Fine-grid Numerical Simulations Research Articles

Comparison of super-resolution deep learning models for flow imaging

The primary goal of this study is to introduce deep learning (DL) methods as a cost-effective alternative to the computationally intensive Direct Numerical Simulation (DNS) simulations. We show that one can obtain a parametric field from a low-resolution input and map it to a fine grid output, significantly reducing the computational burden. We assess five super-resolution models for up-scaling low-resolution flow data into fine-grid numerical simulations’ output for accuracy and efficiency. The proposed architectures employ convolutional neural networks interconnected in encoder/decoder branches. We investigate these models using turbulent velocity fields inside a suddenly expanded channel characterized by complex features, including turbulence, instabilities, asymmetries, separation, and reattachment. Our results reveal that an encoder/decoder model with residual connections delivers the fastest results, a U-Net-based model with skip connections excels at producing sharper edges in regions prone to blurring, while deeper models incorporating maximum and average pooling layers show superior performance in reconstructing velocity profiles. These findings significantly contribute to our understanding of the potential of deep learning in fluid mechanics. The models presented in this study are trained and validated on standard computer hardware and can be easily adapted to other problems. The findings are promising for discovering and analyzing flow physics, highlighting the potential for DL techniques to improve the accuracy of the available fluid mechanics computational tools.

Suitability of hot urea solutions for wettability alteration of bitumen reservoirs – Simulation of laboratory flooding experiments

We report highly fine grid numerical simulation of sandpack flooding experiments conducted using 5 and 10 wt% hot urea solutions for wettability alteration of bitumen reservoirs. Another hot water flooding experiment was also simulated as a baseline for comparison with hot urea solutions. The relative permeability curves have been obtained through history matching of the experimental oil recovery. The results indicated that urea solutions have proved to be efficient in wettability alteration toward more water-wet and attenuation of viscous fingering leading to delayed water breakthrough as was confirmed by the shape of the relative permeability curves. Wettability alteration led to incremental oil recovery compared to the hot water flooding. It was shown that wettability alteration and the in-situ generation of water in oil (W/O) emulsions previously observed in experiments can be captured using the relative permeability curves. These results find application in the design and optimization of thermal bitumen recovery processes.

Propane-Aided Leaching of Bitumen from Oilsands

We report highly fine-grid numerical simulations of solvent-aided bitumen leaching using propane as a solvent. These simulations allowed us to observe the detailed physics of the drainage process, ...

Improved Distance-of-Investigation Model for Rate-Transient Analysis in a Heterogeneous Unconventional Reservoir With Nonstatic Properties

Summary The concept of distance of investigation (DOI) has been widely applied in rate– and pressure–transient analysis for estimating reservoir properties and for optimizing hydraulic fracturing. Despite its successful application in conventional reservoirs, significant errors arise when extending the concept to unconventional reservoirs. This work aims to clearly demonstrate such errors when using the traditional square–root–of–time model for DOI calculations in unconventional reservoirs, and to develop new models to improve the DOI calculations. In this work, the following mechanisms in unconventional reservoirs are first incorporated into the calculation of DOI: (1) pressure–dependency of rock and fluid properties; (2) continuous/discontinuous spatial variation of reservoir properties. To achieve this, pseudopressure, pseudotime, and pseudodistance are introduced to linearize the diffusivity equation. Two novel methods are developed for calculating DOI: one using the concept of continuous succession of steady states, and the other using the concept of dynamic drainage area (DDA). Both models are verified using a series of fine–grid numerical simulations. A production–data–analysis workflow using the new DOI models is proposed to analytically characterize reservoir heterogeneity and fracture properties. The new DOI models compensate for the inability of the traditional square–root–of–time model to capture spatial and temporal variations of reservoir and fluid properties. The pressure–dependency of fluids and reservoirs (i.e., fluid density, fluid viscosity, rock permeability, and rock porosity) and reservoir heterogeneities (i.e., deterioration of reservoir quality from the primary fracture to the reservoir) can significantly retard the propagation of the DOI. Another important outcome of this work is to provide a practical and analytical approach to directly estimate the spatial heterogeneity from the production history of field cases.

Generalized analytical model of Transient linear flow in heterogeneous fractured liquid-rich tight reservoirs with non-static properties

• New analytical transient linear flow is rigorously modeled in heterogeneous fractured liquid-rich tight reservoirs with non-static properties . • Improved rate-transient analysis workflow is proposed to estimate reservoir/fracture properties. • Three liquid-rich shale field examples in North America are analyzed to demonstrate the practical applicability of new model. The industry is increasingly reliant on rate-transient analysis (RTA) to extract valuable information about the reservoir and hydraulic fractures. However, the application of current, commercially-available RTA models can lead to incorrect estimates of reservoir/fracture properties, potentially causing costly mistakes to be made in capital planning and reserve estimation. The root cause of these errors is that currently-available analytical solutions used in RTA models largely ignore reservoir heterogeneities, and assume static reservoir properties. In this work, a new transient linear flow is rigorously modeled in unconventional reservoirs with (1) pressure-dependent rock and fluid properties and (2) both continuous and discontinuous (heterogeneous) porosity and permeability. To achieve this, new transformations of pseudo-pressure, pseudo-time and pseudo-distance are first introduced to reduce the temporal and spatial non-linear diffusivity equation to that with approximately constant coefficients. Both a Laplace-domain solution and approximate analytical solution to the diffusivity equation are verified against a series of fine-grid numerical simulations for the assumption of fractal-based reservoir heterogeneity (over a wide range of stress-dependent rock and fluid properties). The results indicate that reservoir heterogeneity can result in nonlinear square-root-of-time plots. Further, rock and fluid pressure-dependencies act to decrease the slope of the square-root-of-time plot and affect reservoir/fracture property evaluations. Three liquid-rich shale (LRS) field examples in North America are analyzed to demonstrate the practical applicability of the new RTA models. Additional value of new RTA models over the sophisticated numerical simulation is to provide us an improved backforward-analysis workflow that can be used to quantify both effective fracture half-length and non-uniform permeability distribution around the fractures. The major contribution of this work is the introduction of a new analytical model for evaluating the transient linear flow period for the cases of arbitrary reservoir heterogeneity and non-static reservoir properties. This new approach is particularly useful for evaluating the effectiveness of hydraulic fracturing operations by extracting the spatial variability of reservoir quality within the stimulated reservoir volume (SRV).

Effect of Layering on Incremental Oil Recovery From Tertiary Polymer Flooding

Summary It has been demonstrated in both laboratory measurements and field applications that tertiary polymer flooding can enhance oil recovery from heterogeneous reservoirs, primarily through macroscopic sweep (conformance). This study quantifies the effect of layering on tertiary polymer flooding as a function of layer-permeability contrast, the timing of polymer flooding, the oil/water-viscosity ratio, and the oil/polymer-viscosity ratio. This is achieved by analyzing the results from fine-grid numerical simulations of waterflooding and tertiary polymer flooding in simple layered models. We find that there is a permeability contrast between the layers of the reservoir at which maximum incremental oil recovery is obtained, and this permeability contrast depends on the oil/water-viscosity ratio, polymer/water-viscosity ratio, and onset time for the polymer flood. Building on an earlier formulation that describes whether a displacement is understable or overstable, we present a linear correlation to estimate this permeability contrast. The accuracy of the newly proposed formulation is demonstrated by reproducing and predicting the permeability contrast from existing flow simulations and further flow simulations that have not been used to formulate the correlation. This correlation will enable reservoir engineers to estimate the combination of permeability contrast, water/oil-viscosity ratio, and polymer/water-viscosity ratio that will give the maximum incremental oil recovery from tertiary polymer flooding in layered reservoirs regardless of the timing of the start of polymer flooding. This could be a useful screening tool to use before starting a full-scale simulation study of polymer flooding in each reservoir.

Semi-analytical method for matrix diffusion in heterogeneous and fractured systems with parent-daughter reactions

A semi-analytical/numerical method for modeling matrix diffusion in heterogeneous and fractured groundwater systems is developed. This is a significant extension of the Falta and Wang (2017) method that only applied to diffusion in an aquitard of infinite thickness. The current solution allows for the low permeability matrix to be embedded within a numerical gridblock, having finite average thickness, a specified volume fraction and a specified interfacial area with the high permeability domain. The new formulation also allows for coupled parent-daughter decay reactions with multiple species that each have independent retardation factors, decay rates, and yield coefficients in both the high and low permeability parts of the system. The method uses a fitting function to approximate the transient concentration profile in the low permeability part of each gridblock so that the matrix diffusion flux into the high permeability part of the gridblock can be computed as a concentration dependent source-sink term. This approach is efficient because the only unknowns in each gridblock are the concentrations in the high permeability domain, so there is practically no increase in computational effort compared to a conventional transport simulation. The method is shown to compare favorably with an analytical solution for matrix diffusion in fractured media with parallel fractures, with an analytical solution for matrix diffusion with parent-daughter decay reactions, with laboratory experiments of matrix diffusion in a layered system, with a laboratory experiment involving lens shaped inclusions, and with fine grid numerical simulations of transport in highly heterogeneous systems.

THE ROLE OF FRACTURE CAPILLARY PRESSURE ON THE BLOCK-TO-BLOCK INTERACTION PROCESS

Characterization of fractures and the study of multiphase fluid movement through fractures in fractured porous media present difficult challenges to reservoir engineering. Interaction between porous matrix blocks and fractures plays a significant role in oil recovery from double-porosity reservoirs. The block-to-block interaction (or capillary continuity) between porous matrix blocks is a key contributor to the gas-oil gravity drainage mechanism in the gas-invaded zone of naturally fractured reservoirs, which increases the oil recovery. In a continuum scale, fracture is a part of the stack of blocks where there is a pressure difference between the gas and oil phases inside the fracture (called fracture capillary pressure). However, the physics of this capillary pressure and how it affects the gravity drainage mechanism in a stack of porous matrix blocks through the block-to-block interaction process need to be addressed theoretically. In this work a direct fine-grid numerical simulation along with various fracture capillary pressure models, including zero, constant, and saturation-dependent Brooks and Corey (1964), van Genuchten (1980), and Dindoruk and Firoozabadi (1995) functions, are applied to study their influence on oil recovery and therefore the block-to-block interaction process in fractured porous media. Numerical simulation predictions show the positive effect of fracture capillary pressure on oil recovery of a stack porous matrix blocks. The results reveal that the zero fracture capillary pressure model results in a lower ultimate oil recovery factor (23.8%) as compared to the constant (25.8%) and saturation-dependent Brooks and Corey (1964) (28.1%), van Genuchten (1980) (27.5%), and Dindoruk and Firoozabadi (1995) (24.6%) models. These observations are in good agreement with the results in literature. The findings can improve our understanding of the role of fracture capillary pressure on the block-to-block interaction process, which is important in oil recovery from naturally fractured reservoirs using the gravity drainage mechanism.

Analytical modelling of cyclic steam stimulation (CSS) process with a horizontal well configuration

In this study, a new semi‐analytical model has been developed to model heavy oil production using cyclic steam stimulation (CSS) and horizontal wells. In order to predict oil production, the governing equations of heat and fluid flow in cylindrical coordinate were solved. Average temperature in the formation during the soak period was calculated by solving the transient heat transfer equations. The heat transfer model was then coupled to a fluid inflow model to predict the fluid rates. The developed model is validated with the fine grid numerical simulation of the problem using a commercial numerical simulator. The robustness of the model was tested by comparing the model results with the results from the numerical simulator by changing various parameters such as: steam injection rates, reservoir permeability, relative permeability, oil viscosity, and reservoir thermal diffusivity. With some reasonable assumptions, it was seen that the results from the developed model are very close to the results obtained using the numerical simulator. This paper presents the model development in detail and outlines the steps involved in predicting the performance of a CSS process for a horizontal well configuration.

A composite dual-porosity fractal model for channel-fractured horizontal wells

ABSTRACTThe channel fracturing technique has been proven to provide much higher conductive fracture networks by placing discontinuously proppant inside fracture packs. Despite the great success of this new fracturing technique, there is still a lack of models and methods to characterize non-uniform placement of fracture proppant and heterogeneous permeability distribution within stimulated-reservoir-volume (SRV). In this article, a dual-porosity model is coupled with the tri-linear flow model to quantify the production performance of channel fractured horizontal wells. As a consequence of channel fracturing, the non-uniform distribution of fracture networks, and the heterogeneities of both matrix/fracture porosity and permeability are characterized using fractal theory. By implementing the Bessel Function and Laplace transform techniques, analytical solutions are derived by integrating fluid flow across primary hydraulic fractures, SRV and unstimulated reservoir matrices. Through quantitative comparisons of well bottom-hole pressure history, a synthetic fine-grid numerical simulation example is implemented to verify the accuracy of analytical solutions. Sensitivity analysis of fractal dimension, fractal connectivity-index and primary fracture conductivity is carried out to quantify the temporal and spatial consequences of channel fracturing on both reservoir/facture heterogeneity and well productivity throughout well life.

Modified shape factor incorporating gravity effects for scaling countercurrent imbibition

Imbibition is the main oil (or gas) production mechanism in water invaded naturally fractured reservoirs (NFRs). Scaling laboratory imbibition results is important in studying imbibition in actual case of NFRs. Different scaling equations have been presented in the literature for countercurrent imbibition. Important parameters such as gravity effects are required to be considered in the scaling process. Gravity effects are important in field scale where Bond Number, the ratio of gravity to capillary forces, is large enough to influence the fluid exchange between matrix and fracture system. In this study by a theoretical approach, a modified shape factor accounting for the gravity effects is presented by extension of the well-known Kazemi-Gilman-Elsharkawy (KGE) shape factor and can be used in existing countercurrent imbibitions scaling equations. To validate the presented shape factor in scaling countercurrent imbibition results, the fine grid numerical simulation technique has been used to produce a wide range of experiments. Results show that the new shape factor works well when gravity effects are considerable in comparison to capillary forces.

Analytical modeling dynamic drainage volume for transient flow towards multi-stage fractured wells in composite shale reservoirs

This paper presents an analytical solution to predict the expansion of drainage volume for a composite transient linear flow system consisting of stimulated-reservoir volume (SRV) and unstimulated reservoir volume. Correct prediction of dynamic drainage volume (DDV) is essential in production data analysis and well space evaluation. With the prediction of DDV, we can calculate the average pressure and average saturation with different time, which can assist designing different exploitation scenarios.The compound linear flow solution within both SRV and unstimulated matrix is derived under constant-flowing-bottomehole-pressure (FBHP) and constant-rate boundary conditions. Laplace transform and numerical inversion are implemented to obtain the analytical solution. The location of pressure front is calculated using the impulse response concept, which is the maximum rate of pressure response. A multi-variable regression method is applied to determine an empirical equation indicating the relationship between DDV and the square root of time; the empirical equation includes conductivities of both SRV and unstimulated matrix volume, fracture length, and fracture spacing.The results suggest that the DDV demonstrates a linear relationship with square-root-of-time for both linear flow regime within SRV and the compound linear flow in composite fractured reservoirs. The advancement of DDV within stimulated-reservoir volume is much faster than unstimulated matrix, which is approximately 100 times in general. To verify the accuracy of newly derived DDV equations, we analyze the synthetic production data from series of fine-grid numerical simulations. Finally, we make a sensitivity analysis of hydraulic fracture spacing with respect to DDV, which can be implemented to evaluate the fracturing design. In practice, the solution derived can be used to determine optimal fracture spacing.

Impact of Distance-of-Investigation Calculations on Rate-Transient Analysis of Unconventional Gas and Light-Oil Reservoirs: New Formulations for Linear Flow

Summary Long-term transient linear flow of hydraulically fractured vertical and horizontal wells completed in tight/shale gas wells has historically been analyzed by use of the square-root-of-time plot. Pseudovariables are typically used for compressible fluids to account for pressure-dependence of fluid properties. Recently, a corrected pseudotime has been introduced for this purpose, in which the average pressure in the distance of investigation (DOI) is calculated with an appropriate material-balance equation. The DOI calculation is therefore a key component in the determination of the linear-flow parameter (product of fracture half-length and square root of permeability, xfk) and the calculation of contacted fluid in place. Until now, the DOI for transient linear flow has been determined empirically, and may not be accurate for all combinations of fluid properties and operating conditions. In this work, we have derived the DOI equations analytically for transient linear flow under constant-flowing-pressure and -rate conditions. For the first time, rigorous methodologies have been used for this purpose. Two different approaches were used: the maximum rate of pressure response (impulse concept) and the transient/boundary-dominated flow intersection method. The two approaches resulted in constants in the DOI equation that are much different from previously derived versions for the constant-flowing-pressure case. The accuracy of the new equations was tested by analyzing synthetic production data from a series of fine-grid numerical simulations. Single-phase oil and gas cases were analyzed; pseudovariable alteration for pressure-dependent porosity and permeability was required in the analysis. The calculated linear-flow parameters, determined from our new DOI formulations for the constant-flowing-bottomhole-pressure (FBHP) case, and the input values to numerical simulation, are in good agreement. Of the two new DOI-calculation methods provided, the maximum rate of pressure response (unit impulse method) provides more accurate results. Finally, a field case was analyzed to determine the impact of DOI formulations on derivations of the linear-flow parameter from field data. Linear-flow analysis on the basis of the DOI calculations presented in this work is significantly improved over previous formulations for constant FBHP.

Analysis of Near Well-bore Behavior of Gas Condensate Reservoir in Production Stage

The formation of condensate banking in the near well-bore region will lead to the reduction in gas relative permeability and loss of well deliverability. This loss is investigated by conducting a series of fine grid numerical compositional simulation with the gas condensate and dry gas reservoir data. Also, a method for calculation of condensate blockage skin factor is introduced in which the skin factor is calculated using dry gas and gas condensate simulation production data. The authors also found that absolute permeability and pressure drawdown of the producing well have significant influence on the well gas deliverability and condensate blockage skin factor.

Effect of Methane Co-Injection in SAGD--Analytical and Simulation Study

Summary Steam-assisted gravity drainage (SAGD) is a commercially viable recovery method for oil sands of Athabasca used where other methods have been unsuccessful. In one variation of SAGD, a small amount of a noncondensable gas is added to the injected steam to maintain pressure in the chamber while using the energy in place, reducing steam consumption and providing thermal insulation from overburden heat losses. The role of gas during steam-gas co-injection processes, in terms of its effects on chamber development, bitumen flow rates, and heat losses, is not fully understood, and therefore is the main focus of this work. A new analytical model for gas injection in SAGD is derived, taking into account the three-phase flow of gas, oil, and water in the reservoir. The analytical theory is used to predict the fluid flow rates as well as phase mobility, relative permeability, and saturation profiles in the mobile oil region. The theoretical results are replicated by fine-grid numerical simulations. Methane was used as the noncondensable gas for the purpose of this study because it is the main solution gas in most reservoirs. It is, however, believed that the findings of this study are equally applicable to other noncondensable gases such as nitrogen, air, helium, and others. Fine-grid numerical simulations were performed to gain a visual understanding of gas distribution in a SAGD chamber and its effect on in-situ steam quality, overburden heat losses, phase saturations, and fluid-flow rates. The simulation results support the predictions of the mathematical theory. The results of the analytical and numerical study reveal that methane co-injection with steam is in general unfavorable in a SAGD operation. The injected methane tends to accumulate at the steam condensation front, which lowers the heat transfer rate of steam to the adjacent oil, resulting in lower oil production rates and slower growth of the chamber.

Equivalent Reaction Constant for Isothermal Reactive Flow with Power-Law Kinetics

The scale difference between chemical reactions and diffusive processes makes the numerical simulation of reactive flows a challenging task. The subscale effects of reactions are hard to capture unless the computational expenses of very fine-grid numerical simulations can be tolerated, which is not practically feasible. A solution is to develop methodologies to account for small-scale effects using coarse-grid simulations. In this article, an equivalent reaction constant for steady-state and isothermal reactive flow with a power-law kinetics is developed. The effects of Peclet number, Thiele modulus, and reaction order are investigated.

Effect of fracture pressure depletion regimes on the dual-porosity shape factor for flow of compressible fluids in fractured porous media

A precise value of the matrix-fracture transfer shape factor is essential for modeling fluid flow in fractured porous media by a dual-porosity approach. The slightly compressible fluid shape factor has been widely investigated in the literature. In a recent study, we have developed a transfer function for flow of a compressible fluid using a constant fracture pressure boundary condition [Ranjbar E, Hassanzadeh H, Matrix-fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media. Adv Water Res 2011;34(5):627–39. doi:10.1016/j.advwatres.2011.02.012]. However, for a compressible fluid, the consequence of a pressure depletion boundary condition on the shape factor has not been investigated in the previous studies. The main purpose of this paper is, therefore, to investigate the effect of the fracture pressure depletion regime on the shape factor for single-phase flow of a compressible fluid. In the current study, a model for evaluation of the shape factor is derived using solutions of a nonlinear diffusivity equation subject to different pressure depletion regimes. A combination of the heat integral method, the method of moments and Duhamel’s theorem is used to solve this nonlinear equation. The developed solution is validated by fine-grid numerical simulations. The presented model can recover the shape factor of slightly compressible fluids reported in the literature. This study demonstrates that in the case of a single-phase flow of compressible fluid, the shape factor is a function of the imposed boundary condition in the fracture and its variability with time. It is shown that such dependence can be described by an exponentially declining fracture pressure with different decline exponents. These findings improve our understanding of fluid flow in fractured porous media.

A new analytically derived shape factor for gas–oil gravity drainage mechanism

Matrix–fracture exchange term is an important parameter in simulation of naturally fractured reservoirs using double porosity concept. A matrix block which is located in gas-cap is going to be produced under gravity drainage retarded by capillary force. Hence, oil can be produced as long as gravitational force is greater than the capillary force. The interaction of these forces which controls the oil production from a matrix block may be expressed by a factor which is matrix–fracture transfer shape factor. Customarily, this factor has been used as a history matching parameter as it is still not well understood and the formulations used in commercial simulators are not precise. Therefore, a mathematically derived shape factor which reflects the gravity drainage behavior into an exchange term using shape factor concept is essential. Traditionally, a constant shape factor has been used in most of the commercial reservoir simulators. In this paper, we use analytical modeling to develop a time dependent shape factor for gas–oil gravity drainage mechanism for a single matrix block in a gas-cap. The obtained model is verified against several fine-grid numerical simulations as well as two major existing models being used in a commercial simulator. The obtained shape factor starts from a dimensionless shape factor of 2 ( σL z 2 = 2) and demonstrates a transient behavior at early times and then converges to π 2/4 at late times. The capillary force distribution throughout the matrix block is also obtained during the process showing a nonlinear distribution at early times which turns into a linear one at late times. The linear behavior of capillary force distribution corresponds to late time stabilization of shape factor. Moreover, the mathematical derivation predicts very early time constant maximum drainage rate after which drainage rate starts to decline. The results obtained in this paper will improve our understanding of the gas–oil gravity drainage in fractured reservoirs and will find application in more accurate representation of this process in dual-porosity simulators.

A fine resolution parametric study on the numerical simulation of gas–solid flows in a periodic riser section

Fine grid numerical simulations have been performed on a periodic section of a riser using the kinetic theory of granular flows (KTGF) approach. A substantial number of different model settings and constants were evaluated in order to assess their importance on the overall hydrodynamics occurring in the riser. Special care was taken to ensure that sufficient spatial and temporal resolution was implemented so that the discretization error would be minimal. The description of the shear at the walls, the particle–particle restitution coefficient and the models selected for the solids viscosity and the diffusion term in the granular temperature equation were found to be the most influential factors. Model selection for the frictional viscosity and solids pressure also had a marginally significant influence. It was found that the effects of particle–wall restitution coefficient, drag formulation, bulk viscosity and gas phase turbulence were negligible. The final combination of model settings produced results that compare well with published experiments.

Equivalent Rate Constant for Numerical Simulation of Linear Convection-Diffusion-Reaction

Abstract The accurate numerical modelling of reactive flow is essential for the interpretation of experimental measurements and the design of field-scale processes. Sharp reaction fronts are difficult to capture using traditional numerical schemes, unless fine grid numerical simulations are used; however, fine grid simulations are computationally expensive. On the other hand, using coarse grid block simulations leads to excessive front dissipation and inaccurate results. In most practical cases, it is not feasible to choose small grid blocks; therefore, one needs to account for the small-scale gradients that cannot be captured by coarse grid blocks when using traditional methods. As a first step toward the development of upscaling techniques for reaction kinetics, an equivalent reaction constant for a simple steady-state convection-diffusion-reaction (CDR) was determined. This shows how an effective reaction constant can be obtained as a function of the Peclet number, Thiele modulus and size of the reaction zone, such that the coarse grid simulation agrees with the fine grid simulation.