In this dissertation we discuss numerical methods used for uncertainty quantifi- cation applications to flow in porous media. We consider stochastic flow equations that contain both a spatial and random component which must be resolved in our numerical models. When solving the flow and transport through heterogeneous porous media some type of upscaling or coarsening is needed due to scale disparity. We describe multiscale techniques used for solving the spatial component of the stochastic flow equations. These techniques allow us to simulate the flow and transport processes on the coarse grid and thus reduce the computational cost. Additionally, we discuss techniques to combine multiscale methods with stochastic solution techniques, specifically, polynomial chaos methods and sparse grid collocation methods. We apply the proposed methods to uncertainty quantification problems where the goal is to sample porous media properties given an integrated response. We propose several efficient sampling algorithms based on Langevin diffusion and the Markov chain Monte Carlo method. Analysis and detailed numerical results are presented for applications in multiscale immiscible flow and water infiltration into a porous medium.

Reservoir characterization techniques have made possible geological reservoir models with multi-million grid blocks populated with permeability, porosity, and fluid saturations. These geological models are often too large to be simulated because of computational limits. These computational limits mean that typical full-field reservoir simulation models are limited to fewer than 1 million cells - at least two orders of magnitude smaller than the geological models. Upscaling techniques have been used to bridge the gap between these geological models and full-field reservoir simulation. Although there have been significant efforts in developing single-phase and two-phase upscaling algorithms, a limited verification of upscaling methods has been performed on a full-field basis. In addition to upscaling techniques, parallel simulation approaches have been developed to solve multi-million cell models with reasonable computational efficiency. Parallel simulations take up to a few hours of CPU time instead of days to run multi- million cell models. However, when many simulations are to be performed over a large range of parameter values for uncertainty studies, parallel simulations again become prohibitive and upscaling must be employed. On the other hand, the results from these upscaled simulations must be validated with results from fine-scale simulations to give confidence on the reliability of the results. There is really no way of knowing how good the results are unless we are able to perform the fine-scale simulations for verification. Parallel ultra-fine-scale simulations may provide the tool for this verification requirement. In this work, we developed several new single-phase upscaling algorithms, and investigated the verification of these techniques applied to a reservoir model and a synthetic model. For complicated multi-phase flow, the single-phase upscaling may lead to large errors. To overcome the inaccuracy, a new relative permeability upscaling approach was investigated in this dissertation research. The new approach was verified by using three-phase, 3D, and highly heterogeneity reservoir model. Based on case studies, the results from the fine-scale model may appropriately be used to guide the upscaling. The parallel simulation may guide engineers to find appropriate upscaled models through a tuning procedure. This tuning procedure has been explored in the current study to obtain results that are in close agreement with the fine-scale simulation results. The combination of parallel simulation technology and upscaling algorithms can be used to provide a better estimation of the amount of uncertainty in predicted oil recovery for real fields. / text

The term upscaling and determination of pseudo curves, or effective parameters, used on a coarse-scale simulation grid are related to the complex and extensive problems associated with reservoir studies. The primary strategy mainly focuses on having a good physical and practical understanding of the particular processes in question, and an appreciation of reservoir model sensitivities. Thus the building of the reservoir simulation models can be optimally determined.By concentrating on the modelling and upscaling gas injection for Enhanced Oil Recovery (EOR) process, which includes Interfacial Tension (IFT) and the amicability effect, a new effective and efficient algorithm of upscaling will be investigated and determined by using several upscaled parameters. The sensitivities of these determined coarse scale parameters (i.e. porosity, absolute and relative permeability and capillary pressure), will also be studied through a history matching of the existing field.

The development of coarsened reservoir simulation models from high resolution geologic models is a critical step in a simulation study. The optimal coarsening sequence becomes particularly challenging in a fluvial channel environment where the channel sinuosity and orientation can result in pay/non-pay juxtaposition in many regions of the geologic model. The optimal coarsening sequence is also challenging in tight gas sandstones where sharp changes between sandstone and shale beds are predominant and maintaining the pay/non-pay distinction is difficult. Under such conditions, a uniform coarsening will result in mixing of pay and non-pay zones and will likely result in geologically unrealistic simulation models which create erroneous performance predictions. In particular, the upgridding algorithm must keep pay and non-pay zones distinct through a non-uniform coarsening of the geologic model. We present a coarsening algorithm to determine an optimal reservoir simulation grid by grouping fine scale geologic model cells into effective simulation cells. Our algorithm groups the layers in such a way that the heterogeneity measure of an appropriately defined static property is minimized within the layers and maximized between the layers. The optimal number of layers is then selected based on an analysis resulting in a minimum loss of heterogeneity. We demonstrate the validity of the optimal gridding by applying our method to a history matched waterflood in a structurally complex and faulted offshore turbiditic oil reservoir. The field is located in a prolific hydrocarbon basin offshore South America. More than 10 years of production data from up to 8 producing wells are available for history matching. We demonstrate that any coarsening beyond the degree indicated by our analysis overly homogenizes the properties on the simulation grid and alters the reservoir response. An application to a tight gas sandstone developed by Schlumberger DCS is also used in our verification of our algorithm. The specific details of the tight gas reservoir are confidential to Schlumberger's client. Through the use of a reservoir section we demonstrate the effectiveness of our algorithm by visually comparing the reservoir properties to a Schlumberger fine scale model.

Discrete Fracture Networks (DFN) models have long been used to represent heterogeneity associated with fracture networks but all previous approaches have been either in 2D (assuming vertical fractures) or for simple models within a small domain. Realistic representation of DFN on field scale models have been impossible due to two reasons - first because the representation of extremely large number of fractures requires significant computational capability and second, because of the inability to represent fractures on a simulation grid, due to extreme aspect ratio between fracture length and aperture. This thesis presents a hierarchal approach for fracture modeling and a novel random walker simulation to upscale the fracture permeability. The modeling approach entails developing effective flow characteristics of discrete fractures at micro and macrofracture scales without explicitly representing the fractures on a grid. Separate models were made for micro scale and macro scale fracture distribution with inputs from the seismic data and field observations. A random walker simulation is used that moves walkers along implicit fractures honoring the intersection characteristics of the fracture network. The random walker simulation results are then calibrated against high-resolution flow simulation for some simple fracture representations. The calibration enables us to get an equivalent permeability for a complex fracture network knowing the statistics of the random walkers. These permeabilities are then used as base matrix permeabilities for random walker simulation of flow characteristics of the macro fractures. These are again validated with the simulator to get equivalent upscaled permeability. Several superimposed realizations of micro and macrofracture networks enable us to capture the uncertainty in the network and corresponding uncertainty in permeability field. The advantage of this methodology is that the upscaling process is extremely fast and works on the actual fractures with realistic apertures and yields both the effective permeability of the network as well as the matrix-fracture transfer characteristics. / text

This thesis presents different methods that improve the ability to relate the flow properties of heterogeneous reservoirs to equivalent anisotropic flow properties in order to predict the performance of the Steam Assisted Gravity Drainage (SAGD) process. Process simulation using full scale heterogeneous reservoirs are inefficient and so the need arises to develop equivalent anisotropic reservoirs that can capture the effect of reservoir heterogeneity. Since SAGD is highly governed by permeability in the reservoir, effective permeability values were determined using different upscaling techniques. First, a flow-based upscaling technique was employed and a semi-analytical model, derived by Azom and Srinivasan, was used to determine the accuracy of the upscaling. The results indicated inadequacy of flow-based upscaling schemes to derive effective direction permeabilities consistent with the unique flow geometry during the SAGD process. Subsequently, statistical upscaling was employed using full 3D models to determine relationships between the heterogeneity variables: k[subscript italic v]⁄k[subscript italic h] , correlation length and shale proportion. An iterative procedure coupled with an optimization algorithm was deployed to determine optimal k[subscript italic v] and k[subscript italic k] values. Further regression analysis was performed to explore the relationship between the variables of shale heterogeneity in a reservoir and the corresponding effective properties. It was observed that increased correlation lengths and shale proportions both decrease the dimensionless flow rates at given dimensionless times and that the semi-analytical model was more accurate for cases that contained lower shale proportions. Upscaled heterogeneous values inputted into the semi-analytical model resulted in underestimation of oil flow rate due to the inability to fully account for the impact of reservoir barriers and the configuration of flow streamlines during the SAGD process. Statistical upscaling using geometric averaging as the initial guess was used as the basis for developing a relationship between correlation length, shale proportion and k[subscript italic v]⁄k[subscript italic h]. The initial regression models did not accurately predict the anisotropic ratio because of insufficient data points along the regression surface. Subsequently a non-linear regression model that was 2nd order in both length and shale proportion was calibrated by executing more cases with varying levels of heterogeneity and the regression model revealed excellent matches to heterogeneous models for the prediction cases. / text

The relevance, value and multi-dimensional application of soil moisture in many areas such as hydrological, meteorological and agricultural sciences have increased the focus on this important part of the ecosystem. However, due to its spatial and temporal variability, accurate soil moisture determination is an ongoing challenge. In the fall of 2013 and spring of 2014, the accuracy of five soil moisture instruments was tested in heavy clay soils and the Root Mean Squared Error (RMSE) values of the default calibration ranged from 0.027 and 0.129 m3 m-3. However, after calibration, the range was improved to 0.014 – 0.040 m3 m-3. The need for calibration has led to the development of generic calibration procedures such as soil texture-based calibrations. As a result of the differences in soil minerology, especially in clay soils, the texture-based calibrations often yield very high RMSE. A novel approach that uses the Cation Exchange Capacity (CEC) grouping was independently tested at three sites and out of seven different calibration equations tested; the CEC-based calibration was the second best behind in situ derived calibration. The high cost of installing and maintaining a network of soil moisture instruments to obtain measurements at limited points has influenced the development of models that can estimate soil moisture. The Versatile Soil Moisture Budget (VSMB) is one of such models and was used in this study. The comparison of the VSMB modelled output to the observed soil moisture data from a single, temporally continuous, in-field calibrated Hydra probe gave mean RMSE values of 0.052 m3 m-3 at the eight site-years in coarse textured soils and 0.059 m3 m-3 at the six site-years in fine textured soils. At field-scale level, the representativeness of an arbitrarily placed soil moisture station was compared to the mean of 48 data samples collected across the field. The single location underestimated soil moisture at 3 of 4 coarse textured fields with an average RMSE of 0.038 m3 m-3 and at only one of the four fine textured sites monitored with an average RMSE of 0.059 m3 m-3. / February 2017

The impact of the soil structure on flow and transport in partially water saturated soils is currently still a matter of scientific debate. The major aim of this thesis was to investigate the relation between heterogeneity and transport for a natural unsaturated heterogeneous Tertiary sand deposit. In the first part, we analyzed the flow and transport at the scale of an undisturbed monolith. Chloride breakthrough curve experiments were used to derive an apparent dispersion coefficient at the TDR sampling and monolith scale. Application of a Brilliant Blue pulse allowed further the visualization of flow distribution within the monolith. Small undisturbed soil cores were sampled throughout the monolith and the hydraulic characteristic curves at the scale of the cores were determined. Textural variability and structure as inferred from the inspection of the Brilliant Blue pattern and analysis of the small core sampling were subsequently implemented in a 3-D model and transport was simulated. The simulations clearly revealed the importance of the macro-structure on the transport behavior of the soil. We also showed that the micro-variability heterogeneity component was needed to assess the scaling of the effective and local scale dispersivity. In the second part, we studied in-situ chloride transport in the vadose formation separating the bottom of a lake and an unconfined aquifer. First the uncertainty generated by the undersampling of the lake chloride concentration time series were investigated. Subsequently, velocity and dispersivity profiles were assessed by inverse modeling of the soil chloride concentration time series. We observed that the clay layers induced an increase of the dispersivity below them. We hypothesize that fingering flow or convergence phenomena, occurring below sand-clay interfaces, lead to non-representative artificially high dispersivity values. Velocity and dispersivity values just above the clay layers however seem more reliable due to convergence phenomena and better lateral mixing induced by a larger water content. In this formation, the transport behavior could be characterized considering a hierarchical structure of the subsoil heterogeneity. In this model, the flow field micro-variability is influenced by pore structure (possibly characterized by scaling factors). The next complexity level is induced by the slight layering resulting from the sedimentation process (not investigated in this work). Then, the third hierarchical level is assessed by the macro-structure and the sequence of clay layers in the sand. Each of these levels is assumed to have an effect on the solute mixing process and effective macro-dispersivity.

In the past decades, considerable effort has been put into developing high resolution geological models for oil and gas reservoirs. Although the growth of computational power is rapid, the static model size still exceeds the model size for routine reservoir simulation. We develop and apply a variety of grid coarsening and refinement algorithms and single and multiphase upscaling approaches, applied to tight gas and conventional reservoir models. The proposed research is organized into three areas. First the upgridding of detailed three dimensional geologic models is studied. We propose an improved layer design algorithm with considerations of accuracy and efficiency. This involves developing measures of reservoir heterogeneity and using these measures to design an optimal grouping of geologic model layers for flow simulation. The optimal design is shown to be a tradeoff between the desire to preserve the reservoir heterogeneity and a desire to minimize the simulation time. The statistical analysis is validated by comparison with flow simulation results. Accurate upgridding/upscaling of single-phase parameters is necessary. However, it does not always satisfy the accuracy requirements, especially for the model which is aggressively coarsened. We introduce a pseudoization method with total mobility and effective fractional flow as the major targets. This pseudoization method helps to push upgridding/coarsening degree to the limit but still be able to reproduce the fine scale field performance. In practice, it is common to not use a different set of pseudos for every coarse cell; only a limited number of pseudo functions should be generated for different “rock types” or geological zones. For similar well patterns and well control conditions, applying pseudo is able to reproduce the fine scale performance for different simulation runs. This is the second proposed research area. Finally, it is necessary to increase flow resolution for precise field history matching and forecasting. This has received increasing attention, especially when studying hydraulically fractured wells in unconventional reservoirs. We propose a multiscale reservoir simulation model combining local grid refinement (LGR) and pillar-based upscaling for tight gas reservoir performance prediction. Pillar-based coarsening away from the wells is designed for tight gas reservoirs. It compensates for the extra computational cost from LGR, which is used to represent hydraulic fractures. Overall reservoir performances, including the accuracy and efficiency, are evaluated.

Fluids injected during secondary and tertiary floods often leave parts of the reservoir unswept mostly because of large heterogeneity and mobility ratio. Several applications require an analytical scheme that could predict production with as few parameters possible. We develop such an analytical model of volumetric sweep that aims to apply an extension of Koval’s theory where flow is assumed to be segregated under vertical equilibrium conditions for secondary and tertiary displacements. The unified theory for vertical equilibrium (viscous and dispersive) is also derived as a precursor to model development. The original Koval factor is applicable for upscaling secondary miscible floods. The new analytical model for secondary and tertiary floods is applied to provide quick estimates of oil recovery of miscible as well as immiscible displacements, which is then calibrated against field data. The model parameters, Koval factor, sweep efficiency and pore volume, estimated after history matching could be used to make reservoir management decisions. The model is very simple; history matching can be done in a spreadsheet. Single-front, gravity-free, displacements can be modeled using Koval factors. Two-front, gravity-free, displacements can also be modeled using Koval-type factors for both the fronts. These Koval-type factors, coupled with laboratory scale relative permeabilities, allows for scaling the displacement to a larger reservoir system. The new method incorporates by-passed pore volume as a parameter, a difference between this work and that of Molleai, along with Koval factors and local front velocities. For two front displacements, it also accounts for the interaction between the fronts which honors correct mass conservation, another difference with the work of Molleai. The results from new models for secondary and tertiary displacements were verified by comparing them against numerical simulations. The application was also demonstrated on actual field examples. Current techniques for reservoir surveillance rely on numerical models. The parameters on which these numerical models depend on are very large in number, introducing large uncertainty. This technique provides a way to predict performance without the use of computationally expensive fine scale simulation models, which could be used for reservoir management while reducing the uncertainty. / text