Geological Uncertainty Research Articles

Capturing uncertainties through scenario-based integrated static reservoir modeling of lacustrine turbidites in the Barmer Basin, India

Most static modeling workflows deal with stochastic simulations of the uncertain subsurface parameters on a base case model, although recent studies highlighted usefulness of discrete deterministic multiple geological scenario-based modeling. This paper illustrates the benefits of capturing the principal geological uncertainties through discrete subsurface scenarios, through a case study from the Vijaya and Vandana (V&V) field, Barmer Basin, northwest India. The 12 exploration and appraisal wells have established seven stratigraphically trapped oil pools with the maximum resources confined in the V&V mounds, consisting of turbidite sandstones and conglomerates in a shale background, inferred to be deposited in a deep lacustrine environment. Hydraulic fracturing of these sandstones resulted in significant production increase. Detailed subsurface analysis suggests that the V&V mounds consist of two channel complexes represented by a laterally migrating network of turbidite channels with a maximum thickness of 4–5 m for individual sandstones. Multiattribute seismic studies indicate that delineation of these channel sands, controlled mainly by the channel geometries, cannot be resolved by seismic signature/attribute studies alone, which necessitates the iteration of the facies model into five different scenarios. Each of the facies scenarios is further iterated with other key uncertain input parameters for STOIIP calculation (namely saturation, porosity, contact, etc.) to result in 50 deterministic static realizations that capture the wide uncertainty range of in-place volumes, through a cumulative distribution function plot. In the absence of a defined concept, our model highlights the importance of deterministic depiction of subsurface concepts (geologic, geophysical, petrophysical, and dynamic) through a scenario-based approach. This workflow captures a wide range of various high-impact uncertainties in an integrated manner and links discrete, deterministic, scenario-based outcomes to probabilistic reporting. This will help in the decision-making process by linking the model outcome with long-term well testing and ultimately the concept underlying the development plan.

Field-scale history matching with sparse geologic dictionaries

Inference of reservoir flow properties from scattered production data poses a poorly-constrained inverse problem with non-unique solutions. Parsimonious description of reservoir properties can alleviate problem ill-posedness and help with preserving the expected geologic continuity of the solution. Low-rank representations can exploit the prevalent correlations in spatially distributed rock properties to compactly describe them. Recent advances in sparse reconstruction, known as compressed sensing, have inspired efficient and flexible model calibration approaches that offer a departure from classical reduced-order parameterization techniques that only use prior knowledge to select the parameterization subspace. Sparse model calibration allows for dynamic subspace identification by adaptively searching through a large combination of possible relevant subspaces (with different geologic features) to identify an appropriate one that best explains the observed data. We recently presented sparse geologic dictionaries as an effective and flexible method for representation of complex geologic connectivity patterns to facilitate the search for plausible solutions. In this paper, we demonstrate the field-scale application of learned sparse geologic dictionaries for history matching. We use a gradient-based implementation of the history matching problem to investigate the feasibility of adopting the developed method for field-scale application. The paper presents practical considerations, important properties, and alternative implementations of the method. The benefits and limitations of applying the method to reservoir history matching are evaluated and discussed. The performance of the sparse history matching approach with geologic dictionaries is also compared with the classical truncated singular value decomposition parameterization in three field-scale case studies. The results show that sparse model calibration with geologic dictionaries has a superior history matching performance for estimating reservoir properties in realistic settings where the trade-offs between problem dimensionality, model resolution, computation speed, geologic uncertainty, and solution plausibility should be reconciled effectively.

Stochastic Resource Allocation for Well Control With Digital Oil Field Infrastructure

To investigate the flux patterns for a digital oil field system, the modeling and analysis need to include waterflood optimization, which is generated to maintain pressure and sweep oil to production. In this paper, a life-cycle well control optimization is integrated with proactive resource allocation to increase computational efficiency of mature fields. Based on a noninvasive and derivative-free approach, the combinatorial model is developed to address geological uncertainty by modifying injection and production rates at individual wells. With the derived model, the joint well control and field development algorithm is proposed based on their influence on the control variables. The possible benefits from the hybrid method and heuristics are clarified. In addition, the Benders decomposition method is adopted considering the computation burden with the optimal number of wells in a distributed manner at the cost of iteration. With the relaxation technique and acceleration technique, the optimal solutions are achieved to strengthen the cuts generated and to find better feasible solutions in less time. Simulation and experimental results are presented to verify the achievements of the proposed methods.

Surface and borehole geophysics for the rehabilitation of a concrete dam (Penne, Central Italy)

In this paper, surface and borehole geophysical surveys are employed to improve the effectiveness of a rehabilitation intervention (cut-off wall) to be executed on the right abutment of a leaking concrete dam (Penne, Central Italy). In a first step, surface geophysical measurements (electrical resistivity and P-wave velocity data), were collected to provide a geophysical model to be compared to the preliminary geological model available before the geophysical survey. Because significant differences emerged between the geophysical and the preliminary geological models, a second phase of borehole investigations was planned to validate the geophysical reconstruction obtained from surface data. The final geological model resulting from the geophysical results differs significantly from the preliminary model available at the time of the geophysical survey. Therefore, the planned cut-off wall needs to be extended both laterally and in depth beyond the position estimated before the geophysical campaign, so that the low-permeability bedrock is intercepted at a depth equal or lower than the critical one. At the end, the low-cost geophysical campaign (1.5% of the entire costs of the intervention) was highly beneficial to this project, minimizing subsurface geological uncertainty to design an effective anti-seepage intervention and stressing the importance of a representative geological model for engineering design.

Practical use of the concept of geotechnical categories in rock engineering

The aim of the paper is to show how Eurocode 7: Geotechnical Design Part 1: General Rules (EC7) could be developed in order to be in accordance with practise in rock engineering and construction. A main feature is the geological uncertianties, which imply that a risk based approch should be used. The use of Geotechnical Category (GC) has therefore to be improved by (1) combining the consequences of a failure to the geological uncertainties before excavation, and (2) combining the consequences to the ground quality found after excavation. Three GC classes are needed to properly use the GC in rock construction.The paper further describes how GC influences the design, which design method to be applied. It also outlines the types of control, inspection and supervision to be applied in the various GC classes during various stages of a project. An example is presented showing how GC can be determined at various stages of a rock construction.

Utilization of nature-inspired algorithms for gas condensate reservoir optimization

Hydrocarbon production from a gas condensate reservoir has a major concern to the petroleum industry around the world. Due to pressure decline in a reservoir, liquid drops out of solution and cause a significant reduction of well productivity. As a result, huge amount of valuable oil stays underground. The proper production management for a gas condensate reservoir minimizes or eliminates the blocking problems. Optimization of condensate recovery can be achieved through appropriate selection of production and injection rates. However, determination of the optimal rates is a complex problem, involving many factors including geological uncertainty, rock petrophysical characteristics, fluid properties, economic costs, and technical ability. In this research, nature-inspired optimization algorithms are employed to evaluate the potential of production boost. Numerical testing and comparative study revealed that investigated natural-inspired algorithms outperform widely reported in the literature optimization methods and provide a higher quality solutions. The primary concern of this research is to develop an optimal gas condensate production strategy in terms of economic efficiency by utilization of nature-inspired optimization algorithm and providing their ability to generate better quality solution than engineering approach as well as gradient optimization.

RESERVOIR PRESSURE MAINTENANCE BY WATER INJECTION INTO HORIZONTAL WELLS AT THE CONDITION OF THE GEOLOGICAL UNCERTAINTY IN CONTINENTAL DEPOSITS OF HIGH-VISCOSITYOIL OF PK1-3 FORMATION OF THE VOSTOCHNO-MESSOYAKHSKOYE OIL FIELD

The article considers the problem of correct organization of the system of reservoir pressure maintenance by water injection into PK1-3 formation of the Vostochno-Messoyakhskoye oil field that has many geological uncertainties. To remove these uncertainties the authors offered the pilot well program of flooding system and detailed proposals for data diagnosis obtained during this program that will help to determine the most correct approach to the flooding system for this type of reservoirs.

The impact of geological uncertainty on primary production from a fluvial reservoir

Deposition of fluvial sandbodies is controlled mainly by characteristics of the system, such as the rate of avulsion and aggradation of the fluvial channels and their geometry. The impact and the interaction of these parameters have not received adequate attention. In this paper, the impact of geological uncertainty resulting from the interpretation of the fluvial geometry, maximum depth of channels, and their avulsion rates on primary production is studied for fluvial reservoirs. Several meandering reservoirs were generated using a process-mimicking package by varying several controlling factors. Simulation results indicate that geometrical parameters of the fluvial channels impact cumulative production during primary production more significantly than their avulsion rate. The most significant factor appears to be the maximum depth of fluvial channels. The overall net-to-gross ratio is closely correlated with the cumulative oil production of the field, but cumulative production values for individual wells do not appear to be correlated with the local net-to-gross ratio calculated in the vicinity of each well. Connectedness of the sandbodies to each well, defined based on the minimum time-of-flight from each block to the well, appears to be a more reliable indicator of well-scale production.

Effect of mobility and convection-dominated flow on evaluation of reservoir dynamic performance by fast marching method

The determination of optimum well locations and number of wells needed during green field development always comes with unprecedented challenges because of the geological uncertainty, and the non-linear relationship between the input and output variables associated with real reservoirs. These variables are key sources affecting the viability and validity of the results. Reservoir simulation is one of the least uncertain and most reliable prediction tools for dynamic performance of any reservoir. As field development progresses, more information becomes available, enabling us to continually update and, if needed, correct the reservoir description. The simulator can then be used to perform a variety of exercises or scenarios, with the goal of optimizing field development and operation strategies. Optimizing well numbers or locations under such geological uncertainty is achieved by using a reservoir simulator under several geological realizations, and these require multiple reservoir simulations to estimate the field performance for a given well configuration at a given location. Using reservoir simulation becomes impractical when dealing with real field cases incorporating multi-million cells because of the associated CPU demand constraints (Bouzarkouna et al. 2011). For instance, to determine the optimum well locations in a giant field that will result in the most efficient production rate scenario, one requires a large number of simulation runs for different realizations and well configurations. A large amount of runs is technically difficult to achieve even if we have access to super computers. The fast marching method (FMM), which is based on a solution of Eikonal equation, can be used to find the optimum well locations in a green oil field by tracking the pressure distribution in the reservoir. The FMM will enable us to calculate the radius of investigation or pressure front as a function of time without running any simulation and with a high degree of accuracy under primary depletion conditions. The main purpose of this paper is to study the effect of mobility on FMM and extend the investigation of its validity to include two-phase flow and convection-dominated flow and evaluate the ability of the methodology to predict the dynamic performance of the reservoir during pseudo-steady-state flow regime.

Classical and stochastic mine planning techniques, state of the art and trends

Abstract Determination of the best possible ultimate pit for an open pit mine is a fundamental subject that has undergone a highly evolutionary process, reviewed in this study, since the correct choice carries substantial economic impact for the industry. The correct choice can be very beneficial for project analysis, whereas an incorrect choice has the potential to mask huge financial and economic future losses that could render a project unfeasible. The advent of computers in the 1960s allowed sophisticated analysis for the selection of the best ultimate pit determination, under specific modifying factors such as economic, social, environmental, and political, but only in deterministic situations, i.e., when the problem and variables for the ultimate pit determinations were considered deterministically and almost always based on average values. Techniques such as the Lerchs-Grossman algorithm and mixed-integer programming are among many standard tools now used by the mineral industry. Geological uncertainty and the associated risks as well as the need to consider the appropriate time to mine a block during a mine operation have a significant impact on the net present value of the resulting ultimate pits. Stochastic aspects embed a probabilistic component that varies in time and are now under intense investigation by researchers, who are creating algorithms that can be experimented with and tested in real mine situations. One can expect that once these algorithms demonstrate their efficiency and superior results, they will readily dominate the industry.

Incorporation of geometallurgical modelling into long-term production planning

Strategic decisions to develop a mineral deposit are subject to geological uncertainty, due to the sparsity of drill core samples. The selection of metallurgical equipment is especially critical, since it restricts the processing options that are available to different ore blocks, even as the nature of the deposit is still highly uncertain. Current approaches for long-term mine planning are successful at addressing geological uncertainty, but do not adequately represent alternate modes of operation for the mineral processing plant, nor do they provide sufficient guidance for developing processing options. Nonetheless, recent developments in stochastic optimisation and computer data structures have resulted in a framework that can integrate operational modes into strategic mine planning algorithms. A logical next step is to incorporate geometallurgical models that relate mineralogical features to plant performance, as described in this paper.

Robust Optimization of Cyclic CO2 flooding through the Gas-Assisted Gravity Drainage process under geological uncertainties

The purpose of this research is to determine an estimate for actual optimal oil recovery through cyclic Gas-Assisted Gravity Drainage (GAGD) process in a heterogeneous sandstone reservoir under geological uncertainties. A robust optimization approach was adopted to determine the optimal durations of gas injection, soaking, and oil production under geological uncertainties. 100 stochastic reservoir realizations of the 3D permeability and porosity distributions were created to honor geological constraints. Ranking was applied through quantifying of reservoir oil response to select P10, P50, and, P90 that represent the overall reservoir uncertainty. A compositional reservoir flow simulation was used for the GAGD process performance evaluation. Approximately 200 simulation jobs were created, including the aforementioned durations and geological uncertainty parameters, through Design of Experiments (DoE). The Latin Hypercube Sampling was adopted to create these 200 simulation jobs that then evaluated by the compositional reservoir simulation to calculate the cumulative oil production by the end of 10 prediction years. The robust optimization approach was then applied to select the true optimal solution of the highest oil recovery by taking into account the geological uncertainties in permeability, porosity, and anisotropy models. The nominal optimization of one single realization was also adopted for the comparison. The robust optimization has shown its feasibility to increase oil production through the cyclic GAGD process from 4.535 to 4.62547 billion barrels. However, the nominal optimization case increased oil production to 5.9726 billion barrels. The presented robust optimization workflow under geological uncertainties resulted in higher oil recovery and net present value than nominal realization optimization, with providing degrees of freedom for the decision-maker to significantly reduce the project risk. It was specifically concluded that the robust optimal solution represents the most economically feasible solution to obtain the highest NPV through the GAGD process for a range $(30–80) per barrel oil prices. However, the base case and nominal solution (no geological uncertainties) were not economical when the oil price declines to be less than 36 and 32, respectively.

Cost-optimal design of pressure-based monitoring networks for carbon sequestration projects, with consideration of geological uncertainty

Leakage from geologic faults and abandoned wells represents one of the major risks to industrial-scale carbon capture and storage (CCS) projects. Current CCS regulations and best practice guidance suggest that operators emplace risk-informed monitoring, verification, and accounting (MVA) plans to protect public safety and reduce property and environmental damage. Deep subsurface pressure monitoring is regarded as one of the most cost-effective technologies for early leakage detection in CCS projects. In practice, however, the number of deep pressure monitoring wells that an operator can deploy often remains limited because of the high costs associated with drilling, instrumenting, and operating these wells. Thus, optimal design of the pressure monitoring network is essential to minimizing monitoring and liability costs and gaining public support. In this work, we present a general, binary integer programming approach to solve an optimal monitoring well network design problem under multiple constraints. Specifically, our approach helps a CCS operator to design a cost-optimal monitoring network that covers all potentially leaky locations (in a worst-case-scenario sense) while satisfying a prescribed carbon dioxide (CO2) storage performance criterion and considering geological uncertainty. Instead of using cost surrogates as has been done in many other studies, our formulation allows the user to directly assess total costs in terms of monitoring costs and potential economic losses associated with brine and CO2 leakage. Our numerical examples demonstrate that a cost-optimal monitoring network may save millions of dollars in total costs, including well construction and leakage costs. Factors exerting the most impact on the cost-optimal monitoring network design are unit leakage damage costs, pressure threshold for leakage detection, and geological uncertainty.

From axiomatics of quantum probability to modelling geological uncertainty and management of intelligent hydrocarbon reservoirs with the theory of open quantum systems.

As was recently shown by the authors, quantum probability theory can be used for the modelling of the process of decision-making (e.g. probabilistic risk analysis) for macroscopic geophysical structures such as hydrocarbon reservoirs. This approach can be considered as a geophysical realization of Hilbert's programme on axiomatization of statistical models in physics (the famous sixth Hilbert problem). In this conceptual paper, we continue development of this approach to decision-making under uncertainty which is generated by complexity, variability, heterogeneity, anisotropy, as well as the restrictions to accessibility of subsurface structures. The belief state of a geological expert about the potential of exploring a hydrocarbon reservoir is continuously updated by outputs of measurements, and selection of mathematical models and scales of numerical simulation. These outputs can be treated as signals from the information environment E The dynamics of the belief state can be modelled with the aid of the theory of open quantum systems: a quantum state (representing uncertainty in beliefs) is dynamically modified through coupling with E; stabilization to a steady state determines a decision strategy. In this paper, the process of decision-making about hydrocarbon reservoirs (e.g. 'explore or not?'; 'open new well or not?'; 'contaminated by water or not?'; 'double or triple porosity medium?') is modelled by using the Gorini-Kossakowski-Sudarshan-Lindblad equation. In our model, this equation describes the evolution of experts' predictions about a geophysical structure. We proceed with the information approach to quantum theory and the subjective interpretation of quantum probabilities (due to quantum Bayesianism).This article is part of the theme issue 'Hilbert's sixth problem'.

Decision-theoretic sensitivity analysis for reservoir development under uncertainty using multilevel quasi-Monte Carlo methods

At various stages of petroleum reservoir development, we encounter a large degree of geological uncertainty under which a rational decision has to be made. In order to identify which parameter or group of parameters significantly affects the output of a decision model, we investigate decision-theoretic sensitivity analysis and its computational issues in this paper. In particular, we employ the so-called expected value of partial perfect information (EVPPI) as a sensitivity index and apply multilevel Monte Carlo (MLMC) methods to efficient estimation of EVPPI. In a recent paper by Giles and Goda, an antithetic MLMC estimator for EVPPI is proposed and its variance analysis is conducted under some assumptions on a decision model. In this paper, for an improvement on the performance of the MLMC estimator, we incorporate randomized quasi-Monte Carlo methods within the inner sampling, which results in an multilevel quasi-Monte Carlo (MLQMC) estimator. We apply both the antithetic MLMC and MLQMC estimators to a simple waterflooding decision problem under uncertainty on absolute permeability and relative permeability curves. Through numerical experiments, we compare the performances of the MLMC and MLQMC estimators and confirm a significant advantage of the MLQMC estimator.

Stochastic orebody modelling and stochastic long-term production scheduling at the KéMag iron ore deposit, Quebec, Canada

Stochastic optimisation provides a framework that is capable of generating a strategic life-of-mine production schedule that increases net present value while simultaneously reducing the risk associated with geological uncertainty. This paper focuses on the application of the stochastic strategic mine planning for technical risk management in the KéMag iron ore deposit in Quebec, Canada, demonstrating the key steps of the framework. The approach first quantifies both the volumetric and multi-element grade uncertainty of the deposit by generating a set of equally probable scenarios of the orebody. In this case study, the boundaries of the lithologies (volumetric uncertainty) defining the KéMag iron ore deposit are generated using a pattern-based wavelet simulation algorithm. The pertinent grade properties, namely, head iron, Davis Tube weight recovery, Davis Tube concentrate iron and silica content (multi-element grade uncertainty) are jointly simulated using the direct block minimum/maximum autocorrelation factors algorithm. Subsequently, the simulated scenarios of the iron deposit serve as an input to a life-of-mine stochastic integer programming production-scheduling model. The latter stochastic optimisation model is employed to manage and minimize the risk associated with the geological uncertainty of the deposit in terms of meeting production targets while generating a mining sequence of extraction maximising the net present value. The results of the case study quantify the risk associated with the product’s silica content, total iron production and expected discounted annual cash flows.

Issues and Options in the Delineation of Well Capture Zones under Uncertainty.

The delineation of wellhead protection areas (WHPAs) under uncertainty is still a challenge for heterogeneous porous media. For granular media, one option is to combine particle tracking (PT) with the Monte Carlo approach (PT-MC) to account for geologic uncertainties. Fractured porous media, however, require certain restrictive assumptions under this approach. An alternative for all types of media is the capture probability (CP) approach, which is based on the solution of the standard advection-dispersion equation in a backward mode, making use of the analogy between forward and backward transport processes. Within this context, we review the current controversy about the correct form of the conceptual model for transport, finding that the advection-diffusion model, which represents the diffusive interchange between streamtubes with differing velocities, is more physically realistic than the conventional advection-dispersion model. For mildly to moderately heterogeneous materials, stochastic theories and simulation experiments show that this process converges at the field scale to an effective advection-dispersion process that can be simulated with conventional transport models using appropriate macrodispersivity values. For highly heterogeneous materials, stochastic theories do not yet exist but there is no reason why the process should not converge naturally as well. Macrodispersivities appear to be formation-specific. The advection-dispersion model can be used for capture zone delineation in heterogeneous granular media. For fractured porous systems, hybrid equivalent porous medium and discrete fracture network or CP-based approaches may have potential. In general, capture zones delineated by PT without MC will always be too small and should not be used as a basis for land-use decisions.

Reservoir sensitivity analysis for heterogeneity and anisotropy effects quantification through the cyclic CO2-Assisted Gravity Drainage EOR process – A case study from South Rumaila oil field

The effects of reservoir heterogeneity and anisotropy were quantified through the cyclic CO2-Assisted Gravity Drainage (GAGD) process performance in a heterogeneous multi-layering sandstone reservoir. First, an integrating workflow of multiple-point geostatistics (MPS) and Sequential Gaussian Simulation (SGSIM) was adopted for the lithological and petrophysical modeling to capture the fluvial depositional environment and preserve the reservoir heterogeneity, respectively. Next, sensitivity analysis was conducted through compositional reservoir simulation and Design of Experiments (DoE) to eliminate the non-influencing petrophysical parameters on the GAGD process performance. Then, heterogeneity and anisotropy effects were quantified with respect to reservoir permeability and anisotropy ratio, respectively. The effects was attained by generating and incorporating multiple reservoir stochastic images (realizations) that capture the entire geological uncertainty space into the reservoir flow simulation. Based on the reservoir flow responses, it was concluded that the impact of permeability anisotropy on the GAGD process is higher than that of heterogeneity, because the main concept of GAGD process considers vertical fluid movements from the top-layer injection wells to the horizontal producers.

A stochastic optimization method with in-pit waste and tailings disposal for open pit life-of-mine production planning

Environmental responsibility and the sustainable development of mineral resources are a topic of critical importance to the mining industry and at the same time relate to operational and rehabilitation costs to be considered in technical studies. Open pit mining operations impact their local environment in terms of their modification of the landscape and local ecosystems. Many of these impacts are the result of the transportation of large volumes of materials mined and shifted from and to different locations. External stockpiles and waste dumps occupy considerable space as well as involve substantial transportation costs to move materials from open pits to stockpiles and then move them back to the pit for rehabilitation after the end of exploitation. Depending on the shape of the deposit and the intended design of the pit, a desirable option may be to place it directly in the free spaces within the pit, instead of storing all waste and tailings materials in stockpiles and/or waste/tailings dumps. This paper presents a new mathematical formulation integrating to life-of-mine planning and the maximization of net present value, with the related waste and tailings disposal kept within the mined-out parts of a pit, using a stochastic integer program that manages geological uncertainty including metal grades, material types and related chemical compositions. In addition to the traditional variables related to the materials being extracted from the ground in the form of mining blocks, strips of ground following the dip of a pit are considered within the pit as decision variables, and the optimization process aims to optimally define both the sequence of extraction of mining blocks and the reservation of strips needed to store waste materials. An application at an iron ore mine demonstrates the feasibility, applied aspects and advantages of the proposed method.

Development of robust surrogate model for economic performance prediction of oil reservoir production under waterflooding process

Geological uncertainty imposes significant challenge in modeling oil reservoirs. In this paper, a new robust modeling methodology is presented to identify a set of robust surrogate models with unstructured uncertainty for economic performance prediction of an uncertain oil reservoir under waterflooding process. The oil reservoir, being treated as a multi-input, multi-output (MIMO) system in terms of injection inputs and production outputs, is identified within a robust surrogate modeling framework using a frequency-based system identification approach. For this purpose, the concept of unstructured uncertainty model is incorporated with the surrogate modeling in the context of a function block having a norm-bounded uncertainty profile. The identified MIMO surrogate model is integrated with a desired nonlinear net present value (NPV) objective function to synthesize a new modified robust surrogate model in a multi-input, single-output (MISO) configuration form. This new modeling strategy enables direct calculation of economic performance prediction for the target oil reservoir. Finally, applicability of the proposed method is evaluated on “Egg Model” as a three-dimensional synthetic oil reservoir with 8 water injection wells and 4 oil production wells. The results clearly demonstrate that economic performance prediction of the oil reservoir, having uncertain permeability field, lies in the bounds corresponding to the worst-case scenarios of the uncertainty model set.