Reservoir permeability is one of the important parameters derived from well test analysis. Small-scale permeability measurements in wells are usually made using core plugs, or more recently, probe permeameter measurements. Upscaling of these measurements for comparisons with permeability derived well tests (Pressure Build-Up) can be completed by statistical averaging methods. Well Test permeability is often compared with one of the core plug averages: arithmetic, geometric and harmonic. A question that often arises is which average does the well test-derived permeability represent and over what region is this average valid? A second important question is how should the data sets be reconciled when there are discrepancies? In practice, the permeability derived from well tests is often assumed to be equivalent to the arithmetic (in a layered reservoir) or geometric (in a randomly distributed permeability field) average of the plug measures. These averages are known to be members of a more general power-average solution. This pragmatic approach (which may include an assumption on the near-well geology) is often flawed due to a number of reasons, which is tried to be explained in this study. The assessment of in-situ, reservoir permeability requires an understanding of both core (plug and probe) and well test measurements &amp / #8211 / in terms of their volume scale of investigation, measurement mechanism, interpretation and integration. Pressure build-up tests for 26 wells and core plug analysis for 32 wells have valid measured data to be evaluated. Core plug permeabilities are upscaled and compared with pressure build-up test derived permeabilities. The arithmetic, harmonic and geometric averages of core plug permeability data are found out for each facies and formation distribution. The reservoir permeability heterogeneities are evaluated in each step of upscaling procedure by computing coefficient of variation, The Dykstra-Parson&amp / #8217 / s Coefficient and Lorenz Coefficients. This study compared core and well test measurements in South East of Turkey heavy oil carbonate field. An evaluation of well test data and associated core plug data sets from a single field will be resulting from the interpretation of small (core) and reservoir (well test) scale permeability data. The techniques that were used are traditional volume averaging/homogenization methods with the contribution of determining permeability heterogeneities of facies at each step of upscaling procedure and manipulating the data which is not proper to be averaged (approximately normally distributed) with the combination of Lorenz Plot to identify the flowing intervals. As a result, geometrical average of upscaled core plug permeability data is found to be approximately equal to the well test derived permeability for the goodly interpreted well tests. Carbonates are very heterogeneous and this exercise will also be instructive in understanding the heterogeneity for the guidance of reservoir models in such a system.

After the injection of CO₂ into a subsurface formation, various storage mechanisms help immobilize the CO₂. Injection strategies that promote the buoyant movement of CO₂ during the post-injection period can increase immobilization by the mechanisms of dissolution and residual phase trapping. In this work, we argue that the heterogeneity intrinsic to sedimentary rocks gives rise to another category of trapping, which we call local capillary trapping. In a heterogeneous storage formation where capillary entry pressure of the rock is correlated with other petrophysical properties, numerous local capillary barriers exist and can trap rising CO₂ below them. The size of barriers depends on the correlation length, i.e., the characteristic size of regions having similar values of capillary entry pressure. This dissertation evaluates the dynamics of the local capillary trapping and its effectiveness to add an element of increased capacity and containment security in carbon storage in heterogeneous permeable media. The overall objective is to obtain the rigorous assessment of the amount and extent of local capillary trapping expected to occur in typical storage formations. A series of detailed numerical simulations are used to quantify the amount of local capillary trapping and to study the effect of local capillary barriers on CO₂ leakage from the storage formation. Also, a research code is developed for finding clusters of local capillary trapping from capillary entry pressure field based on the assumption that in post-injection period the viscous forces are negligible and the process is governed solely by capillary forces. The code is used to make a quantitative assessment of an upper bound for local capillary trapping capacity in heterogeneous domains using the geologic data, which is especially useful for field projects since it is very fast compared to flow simulation. The results show that capillary heterogeneity decreases the threshold capacity for non-leakable storage of CO₂. However, in cases where the injected volume is more than threshold capacity, capillary heterogeneity adds an element of security to the structural seal, regardless of how CO₂ is accumulated under the seal, either by injection or by buoyancy. In other words, ignoring heterogeneity gives the worst-case estimate of the risk. Nevertheless, during a potential leakage through failed seals, a range of CO₂ leakage amounts may occur depending on heterogeneity and the location of the leak. In geologic CO₂ storage in typical saline aquifers, the local capillary trapping can result in large volumes that are sufficiently trapped and immobilized. In fact, this behavior has significant implications for estimates of permanence of storage, for assessments of leakage rates, and for predicting ultimate consequences of leakage. / text

Many engineering and scientific applications of flow in porous media are characterized by transport phenomena at multiple spatial scales, including pollutant transport, groundwater remediation, and acid injection to enhance well production. Carbon sequestration in particular is a multiscale problem, because the trapping and leakage mechanisms of CO2 in the subsurface occur from the sub-pore level to the basin scale. Quantitative and predictive pore-scale modeling has long shown to be a valuable tool for studying fluid-rock interactions in porous media. However, due to the size limitation of the pore-scale models (10-4-10-2m), it is impossible to model an entire reservoir at the pore scale. A straightforward multiscale approach would be to upscale macroscopic parameters (e.g. permeability) directly from pore-scale models and then input them into a continuum-scale simulator. However, it has been found that the large-scale models do not predict in many cases. One possible reason for the inaccuracies is oversimplified boundary conditions used in this direct upscaling approach. The hypothesis of this work is that pore-level flow and upscaled macroscopic parameters depends on surrounding flow behavior manifested in the form of boundary conditions. The detailed heterogeneity captured by the pore-scale models may be partially lost if oversimplified boundary conditions are employed in a direct upscaling approach. As a result, extracted macroscopic properties may be inaccurate. Coupling the model to surrounding media (using finite element mortars to ensure continuity between subdomains) would result in more realistic boundary conditions, and can thus improve the accuracy of the upscaled parameters. To test the hypothesis, mortar coupling is employed to couple pore-scale models and also couple pore-scale models to continuum models. Flow field derived from mortar coupling and direct upscaling are compared, preferably against a true solution if one exists. It is found in this dissertation that pore-scale flow and upscaled parameters can be significantly affected by the surrounding media. Therefore, using arbitrary boundary conditions such as constant pressure and no-flow boundaries may yield misleading results. Mortar coupling captures the detailed variation on the interface and imposes realistic boundary conditions, thus estimating more accurate upscaled values and flow fields. An advanced upscaling tool, a Super Permeability Tensor (SPT) is developed that contains pore-scale heterogeneity in greater detail than a conventional permeability tensor. Furthermore, a multiscale simulator is developed taking advantage of mortar coupling to substitute continuum grids directly with pore-scale models where needed. The findings from this dissertation can significantly benefit the understanding of fluid flow in porous media, and, in particular, CO2 storage in geological formations which requires accurate modeling across multiple scales. The fine-scale models are sensitive to the boundary conditions, and the large scale modeling of CO2 transport is sensitive to the CO2 behavior affected by the pore-scale heterogeneity. Using direct upscaling might cause significant errors in both the fine-scale and the large-scale model. The multiscale simulator developed in this dissertation could integrate modeling of CO2 physics at all relevant scales, which span the sub-pore or pore level to the basin scale, into one single simulator with effective and accurate communication between the scales. The multiscale simulator provides realistic boundary conditions for the fine scales, accurate upscaled information to continuum-scale, and allows for the distribution of computational power where needed, thus maintaining high accuracy with relatively low computational cost. / text

Dispersion has been shown to degrade miscibility in miscible displacements by lowering the concentration of the injected solute at the displacement fronts. Dispersion can also improve oil recovery by increasing sweep efficiency. Either way, dispersion is an important factor in understanding miscible displacement performance. Conventionally, dispersion is measured in the laboratory by fitting the solution of one-dimensional convection-dispersion equation (CDE) to the effluent concentration from a core flood. However dispersion is anisotropic and mixing occurs in both longitudinal and transverse directions. This dissertation uses the analytical solution of the two-dimensional CDE to simultaneously determine longitudinal and transverse dispersion. The two-dimensional analytical solution for an instantaneous finite volume source is used to investigate anisotropic mixing in miscible displacements. We conclude that transverse mixing becomes significant with large a concentration gradient in the transverse direction and significant local variation in flow directions owing to heterogeneity. We also utilized simulation models similar to Blackwell's (1962) experiments to determine transverse dispersion. This model coupled with the analytical solution for two-dimensional CDE for continuous injection source is used to determine longitudinal and transverse dispersivity for the flow medium. The validated model is used to investigate the effect of heterogeneity and other first contact miscible (FCM) scaling groups on dispersion. We derive the dimensionless scaling groups that affect FCM displacements and determine their impact on dispersion. Experimental design is used to determine the impact and interactions of significant scaling groups and generate a response surface function for dispersion based on the scaling groups. The level of heterogeneity is found to most significantly impact longitudinal dispersion, while transverse dispersion is most significantly impacted by the dispersion number. Finally, a mathematical procedure is developed to use the estimated dispersivities to determine a-priori the maximum grid-block size to maintain an equivalent level of dispersion between fine-scale and upscaled coarse models. Non-uniform coarsening schemes is recommended and validated for reservoir models with sets of different permeability distributions. Comparable sweep and recovery are observed when the procedure was extended to multi-contact miscible (MCM) displacements. / text

Physical and chemical heterogeneities cause the porous media transport parameters to vary with scale, and between these two types of heterogeneities geological heterogeneity is considered to be the most important source of scale-dependence of transport parameters. Subsurface processes associated with chemical alterations result in changing reservoir properties with interlinked spatial and temporal scale, and there is uncertainty in the evolution of those properties and the chemical processes. This dissertation provides a framework and procedures to quantify the spatiotemporal scaling characteristics of reservoir attributes and transport processes in heterogeneous media accounting for chemical alterations in the reservoir. Conventional flow scaling groups were used to assess their applicability in scaling of recovery and Mixing Zone Length (MZL) in presence of chemical reactivity and permeability heterogeneity through numerical simulations of CO₂ injection. It was found out that these scaling groups are not adequate enough to capture the scaling of recovery and transport parameters in the combined presence of chemical reactivity and physical heterogeneity. In this illustrative example, MZL was investigated as a function of spatial scale, temporal scale, multi-scale heterogeneity, and chemical reactivity; key conclusions are that 1) the scaling characteristics of MZL distinctly differ for low permeability and high permeability media, 2) heterogeneous media with spatial arrangements of both high and low permeability regions exhibit scaling characteristics of both high and low permeability media, 3) reactions affect scaling characteristics of MZL in heterogeneous media, 4) a simple rescaling can combine various MZL curves by merging them into a single MZL curve irrespective of the correlation length of heterogeneity, and 5) estimates of MZL (and consequently predictions of oil recovery) will fluctuate corresponding to displacements in a permeable medium whose lateral length is smaller than the correlation length of geological formation. We illustrate and extend the procedure of estimating Representative Elementary Volume (REV) to include temporal scale by coupling it with spatial scale. The current practice is to perform spatial averaging of attributes and account for residual variability by calibration and history matching. This results in poor predictions of future reservoir performance. The proposed semi-analytical technique to scale-up in both space and time provides guidance for selection of spatial and temporal discretizations that takes into account the uncertainties due to sub-processes. Finally, a probabilistic particle tracking (PT) approach is proposed to scale-up flow and transport of diffusion-reaction (DR) processes while addressing multi-scale and multi-physics nature of DR mechanisms and also maintaining consistent reservoir heterogeneity at different levels of scales. This multi-scale modeling uses a hierarchical approach which is based on passing the macroscopic subsurface heterogeneity down to the finer scales and then returning more accurate reactive flow response. This PT method can quantify the impact of reservoir heterogeneity and its uncertainties on statistical properties such as reaction surface area and MZL, at various scales. / text

In this dissertation, we present multiscale analytical solutions, in the weak sense, to the generalized Laplace's equation in Ω ⊂ Rⁿ, subject to periodic and nonperiodic boundary conditions. They are called multiscale solutions since they depend on a coefficient which takes a wide possible range of scales. We define forms of nonseparable coefficient functions in Lᵖ(Ω) such that the solutions are valid for the periodic and nonperiodic cases. In the periodic case, one such solution corresponds to the auxiliary cell problem in homogenization theory. Based on the proposed analytical solution, we were able to write explicitly the analytical form for the upscaled equation with an effective coefficient, for linear and nonlinear cases including the one with body forces. This was done by performing the two-scale asymptotic expansion for linear and nonlinear equations in divergence form with periodic coefficient. We proved that the proposed homogenized coefficient satisfies the Voigt-Reiss inequality. By performing numerical experiments and error analyses, we were able to compare the heterogeneous equation and its homogenized approximation in order to define criteria in terms of allowable heterogeneity in the domain to obtain a good approximation. The results presented, in this dissertation, have laid mathematical groundwork to better understand and apply multiscale processes under a deterministic point of view.

Quantifying carbon emissions of water bodies at regional scale is required as recent studies revealed their contribution in carbon cycling is significant. This demands to scale up water bodies carbon emissions from local to regional scale using as accurate approach as possible. In this study data of carbon (CO2-C) fluxes for 80 sampled lakes were used to scale up to more than 3000 lakes and ponds over the catchment. The most appropriate method for upscaling was the one in which two factors of water body size and location (altitude) were involved and the uncertainties were quantified in an advanced approach (Monte Carlo model). Based on the estimates obtained in this method, the annual carbon emission from all water bodies (~ 500 km2) was about 2900 ton C yr-1 . About 62% of this annual emission was related to the large lake Torneträsk (334 km2) and another 38% to all other lakes and ponds (166 km2). Water bodies in subalpine region dominated (90%) total water bodies area and were the major contributor (97%) to the total carbon emissions of all water bodies. The remaining small contribution (3%) was for water bodies in the alpine region, which contains only 10% of total water bodies area. These data indicate that all water bodies smaller than the large lake Torneträsk especially the ones in the subalpine region have considerable contribution to the annual carbon emission of all water bodies. Considering water body size and altitude factors in the advanced upscaling method was appropriate to obtain accurate estimates.

A growing demand for more detailed modeling of subsurface physics as ever more challenging reservoirs - often unconventional, with significant geomechanical particularities - become production targets has moti-vated research in coupled flow and geomechanics. Reservoir rock deforms to given stress conditions, so the simplified approach of using a scalar value of the rock compressibility factor in the fluid mass balance equation to describe the geomechanical system response cannot correctly estimate multi-dimensional rock deformation. A coupled flow and geomechanics model considers flow physics and rock physics simultaneously by cou-pling different types of partial differential equations through primary variables. A number of coupled flow and geomechanics simulators have been developed and applied to describe fluid flow in deformable po-rous media but the majority of these coupled flow and geomechanics simulators have limited capabilities in modeling multiphase flow and geomechanical deformation in a heterogeneous and fractured reservoir. In addition, most simulators do not have the capability to simulate both coarse and fine scale multiphysics. In this study I developed a new, fully implicit multiphysics simulator (TAM-CFGM: Texas A&M Coupled Flow and Geomechanics simulator) that can be applied to simulate a 2D or 3D multiphase flow and rock deformation in a heterogeneous and/or fractured reservoir system. I derived a mixed finite element formu-lation that satisfies local mass conservation and provides a more accurate estimation of the velocity solu-tion in the fluid flow equations. I used a continuous Galerkin formulation to solve the geomechanics equa-tion. These formulations allowed me to use unstructured meshes, a full-tensor permeability, and elastic stiffness. I proposed a numerical upscaling of the permeability and of the elastic stiffness tensors to gener-ate a coarse-scale description of the fine-scale grid in the model, and I implemented the methodology in the simulator. I applied the code I developed to the simulation of the problem of multiphase flow in a fractured tight gas system. As a result, I observed unique phenomena (not reported before) that could not have been deter-mined without coupling. I demonstrated the importance and advantages of using unstructured meshes to effectively and realistically model a reservoir. In particular, high resolution discrete fracture models al-lowed me to obtain more detailed physics that could not be resolved with a structured grid. I performed numerical upscaling of a very heterogeneous geologic model and observed that the coarse-scale numerical solution matched the fine scale reference solution well. As a result, I believed I developed a method that can capture important physics of the fine-scale model with a reasonable computation cost.

Despite increasing computer power, the requirement to upscale 3D geological models for dynamic reservoir simulation purposes is likely to remain in many commercial environments. This study established that there is a relationship between sandbody size, cell size and changes to predictions of reservoir production as grids are upscaled. The concept of a cell width to sandbody width ratio (CSWR) was developed to allow the comparison of changes in reservoir performance as grids are upscaled. A case study of the Flounder Field in the Gippsland Basin resulted in the interpretation of three depositional environments in the intra-Latrobe reservoir interval. The sandbody dimensions associated with these depositional environments were used to build a series of 3D geological models. These were upscaled vertically and horizontally to numerous grid cell sizes. Results from over 1400 dynamic models indicate that if the CSWR is kept below 0.3 there will be a strong correlation between the average production from the upscaled grids compared to those of a much finer grid, and there will be less than 10% variation in average total field production. If the CSWR is between 0.3 and 1, there could be up to 30% difference, and once the CSWR exceeds 1.0 there is only a weak relationship between the results from upscaled grids and those of finer grids. As grids are upscaled the morphology of bodies in facies models changes, the distribution of petrophysical properties is attenuated and the structure is smoothed. All these factors result in a simplification of the fluid flow pathways through a model. Significant loss of morphology occurs when cells are upscaled to more than a half the width of the reservoir body being modelled. A simple rule of thumb is established — if the geological features of a model cannot be recognised when looking at a layer in the upscaled grid, the properties of the upscaled grid are unlikely to be similar to those of the original grid and the predictions of dynamic models may vary significantly from those of a finer grid. This understanding of the influence of sandbody size on the behaviour of upscaled dynamic models can be used in the planning stages of a reservoir modelling project. Two simple charts have been created. The first chart is for calculating the approximate number of cells in a model before it is built. The second chart is for comparing the proposed cell size against the CWSR, so that the predicted discrepancy between the ultimate production from the upscaled grid and one with much smaller cells can be assessed. These two charts enhance discussion between all interested disciplines regarding the potential dimensions of both static and upscaled dynamic models during the planning stage of a modelling project, and how that may influence the results of dynamic modelling. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1375309 / Thesis (Ph.D.) - University of Adelaide, Australian School of Petroleum, 2009

Despite increasing computer power, the requirement to upscale 3D geological models for dynamic reservoir simulation purposes is likely to remain in many commercial environments. This study established that there is a relationship between sandbody size, cell size and changes to predictions of reservoir production as grids are upscaled. The concept of a cell width to sandbody width ratio (CSWR) was developed to allow the comparison of changes in reservoir performance as grids are upscaled. A case study of the Flounder Field in the Gippsland Basin resulted in the interpretation of three depositional environments in the intra-Latrobe reservoir interval. The sandbody dimensions associated with these depositional environments were used to build a series of 3D geological models. These were upscaled vertically and horizontally to numerous grid cell sizes. Results from over 1400 dynamic models indicate that if the CSWR is kept below 0.3 there will be a strong correlation between the average production from the upscaled grids compared to those of a much finer grid, and there will be less than 10% variation in average total field production. If the CSWR is between 0.3 and 1, there could be up to 30% difference, and once the CSWR exceeds 1.0 there is only a weak relationship between the results from upscaled grids and those of finer grids. As grids are upscaled the morphology of bodies in facies models changes, the distribution of petrophysical properties is attenuated and the structure is smoothed. All these factors result in a simplification of the fluid flow pathways through a model. Significant loss of morphology occurs when cells are upscaled to more than a half the width of the reservoir body being modelled. A simple rule of thumb is established — if the geological features of a model cannot be recognised when looking at a layer in the upscaled grid, the properties of the upscaled grid are unlikely to be similar to those of the original grid and the predictions of dynamic models may vary significantly from those of a finer grid. This understanding of the influence of sandbody size on the behaviour of upscaled dynamic models can be used in the planning stages of a reservoir modelling project. Two simple charts have been created. The first chart is for calculating the approximate number of cells in a model before it is built. The second chart is for comparing the proposed cell size against the CWSR, so that the predicted discrepancy between the ultimate production from the upscaled grid and one with much smaller cells can be assessed. These two charts enhance discussion between all interested disciplines regarding the potential dimensions of both static and upscaled dynamic models during the planning stage of a modelling project, and how that may influence the results of dynamic modelling. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1375309 / Thesis (Ph.D.) - University of Adelaide, Australian School of Petroleum, 2009