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Zhao, Xiaoxi (University of Southern California) | Popa, Andrei S. (Chevron Corporation) | Ershaghi, Iraj (University of Southern California) | Aminzadeh, Fred (University of Southern California) | Li, Yuanjun (Stanford University) | Cassidy, Steve D. (Chevron Corporation)
Summary This paper presents a methodology for the geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. Given certain known well‐log data in a geological region, the Kriging methodology is used to estimate or predict spatial phenomena at nonsampled locations from the estimated random function. The approach assumes that the data are accurate and precise, and the random function is generated from a thorough descriptive analysis of the known data set. Regarding the assumptions considered in classic Kriging, it is realistic to assume that spatial data contain a certain amount of imprecision, mostly because of measurement errors, and information is lacking to properly assess a unique random‐function model. A methodology is presented for the geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. A combination of regular, or classic, Kriging and the fuzzy‐logic method is proposed. As such, imprecise input data and variogram parameters are modeled on the basis of fuzzy‐logic theory, while the predictions and variances are computed from Kriging analysis characterized by membership functions. Last, an optimization method is included to solve the constrained fuzzy‐nonlinear‐equation system. The proposed methodology was implemented, and a user‐friendly integrated tool was developed, which enables the user to create a grid structure on the basis of the input data, conduct statistical analysis, and run fuzzy Kriging for various problems. We used the tool to run a test case using the SPE 10 (SPE Comparative Solution Project, Model-II 2000) porosity data. With the fuzzy‐Kriging methodology, two maps are generated with upper‐bound values and lower‐bound values. Compared with true data, the upper‐bound map trends to include higher values better, while the lower‐bound map trends to include lower‐value parts better. In addition, a case study has been conducted using measured core‐permeability data in a heterogeneous reservoir to demonstrate the viability of the technology.
Summary To estimate the production potential at a new, prospective field site by means of simulation or material balance, one needs to collect various forms of costly field data and make assumptions about the nature of the formation at that site. Decline–curve analysis (DCA) would not be applicable in this scenario, because producing wells need to pre–exist in the target field. The objective of our work was to make first–order forecasts of production rates at prospective, undrilled sites using only production data from existing wells in the entire play. This is accomplished through the co–Kriging of decline–curve parameter values, where the parameter values are obtained at each existing well by fitting an appropriate decline model to the production history. Co–Kriging gives the best linear unbiased prediction of parameter values at undrilled locations, and also estimates uncertainty in those predictions. Thus, we obtained production forecasts at P10, P50, and P90, and we calculated the estimated ultimate recovery (EUR) at those same levels across the spatial domain of the play. To demonstrate the proposed methodology, we used monthly gas–flow rates and well locations from the Marcellus shale–gas play in this research. Monitoring only horizontal and directional wells, the gas–production rates at each well were carefully filtered and screened. Also, we normalized the rates by perforation–interval length. We only kept production histories of 24 months or longer to ensure good decline–curve fits. Ultimately, we were left with 5,637 production records. Here, we chose Duong's decline model (Duong 2011) to represent the production decline in this shale–gas play, and fitting this decline curve was accomplished through ordinary least–squares (OLS) regression. Interpolation was done by universal co–Kriging while considering the correlation between the four parameters in Duong's model, which also showed linear trends (the parameters showed dependency on the x and y spatial coordinates). Kriging gave us the optimal decline–curve coefficients at new locations (P50 curve), as well as the variance in these coefficient estimates (used to establish P10 and P90 curves). We were also able to map EUR for 25 years across the study area. Finally, the universal co–Kriging model was cross validated with a leave–one–out scheme, which showed significant, but not unreasonable, error in the decline–curve–coefficient prediction. The methods proposed were implemented and did not require various costly data, such as permeability and bottomhole pressure, thus giving operators a risk–based analysis of prospective sites. While we demonstrated the procedure on the Marcellus shale–gas play, it is applicable to any play with existing producing wells. We also made this analysis available to the public in a user–friendly web application (Xi and Morgan 2018).
Summary Decline curves are the simplest type of model to use to forecast production from oil and gas reservoirs. Using a selected decline model and observed production data, a trend is projected to predict future well performance and reserves. Despite capturing general trends, these models are not sufficient at describing the underlying physics of complex multiphase porous-media flow phenomena and at explaining variations in production caused by changes in operational conditions. The application of these models within a Bayesian framework is a feasible alternative to mitigate this issue and obtain more‐robust forecasts by representing the possible outcomes with probability distributions. However, one important aspect that conditions the production forecasts and their uncertainty is the design of a suitable prior distribution, which can be subjective. To address the aforementioned issue, this paper presents a workflow for the development of a localized prior distribution for new wells drilled in shale formations that combines production data from pre‐existing surrounding wells and spatial data, specifically well‐surface/bottom coordinates. This workflow aims to establish engineering criteria to reduce the subjectivity in the design of a prior distribution, assessing spatial continuity of the parameters of a physics‐based decline‐curve model (θ2 model), automatically identifying regions where uncertainty can be reduced a priori, and reliably quantifying the uncertainty. A case study of 814 gas wells in the Barnett Shale is presented, and several maps are generated for the analysis of important properties to be considered during field development. The dry‐gas window presented more‐continuous decline‐curve parameters than the wet‐gas and gas/condensate windows, which resulted in lower uncertainty with the localized prior approach. As more data are acquired with time, the uncertainty in the production forecasts is further reduced and the localized prior becomes more informative, especially in the dry‐gas window. The localized prior can then serve as an indicator for the performance of new infill wells in different locations. Portions of the content of this paper were initially presented in Holanda et al. (2018b), and are further developed and reviewed here.
Correia, Pedro (Geovariances) | Geffroy, François (Geovariances) | Binet, Héléne (Geovariances) | Chautru, Jean-Marc (Geovariances) | Renard, Didier (Mines ParisTech) | Yewgat, A. (Mines ParisTech) | Huguet, Frédéric (Storengy SAS) | Formento, Catherine (Storengy SAS)
ABSTRACT Geophysical surveys are often composed of several different datasets, typically with multiple crossover points, and generally contaminated by unknown and systematic error. Mis-ties are those intersections where two or more different measures are available at the same location. The problem is old and has different solutions. A new geostatistical method is presented, which is based on a variation of Kriging with variance of measurement errors. The variance of measurement error is here defined for each profile, which allows reducing mis-ties and uncertainty. The generated Time maps are significantly enhanced and can be used as reliable drift maps in Time-to-depth conversion operations. Thanks to the use of editable variances assigned to each profile, geophysicists can control local uncertainties and get enhanced Time maps after a rapidly converging trial-and-error approach. Such variances must be editable, as default values are calculated from profilesâ€™ intersections, which are rarely numerous enough to lead to robust statistics. For the same reason, the covariance model used in the Kriging system can be modified. By default, an automatic fitting is proposed for the sake of simplicity, but it might be valuable in some cases to modify default fitting, making use in such a case of the geophysicistâ€™s experience. Presentation Date: Wednesday, September 18, 2019 Session Start Time: 9:20 AM Presentation Time: 11:00 AM Location: Poster Station 3 Presentation Type: Poster
Zhao, X.. (University of Southern California) | Popa, A. S. (Chevron Corporation) | Ershaghi, I.. (University of Southern California) | Aminzadeh, F.. (University of Southern California) | Li, Y.. (University of Southern California) | Cassidy, S. D. (Chevron Corporation)
Abstract This paper presents a methodology for geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. Given certain known data in a geological region, the Kriging methodology is used to estimate or predict spatial phenomenon at non-sampled locations from the estimated random function. The approach assumes that the data is accurate and precise and the random function is generated from a thorough descriptive analysis of the known dataset. Regarding the assumptions considered in classic Kriging, it is realistic to assume that spatial data contains a certain amount of imprecision mostly due to measurement errors, and the information is lacking to properly assess a unique random function model. This paper presents a methodology for geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. Combination of regular or classic Kriging and Fuzzy Logic method is proposed. As such, imprecise variogram parameters are modeled based on Fuzzy Logic theory, while the predictions and variances are computed from Kriging analysis characterized by membership functions. Lastly, an optimization method is included to solve the constrained fuzzy non-linear equation system. The proposed methodology was implemented into a friendly integrated tool, which enables the user to create a preferred grid, conduct statistical analysis and run fuzzy kriging for various problems. The tool was validated using the SPE-10 porosity data. Additionally, a case study has been conducted using measured core permeability data in a heterogeneous reservoir to demonstrate the viability of the technology.
Abstract Calibrating complex subsurface geological models against dynamic well observations yields to a challenging inverse problem which is known as history matching in oil and gas literature. The highly nonlinear nature of interactions and relationships between reservoir model parameters and well responses demand automated, robust and geologically consistent inversion techniques. The ensemble of calibrated and history matched models quality determines the reliability of production uncertainty assessment. Reliable production forecasting and uncertainty assessment are essential steps toward reservoir management and field development. The Bayesian framework is a widely accepted approach to incorporate dynamic production data to the prior probability distribution of reservoir models and obtain the posterior distribution of reservoir parameters. Uncertainly assessment is performed by sampling the posterior probability distribution which is a computationally challenging task. Markov-Chain Monte Carlo (MCMC) algorithm has shown successful application in reservoir model calibration and uncertainty quantification is recent years. MCMC can efficiently sample the high-dimensional and complex posterior probability distribution of reservoir parameters and generate history matched reservoir models that consequently can be used for production forecasting uncertainty assessment. MCMC method is a gradient-free approach which makes is favorable when gradient information is not available through reservoir simulation. In MCMC method normally to march to next iteration the new sample is independent of the previous sample and the proposal distribution is rather random. To improve the sampling procedure and make MCMC process more efficient we propose an approach based on locally varying mean (LVM) Kriging to base the new sample generation on the previous iteration sample. In this method, the previous sample is used as the varying mean map in the geostatistical simulation approach to generate the new proposal for the next iteration. Using LVM Kriging to relate the new sample to previous iteration sample, make the chain of samples in MCMC more correlated and geologically consistent. Also this new proposal distribution makes the sampling procedure more efficient and avoids random and arbitrary movements is the parameter space. We applied MCMC with LVM Kriging to a suite of 2D and 3D reservoir models and obtained the calibrated model. We observed that the application of the new proposal distribution based on LVM Kriging along with MCMC improved the quality of the samples and resulted in promising uncertainty quantification. We also observed meaningful improvement in calibrated reservoir models quality and uncertainty interval while utilizing LVM comparing to random proposal or transition distribution in MCMC. MCMC with LVM Kriging as proposal distribution results in improved uncertainty assessment through enhancing the quality of the generated samples from posterior probability distribution of reservoir model parameters. Traditional random or independent proposal distribution does not represent the dependency of the samples through MCMC chain and iterations while this challenge is addressed by combining MCMC with LVM.
Spatial petrophysical property modeling is a crucial step in reservoir characterization as it directly affects heterogeneity and flow modeling. So it is essential to look for an optimal algorithm that leads to capture the most realistic spatial modeling. Many conventional kriging algorithms have been adopted for spatial permeability modeling such as simple, ordinary, and universal kriging. All these approaches are linear unbiased estimators as covariance structure is estimated first, and then used for interpolation leading to ignore the effect of uncertainty in the covariance structure on subsequent predictions. To overcome the restrictions of unbiased prediction in conventional approaches, Bayesian Kriging has been recently suggested to take into account the uncertainty about Variogram parameters on subsequent predictions. Bayesian Kriging incorporates a prior knowledge about observations such as expert grasp and outcome from neighboring data to be considered as a qualified guess in spatial estimation procedure. Commonly, the prior distribution is classified in term of Variogram parameters such as coefficients, data variance, range, and nugget to be adopted as a qualified guess in the spatial estimation. The qualified guess allows uncertainty estimation reduction to achieve more realistic spatial modeling and improved reservoir characterization. The observation uncertainty is represented as a posterior distribution and predictive parameter distribution avoiding unrealistic small regions within the observations to attain optimal unbiased linear interpolation through Bayesian kriging algorithm. Due to the some similarity between Bayesian Kriging and Universal Kriging, which incorporates 2D trend in the spatial modeling, the two algorithms were considered for comparative spatial modeling of formation permeability in a real heterogeneous sandstone reservoir. The spatial modeling was also done through simple and ordinary kriging for extensive comparison. A statistical sampling approach was considered to rank and select the three quantiles P10, P50, and P90 of the created equiprobable reservoir stochastic images. The entire work was done through R, the most open-source statistical computing language.
Abstract We present a new methodology for improving the economic returns of shale gas plays. The development of an economically efficient drilling programme in such plays is a challenging task, requiring a large number of wells. Even after a relatively large number of wells have been drilled, the average well production and the variation of well performance (economics) remains highly uncertain. The ability to delineate a shale play with the fewest number of wells and to focus drilling in the most productive areas is an important driver of commercial success. The importance of probabilistic modelling in managing uncertainty in shale gas plays has been explicitly emphasised in a number of studies. The objective of this study is to develop a practical valuation methodology that addresses these complexities and is dynamic, in the sense that the optimal drilling strategy can be continually updated as we learn the outcome of each well drilled. Maximizing the returns from a shale gas play is essentially a problem of choosing well locations and numbers to optimize production volumes & rates. Drilling policies have to take account of a large number of already-drilled locations, possible new drilling locations, spatial dependencies between performance at those different (possible) well locations and the extent of uncertainty as to whether or not a well will be economic. These factors cause typical valuation methodologies to be impractical due to the "curse of dimensionality". In this study an unconventional play is divided into cells. In each cell a fixed number of wells can be drilled. The chance of success (of a well having an NPV greater than zero) in any given cell is itself considered to be an uncertain variable. An initial probability distribution for the chance of success of each cell is derived from analogous plays plus any available information about the specific play. The methodology proceeds as follows. First, as each new well (or group of new wells) is drilled, the outcome is used in combination with the prior probability distribution (using Bayes Theorem) to create an updated probability distribution for the chance of success of the relevant grid cell. Thus, our initial estimate can be continuously updated as we get more and more actual outcomes. Second, the influence of the new chance of success on the surrounding cells, due to spatial correlation, is updated using indicator kriging, a geostatistical technique. The methodology proposed in this study informs the development of drilling policies for shale gas opportunities by using a probabilistic model that accounts for the uncertainty in the chance of success and its spatial dependency. The use of cells to represent a set of wells simplifies the analysis and greatly reduces the computing requirements. The methodology has been applied to a well set from the Barnett Shale, Texas, United States of America.
Abstract Number of estimators and simulators is used in petroleum engineering applications in order to model the static and dynamic characteristics of hydrocarbon reservoirs. However, the fundamental concepts used in estimators and simulators do vary from each other leading to the necessity of choosing the optimal approach with an extensive care. Geostatistics is perhaps the most widely used technique in simulation which is popularly used in different engineering applications and specifically in reservoir simulation. In this paper, three shortcomings of geostatistics in estimation and simulation are highlighted and alternative methods are proposed. Firstly, Kriging estimator, as is used in geostatistics, guarantees minimum estimation variance if the variogram model is the optimum fit and stationary conditions are met. However, in real situation, these conditions are not always satisfied. Geostatistics is prone to number of deficiencies some of which are discussed in this paper in the context of oil and gas applications and alternative solutions are presented. As such, the incapability of variogram, as being the fundamental tool in geostatistical analysis, in capturing the periodic structure of reservoir formation is shown. Then, it is discussed why the use of signal decomposition tools such as Fourier, fast Fourier and wavelet transform is more advantageous than variogram in this context. Secondly, to overcome the smoothing property associated with geostatistics fractal simulation is recommended as an alternative approach. Finally, it is discussed how intrinsic properties of reservoir can be included in multi-fractal or neural network to improve the estimation and simulation process. This appears to be difficult to apply if geostatistical approaches are used for estimation or simulation purposes.