP estimation always leads to parameter estimates with a precision (sometimes significantly) higher than that obtained through ML, at the cost of only a slightly worse fit. Thus, a three exponential model can be adopted to describe the CP impulse response model in place of the two exponential model usually identified in the literature by the ML/LS approach. Simulated case studies are also reported to evidence the importance of taking into account a priori information in a data poor situation, e.g., when a few or too noisy measurements are available. In conclusion, our results show that, when a priori information on the unknown model parameters is available, Bayes estimation can be of relevant interest, since it can significantly improve the precision of parameter estimates with respect to Fisher estimation. This may also allow the adoption of more complex models than those determinable by a Fisherian approach.

A typical goal of a geostatistical analysis is to perform spatial prediction. Geostatistical models traditionally take the form of regression models with spatially correlated errors, employing covariate information in the mean function. The random process in such models is indexed in terms of the locations of the study region. The machine learning community employs Gaussian processes for many tasks, and these processes may be indexed on information that statisticians would term as covariates. We compare the predictive ability of a traditional geostatistical model with that of a non‐traditional Gaussian process model, which has a constant mean and whose random process is indexed on orographic covariates. The models we compare are both simply parametrized, fit with straightforward inference methods, and use classical kriging predictors. The non‐traditional model achieves non‐stationarity in the geographic space. We apply the two models to soil moisture data sets, which have extensive spatial sampling, and we apply the models to six separate days of data. In terms of quantitative measures, we find that the models' predictive abilities are comparable, with the non‐traditional outperforming the traditional model perhaps slightly. Qualitatively, the non‐traditional model's predictions may be more closely linked to the orographic covariates. Copyright © 2014 John Wiley & Sons, Ltd.

For more than half a century, geostatistics has been developed and has increasingly been used for modeling subsurface heterogeneity. Traditionally, geostatistical simulations are based on a random function model defined according to the specificities of the geological formation under investigation. Unlike traditional geostatistics, multiple‐point (MP) geostatistics avoids the explicit definition of a random function but directly infers the necessary multivariate distributions from training images. This confers on MP geostatistics a potential applicability to any geological environment, provided that there is a training image representative of the geological heterogeneity and that the essential features of this training image can be characterized by statistics defined on a limited point configuration. This paper presents a comprehensive review of MP geostatistics. If the principle of MP geostatistics is straightforward and attractive, its industrial applicability largely depends on the implementation methods. The use of a search tree for storing MP statistics is a great step that made MP geostatistics actually applicable in industry. In the meantime, other methods such as multiple grids and image postprocessing are introduced and allow enhancing the reproduction of patterns observed in training images. Because of the sequential procedure, MP simulations can easily honor local hard data. There are methods available for constraining MP simulations to target global statistics (facies proportions) and also for integrating spatial auxiliary constraints that can tremendously improve the spatial features of MP simulations. Furthermore, both the gradual deformation method and the probability perturbation method are compatible with MP geostatistics and allow integrating hydrodynamic data into MP simulations. More recent developments of MP geostatistics include sequentially simulating patterns instead of points and using different geological scenarios (training images) for dynamic data inversion.

An approach to model the detailed 3-D distribution of lithofacies and porosity constrained to seismic data is presented. The simulated annealing-based approach explicitly honors the relatively coarse vertical resolution, from a reservoir modeling perspective, and the less than perfect correlation of seismic with lithofacies proportions and effective porosity. Conventional geostatistical procedures such as co-located cokriging or the Markov-Bayes model assume that the seismic attribute has the same volumetric support as the geological modeling cells. The conventional techniques are reviewed, details of the proposed methodology are presented, and a reservoir case study is shown. Introduction Due to geological complications and inherent limitations in seismic data acquisition, seismic inexactly measures lithofacies proportions and average porosity. Typically, the seismic-derived lithofacies proportion and porosity may be correlated with the true values with a correlation of 0.5 to 0.7. The specific seismic attribute and degree of correlation must be calibrated for each reservoir. Geostatistical techniques to integrate seismic data must account for this precision. Another consideration, when using seismic data, is that seismic-derived proportions and porosity represent a volume significantly larger than the typical geological modeling cell. The areal resolution is often comparable. The vertical resolution; however, is 10 to 100 times the resolution of the geological modeling cells. Current geostatistical models are built at a vertical resolution of 1-3 feet and current seismic data informs a 30-100 foot vertical average. The detailed resolution of geostatistical models is considered necessary to transfer the effect of heterogeneities into flow simulators. There is a need for geostatistical modeling tools to explicitly account for the precision and scale of seismic data. Most conventional geostatistical techniques account for the precision of the seismic data by treating it as soft or secondary data. At times, the seismic data is considered to represent an arithmetic volume average of the lithofacies indicator or the porosity. A review of conventional geostatistical techniques will show that they do not simultaneously handle the precision and scale of seismic data. An approach will be proposed based on an extension of the simulated-annealing approach to geostatistical model construction. An objective function is constructed that constrains the model to the seismic data accounting for both the predefined precision and scale of the seismic data. An example from a West Texas Permian Basin reservoir is developed to demonstrate the practical applicability of the proposed approach. The 3-D distribution of porosity is constrained to a seismic attribute representing an imprecise (= 0.5) 50-foot vertical average. In an increasing number of cases, seismic data or hand-drawn geological trend maps are available and we want to build them into detailed 3-D geostatistical models. The methodology proposed in this paper is suited to those situations. The importance of simultaneously accounting for the precision and scale of seismic data could only be assessed by a number of comparative reservoir case studies. These comparative studies are considered important but are not included in this paper. For the time being we must argue that better reservoir models are obtained when they are constrained to the maximum amount of data properly accounting for the precision and scale of each data source. P. 9

This paper investigates the tracking control problem of underactuated surface vessels with non-diagonal inertia and damping matrices. Based on the inherent cascaded structure of vessel dynamics and the cascaded system theory, the tacking control problem of vessels is converted to the stabilization control problem of two cascaded subsystems. Two trajectory tracking control laws are designed respectively for known and unknown model parameters. The first feedback control law is derived with the aid of Lyapunov method, realizing the global $${\mathscr {K}}$$ -exponential trajectory tracking of vessels with accurate model parameters. As all the state variables in the first law are independent of model parameters, the sliding mode technique is applied to extend the first control scheme to the case of unknown model parameters, resulting into an another tracking control law, which ensures the global $${\mathscr {K}}$$ -exponential stability of the closed-loop error system despite unknown model parameters. Effectiveness of the proposed controllers is demonstrated by numerical simulations.

Geostatistical 3D geological model construction to estimate the capacity of commercial scale injection and storage of CO2 in Jacksonburg-Stringtown oil field, West Virginia, USA

BackgroundWhen creating mechanistic mathematical models for biological signaling processes it is tempting to include as many known biochemical interactions into one large model as possible. For the JAK-STAT, MAP kinase, and NF-κB pathways a lot of biological insight is available, and as a consequence, large mathematical models have emerged. For large models the question arises whether unknown model parameters can uniquely be determined by parameter estimation from measured data. Systematic approaches to answering this question are indispensable since the uniqueness of model parameter values is essential for predictive mechanistic modeling.ResultsWe propose an eigenvalue based method for efficiently testing identifiability of large ordinary differential models and compare this approach to three existing ones. The methods are benchmarked by applying them to models of the signaling pathways mentioned above. In all cases the eigenvalue method proposed here and the orthogonal method find the largest set of identifiable parameters, thus clearly outperforming the other approaches. The identifiability analysis shows that the pathway models are not identifiable, even under the strong assumption that all system state variables are measurable. We demonstrate how the results of the identifiability analysis can be used for model simplification.ConclusionWhile it has undoubtedly contributed to recent advances in systems biology, mechanistic modeling by itself does not guarantee unambiguous descriptions of biological processes. We show that some recent signal transduction pathway models have reached a level of detail that is not warranted. Rigorous identifiability tests reveal that even if highly idealized experiments could be carried out to measure all state variables of these signaling pathways, some unknown parameters could still not be estimated. The identifiability tests therefore show that the level of detail of the investigated models is too high in principle, not just because too little experimental information is available. We demonstrate how the proposed method can be combined with biological insight, however, to simplify these models.

The series elastic actuator (SEA) has been widely used in the exoskeletons. However, its mathematical model contains both unknown parameters and states, which need to be jointly estimated online via the measurable input and output. To address this issue, an adaptive observer is developed to reconstruct both unknown model parameters and states simultaneously. In particular, a constructive approach is proposed to extract the parameter estimation error, which is used to design a new adaptive law decoupled from the observer error dynamics. A high-gain modification is also incorporated into the observer to handle the dependency on the unknown system states in the regressor. As a result, the convergence of both the observation error and the estimation error can be proved under the standard excitation condition. Moreover, since the SEA used in the exoskeletons may operate under a weak excitation due to the slow training of rehabilitation, a further tailored adaptive law with the eigenvalue decomposition method is introduced to adapt the weak excitation. Finally, the effectiveness of the proposed methods is validated via experiments on a real-world SEA test rig.

Mixed therapy in cancer treatment for personalized drug administration using model reference adaptive control

A compartmental model for the in vitro uptake kinetics of the anti‐cancer agent topotecan is proposed. This model provides a description of the activity of the drug, and subsequent delivery of active form to the nuclear DNA target. The unknown model parameters are estimated from two‐photon laser‐scanning microscopy data, which provide concentrations of topotecan (active plus inactive forms) in the extracellular region containing live human breast tumour cells (MCF‐7 cell line), the cytoplasm and the nucleus. This determines an output structure for which the model is uniquely identifiable, that is, the unknown parameters are uniquely determined from noise‐free, continuous and perfect data. The model allows in silico predictions of the dose dependence of target binding. Copyright © 2005 John Wiley & Sons, Ltd.