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ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery
- Conference date: 29 Aug 2016 - 01 Sep 2016
- Location: Amsterdam, Netherlands
- ISBN: 978-94-6282-193-4
- Published: 29 August 2016
21 - 40 of 163 results
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XFEM-EDFM-MINC for Coupled Geomechanics and Flow in Fractured Reservoirs
Authors G.T. Ren, J.M. Jiang and R.M. YounisUnconventional reservoirs are often naturally and hydraulically fractured with characteristically small pores and low permeability within the matrix. The underlying fracture networks can have a wide range of length scales and complex geometries. While hydraulic fractures may be propped, natural fractures are predominantly supported by pore pressure. A timely topic in the simulation of unconventional petroleum resources is in devising models that can accurately capture the coupling between the geomechanics of the fractured media and the multiphase fluid flow and transport. We develop a mixed discretization approach to adequately resolve the fracture system while accurately and efficiently modeling both flow and geomechanics. An extended finite element method (XFEM) is applied to approximate the geomechanics, and an embedded-discrete-fracture model (EDFM) is used for the multiphase flow equations. The two schemes are fully coupled, and the time discretization for flow is fully-implicit. Moreover, a hybrid fracture representation concept is employed where the multiple interacting continua (MINC) approach is used in conjunction with the embedded discrete representation in order to capture small-scale fracture networks efficiently. Several validation and computational results are presented. We also apply the proposed method to production scenarios with horizontal wells and hydraulic fractures in reservoirs with secondary fractures.
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Modelling Geomechanical Impact of CO2 Injection and Migration Using Precomputed Response Functions
Authors O.A. Andersen, H.M. Nilsen and S.E. GasdaWhen injecting CO2 or other fluids into a geological formation, pressure plays an important role both as a driver of flow and as a risk factor for mechanical integrity. The full effect of geomechanics on aquifer flow can only be captured using a coupled flow-geomechanics model. In order to solve this computationally expensive system, various strategies have been put forward over the years, with some of the best current methods based on sequential splitting. In this present work, we seek to approximate the full geomechanics effect on flow without the need of coupling with a geomechanics solver during simulation. We do this by means of precomputed pressure response functions. At grid model generation time, a geomechanics solver is used to compute the mechanical response of the aquifer for a set of pressure fields. The relevant information from these responses is then stored in a compact form and embedded with the grid model. We test the accuracy and computational performance of our approach on a simple 2D model and a more complex 3D model, and compare the results with those produced by a fully coupled approach as well as from as simple decoupled method based on Geertsma's uniaxial expansion coefficient.
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Micro-continuum Formulation for Modelling Dissolution in Natural Porous Media
Authors CS Soulaine and H.A. TchelepiAdvances in imaging technologies and high-performance computing are making it possible to perform Direct Numerical Simulation (DNS) of flow processes at the pore scale; nevertheless, the restrictions on the physical size of the sample (porous rock) that can be fully resolved using Navier-Stokes-based DNS are quite severe. Even for samples on the order of a cm3, the complexity of the spatial heterogeneity of the pore space precludes Navier-Stokes-based DNS. To deal with this challenge of having a wide range of length scales – even for `small’ systems, we describe a micro-continuum formalism, whereby locally averaged equations and associated coefficients can be used to model the effects of scales that are below instrument resolution and/or DNS capability. A hybrid modeling framework based on the Darcy-Brinkman-Stokes (DBS) equation is employed. In this approach, a single equation is applicable for flow in `channels’ (so-called `free flow') and in porous media (solid-fluid aggregates). A unified simulation framework for multi-physics problems of mass and heat transport in natural porous media based on a hybrid Darcy-Brinkman-Stokes approach is described. We discuss two specific applications: minerals dissolution in CO2-brine systems, and dissolution instabilities (worm-holing phenomena) associated with the acidizing treatment of carbonate formations.
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Estimation of In-situ Compositions in Lean Gas Condensate Reservoirs
Authors H.R. Nasriani, E. Asadi, C. Johnson, M. Nasriani and A. ChamchineThere is a high degree of complexity in both fluid flow and phase-behavior of gas condensate reservoirs during the depletion period due to retrograde condensation. Natural depletion in gas condensate reservoirs results in low condensate recoveries at surface due to in-situ condensation and accumulation of condensate in the reservoir especially in vicinity of wellbore. During reservoir depletion, the overall composition of reservoir fluid varies and becomes different from initial reservoir composition as pressure decline to values less than dew-point pressure. In-situ gas and condensate composition in the reservoir are changing accordingly. This paper develops a novel approach to obtain initial gas condensate reservoir composition from gas and liquid compositions taken from separator tests during several depletion stages. Based on the composition of mixed sample and initial reservoir composition, a set of novel correlations is developed for estimating initial gas condensate reservoir composition. The generalized reduced gradient (GRG) algorithm of iteration was used to tune the constant and exponents of the correlation based on available field data. The convergence criteria were to minimize the value of the squared sum, SS, of the difference between the real data and the estimated one.
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The Role of the Y-function for Checking the Reliability of PVT-data
Authors K. Potsch, P. Toplack and T. GumpenbergerPVT experiments are not free of systematic and random errors, therefore checking the validity of the results is a must. Textbooks use the Y-function as a consistency checks for constant compositions experiment (CCE) of black oil (BO). This paper extends the application of the Y-function. As it turns out, the Y-function can not only be applied for the CCE of BO, but also for the same experiment with a gas-condensate (GC). Furthermore, it will be demonstrated that the Y-function is a useful tool for evaluation of the differential liberation experiment (DLE, black oil) and the constant volume depletion (CVD, gas-condensate) as well. So far this aspect has not been used in the consistency checks of laboratory measurements. Textbooks claim that the Y-function is a straight line. Modeling the fluid system (BO or GC) with an equation of state (EOS) shows that the shape of this function is only close to a straight line. Having set up the Y-function properly, it is possible to estimate the quality of the CCE and the DLE for the black oil case and CCE and CVD for the gas-condensate by comparing the results of the experiments with the EOS calculations.
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Compositional Numerical Simulation of CO2-enhanced Gas Recovery in very Low-pressure Partially-depleted Reservoirs
Authors S. Goudarzi, S.A. Mathias and J.G. GluyasInjection of CO2 into depleted gas reservoirs offers the potential to securely store carbon dioxide while enhancing hydrocarbon recovery. Mathias et al. (2014) developed a two-layer vertical-equilibrium model for the injection of carbon dioxide into a low-pressure (<1 MPa) reservoir. In contrast to previous vertical-equilibrium models, the compressibility of all components was fully accounted for and non-Darcy effects were also considered. However, they ignored the effects of compositional changes and mixing between phases. This study seeks to extend the study of Mathias et al. (2014) by incorporating compositional effects, using a Method of Lines (MOL) three-component two-phase numerical simulator. MOL requires formulation of derivatives of the Primary Variables (PV) with respect to time. This often gives rise to the need for evaluating partial derivatives of some of the flow variables (in particular the bulk fluid density per unit volume of rock) with respect to the PVs. In this work, it will be shown how it is possible to evaluate these partial derivatives analytically (as opposed to e.g. conventional finite differencing). Moreover, following an assumption of constant equilibrium ratios for CO2 and CH4, a ternary flash calculator is developed providing closed-form relationships for exact interpolation between equations-of-state for CO2+H2O and CH4+H2O binary mixtures. Overall, the numerical results confirm the finding of Mathias et al. (2014), that pressure build-up, during CO2 injection into low pressure depleted gas reservoirs, can be expected to be largely unaffected by heat transport and other associated thermodynamic effects. Reference: Mathias, S. A., McElwaine, J. N., & Gluyas, J. G. (2014). Heat transport and pressure buildup during carbon dioxide injection into depleted gas reservoirs. Journal of Fluid Mechanics, 756, 89-109.
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Computer Modelling of Non-isothermal, Multiphase and Multicomponent Flow by Using Combined EOR Technologies
Authors T.S. Imankulov and D. Akhmed-ZakiDevelopment of new highly efficient oil production technologies require a deep analysis of complex mechanisms in real reservoir processes. To improve existing methods of enhanced oil recovery (EOR), including chemical, microbial (MEOR) and thermal methods, it is necessary to study the processes that are accompanied by phase state change of reservoir fluids. In the development of oil and gas fields using various EOR methods occurs complex three-phase flow of multicomponent mixtures with intensive phase transformations. Using known simplified models, which are not takes into account the phase transitions, to describe such processes can lead to significant quantitative errors and qualitative distortion of real process nature in reservoir. Similar changes in reservoir flow can occur when using thermal methods. This paper presents a numerical study of EOR methods by chemical (polymer or surfactant, polymer and surfactant) injection, MEOR and thermal methods, which considers, that flow is multiphase and fluids are multicomponent. Developed numerical algorithm to solve the problem using a fully implicit method. Considered different flooding system, different sequence of injection agents in combined flooding case and different temperature of injected water. Distributions of main technological parameters are obtained and efficiency of methods mentioned before is shown. The main results of numerical experiments are compared with calculations of reservoir simulator Eclipse 300 (PVTi).
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Shale Gas Lithofacies Classification and Gas Content Prediction Using Artificial Neural Network
Authors S. Ouadfeul and L. AliouaneHere, we show the contribution of the artificial intelligence such as neural network to predict the lithofacies in the lower Barnett shale gas reservoir. The Multilayer Perceptron (MLP) neural network with Hidden Weight Optimization Algorithm is used. The input is raw well-logs data recorded in a horizontal well drilled in the Lower Barnett shale formation, however the output is the concentration of the Clay and the Quartz calculated using the ELAN model and confirmed with the core rock measurement. After training of the MLP machine weights of connection are calculated, the raw well-logs data of two other horizontal wells drilled in the same reservoir are propagated though the neural machine and an output is calculated. Comparison between the predicted and measured clay and Quartz concentrations in these two horizontal wells shows the ability of neural network to improve shale gas reservoirs characterization. The present paper is limited only to the lithofacies prediction however the MLP neural network will be used also for the prediction of the gas content form well-logs data in the Barnett shale gas reservoirs which is an extended work of the present research.
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Analytical Modelling of Low-salinity Waterflooding with Fines Migration
Authors S. Borazjani, A. Behr, L. Genolet, A. Van der net, Z. You and P. BedrikovetskyLow-salinity waterflood is presently one of the most cost-effective EOR methods. The governing system of equations for oil and water flow with changing water salinity is derived. So-called mechanistic model is developed. This model accounts for two EOR factors: a) fines migration, which is induced by salinity variation and reduces the relative permeability of water, and b) wettability alteration, which affects the relative permeability and capillary pressure. The “lump-salt” description is used, i.e. the ionic water composition is represented by the total ionic concentration. The basic equations are simplified for three basic asymptotic cases: 1) large length scales, 2) low-velocities, 3) high-velocities. An exact analytical solution is derived for 1d displacement large-scale-approximation problem of oil using high-salinity water followed by the injection of low-salinity-slug and high-salinity water chase drive. The solution is obtained by so-called splitting method, where the Lagrangian coordinate is used as a free variable instead of time. The effects of wettability alteration and fines migration on oil recovery as two distinct physical mechanisms are analyses using the analytical model. The significant EOR-effects of both mechanisms are observed for the typical oil-reservoir conditions.
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Modelling Bio-reactive Transport in Underground Hydrogen Storages - Spatial Separation of Gaseous Components
Authors B. Hagemann, M. Panfilov and L. GanzerIn the context of energy revolution large quantities of storage capacity are required for the integration of strongly fluctuating energy production from wind and solar power plants. The conversion of electrical energy into chemical energy in the form of hydrogen is one of the technical possibilities. This technology, where hydrogen is stored in subsurface formations similar to the storage of natural gas, is currently in the exploratory focus of several European countries. Despite the deviating hydrodynamic behavior of hydrogen compared to natural gas, bio-chemical reactions can have an imported role in underground hydrogen storage. The fact that hydrogen is a favored substrate for several anaerobic microorganisms induces their growth and results in a degradation of hydrogen. In particular the activity methanogenic archaea can lead to drastic variations in the gas composition which were observed in some former town gas storages. To describe this behavior a mathematical and numerical model was developed in preliminary work which couples compositional two-phase flow with bio-chemical reactions and the dynamics of microbial growth and decay. In the present paper the stability of the coupled dynamic system was analyzed. Dependent on the parameter space it was shown that the system can undergo a limit cycle behavior or diffusion-driven (Turing) instability. The numerical solutions within these parameter regions show different oscillatory regimes. The instability leads to the formation of alternating spots with either a high concentration of H2 or a high concentration of CH4. The injection rate is a decisive factor which controls the behavior of the dynamic system.
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A Network Model for the Kinetics of Bioclogged Flow Diversion for Enhanced Oil Recovery
Authors L.A. López Peña, B. Meulenbroek and F.J. VermolenAfter the primary extraction in oil reservoirs up to 60 % of the oil remains trapped in the reservoir (Sen, 2008). Therefore, different mechanisms have been developed to get the oil out to the reservoir. One of these techniques is Microbial Enhanced Oil Recovery (MEOR) which is a technique used to produce more oil in a secondary extraction by using microbes in the reservoir. The main effects caused by microbes in oil recovery is the reduction of the interfacial tension between oil and water, wettability change of the rock and bioclogging caused by the growth and development of biofilm. Among these mechanisms, interfacial tension reduction and biclogging is thought to have the greatest impact on recovery (Sen, 2008). In this work, we describe the growth of biofilm, the growth of the microbial population and the transport of nutrients using a pore network model. We follow the previous models of Thullner et al. (Thullner, 2008) and Ezeuko et al. (Ezeuko, 2011) in which the biofilm is considered as a permeable layer. We consider the biofilm and the bacteria separately. Additionally, we assume that once a tube is full with biofilm, this biofilm can spread to the neighboring tubes. Finally, we study the changes in the hydrodynamic properties of the medium caused by the plugging of the pores and we study the flow diversion of water caused by plugging of the high permeability zones.
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The HLD Model for Optimum Phase Behavior Formulation of Ternary Surfactant Mixtures
Authors M. Mavaddat, S. Riahi and A. BahramianBecause many surfactant flooding formulations involve surfactant mixtures, mixtures are generally used to obtain the best formulation to increase the oil recovery. This study is focused on an experimental study to investigate the phase behavior of the mixture of two and three anionic surfactants in the Winsor type III condition. The HLD model has been used as an equation of state to predict the properties of the water-oil-surfactant systems. The model was applied to conduct single and double salinity scans to find the optimal salinity for an unknown surfactant and for the mixture of two and three surfactants at the same total surfactant concentration. Results show that in a binary anionic surfactant mixture, the concentration of the second anionic surfactant (in molar fraction) has a linear relation to the logarithm of the optimal salinity at the same system, which follows the ideal mixing rule in the system. Ternary anionic surfactant mixtures are found to exhibit a slight deviation from ideal at some surfactant compositions. The deviations were fitted using a three-parameter Margules equation. The results suggest that the deviations are due to the change in the surfactant molar fraction on the surface, which is caused by presence of highly branched surfactants.
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A Model for Non-Newtonian Flow in Porous Media at Different Flow Regimes
Authors O.M. Nødland, A. Lohne, A. Stavland and A. HiorthThe EOR potential of polymer flooding is well documented in the scientific literature. However, it has remained a challenge to create good simulation tools that can be used for predictive purposes. A main limitation with the current models is the insufficient description of the transition between the different flow regimes that characterize the polymer rheology. Typically, Newtonian behaviour is observed at low shear rates, followed by shear-thinning, shear-thickening and shear-degradation regimes at increasing shear rates. Furthermore this is complicated by the fact that the apparent viscosity of the polymer is influenced by a combination of factors, such as adsorption, brine salinity, polymer concentration and molecular weight. In this work we present a core scale simulation model that is capable of describing all the aforementioned flow regimes. The novel feature of the proposed model is the inclusion of an equation to describe polymer (mechanical) degradation. The polymer degradation rate is linked to the effective pore radius (via permeability through a Kozeny-Carman type equation), wall shear stress, and polymer molecular weight, Mw. The degradation results in a lower Mw, while the polymer volumetric concentration is unaffected. The change in Mw over a time step is found using an implicit chord method at the end of each transport time step, and the solution is then used to update the effective polymer properties. The main flow field is computed using a standard sequential algorithm, where a linear pressure equation is solved first, followed by an implicit saturation equation formulated in a fractional flow approach. The model is applied to a series of laboratory experiments. Our model explains the core data very well, taking into account that several experimental factors have been varied such as synthetic polymer types, core length and permeability.
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Wettability Alteration with Smart Water on Carbonate Reservoirs - Effect of Phosphate and Sulfate Multivalent Anions
Authors H. Moradi, A.H. Saeedi Dehaghani, A. Bahramian and M.N. KardaniIncreasing evidence has shown that the adjusting of ionic composition of water injection in the reservoir can improve oil recovery. The carbonate reservoirs comprise about the 50% of oil reservoirs and most of them are neutral to oil wet. More than 65% of carbonate reservoirs are oil wet and 12% are intermediate wet. Wettability is a very important factor in the oil recovery mechanisms. Because it has drastic effect on the distribution and flowing of the oil and water in the reservoirs. Ionic composition adjusting of water which is referred as smart water is known as one of wettability alteration method in the carbonate reservoirs. Although the mechanisms behind of this technic don’t have recognized as well but wettability alteration is a result of these mechanisms. The effect of individual ions must be investigate for better recognition of mechanisms causing improving oil recovery. For complete investigation of mechanisms of smart water technic it must be studied about the crude oil/formation water/injection water/rock system but this system is very complex and for better understanding it must become simpler. In this study, the aim is the wettability alteration deal with smart water using sessile drop method for contact angle measurement. The effect of phosphate and sulfate multivalent anions in different temperature, pH and concentration of them was studied. Modified brines containing phosphate salt were more effective than with sulfate salt. In acidic pH both anions have not good effect but in upper pH they can alter wettability to more water wetting. The phosphate was more soluble in upper temperatures and both anions have better wettability alteration properties to water wetting.
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Numerical Modelling of Non-newtonian Fluid Flow in Fractures and Porous Media
Authors K. Bao, A. Lavrov and H.M. NilsenNon-Newtonian fluids having Bingham or power-law rheology are common in many applications within drilling and reservoir engineering. Examples of such fluids are drilling muds, foams, heavy oil, hydraulic-fracturing and other stimulation fluids, and cement slurries. Despite the importance of non-Newtonian rheology, it is rarely used in reservoir simulators and fracture flow codes. We study two types of non-Newtonian rheology: the truncated power-law (Ostwald-de Waele) fluid and the Bingham fluid. For either of the two types of non-Newtonian rheology, we construct relationships between the superficial fluid velocity and the pressure gradient in fractures and porous media. The Bingham fluid is regularized by means of Papanastasiou-type regularization for porous media and by means of a simple hyperbolic function for fracture flow. Approximation by Taylor expansion is used to evaluate the fluid velocity for small pressure gradients to reduce rounding errors. We report simulations of flow in rough-walled fractures for different rheologies and study the effect of fluid parameters on the flow channelization in rough-walled fractures. This effect is known from experiments and from previous numerical studies. We demonstrate how different rheologies on different domains can be included in a fully-unstructured reservoir simulation that incorporates discrete fracture modeling (DFM). The above formulation was implemented in the open-source Matlab Reservoir Simulation Toolbox (MRST), which uses fully implicit discretization on general polyhedral grids, including industry-standard grids with DFM. This robust implementation is an important step towards hydro-mechanically coupled simulation on hydraulic fracturing on reservoirs with realistic non-Newtonian fluid rheology.
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On the Coupling of Two-phase Free Flow and Porous Flow
Authors Z.Q. Huang, B. Gao, X.Y. Zhang and J. YaoThe coupling of free flow with porous flow is of special interest in a wide range of environmental phenomena and industrial applications. In this work, we extend the classical single-phase two-domain model to a laminar two-phase coupling flow system. The free fluid region can be considered as separated two-phase flow for simplicity, which is modeled by using the Navier-Stokes and Cahn-Hilliard equations. And the mathematical model of two-phase flow in porous media is based on Darcy’s theory. The main challenge is how to introduce specific interface conditions to couple these two models. To this end, the normal continuity conditions of flux and forces are developed, and an extended Beavers-Joseph-Saffman condition for two-phase flow system is also proposed as a Cauchy boundary condition based on consistent phenomenological explanations. These lead to a simple and solvable coupling model, and an efficient finite element numerical scheme is developed. The numerical results show that the developed model is capable to capture the macroscopic flow characteristics of laminar two-phase coupling flow system by comparing to the experimental results. Our model can be used to model the related two-phase flow process in karstic aquifers and fractured reservoirs, and wind-driven evaporation from soil.
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Pseudo-3D Hydraulic Fracture Model with Complex Mechanism of Proppant Transport and Tip Screen Out
Authors A. Bochkarev, S. Budennyy, R. Nikitin and D. Mitrushkintip screen-out (TSO), packed (final) fracture geometry, fluid rheology changing caused by proppant presence. Proppant transport governs fracture shape and effective fracture area. The key objective of present work is to develop a joint solution of fracture growth within the multi-layered lithology and multiphase flow inside to present a complete fracture model, full enough for design-execution-evaluation cycle. The project is carried-out with financial support of the Ministry of Education and the Ministry of Education and Science of the Russian Federation in the framework of the project №14.581.21.0008 from 03.10.2014 г (ID RFMEFI58114X0008)
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Accelerating Coupled Dynamic Flow and Geomechanical Systems for Complex Reservoir Simulations
Authors A. Pearce, H. Mustapha, S. Kisra, P. Welsh and K.H. LeeCoupled reservoir and geomechanical simulations are required to account accurately for pressure variations and induced stress changes during the lifecycle of oil and gas fields. This understanding is critical for mitigating drilling and completion hazards and for predicting compaction or subsidence in reservoirs containing weak and fractured rocks. 3D full-field models with accurate rock and fluid properties have become increasingly possible due to advances in the hardware and software involved in determining property distribution. This has subsequently led to a rise in the size and complexity of the simulations and computational efforts. Typically, geomechanics simulations are computationally more expensive than reservoir simulations due to the grid size required to model a geomechanical problem. This paper introduces a new workflow based on the latest multiparallel high-resolution unstructured reservoir simulator coupled to a geomechanics simulator using enhanced linear solvers with advanced deflation algorithms. A performance analysis reveals that the proposed coupled system is significantly faster than the existing workflow. The accelerated coupled system brings the solution of full 3D geomechanics problems within the reach of a wider community of engineers using laptop or desktop computers, enabling them to deliver answers to field problems faster and without the need to access high-end supercomputers.
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The Impact of Layering and Heterogeneity on Stresses around Boreholes
Authors H. Agheshlui and S.K. MatthaiAn accurate understanding of orientation and magnitude of the stresses surrounding a borehole is decisive for the identification of a stable well path and the successful design of completions and stimulation measures including fracking stages. However, measurement of the in situ stress is challenging: Current engineering practice favours two approaches: 1) borehole break-out – drilling-induced fracture interpretation complemented by extended leak-off tests; 2) stress estimation from shear-wave slowness measured with advanced wireline tools. Both methods rely on the applicability of Kirsch’s (1898) stress perturbation equations and laboratory measurements of elastic moduli that are correlated with rock properties that can be logged. Method (2) can be applied where breakouts are absent. Yet little is known how lithologic layering and spatial variations in the mechanical properties of the rock affect its results and the detection limit of stress anisotropy. Here we present finite-element simulations of borehole-related stress perturbations in multi-layer composites with realistic scattering and spatial variations in elastic moduli. Using stress magnitude-shear wave correlations from the literature, travel times are calculated for borehole-parallel trajectories. These results are interpreted in terms of the minimum differential stress that needs to exist in order to be able to detect stress directions. The uncertainty of measurements of stress magnitudes obtained with this method is analysed as well. Our results show that the stress field around boreholes is strongly affected by lithological variations. “Ideal” Kirsch-compatible conditions where the well is aligned with one of the eigenvectors of the stress field, layers are perpendicular to the well, and far-field stresses are Andersonian is rare. More common scenarios are going to be illustrated with a series of simulations including deviated wells. These will also elucidate how rock stress responds to fluid pressure changes in the well.
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Virtual Element Method for Geomechanical Simulations of Reservoir Models
Authors X. Raynaud, H. Nilsen and O. AndersenIn this paper we study the use of Virtual Element method for geomechanics. Our emphasis is on applications to reservoir simulations. The physical processes that form the reservoirs, such as sedimentation, erosion and faulting, lead to complex geometrical structures. A minimal representation, with respect to the physical parameters of the system, then naturally leads to general polyhedral grids. Numerical methods which can directly handle this representation will be highly favorable, in particular in the setting of advanced work-flows. The Virtual Element method is a promising candidate to solve the linear elasticity equations on such models. In this paper, we investigate some of the limits of the VEM method when used on reservoir models. First, we demonstrate that care must be taken to make the method robust for highly elongated cells, which is common in these applications, and show the importance of calculating forces in terms of traction on the boundary of the elements for elongated distorted cells. Second, we study the effect of triangulations on the surfaces of curved faces, which also naturally occur in subsurface models. We also demonstrate how a more stable regularization term for reservoir application can be derived.
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