No-flow Boundary Conditions Research Articles

On a Class of p(z)-Biharmonic Kirchhoff Type Problems with Indefinite Weight and No-Flow Boundary Condition

On a Class of p(z)-Biharmonic Kirchhoff Type Problems with Indefinite Weight and No-Flow Boundary Condition

Core-scale numerical simulation and comparison of breakdown of shale and resulting fractures using sc-CO2 and water as injectants

Supercritical carbon dioxide (sc-CO2) is an alternative to water for stimulation of low permeability systems such as shale gas and geothermal resources. Previously core-scale experimental studies have compared the behavior of CO2 to water injection for sample breakdown. Due to differences in experimental setup and core sample preparation, inconsistent or even apparently contradictory conclusions have resulted. To reconcile this contradiction, a phase-field numerical model is applied to understand hydraulic fracturing experiments using Green River shale found in the literature. The finite element numerical model incorporates a rate-dependent phase-field fracture model developed separately to describe fracture initiation and growth. We investigate the impact of various material and fluid properties on the resulting fractures. Most importantly, we study the effect of fluid properties and boundary conditions on the breakdown pressure, including the direction of the resulting fracture plane. Model results predict that (1) sc-CO2 injection in the laboratory may result in greater breakdown pressure than that of water under no-flow boundary conditions because lower viscosity sc-CO2 may result in pressure build up at the core boundary that opposes fracture initiation and (2) lower viscosity sc-CO2 also produces fast-propagating fractures that are less influenced by the bedding plane on their resulting fracture topology. Our model offers a straightforward explanation and reconciliation of existing experimental observations, as well as a means to extrapolate to new conditions. Exploration of field-scale conditions suggests less pronounced or no elevation in breakdown pressure when sc-CO2 is injected because the pressure build up effect at the system boundary is significantly less or absent at field length scales.

Modelling CO2 plume spreading in highly heterogeneous rocks with anisotropic, rate-dependent saturation functions: A field-data based numeric simulation study of Otway

In this field-data-based simulation study, we apply novel rate-dependent, anisotropic saturation functions curve-fitted to match the flow behaviour of laminated sand- and siltstones from the CO2CRC's Otway International Test Centre (Australia), in well-spot simulations of plume spreading at the site. Scenario analysis is conducted by performing simulations on various stochastic model realisations under different conditions, investigating what controls plume migration. Results indicate that high-permeability streaks and intraformational baffles determine plume spreading in flat-lying stratigraphy. Displacement of CO2 is unstable and focussed by high-permeability streaks, leading to multi-layer plumes. In inclined stratigraphy, buoyancy forces control plume migration channelling it up dip. However, high-permeability streaks still influence plume shape. Comparisons for different injection rates show that heterogeneity-induced fingering dominates at high injection rates, whereas gravity override and vertical movement of the plume are prominent at low injection rates. Continuous build-up of fluid pressure in the storage horizon influences the properties of CO2 when no-flow (closed) lateral boundary conditions are applied. By contrast, a marked pressure drop occurs in the near-well region for hydrostatic (open) lateral boundary conditions. These also foster uneven heterogeneity-fingering of the plume. Since flow velocity in the highly heterogeneous Paaratte formation varies over many orders of magnitude, force balances range between the capillary and the viscous limit even at some distance to the injector. A comparison with simulations based on standard Brooks-Corey saturation functions shows that taking flow-rate dependence and anisotropy into account leads to significant differences in the shape of the evolving CO2 plume and the saturation within.

Pore-scale study of the anisotropic effect on immiscible displacement in porous media under different wetting conditions and capillary numbers

This article investigates the effect of anisotropy on two-phase flows in porous media (both drainage and imbibition) under various wetting angles and capillary numbers. The focus is on the applied results useful to petroleum engineering. Mathematical modeling based on lattice Boltzmann equations in combination with multi-relaxation time and color-gradient model is used as a research tool. This paper proposes a new method for creating digital models of porous media, based on a combination of a quartet structure generation set and Monte-Carlo algorithms. The advantage is the ability to control both heterogeneity and anisotropy. This paper presents visual phase diagrams of the invasion patterns in anisotropic samples as a function of wetting angle and capillary number. The results show that the ratio between capillary and viscous forces is highest for the sample with favorable anisotropy, which has the biggest permeability, and decreases on going to unfavorable anisotropy (lowest permeability). This effect leads to the finding that boundary capillary numbers, which determine the transition between crossover mode and invasion patterns (viscous and capillary fingering, and compact displacement), are shifted towards lower values as anisotropy increases. The images of fluid distributions are numerically described using fractal dimension and displacement efficiency. It was found that the anisotropic effect is most pronounced at capillary fingering and does not significantly affect the invasion characteristics in the imbibition mode. The sensitivity of the fractal dimension and displacement efficiency to a change in the wetting angle is maximal at favorable anisotropy and decreases on going to unfavorable anisotropy. During drainage, the fractal dimensions, estimated for favorable and unfavorable anisotropy, show opposite trends depending on the capillary number. In the imbibition mode, there is an increase in the fractal dimension with an increase in the capillary number for all porous structures. In addition, this paper investigates the anisotropy effect on the «injected fluid - skeleton » interfacial length under different wetting and dynamic conditions. It was shown that the anisotropy has no significant effect on this parameter, and the best interaction efficiency was found with a compact displacement, and the least with a viscous fingering. Also, the influence of the no-flow and periodic boundary conditions on the invasion characteristics is examined. The results show that boundary conditions do not significantly affect the invasion in the sample with favorable anisotropy for any wetting angles. Periodic boundary conditions negatively influence the displacement efficiency for samples with isotropic and unfavorable structures at drainage, but no noticeable effect was found during imbibition. • The anisotropic effect is most pronounced with capillary fingering. • Anisotropy does not significantly affect invasion during imbibition. • Anisotropy has no significant effect on the «injected fluid - skeleton » interface. • Boundary conditions do not affect invasion with favorable anisotropy for any wettability.

Numerical modeling of fluid flow and heat transfer in Kurşunlu geothermal field-KGF (Salihli, Manisa / Turkey)

Nowadays, the need for energy is increasing more and more. It is more difficult to acquire new resources in various fields than to preserve existing energy resources. Although Turkey is a very rich state in terms of various energy resources, misuse of these resources can even lead to conflicts that may occur between the states in forthcoming years. In today's economic conditions, we can only protect our energy resources with the correct way of management. In this context, it is very important to reveal the mechanisms that make up the geothermal systems, which are very common in Western Anatolia. In this study, how the fluid circulation mechanism in the geothermal system takes place, and under which conditions the infiltrating water is heated in the Kurşunlu geothermal field (KGF) have been examined. FEFLOW software was used in numerical modeling. Fluid flow and heat transfer equations are solved on a twodimensional vertical model using FEFLOW software. A variable-width finite element mesh consisting of 55,590 elements was created in this scope. Since triangular meshes are preferred in vertical models, the mesh produced according to the Delaunay method was used. All lateral boundaries are designed as a no-flow boundary condition. For boundary conditions, hydraulic heads on top of the model and temperature values at both the top and bottom of the model are defined. Additionally, initial values were produced for the entire Kurşunlu geothermal system under steady-state conditions, and a transient model was built to run for 700,000 days. The regional flow direction is towards to the North. The fluids are transmitted deeply and heated through fault zones and transported towards the surface. Convective flows start to form below -1000 m altitudes in the fault zones and in the geothermal aquifer widespread convective flows in deeper regions were formed, while smaller spread convective flows were formed near the surface and shallow depths of the aquifer. In the process of convective flow, heated fluids reach to Kurşunlu region and forms the spring. Finally, two more possible high-temperature areas have been identified, indicating that the flow vectors point to the surface.

Large time dynamics of 2D semi-dissipative Boussinesq equations

In this paper, we construct an alternative proof for the long-time behavior of large-data classical solutions to the 2D semi-dissipative Boussinesq equations without thermal diffusion on a bounded domain subject to the stress-free boundary conditions, which was previously studied in Doering et al (2018 Physica D 376/7 144–59). To demonstrate the effectiveness of the new approach, we study the long-time behavior of large-data classical solutions to the initial-boundary value problem of a related model with density variance and subject to the no-flow boundary condition, for which the analytic technique utilized in Doering et al (2018 Physica D 376/7 144–59) is not directly accessible.

Three-dimensional response to a partially penetration well pumping in a general anisotropic three-layer aquifer system

Three-dimensional response to well pumping in a multilayer aquifer system with rigorous treatment of interfacial flow attracts great attention among the subsurface hydrologists because of its theoretical importance and practical merits. In this study, a general theory of three-dimensional groundwater flow to a well of infinitesimal radius in an anisotropic three-layer aquifer system is developed with a constant-rate pumping in the lower layer, and associated semi-analytical solutions of drawdowns in individual layers are obtained. This theory discards many previous assumptions for studying a multilayer aquifer system such as simplified flow configuration in the aquitard, treatment of cross-formation flow as a volumetric averaged term adding to the governing equations of aquifer flows, and sometimes negligence of vertical flow components in the aquifers. Additionally, this theory considers three types of commonly used top boundary conditions: A water table with delayed gravity drainage boundary condition (Case 1), a constant-head top boundary condition (Case 2), and a no-flow top boundary condition (Case 3). The dimensionless drawdown solutions in Laplace domain are obtained and inverted numerically to calculate the time-domain solutions. The solution encompasses existing solutions for a two-layer or a single-layer (e.g. unconfined or confined) aquifer system as subsets. The developed solutions are tested extensively with a finite-element numerical solution using COMSOL and other existing solutions, and a sensitivity analysis is performed to prioritize the influences of different parameter groups. The results of this study can be used to predict the pumping induced drawdown at any position with an observation well, and to estimate the hydraulic parameters of the lower pumped layer and upper layer, as well as the middle layer. The developed theory provides a basis for other potential applications such as geotechnical engineering and groundwater management.

Diffusion of interacting particles in a channel with reflection boundary conditions.

The diffusive transport of biased Brownian particles in a two-dimensional symmetric channel is investigated numerically considering both the no-flow and the reflection boundary conditions at the channel boundaries. Here, the geometrical confinement leads to entropic barriers which effectively control the transport properties of the particles. We show that compared to no-flow boundary conditions, the transport properties exhibit distinct features in a channel with reflection boundary conditions. For example, the nonlinear mobility exhibits a nonmonotonic behavior as a function of the scaling parameter f, which is a ratio of the work done to the particles to available thermal energy. Also, the effective diffusion exhibits a rapidly increasing behavior at higher f. The nature of reflection, i.e., elastic or inelastic, also influences the transport properties firmly. We find that inelastic reflections increase both the mobility and the effective diffusion for smaller f. In addition, by including the short range interaction force between the Brownian particles, the mobility decreases and the effective diffusion increases for various values of f. These findings, which are a signature of the entropic nature of the system, can be useful to understand the transport of small particles or molecules in systems such as microfluidic channels, membrane pores, and molecular sieves.

Dynamic coupling of analytical linear flow solution and numerical fracture model for simulating early-time flowback of fractured tight oil wells (planar fracture and complex fracture network)

Abstract Quantitative analysis of flowback data provides an early opportunity to interpret hydraulic fracture properties of fractured wells in unconventional reservoirs. In recent years, a number of studies have been dedicated to the analysis of single-phase flow data prior to hydrocarbon breakthrough and multi-phase flow data after hydrocarbon breakthrough during flowback. This study provides a new model to simulate flowback of fractured tight oil wells with planar fractures or a complex fracture network. In the model, fractures are discretized into a number of segments and the fully numerical method is used to model fracture flow. It is assumed that the pressure interference in the matrix caused by these fracture segments can be accounted using a no-flow boundary condition and that the drainage volumes of fracture segments form isolated regions to allow for linear flow from the matrix to each segment. To model this behavior, matrix flow is simulated using the analytical transient linear flow solution which is dynamically coupled with the fracture flow model by imposing continuity of pressure and flux on the fracture surfaces. The model is simple, but rigorous enough to take into account the important physics of the fracture system, including arbitrary fracture geometries and fracture conductivity distributions. The pressure and saturation gradients of each phase in fractures are accounted for. The ease of model setup and improved computational performance makes it convenient for practical application. The new model is verified with a commercial reservoir simulator. Investigation of the assumption of no-flow boundary condition for the pressure interference in the matrix caused by the fracture segments reveals that the assumption becomes poorer with increasing matrix permeability and decreasing fracture permeability. Flow regime analysis demonstrates that, for planar fractures, water exhibits an early transient linear flow period (first linear flow in the fracture) followed by boundary-dominated flow (first boundary-dominated flow in the fracture) prior to hydrocarbon breakthrough, after which a second transient linear flow in the fracture develops. The final flow period is boundary-dominated flow (second boundary-dominated flow in the fracture), indicating the end time of the flowback period. At this stage, oil will be the dominant phase in the fracture, and will exhibit transient linear flow in the matrix, which is the first flow-regime typically observed during the long-term production period. The case of a complex fracture network exhibits a similar sequence of flow regimes. The second transient linear flow for this case has not been discussed in the literature. The development of this flow period may be caused by the maintenance of fracture pressure, and decrease of water relative permeability due to hydrocarbon contribution from the matrix. A field example from western Canada exhibits the flow regimes noted above and is history-matched successfully using the new model. The results demonstrate the practical application of the new model for deriving reservoir/fracture properties from flowback data and for forecasting.

A Laplace-transform boundary element model for pumping tests in irregularly shaped double-porosity aquifers

In this paper, a novel Laplace-transform boundary element model for pumping tests in irregularly shaped double-porosity aquifers is presented. The aquifer could be associated with various boundary conditions, such as the general Robin boundary conditions or even the mixed boundary conditions with no-flow boundary on one side and constant head boundary on the other side. The derived solution has analytical characteristics since it is obtained through the Green’s function method within the domain. Unlike traditional numerical methods, the proposed solution using the Laplace-transform boundary element method does not require discretization in spatial and temporal dimensions. In this study, boundary integral equation using Green's second theorem and the fundamental solution of Green function for dual-porosity models are given. A boundary element matrix in Laplace space is established, which allow us to consider irregularly shaped boundary. The drawdown and flux on the boundary can be obtained through solving the boundary element matrix by Gauss's elimination method. The final solution for wellbore drawdown is evaluated using numerical Laplace inversion algorithm of Stehfest (1970). Many previous solutions for transient flow in finite single-porosity aquifers with no-flow outer boundary condition are shown to be special cases of the present solution. Furthermore, the solution with a mixed outer boundary condition is used to investigate the effect of important parameters of aquifers on wellbore drawdown, including fracture storage coefficient, inter-porosity flow coefficient from matrix to fissures, boundary shape, boundary location, and interference of multiple wells.

Innovative Analytical Solutions to 1, 2 and 3D Water Infiltration into Unsaturated Soils for Initial-Boundary Value Problems

Fluid infiltration into unsaturated soil is of vital significance from many perspectives. Mathematically, such transient infiltrations are described by Richards' equation; a nonlinear parabolic Partial Differential Equation (PDE) with limited analytical solutions in the literature. The current study uses separation of variables and Fourier series expansion techniques and presents new analytical solutions to the equation in one, two, and three dimensions subject to various boundary and initial conditions. Solutions for 1-D horizontal and vertical water infiltration are derived and compared to numerical finite difference method solutions, whereby both solutions are shown to coincide well with one another. Solutions to 2- and 3-D vertical water infiltration are derived for constant, no-flow, and sinusoidal boundary and initial conditions. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed form solution. The solutions may be utilized to test numerical models that use different computational techniques

Mixing and un-mixing by incompressible flows

We consider the questions of efficient mixing and un-mixing by incompressible flows which satisfy periodic, no-flow, or no-slip boundary conditions on a square. Under the uniform-in-time constraint \|\nabla u(\cdot,t)\|_p\le 1 we show that any function can be mixed to scale \epsilon in time O(|\mathrm {log}\:\epsilon|^{1+\nu_p}) , with \nu_p=0 for p<\tfrac{3+\sqrt 5}2 and \nu_p\le \tfrac 13 for p\ge \tfrac{3+\sqrt 5}2 . Known lower bounds show that this rate is optimal for p\in(1,\tfrac{3+\sqrt 5}2) . We also show that any set which is mixed to scale \epsilon but not much more than that can be un-mixed to a rectangle of the same area (up to a small error) in time O(|\mathrm {log}\:\epsilon|^{2-1/p}) . Both results hold with scale-independent finite times if the constraint on the flow is changed to \|u(\cdot,t)\|_{\dot W^{s,p}}\le 1 with some s<1 . The constants in all our results are independent of the mixed functions and sets.

A simulation investigation of performance of polymer injection in hydraulically fractured heterogeneous reservoirs

Polymer flooding is a popular enhanced oil recovery method because of its impact on improvement of sweep efficiency. Polymers are non-Newtonian fluids with different behavior at different flow rates. At high shear rate, they reveal shear-thinning behavior, which is the apparent viscosity reduction by increasing shear rate. However, higher shear rate let them become dilatant. Consequently, an increase in shear rate contributes to increase in apparent viscosity and thus decreases the well injectivity of polymer. Hydraulic fracturing reduces mechanical shearing in the vicinity of the wellbore and plays an important role in feasibility of polymer flooding scenarios due to injectivity enhancement and lowering shear rate. A three-dimensional simulator was used to construct a synthetic model of the reservoir with no-flow boundary condition. Results show that induced hydraulic fractures improve recovery factor of homogeneous and heterogeneous reservoirs due to an increase in injectivity of polymer flooding, although in vertically heterogeneous reservoirs, induced fractures are not very effective. The results also show that the induced fractures are more successful in reservoirs with viscous oil.

A Globally Coupled Pressure Method for the Discretization of the Tensor-Pressure Equation on Non-K-orthogonal Grids

Summary Complex permeability tensors together with general nonorthogonal and unstructured grids pose great challenges to reservoir simulation. The widely used two-point flux approximation (TPFA) is inadequate for a rigorous discretization of the flow equations on such challenging grids. Multipoint flux approximation (MPFA) methods have been proposed to meet the challenges and are currently being deployed in next-generation simulators. In this work, we propose an alternative flux-continuous cell-centered finite-volume method called the globally coupled pressure (GCP) method to discretize the pressure equation on general grids with full permeability tensors. To accurately construct fluxes through control-volume interfaces, pressure at the centroid of all interfaces is introduced as auxiliary unknowns. Flux continuity across each interface gives one equation. Assembling all the flux-continuity equations together gives a system of linear equations that can be solved simultaneously for all the auxiliary unknowns. Flux across control-volume interfaces can then be approximated with the pressure values at control-volume centers only. The fundamental difference between the GCP method and MPFA methods is that, in the latter, auxiliary unknowns are locally coupled within an interaction region and then eliminated in a local stencil by imposing flux-continuity conditions, whereas in the former, all the auxiliary unknowns are globally coupled and can only be eliminated in a global stencil. Consequently, control volumes in our GCP method are directly associated with the edges of the original grid and not by means of a dual grid overlaid and allied with the centers of the grid. Two variants of the GCP method are presented here, and extensive numerical experiments are conducted to test the performance of the GCP methods. The results show that both variants of our GCP method are in good agreement with the classical MPFA-O method on non-K-orthogonal grids for less-challenging problems. Convergence studies reveal that the first variant of our GCP method has slower convergence rates than the MPFA-O method for some problems. However, the second variant of GCP has comparable, and in some cases, better convergence properties compared with the MPFA-O method. With numerical experiments, we further investigate monotonicity properties of our GCP method on highly anisotropic media. For Dirichlet boundary conditions, our GCP methods also suffer from nonphysical oscillations, with some degrees of improvement over the MFPA-O method. When no-flow boundary conditions are used, our GCP method is much more robust and does not produce spurious boundary extrema as MPFA methods do. Finally, we extend our GCP method to fully unstructured grids, and the results show that it is also more robust than the MPFA-O method on unstructured grids.

New Analytical Solutions to 2-D Water Infiltration and Imbibition into Unsaturated Soils for Various Boundary and Initial Conditions

Fluid infiltration and imbibition into unsaturated soil are of vital significance from many perspectives. Mathematically, such transient flows are described by Richards’ equation, a nonlinear parabolic partial differential equation with limited analytical solutions in the literature. However, the choice of exponential model for water content and hydraulic conductivity linearizes the nonlinear Richards’ equation, making it possible to obtain an analytical solution via classical approaches. In this study, separation of variables and Fourier series expansion techniques are used to derive new analytical solutions to 2-D vertical and horizontal infiltration and imbibition into unsaturated soils for nonsymmetrical boundary and nonuniform initial conditions. A total of 11 cases are considered, where high water content is imposed on the top, side, or bottom edges of the sample and water is infiltrated (from the top and/or side boundaries) and imbibed (from the bottom boundary) into the sample. Residual water content and/or no-flow boundary condition are assumed on other edges of the sample. Initial conditions include 7 cases of constant residual water content, 2 cases of sinusoidal, and 2 cases of exponential water content functions over the sample. Presented analytical solutions are such that both steady and unsteady solutions may be obtained from a single closed-form solution. Two-dimensional and 3-D plots of water content are presented for the transient as well as steady-state conditions. To illustrate the use of the derived equations, water content values from numerical solutions are compared to those from analytical solutions for four cases, showing a maximum error of <2 %. The presented analytical solutions may be used as a benchmark for verification and accuracy assessment of numerical approaches where nonsymmetrical boundary and/or nonuniform initial conditions exist.

Saltwater upconing zone of influence

In this study, we define and characterize the saltwater upconing zone of influence (SUZI). The SUZI is the region around a pumping well within which significant rise in the saltwater-freshwater interface occurs. While the zone of influence of a pumping well can be clearly defined in terms of hydraulics (e.g., drawdown), the SUZI has not been recognised and characterised, despite its importance for groundwater decision-making in coastal regions. We explore the SUZI under various conditions and compare common methods of investigation using both axisymmetric (1D and 2D vertical cross-section) and 3D simulations of saltwater upconing at the field scale, based on a combination of numerical and analytical approaches. The SUZI was found to be dependent on the relative magnitudes of pumping, regional flow, distance of the well from the coast, and position of the well above the interface, as expected. The three-dimensional coastal setting simulations revealed an asymmetric shape of the lateral extent of the SUZI, which is largest in the direction parallel to the coast. This occurs because the ocean and the inland extent of the seawater wedge limit the propagation of the SUZI perpendicular to the coast. Predictions of the SUZI using the Ghyben–Herzberg approximation, including cases where sloping interfaces occur (i.e., in contrast to the artificiality of horizontal interfaces used in axisymmetric approaches), provide reasonable first approximations of the SUZI. Numerical modelling of dispersive upconing in the 3D inclined interface case is influenced by practical limits to the model domain size and grid resolution. For example, the no-flow boundary condition at 1500m from the pumping well elongates the SUZI in the direction parallel to the coast. This study extends previous concepts of well interference, which have previously been based on hydraulics only, by introducing the SUZI and characterising its extent, with consideration given to differences in commonly adopted methods of upconing quantification.

Helical Boundary Conditions To Capture Interpattern Flow in In-Situ-Upgrading-Process Pattern Simulations

Summary The in-situ upgrading process (IUP) involves very complicated multiphase thermal transport and chemical reactions. Numerical simulation of IUP is computationally expensive, and direct simulation of field-scale IUP models becomes prohibitive. A practical way is to simulate a sector model under proper boundary and initial conditions and then upscale the simulation results to the full field scale with superposition techniques. Because of nonsymmetric pattern configuration and time delay of developing patterns sequentially, interpattern flow may become significant, and its impact on the simulation results cannot be neglected. Therefore, no-flow boundary conditions become inappropriate for such IUP sector models. In this paper, we proposed a new type of “helical” boundary conditions (HBCs) in which pressures and temperatures are periodic in space, except for a shift in time. The HBCs are specifically designed for a field-scale IUP development in which patterns are developed sequentially in time, along a long strip. In such a development, each pattern has exactly the same well configuration and operational schedule, except for a time delay. Because of high viscosity of heavy oil and low heat conductivity of formation rock, the impact of field boundary conditions on interpattern flow will be dampened quickly within only a few patterns, and a repetitive “pseudosteady state” of interpattern flow develops, in which the energy and mass fluxes from the previous pattern to the current pattern are the same as those from the current pattern to the next pattern, except for the delay time. By use of a 1D heat-transfer model, we analytically demonstrate that the pseudosteady state and therefore the HBCs hold for a long strip of patterns. A practical procedure to implement these HBCs in numerical simulation is developed, in which the state variables (pressure, temperature, and fluid composition) are iteratively updated in gridblocks on both edges of a sector model that is composed of two patterns. This iterative approach to impose HBCs was implemented in our in-house simulator. This approach was tested and validated by simulation results of an IUP model that is composed of 59 patterns. Our results show that the full-field boundary conditions only affect the production-rate profiles of the first and the last patterns. Production-rate profiles generated from all other patterns are almost identical except for the interpattern time delay, which also validates the pseudosteady state of interpattern flow for a more-complicated IUP model. The two-pattern sector model with the HBCs converges in three to four iterations. The production-rate profiles of oil, gas, and water generated by the sector model with HBCs are almost identical to those produced from one of those inner patterns in the 59-pattern model. With the 1D example, we also analytically demonstrate the convergence of our numerical implementation of HBCs. In terms of clock-time used, it is possible to achieve 5N time speedup through application of the HBCs, in which N is the number of patterns in a field-scale model. Therefore, the new approach is proved a key enabler for field-scale IUP pattern optimization. Provided that the interpattern-pressure communication that is induced by the pattern delay time is not too severe, we expect that one can also apply HBCs to the simulation of other field-scale thermal processes, such as the in-situ conversion process and steamfloods.

Convective mixing influenced by the capillary transition zone

Convective mixing of dissolved carbon dioxide (CO2) with formation brine has been shown to be a significant factor for the rate of dissolution of CO2 and thus for determining the viability of geological CO2 storage sites. In most previous convection investigations, a no-flow boundary condition was used to represent the interface between an upper region with CO2 and brine and the single-phase brine region beneath. However, due to interfacial tension between the phases, the water phase is partly mobile in the upper region and advection may occur. Based on linear stability analysis and numerical simulations, we show that advection across the interface leads to considerable destabilization of the system. In particular, the time of onset of instability is reduced by a factor of two and the rate of dissolution is enhanced by a factor of two for three of four formations we consider, and by 40 % for the fourth formation. It is found that exponential decay of the relative permeability away from the interface provides a useful approximation to the real system. In addition, the exponential decay also simplifies the linear stability analysis. Interestingly, formations with large absolute permeability and small porosity have the largest impact from the transition zone, despite the fact that the relative permeability decays quickly above the interface in these formations. This is because the length-scale of instability is smallest in these formations.

Improvements on the normal mode decomposition method used in harbor resonance

In the article by Sobey (Rodney J. Sobey, 2006. Normal mode decomposition for identification of storm tide and tsunami hazard. Coastal Engineering 53, 289–301), the author proposed a normal mode decomposition method to calculate the eigenfrequencies, the eigenmodes and the response amplitudes of different resonant modes in natural harbors that are subjected to storm tides and tsunamis. However, the numerical method to address the no-flow boundary condition in that article is imprecise, which would lead to inexact eigenfrequencies and eigenmodes. In this article, the mirror-image method was proposed to improve this handling process. The accuracy of the improved normal mode decomposition method was verified using three verification tests. With a set of numerical experiments, it was determined that during the process of decomposing the response amplitudes of different resonant modes, the numerical fitting error between the simulated free surfaces and the corresponding fitted ones gradually increases with the wave nonlinearity inside the harbor. This article sought to identify the critical wave condition under which the normal mode decomposition method can accurately decompose the response amplitudes of different modes.

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.