Reservoir Engineering Applications Research Articles

Thermodynamically consistent numerical modeling of immiscible two‐phase flow in poro‐viscoelastic media

AbstractNumerical modeling of immiscible two‐phase flow in deformable porous media has become increasingly significant due to its applications in oil reservoir engineering, geotechnical engineering and many others. The coupling between two‐phase flow and geomechanics gives rise to a major challenge to the development of physically consistent mathematical models and effective numerical methods. In this article, based on the concept of free energies and guided by the second law of thermodynamics, we derive a thermodynamically consistent mathematical model for immiscible two‐phase flow in poro‐viscoelastic media. The model uses the fluid and solid free energies to characterize the fluid capillarity and solid skeleton elasticity, so that it rigorously follows an energy dissipation law. The thermodynamically consistent formulation of the pore fluid pressure is naturally derived for the solid mechanical equilibrium equation. Additionally, the model ensures the mass conservation law for both fluids and solids. For numerical approximation of the model, we propose an energy stable and mass conservative numerical method. The method herein inherits the energy dissipation law through appropriate energy approaches and subtle treatments for the coupling between two phase saturations, the effective pore pressure and porosity. Using the locally conservative cell‐centered finite difference methods on staggered grids with the upwind strategies for saturations and porosity, we construct the fully discrete scheme, which has the ability to conserve the masses of both fluids and solids as well as preserve the energy dissipation law at the fully discrete level. In particular, the proposed method is an unbiased algorithm, that is, treating the wetting phase, the non‐wetting phase and the solid phase in the same way. Numerical results are also given to validate and verify the features of the proposed model and numerical method.

Toward real-time fracture detection on image logs using deep convolutional neural network YOLOv5

Fractures in reservoirs have a profound impact on hydrocarbon production operations. The more accurately fractures can be detected, the better the exploration and production processes can be optimized. Therefore, fracture detection is an essential step in understanding the reservoir’s behavior and the stability of the wellbore. The conventional method for detecting fractures is image logging, which captures images of the borehole and fractures. However, the interpretation of these images is a laborious and subjective process that can lead to errors, inaccuracies, and inconsistencies, even when aided by software. Automating this process is essential for expediting operations, minimizing errors, and increasing efficiency. Although there have been some attempts to automate fracture detection, this paper takes a novel approach by proposing the use of YOLOv5 as a deep-learning (DL) tool to detect fractures automatically. YOLOv5 is unique in that it excels at speed, training, and detection while maintaining high accuracy in fracture detection. We observe that YOLOv5 can detect fractures in near real time with a high mean average precision of 98.2, requiring significantly less training than other DL algorithms. Furthermore, our approach overcomes the shortcomings of other fracture detection methods. Our method has many potential benefits, including reducing manual interpretation errors, decreasing the time required for fracture detection, and improving fracture detection accuracy. Our approach can be used in various reservoir engineering applications, such as hydraulic fracturing design, wellbore stability analysis, and reservoir simulation. By using this technique, the efficiency and accuracy of hydrocarbon exploration and production can be significantly improved.

Electrohydrodynamic peristaltic microfluidic flow with thermal and molecular analysis featuring sloped configurations

This study examines the simultaneous impacts of thermal and molecular transfer of non-Newtonian fluid propelled by the collective impact of electroosmosis and peristaltic pumping via inclined channel. Internal frictional heating and heat generation are also taken into consideration. The electrohydrodynamic phenomenon is described by the nonlinear Poisson–Boltzmann equation. The problem is modulated mathematically through a system of coupled non-linear differential equations in wave frame of reference. Using the regular perturbation method around a small wave number δ, we obtain sequential solutions for stream function ψ, electric potential ϕ, velocity u, pressure gradient dpdx, temperature θ and concentration Ω distributions. The influences of newly arising parameters such as the electroosmotic parameter me, relaxation time parameter λ1, retardation time parameter λ2, wave number δ, Reynold’s number Re, heat generation parameter β, Prandtl number Pr, Schmidt number Sc, Froude number Fr and Soret number Sr on physical characteristics are analyzed graphically. There is an inverse correlation between the parameters of relaxation time and retardation time and their effects on axial velocity, concentration, and temperature distributions. In particular, the temperature of the fluid increases as the EDL parameter, relaxation time parameter, viscous dissipation parameter, and Reynolds number increase, but decreases as the retardation time parameter increases. Moreover, the axial pressure gradient increases as Reynolds number and inclination angle increase, while it decreases as Froude number increases. The current model has applicability in the chemical industry, reservoir engineering, micro-fabrication, and biomedical applications, where electroosmotic effects on heat and mass movement are crucial.

A new nuclear magnetic resonance-based permeability model based on two pore structure characterization methods for complex pore structure rocks: Permeability assessment in Nanpu Sag, China

The nuclear magnetic resonance (NMR) estimate of permeability is a fundamental method that has numerous applications in reservoir engineering and petrophysics. To improve the accuracy of the NMR-based permeability model, many variables are introduced into NMR-based permeability prediction models due to geometric complexity and pore structure heterogeneity. In this paper, two pore structure characterization methods are investigated based on the Kozeny-Carman model and equivalent component model. Furthermore, an NMR-based permeability model accounting for the effect of pore structure is developed based on the analysis of the relationship between two pore structure parameters, and it is applied to practically predict permeability. Results indicate that the new model-calculated permeability has good agreement with experimental data; moreover, the adaptability of the new NMR-based permeability prediction model is highly improved through reducing undetermined variables, and key parameters can be measured directly using NMR. The new model provides a valuable scientific resource and assists in the evaluation of hydrocarbon-bearing reservoirs with complex pore structure, such as tight sandstone, shale, and carbonate rock.

Radius of investigation and impact of inter-porosity flow in naturally-fractured reservoirs

The radius of investigation (ROI) is an essential parameter in many reservoir engineering applications. Inconsistencies and ambiguities still exist between different ROI evaluation methods, especially for the dual-porosity (DP) system representing naturally fractured reservoirs. This paper revisited the impact of inter-porosity flow in the DP system and presented ROI equations in the early and late flow periods based on the Laplace space solution of the Warren-Root dual-porosity model. A new comprehensive ROI expression for the DP system is derived which accounts for both storativity ratio () and inter-porosity flow coefficient (). The new method in this paper reveals that the log-log plot of ROI vs. time for DP models exhibits two parallel straight lines linked by a smooth transition curve. The distance of these two straight lines is dependent on and ROI in the intermediate flow period is highly dependent on the value of The new ROI equation for the DP system can degenerate to similar ROI formula for the single-porosity system in early and late flow periods. Comparison analysis of the new method with other approaches is conducted and the validation of the new method is realized by analytical and numerical matching.

UNRAVELING THE COMPLEX DYNAMICS OF FLUID FLOW IN POROUS MEDIA: EFFECTS OF VISCOSITY, POROSITY, AND INERTIA ON THE MOTION OF FLUIDS

This study investigates novel solitary wave solutions of the Gilson–Pickering ([Formula: see text]) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The [Formula: see text] equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences.

Experimental Investigation of Trapped Oil Mobilization with Ferrofluid

Summary Nanoparticles have great potential to mobilize trapped oil in reservoirs because of their chemical, thermal, and interfacial properties. However, the direct application of magnetic forces on superparamagnetic nanoparticles in reservoir engineering applications has not been extensively investigated. We demonstrate the enhanced oil recovery (EOR) potential of hydrophilic superparamagnetic nanoparticles in oil production by direct observation using microfluidics. We studied the mobilization of oil blobs by a ferrofluid (a suspension of hydrophilic superparamagnetic nanoparticles in water) both in a converging/diverging micromodel channel and in a foot-long pore network micromodel, both with varying depth (so-called 2.5D micromodels). The water-based ferrofluid in all cases was the wetting fluid. Initial ferrofluid flooding experiments in single channels were performed without and then with a static magnetic field. This magnetic field caused oil droplet deformation, dynamic breakup into smaller droplets, and subsequent residual oil saturation reduction. During the flooding, after the magnetic field was applied, significant oil displacement was observed within 2 hours [6 pore volumes injected (PVI)], and 86.2% of the oil that was not mobilized without a magnetic field was mobilized within 64 hours (192 PVI). Then, in experiments in the micromodel and in a Hele-Shaw cell without flooding, we observed self-assembly of oil droplets, indicating the formation of the hydrophilic magnetic nanoparticle microstructures (chains under the magnetic field) and their interaction with the oil blobs. Further ferrofluid flooding experiments were performed in a foot-long micromodel under a rotating magnetic field. The oil saturation was reduced from 44.6 to 33.3% after 17 hours (8.5 PVI) of ferrofluid flooding after the rotating magnetic field was applied. Finally, a discussion of field application of ferrofluid flooding is presented.

The Efficacy and Superiority of the Expert Systems in Reservoir Engineering Decision Making Processes

In the process of making a critical decision in reservoir engineering, most of the time we find ourselves in a quandary. Like in any other scientific or technical field, when we find ourselves having to make a critical decision at a juncture, we cannot go ahead with our gut feelings, but rather must figure out what knowledge and information is lacking. In generating the missing knowledge and understanding, the depth and the rapid nature of the search will surface as two critical parameters. In other words, most of the time, a shallow search that can be conducted in a short period of time will not produce the missing information and the knowledge and more often, possibly, it will provide misguidance. When a large volume of sources of information is reviewed and the missing knowledge is generated using unbiased deductive methodologies, then, one can make an informed decision based on facts rather than intuition. In achieving such a desired result, it will be necessary to use fast algorithmic protocols to not sacrifice the wide nature of the search domain, to ensure that it is possible to generate the desired solution. In this paper, it is shown how in reservoir engineering desirable decisions can be reached in a timely manner choosing the most appealing course of action. It is true that in reservoir engineering applications, the decision-making process may involve a blend of intuition and scientific and rational thinking, critical factors such as blind spots, and the use of conventional methodologies that make decision-making hard to fully operationalize or to get a handle on. Luckily, there are mathematical and computational tools to ensure that scientists/engineers consistently make correct decisions, which include gathering as much information as possible and considering all possible alternatives (like combinatorial analysis protocols). The tool (model) proposed in this paper for making critical reservoir engineering decisions is a new computational platform/protocol that exploits the advantages of mathematically developed formulations and of the models that are based on the data/information collected. It is furthermore shown that the analyses conducted, and critical decisions reached, represent more thorough and far-reaching solutions that are structured using less computational overhead, thereby increasing the quality of the decision even further.

MODELLING OF WITHDRAWAL OF A STRATIFIED FLUID FROM A POROUS MEDIUM

The problem of withdrawal of fluid from a porous medium has many applications in reservoir engineering and aquifer sustainability...

Artificial neural network models for reservoir-aquifer dimensionless variables: influx and pressure prediction for water influx calculation

Calculation of water influx into petroleum reservoir is a tedious evaluation with significant reservoir engineering applications. The classical approach developed by van Everdingen–Hurst (vEH) based on diffusivity equation solution had been the fulcrum for water influx calculation in both finite and infinite-acting aquifers. The vEH model for edge-water drive reservoirs was modified by Allard and Chen for bottom-water drive reservoirs. Regrettably, these models solution variables: dimensionless influx (WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document}) and dimensionless pressure (PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document}) were presented in tabular form. In most cases, table look-up and interpolation between time entries are necessary to determine these variables, which makes the vEH approach tedious for water influx estimation. In this study, artificial neural network (ANN) models to predict the reservoir-aquifer variables WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document} was developed based on the vEH datasets for the edge- and bottom-water finite and infinite-acting aquifers. The overall performance of the developed ANN models correlation coefficients (R) was 0.99983 and 0.99978 for the edge- and bottom-water finite aquifer, while edge- and bottom-water infinite-acting aquifer was 0.99992 and 0.99997, respectively. With new datasets, the generalization capacities of the developed models were evaluated using statistical tools: coefficient of determination (R2), R, mean square error (MSE), root-mean-square error (RMSE) and absolute average relative error (AARE). Comparing the developed finite aquifer models predicted WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} with Lagrangian interpolation approach resulted in R2, R, MSE, RMSE and AARE of 0.9984, 0.9992, 0.3496, 0.5913 and 0.2414 for edge-water drive and 0.9993, 0.9996, 0.1863, 0.4316 and 0.2215 for bottom-water drive. Also, infinite-acting aquifer models (Model-1) resulted in R2, R, MSE, RMSE and AARE of 0.9999, 0.9999, 0.5447, 0.7380 and 0.2329 for edge-water drive, while bottom-water drive had 0.9999, 0.9999, 0.2299, 0.4795 and 0.1282. Again, the edge-water infinite-acting model predicted WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and Edwardson et al. polynomial estimated WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{eD}$$\\end{document} resulted in the R2 value of 0.9996, R of 0.9998, MSE of 4.740 × 10–4, RMSE of 0.0218 and AARE of 0.0147. Furthermore, the developed ANN models generalization performance was compared with some models for estimating PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document}. The results obtained for finite aquifer model showed the statistical measures: R2, R, MSE, RMSE and AARE of 0.9985, 0.9993, 0.0125, 0.1117 and 0.0678 with Chatas model and 0.9863, 0.9931, 0.1411, 0.3756 and 0.2310 with Fanchi equation. The infinite-acting aquifer model had 0.9999, 0.9999, 0.1750, 0.0133 and 7.333 × 10–3 with Edwardson et al. polynomial, then 0.9865, 09,933, 0.0143, 0.1194 and 0.0831 with Lee model and 0.9991, 0.9996, 1.079 × 10–3, 0.0328 and 0.0282 with Fanchi model. Therefore, the developed ANN models can predict WeD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{{{\ ext{eD}}}}$$\\end{document} and PD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{D}$$\\end{document} for the various aquifer sizes provided by vEH datasets for water influx calculation.

Integration of optimization algorithm for component lumping in reservoir engineering applications

Reservoir oil and gas are ordinarily characterized with quite a lot of components which mainly differ in number of hydrogen and carbon atoms. Dealing with such large number of components requires considerable amount of computation time. It is required to lump these components into pseudo components in order to reduce the CPU computation timings. The main objective is to determine a proper lumping scheme in order to minimize the associated errors as a result of the lumping process. In accordance with this purpose, the optimization algorithm known as genetic algorithm is incorporated to the objective function developed to specify the deviation in the properties of oil, and gas. The variables utilized to figure out lumping scheme were defined in two different ways. The bounds on the input variables for the objective function were narrowed in line with the relevant results obtained from the previous studies. The variables indicating the group of non-hydrocarbon components were also included in the input set. Implementation of the proposed methodology resulted in having a proper lumping scheme with only a few iterations.

Immersive diagnostics of reservoirs under WAG injection, Part II – Effect of depositional settings and dynamic spatial correlations

Immersive visualization and diagnostics have become essential tools in the arsenal of engineers and scientists in developing innovative concepts, products, and techniques. In reservoir engineering applications, these approaches can be immensely helpful in detecting the dynamics occurring within the reservoir under complex and evolving conditions. In Part I of this study, we presented several novel immersive diagnostics quantifying displacement measures and unraveled some interesting dynamics around the injector in reservoirs undergoing Water-Alternating-Gas (WAG) injection. This article is Part II of the study on the utility of immersive diagnostics to ascertain the effect of geological formation characteristics on WAG reservoir dynamics.WAG injection can yield incremental recovery around 10% in mature oilfields. In this study, using immersive metrics, we critically investigated WAG performance of four different injection-fluid streams (pure CO2, pure methane, lean gas, and enriched gas) and three WAG injection-cycle strategies in several depositional systems. Spatial clusters of displacement behavior appeared. The relationship of oil displacement efficiency with various driving forces may depend on multiple elements. These elements include compositional variation in the residual oil close to the injector and the producers and the corresponding changes in thermophysical and multiphase flow properties. The pressure level, distance to the injector, formation permeability, and depositional units contribute to the flood performance. Over time with pressure depletion and frontal movement from the injector to producers, the distinction of these clusters dissipates.Spatial correlational analyses of displacement measures reveal interesting observations. The fluid flow of the oleic phase in channelized/turbiditic formations tends to be more inertial than diffusive. Oil displacement efficiency in the channelized/turbiditic formations depends strongly on the injector-producer connectivity via these flow conduits. In most formations, we can expect a reduction in water viscosity or an increase in the inertial force in the water, yielding more efficient oil displacement.At late times, diffusive flow dictates oil displacement in oil-water zones, wherein the advective flow of oil predominates in pockets where all three phases (oil, gas, and water) are present. In general, reservoir locations with high water-phase inertial to viscous forces wield higher oil-displacement efficiency. However, in high-permeable formations at late times, reservoir locations closer to the producers having low-permeability experience low-water inertial-to-viscous forces and relatively less oil displacement efficiency, which gets worse with the presence of all three phases.

Contact between rough rock surfaces using a dual mortar method

The mechanical behavior of fractures in rocks has strong implications for reservoir engineering applications. Deformations, and the corresponding change in contact area and aperture field, impact rock fracture stiffness and permeability, thus altering the reservoir properties significantly. Simulating contact between fractures is numerically difficult as the non-penetration constraints lead to a nonlinear problem and the surface meshes of the solid bodies on the opposing fracture sides may be non-matching. Furthermore, due to the complex geometry, the non-penetration constraints must be updated throughout the solution procedure. Here we present a novel implementation of a dual mortar method for contact. It uses a non-smooth sequential quadratic programming method as solver, and is suitable for parallel computing. We apply it to a two body contact problem consisting of realistic rock fracture geometries from the Grimsel underground laboratory in Switzerland. The contributions of this article are: (1) a novel, parallel implementation of a dual mortar method with a non-smooth sequential quadratic programming method, (2) realistic rock geometries with rough surfaces, and (3) numerical examples, which prove that the dual mortar method is capable of replicating the nonlinear closure behavior of fractures observed in laboratory experiments.

Dynamics anisotropy in a porous solid with aligned slit fractures

Crustal rocks are commonly permeated by aligned fractures which may control the wave anisotropy and permeability pattern. In the presence of pore fluid the mechanical and hydraulic features of such rocks become more complex. Understanding the dynamic anisotropy in fluid-saturated fractured rocks is important for detecting and characterizing fractured reservoirs and fault zones with applications in geomechanics, hydrogeology, exploration geophysics and reservoir engineering. For waves propagating normal to the fractures, the effects of wave-induced fluid flow (WIFF) due to the presence of permeable fractures on seismic dispersion and attenuation are significant and have been quantified in earlier studies. But previous literatures are restricted to low frequency range within which the fracture size is much smaller than the incident wavelength. In this paper, we extend low-frequency normal incidence results to full-frequency oblique incidence. We first derive exact solutions of the scattering problem of obliquely incident plane waves by a single slit fracture in a poroelastic solid. Based on previous analysis, for ideal fractures with infinitesimal thickness, the fracture fluid can be modelled as an incompressible one. Then, based on the solutions and Foldy's scattering theorem we develop a dynamic-effective-medium model to estimate frequency-dependent anisotropy of wave propagation in a fluid-saturated poroelastic rock with a sparse set of aligned fractures. We find that for the oblique incidence problem apart from WIFF there exist another two important attenuation mechanisms, i.e., the elastic scattering (scattering into fast P and S waves via mode conversion at the fracture faces) and Biot's global flow, in causing velocity dispersion and attenuation. The mixed-boundary problem reveals that the WIFF is controlled by the normal displacement discontinuity that is determined by effective normal stress applied on the fracture faces, while the scattering effects by the tangential displacement discontinuity that is determined by effective shear stress. Because the effective normal and shear stresses depends on incident angles and frequency, the dispersion and attenuation of both P and S waves are anisotropic and frequency-dependent. In contrast, Biot's global flow is an intrinsic energy loss mechanism that can play a role in causing velocity dispersion and attenuation at higher frequency range but it is independent of the presence of fractures or incident angle.

Effective properties of a porous medium with aligned cracks containing compressible fluid

SUMMARYUnderstanding the wave propagation in fluid-saturated cracked rocks is important for detecting and characterizing cracked reservoirs and fault zones with applications in geomechanics, hydrogeology, exploration geophysics and reservoir engineering. In sedimentary rocks, microscopic-scale pores are usually filled with fluid. One logical means of modelling the essential features of such rocks is to use poroelasticity theory. But previous models of wave propagation in cracked porous medium are either restricted to low frequencies at which effects of the elastic scattering (scattering into fast-P and S waves via mode conversion at the crack faces) are negligible or to the case that the crack-filling fluid is assumed to be incompressible. To overcome these restrictions, we consider the effects of crack fluid compressibility by extending spring condition into poroelasticity and derive exact solutions of the scattering problem of an incident P wave by a circular crack containing compressible fluid in a porous medium. Based on the solutions, we develop two different effective medium models to estimate frequency-dependent effective velocity and attenuation in a fluid-saturated porous rock with a set of aligned cracks. The mixed-boundary value problem reveals that both the wave-induced fluid flow (WIFF) and elastic wave scattering can cause important velocity dispersion and attenuation. The diffusion-type WIFF dominates the velocity change and attenuation for the low frequency range, while the elastic scattering dominates them for the relatively higher frequency range. The dependences of the P-wave velocity on the crack fluid compressibility are different at different frequencies. For the WIFF-dominated frequency range and Rayleigh-scattering frequency range, the P-wave velocity decreases with the crack fluid compressibility. In contrast, for the Mie scattering frequency range, the opposite occurs (the P-wave velocity increases with the crack fluid compressibility).

Bubble Pressure Prediction of Reservoir Fluids using Artificial Neural Network and Support Vector Machine

Bubble point pressure is an important parameter in equilibrium calculations of reservoir fluids and having other applications in reservoir engineering. In this work, an artificial neural network (ANN) and a least square support vector machine (LS-SVM) have been used to predict the bubble point pressure of reservoir fluids. Also, the accuracy of the models have been compared to two-equation state-based models, i.e. SRK-EOS and PR-EOS and four empirical equations, i.e. Whitson, Standing, Wilson and Ghafoori et al. Compared to the experimental data, the average relative deviations (ARD) of bubble pressure prediction for these equations were obtained to be 14%, 29%, 66%, 30%, 38%, and 11%, respectively. The best semi-empirical equation has an ARD of about 11% while, the ANN and LS-SVM models have an ARD of 8% and 4.68%, respectively. Thus, it can be concluded that generally, these soft computing models appear to be more accurate than the empirical and EOS based methods for prediction of bubble point pressure of reservoir fluids.

Water footprint of hydraulic fracturing in Northeastern British Columbia, Canada

The method of hydraulic fracturing used to exploit unconventional shale gas has raised public concerns over the volumes of freshwater that are extracted for injection operations as well as the volumes of wastewater generated as a by-product of gas production. Using data from the British Columbia Oil and Gas Commission, this paper examines the volumes of produced and injected water from hydraulically fractured gas wells in Northeastern British Columbia. The two major producing shale gas basins in the province are the Montney and the Horn River. In this study, these are divided into several sub-basins based on existing geological and reservoir engineering applications. For each sub-basin the average volumes of wastewater- and injected water per well are calculated and then normalized to cumulative gas production. Ratios of injected water: gas production and wastewater: gas production are then applied to estimated volumes of remaining gas reserves in each sub-basin in order to calculate a total water footprint of future exploitation. These extrapolated water footprints were further elaborated into three scenarios of wastewater recycling rates: 0, 40, and 100% re-use. This study also compares the quality and quantity of wastewater produced from hydraulically fractured wells to their conventional counterparts in the province. Based on these calculations, the total future freshwater withdrawal and wastewater production volumes for all basins range from 1.65 to 3 billion, and 0 to 1.35 billion cubic metres, respectively. Volumes of freshwater withdrawal are relatively modest compared to other industries when considering the size of Northeastern British Columbia and the time-scale of extraction. In general, hydraulically fractured wells in Northeastern British Columbia produce volumes of wastewater that are equal to or lower than those required for injection. Unconventional gas wells often produce far less wastewater than their conventional counterparts.

A Gaussian Mixture Model as a Proposal Distribution for Efficient Markov-Chain Monte Carlo Characterization of Uncertainty in Reservoir Description and Forecasting

SummaryGenerating an estimate of uncertainty in production forecasts has become nearly standard in the oil industry, but is often performed with procedures that yield at best a highly approximate uncertainty quantification. Formally, the uncertainty quantification of a production forecast can be achieved by generating a correct characterization of the posterior probability-density function (PDF) of reservoir-model parameters conditional to dynamic data and then sampling this PDF correctly. Although Markov-chain Monte Carlo (MCMC) provides a theoretically rigorous method for sampling any target PDF that is known up to a normalizing constant, in reservoir-engineering applications, researchers have found that it might require extraordinarily long chains containing millions to hundreds of millions of states to obtain a correct characterization of the target PDF. When the target PDF has a single mode or has multiple modes concentrated in a small region, it might be possible to implement a proposal distribution dependent on a random walk so that the resulting MCMC algorithm derived from the Metropolis-Hastings acceptance probability can yield a good characterization of the posterior PDF with a computationally feasible chain length. However, for a high-dimensional multimodal PDF with modes separated by large regions of low or zero probability, characterizing the PDF with MCMC using a random walk is not computationally feasible. Although methods such as population MCMC exist for characterizing a multimodal PDF, their computational cost generally makes the application of these algorithms far too costly for field application. In this paper, we design a new proposal distribution using a Gaussian mixture PDF for use in MCMC where the posterior PDF can be multimodal with the modes spread far apart. Simply put, the method generates modes using a gradient-based optimization method and constructs a Gaussian mixture model (GMM) to use as the basic proposal distribution. Tests on three simple problems are presented to establish the validity of the method. The performance of the new MCMC algorithm is compared with that of random-walk MCMC and is also compared with that of population MCMC for a target PDF that is multimodal.

An efficient hybrid-grid crossflow equilibrium model for field-scale fractured reservoir simulation

Multiphase flow simulation in fractured reservoirs at field scale is a significant challenge. Despite recent advances and a wide range of applications in both hydrology and hydrocarbon reservoir engineering, discussing efficient methods that cover computational complexity, accuracy, and flexibility aspects is still of paramount importance for a better understanding of these complex media. In this work, we present a new method that handles both the topological and computational complexities of fractures, taking into consideration advantages of various existing approaches, which include hybrid-grid and crossflow equilibrium models. The hybrid-grid (HG) model consists of representing fractures as lower-dimensional objects that still are represented as control volumes in a computational grid. The HG model is equivalent to a single-porosity model with a practical solution for the small control volumes at the intersection between fractures; however, the overall simulation run time is still dominated by the remaining fracture small control volumes. To overcome single-porosity computational challenges, a crossflow equilibrium (CFE) concept between discrete fractures and a small neighborhood in the matrix blocks can be employed. The CFE model consists of combining fractures with a small fraction of the neighborhood matrix blocks on either side in larger elements to achieve a better computational efficiency than conventional single-porosity models. The implementation of a CFE model at field scale is not practical because of the fracture topological challenges associated with the construction of an accurate computational grid for the CFE elements. In this work, we propose a method based on a combination of HG and CFE models to overcome the challenges associated with the HG fracture small control volumes and field-scale CFE computational grid construction. First, we assess the performance of the existing CFE model, and we propose an improved model. In addition, we suggest an input data handling method that is sufficient to account for fractional flow inside the CFE elements for flow in homogeneous fractured reservoirs without the need of any change in the simulator. Second, we describe the uniqueness of the proposed method, and we discuss different numerical examples to assess both the accuracy and computational efficiency. The results obtained are very accurate, and, computationally, one to two orders of magnitude speedup can be achieved. The improved CFE results are superior over the traditional CFE model. Combined with the HG model, the results are significantly improved while retaining a very good performance.

A two-stage preconditioner for multiphase poromechanics in reservoir simulation

Many applications involving porous media–notably reservoir engineering and geologic applications–involve tight coupling between multiphase fluid flow, transport, and poromechanical deformation. While numerical models for these processes have become commonplace in research and industry, the poor scalability of existing solution algorithms has limited the size and resolution of models that may be practically solved. In this work, we propose a two-stage Newton–Krylov solution algorithm to address this shortfall. The proposed solver exhibits rapid convergence, good parallel scalability, and is robust in the presence of highly heterogeneous material properties. The key to success of the solver is a block-preconditioning strategy that breaks the fully-coupled system of mass and momentum balance equations into simpler sub-problems that may be readily addressed using targeted algebraic methods. Numerical results are presented to illustrate the performance of the solver on challenging benchmark problems.