Embedded Response Surfaces Approach for Uncertainty Quantification

Quantifying uncertainty in petroleum resources is important for development planning and decision making. Increasingly, geostatistical techniques are used to integrate diverse data sources and provide a defensible model of uncertainty. Petroleum resources are calculated from a combination of variables including thickness, porosity and saturation. Uncertainty in global petroleum resources are calculated stepwise:establish the local uncertainty in each variable using a conventional Gaussian geostatistical model;sample the local distributions with spatial correlation using a p-field based technique;modify the p-field samples to have the correct multivariate variability using the LU technique; and,assemble the distribution of uncertainty over any volume using the joint spatial/multivariate realizations. The alternatives to this technique are a simplistic Monte Carlo simulation without spatial correlation or a more complex high resolution geostatistical model. Speed and mathematical consistency are the main advantages of the proposed technique. The theoretical basis of the spatial/multivariate decomposition approach is developed with the assumptions and implementation details. A synthetic example from a realistic case study is presented showing the global uncertainty in oil-in-place over arbitrarily large areas. Introduction Geostatistical techniques have been increasingly used for reservoir characterization for two main reasons:different data sources can be integrated to predict a reservoir property between wells; and,an assessment of uncertainty in the estimation can be obtained(1). Quantifying global uncertainty in petroleum resources is important for reservoir development planning and decision making. There are two challenges that must be addressed:the scale of uncertainty ---local uncertainty must be scaled to global uncertainty; and,the multivariate relationship between the variables that go into resource calculations ---predictions of uncertainty must account for the correlation between variables. Conventional geostatistical techniques predict local uncertainty at the scale of the data. Global uncertainty refers to the petroleum resource, or the oil-in-place (OIP), for an arbitrarily large area. The global uncertainty in OIP cannot be calculated by simply summing the local uncertainties. The spatial continuity of the variables must be considered in scaling local uncertainty to global uncertainty. If the variable is very discontinuous, then the uncertainty decreases quickly with scale. If the variable is continuous, then the uncertainty decreases slowly, but fewer data are needed to constrain the uncertainty. Simulation must be used to combine uncertainty reconciling these two notions. OIP is calculated from several reservoir properties, such as net pay thickness, porosity and oil saturation. Treating the variables independently has a risk of underestimating the global resource uncertainty. High values average out with low values. The correlation between the multiple constituent variables of OIP must be calculated. A spatial/multivariate decomposition approach is proposed for assessing the global uncertainty in OIP from local uncertainties. The joint spatial and multivariate correlations are taken into account. The key idea is to simulate a set of spatially correlated probability values (a ‘p-field’) and then simultaneously draw the variable of interest at multiple locations. The LU decomposition is used to account for the multivariate correlations at each location.

Turbo-machinery gradually has expanded its business into the automotive and aircraft industries. A core part of turbo-machinery is the impeller which can lead to manufacturing problems because it has twisted surfaces. Therefore, impeller machining requires five-axis machining technology and expert knowledge. Five-axis machining has the advantages of being able to select a variety of tool axis in the machining and remove uncut region which are impossible in the three-axis machining, which could obtain high productivity and good surface quality. Rough cutting is one very important operation as it affects productivity in the impeller machining and it is necessary to determine cutting strategies and select optimal cutting condition. This paper proposes a statistical method to optimize the rough cutting parameters in impeller machining by response surface methodology and efficient strategy to divide cutting region. Firstly, the rough operation was divided into three steps to remove volume from inducer to exducer and two steps were also added to remove the fillets between blade surfaces and hub surfaces. These machining strategies are selected as the qualitative factors when the response surface method is used. Secondly, cutting time was set as the response factor for productivity, and step-down, step over, and feed rate were determined as independent factors. Finally, the response surface model was estimated by a single surface in order to predict rough cutting time and the optimum cutting conditions were searched by the estimated model.

The span-wise obstacle on the suction surface of a hydrofoil has been verified to be an effective passive control method for cloud cavitation. The position of obstacle significantly influences the performance of cavitation control. In this research, we investigated the effect of obstacle position on attached cavitation control on the suction surface of a NACA0015 hydrofoil through response surface methodology. The cavitation types covered from sheet cavitation to partial and transitional cavity oscillations. We derived regression equations and built response surfaces to illustrate the quantitative relationship between individual factors (obstacle position, cavitation number, and angle of attack) and cavitation dynamic response parameters (cavity length, acoustic intensity, and energy flux). Sheet cavitation was effectively suppressed because the obstacle increased the pressure at the near-wall region. However, the obstacle would induce a shear cavitation when its position was too close to the leading edge of the hydrofoil. Under partial cavity oscillation conditions, the obstacle was consistently performed well in cloud cavitation control. The cavitation dynamic response parameters significantly decreased. Under transitional cavity oscillation conditions, the obstacle cannot suppress the cavitation because the transitional cavity oscillation was likely a system-inherent instability. This research is beneficial for a comprehensive understanding of cavitation control mechanism using an obstacle and for further industrial application of obstacle in hydraulic machinery to control cavitation.

A statistical screening methodology is presented to address uncertainty related to main geological assumptions in green field modeling. The goals are the identification of the entire range of uncertainty on production, learning which are the most impacting geological uncertain inputs and understanding the relationships between geological scenarios and classes of dynamic behavior. The paper presents the methodology and an example application to a green field case study. The method is applied on an ensemble of reservoir models created by combining geological parameters across their range of uncertainty. The ensemble of models is then simulated with a selected development strategy and dynamic responses are grouped in classes of outcome through clustering algorithms. Ensemble responses are visualized on a multidimensional stacking plot, as a function of the geological input, and the most influential parameters are identified by axes sorting on the plot. Geological scenarios are then classified on dynamic responses through classification tree algorithms. Finally, a representative set of models is selected from the geological scenarios. The example study application shows a final oil recovery uncertainty range of 4:1, which is reasonable for a green field in lack of data. Such high range of uncertainty could hardly be found by common risk assessment based on fixed geological assumptions, which often tend to underestimate uncertainty on forecasts. Ensemble outcomes are grouped in four classes by oil recovery, plateau strength, produced water, and breakthrough time. The adoption of such clustering features gives a broad understanding of the reservoir dynamic response. The most influential geological inputs among the examined structural and sedimentological parameters in the example application result to be the fault orientation and channel fraction. This screening result highlights the main drivers of geological uncertainty and is useful for the following scenario classification phase. Classification of the geological scenarios leads to five classes of geological parameter sets, each linked to a main class of dynamic behavior, and finally to five representative models. These five models constitute an effective sampling of the geological uncertainty space which also captures the different types of dynamic response. This paper will contribute to widen the engineering experience on the use of machine learning for risk analysis by presenting an application on a real field case study to explore the relationship between geological uncertainty and reservoir dynamic behavior.

New Perspectives for 3D Visualization of Dynamic Reservoir Uncertainty

The parallel mechanism exhibits high stiffness and excellent dynamic response, making it ideal for high-precision applications. In our early work, a novel 6-DOF redundant parallel posture mechanism with four limbs for solar wing docking has been proposed; each limb consists of three links and four joints. This paper primarily focuses on optimization design of the mechanism. The calculation of workspace volume reveals that factors influencing the range of posture adjustment include dynamic platform parameters, static platform parameters, the drive trajectory of each kinematic pair, and the angles between each kinematic pair. A sensitivity analysis was conducted to examine the impact of each parameter on the range of posture adjustment. To reduce computational complexity and improve analysis efficiency, a combined approach of single-factor analysis and response surface methodology (RSM) is used in the paper. Single-factor analysis is utilized to evaluate the effect of each parameter on the posture adjustment range. Based on these results, RSM is used to establish a regression model for parameters; thereby, the optimal parameter combination for the mechanism is determined. The regression coefficient R2 = 0.9374 attests to the validity of the proposed model. Finally, a comparison of the posture adjustment range before and after optimization is presented, providing a foundation for the practical application of the redundant parallel mechanism. This paper introduces a novel structural design concept aimed at resolving the conflict between heavy loads and compact sizes in redundant parallel mechanisms while providing valuable insights for miniaturized design.

Response surface methodology provides a powerful tool for finding approximations to complex response functions. In many applications the evaluation of each design point required to construct the response surface approximations, is expensive and/or time consuming. The cost associated with the evaluation of design points generally motivates the use of statistical design of experiments to select a small number of 'optimal' points at which to evaluate the response quantity. Response surface approximations have two types of error; the first is random error or noise and the second is bias or modeling error. Traditionally points are selected so as to minimize the random error using minimum variance based designs, for example the Z?-Optimal design. However, when using a low order polynomial to approximate a response quantity with unknown behavior, large bias errors may be introduced. In the present work, a genetic algorithm is developed for obtaining a minimum-bias based design. Two example problems are evaluated and the results as obtained from both .D-Optimal as well as minimum-bias designs are compared.

Multi-level numerical and statistical analysis of the hygrothermal behavior of a non-vegetated green roof in a mediterranean climate

Time-variant reliability analysis can effectively estimate the safe state of structures under dynamic uncertainties during their lifecycle. However, one of its key challenging issues is computational efficiency. To improve the efficiency, this paper develops an approximation of the first-crossing probability density function (PDF) method, termed AFC-PDFM, based on the first-crossing theory. The response surface of the first-crossing time point (FCTP) about input variables is first estimated by using the moving least squares method (MLSM) and uniform design. Furthermore, an iterative algorithm is developed to compute the origin moments of the FCTP based on the MLSM response surface. Finally, the PDF of the FCTP is obtained by combining the origin moments and the maximum entropy method to avoid direct calculations of the first-crossing rate, which is often intractable, and then the time-variant reliability is estimated. Three examples are implemented to testify and validate the effectiveness of the proposed method and potential applications to engineering problems.

Nowadays integrated uncertainty studies have become a necessary step of any reservoir development project. The main challenge is then not to evaluate the uncertainty for a particular variable but to integrate the uncertainties of all variables and all modelling steps. Despite the advantages of full Monte-Carlo approach and the increasing computation power there are still the limits of applying globally this approach from the seismic up to the dynamic simulation. For this purpose different approaches were designed, combining screening results, Experimental Design and Monte-Carlo methods (Milliken et al. 2008; Li et al. 2010). In this paper we compare two approaches of integrated uncertainty study applied to a real case study. Both consist of three main steps: structural, geological and dynamic uncertainties with stepwise integration of uncertainties. The first method evaluates geological uncertainty with a full Monte-Carlo method; the second is based on the mono-parameter sensitivity study and its impacts combination by scalar Monte-Carlo method. For both approaches structural and dynamic uncertainties are evaluated using the same workflows. The comparison is twofold: firstly on static volume, secondly on the transfer of the static uncertainty to the dynamic part.

The quality of the surface plays a very important role performance of milling as a good-quality milled surface in a variety of manufacturing industries including the aerospace and automotive sectors where good quality surface significantly improves fatigue strength, corrosion resistance, or creep life. This study discussed the issue of surface machined quality and the effort taken to predict surface roughness. For thus purpose , hardened materials AISI H13 tool steel with hardness of 48 Rockwell Hardness (HRC) was chosen for work material. Machining was done at High Cutting speed (Vc) from 150 up to 250 m/min, feedrate (Vf) 0005-0.15 mm/rev and depth of cut (DOC) 0.1-0.5mm. The analysisi and observation of the surface roughness were done by using optical surface roughness machine. Response Surface Methodology (RSM) Model was used to design the prediction model with parameters generated by using Central Composite Face (CCF) methods. A prediction model developed with 90% accuracy with the conclusion of feedrate as the main contributor to surface roughness followed by cutting speed. Therefore, RSM has been proven to be an efficient method to predict the surface finish during end-milling of H13 tool steel using TiAIN coated carbide tool inserts under dry conditions.

Technical evaluation of a resource play is critical to understanding long term economics and generating meaningful forecasts of future production. Producing wells often exhibit transient flow behavior for extended periods, have limited drainage areas, and produce a majority of their recoverable reserves at low rates. Often closeology or rules of thumb are used with a depressing lack of science in companies' development plans. These mistakes can result in significant economical damage that may take years to materialize. Equally, resource plays can provide a long-term supply of oil and gas to help meet energy demands while providing oil companies with a predictable return and production forecast. The initial production from the Bakken Formation of North Dakota began in 1950's as a back up zone should other primary targets fail to be productive. The play experienced two further development periods prior to the most recent burst of activity as a resource play. Examples of evaluation from the Bakken shale will be considered in this paper. Why conduct technical analysis? Several myths permeate the industry regarding the technical evaluation of resource plays. The first and most often heard is that statistical plays do not require technical evaluation. The fallacy of this logic is that a statistical distribution with the mean below what is considered economic will result in a obviously failed development. Evaluating the likely distribution then becomes key, which can only be done intelligently using technical analysis. The next often heard objection is that in an absence of unique solutions, technical evaluation is unable to give a meaningful guide to critical reservoir and completion properties. This argument has a grain of truth, as technical evaluation technique's will not yield a 100% unique result. However, consistent application of techniques from pressure transient well testing, analytical production analysis, and numerical simulation will narrow that range of uncertainty significantly. It must also be remembered that more conventional reservoir development have historically provided plenty of surprises for operators during the development phase. This indicates that a trait of working under limited data is the potential to fail to correctly identify the extremes of the reservoir property ranges, a constant hazard in our industry. The author proposes in this paper that consistent application of a range of modeling methods can reduce the lack of uniqueness issue to a tolerable level in resource play developments. This will allow for economic decisions to be made intelligently. Technical analysis is also key to achieving realistic reserve estimations and production forecasts based on a knowledge of the uncertainty in future production. Without understanding the reservoir properties and the physical ranges associated with them, an informed stochastic range can never be generated. The technical analysis approach for resource plays is divided into the following three levels in this paper. The well level, conducting analytical and numerical simulation analysis to understand reservoir and completion properties for further optimization and reserve estimation. The field level analysis considering trends in data pertaining to development options and finally the play level, considering analogue and historic play data. Bakken Shale Case Study The recent resource play development of the Bakken shale is located primarily in the North Dakotan and Eastern Montanan section of the Williston Basin1 (See Figure 1). Like many formations currently under development as a resource play previous phases of activity have occurred associated with oil price or technological changes.

Determination of in-situ engineering properties of soil using response surface methodology

The risk of economic failure of infill drilling campaigns increases with the increasing maturity of an oil field. The increased risk is due to the lower remaining movable oil in the late stages of a field's life compared to earlier phases. Despite the additional information owing to drilling of wells and production history, substantial geological and dynamic uncertainties remain. In this paper, an approach is presented for selecting an infill well portfolio of 10 wells from a total of 96 that explicitly accounts for these uncertainties. The geological uncertainty can be exposed by generating a multitude of geological models conditioned to known log/correlation data. These models need to be classified and a subset of representative models extracted that in turn are calibrated (history matched) to past field performance and then used for forecasting. Calibration using a suitable ensemble of representative geological models aims to maintain geological diversity while reducing the error to historical data through an objective function. Probabilistic Property Maps can then be generated to identify potential infill well locations. Probabilistic maps offer a major improvement compared to selection of infill well locations using a single geological realization. For mature assets, forecasting requires simulation of incremental oil recovery since infill wells represent incremental projects over a base case. Using an appropriate ensemble of models allows probabilistic representation of the incremental oil production and associated economics. First, individual infill well performances are forecasted. Next, the individual infill well locations are evaluated taking the history matching error into account and using utility theory to cover the risk adverseness attitude of the company. Doing so enables selection of an infill well portfolio under uncertainty and leads to a selection of well locations carrying lower risk compared to a selection of locations neglecting history match error and/or disregarding the risk attitude of a company.

Steam chamber conformance in Steam Assisted Gravity Drainage (SAGD) influences the efficiency and economic performance of bitumen recovery. Conventional SAGD well completion designs provide limited control points in long horizontal well pairs leading to development of a non-ideal steam chambers. Developing advanced wellbore completions and optimizing downhole tool settings is critical to achieve optimal steam distribution in heterogeneous reservoirs for optimal recovery. This paper presents a workflow to optimize SAGD well completion design by using flow control devices (FCDs). Optimum FCD placement, and specifications are determined in consideration of reservoir heterogeneity. Uncertainties in spatial distribution of facies and rock types, reservoir rock and fluid properties are represented by multiple equiprobable deterministic and stochastic geological realizations using Monte-Carlo simulation. The methodology is based on constrained nonlinear optimizationtomaximize the net present value (NPV) as the objective function. A coupled wellbore/reservoir simulation model of a well pad is implemented in the study, and the efficacy of different scenarios with varied well designs are assessed from evaluating bitumen production, steam injection, and well completion expenses. Results indicate superior performance of the wells equipped with FCDs compared to conventional concentric and parallel dual string well completion designs. For the cases examined, this translates to an average 7% increase of the expected NPV for different well completion designs when using FCDs. Furthermore, results show using zonal isolation in the well design is essential for compartmentalized reservoirs such aspoint bar deposits with their significant heterogeneity. Advanced wellbore completions provide sufficient tools to constrain steam injection and liquid production into and from different well segments, and manage steam chamber conformance along the horizontal well pairs, improve production efficiency, increase bitumen recovery, and reduce operating costs. A novel workflow is presented to optimize advanced wellbore completions utilizing flow control devices. This integrated assisted optimization approach considers uncertainties in geological properties, and determines the optimal FCD parameters and well completion design with acceptable computational effort. This integrated workflow allowed us to undertake a thorough evaluation of the key subsurface uncertainties, and design an overall development plan. The probabilistic nature of the results legitimize quantifying the uncertainties and identify associated risks for different completion strategies.