Flow Equations Research Articles

Study Explores Fracture-Driven Interaction Mitigation Strategies for Field Development

_ This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper URTeC 4055561, “Numerical Investigation of Fracture-Driven-Interaction Mitigation Strategies and Field Development Optimization,” by Ruiting Wu, SPE, Xiaolong Liu, and Lili Xu, Chevron, et al. The paper has not been peer reviewed. _ Numerical modeling of fracture-driven interactions (FDI) using a coupled hydraulic-fracturing-propagation, reservoir-flow, and geomechanics tool has been conducted. The numerical study showed that, for the basins studied, FDI and some mitigation strategies have minimal effect on long-term parent-well productions. Parent depletion (volume and areal extent), well spacing, and child-fracture asymmetry, however, greatly affect child-well production as the result of FDI. Introduction Interwell FDI, also known as frac hits, has garnered recent attention given that more than 60% of new wells drilled in US unconventional reservoirs are infill drilling. Whereas uncontrollable parameters, such as natural fractures and matrix permeability, do contribute to the effect of FDI on well production, the authors’ primary focus lies on controllable factors. Field observations have shown that tighter well spacing and smaller completion sizes tend to result in fewer negative effects on both parent- and child-well production. Additionally, fracture-fluid injection rates, the number of clusters, and cluster spacing all influence FDI effects. Simulation Tool and Workflow A coupled hydraulic-fracture, reservoir, and wellbore simulator was used to simulate the fracturing and production stage of the parent and child wells and related fracture-driven well interaction. The simulator simultaneously solves the equations of fluid flow, proppant transport, water/solute transport, fracture mechanics, and poroelasticity in the well, fracture, and matrix domains. Fracture propagation is calculated using the multilayer-tip-elements algorithm. Fracture elements are represented by uniformly sized rectangles. Fracture-to-fracture stress shadow is calculated with the 3D displacement discontinuity method, and the poroelastic stress responses during fracture propagation and depletion are calculated with a finite-volume method. The tool is designed to simulate the entire life cycle of the FDI interaction. The modeling workflow also includes calibration with field observations of fracture geometry, the pressure response at the parent well during fracturing of the child well, the water cut change at the parent well as the result of FDI, and production performance. Field Case Study: Permian Basin The primary objective of this study is to identify the key parameters that contribute to FDI and mitigate the negative effects of FDI. Two field case studies were conducted in two basins, the Permian Basin and the DJ Basin. In both field studies, a history match of parent and child well performance was first conducted. Then, a sensitivity study was performed on various key parameters to understand their effects on the production performance at both parent and child wells. Both field case studies are focused on the lateral FDI. The first case study is included in this synopsis.

Relational Lorentzian Asymptotically Safe Quantum Gravity: Showcase Model

In a recent contribution, we identified possible points of contact between the asymptotically safe and canonical approaches to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity, which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantized, and one can construct the generating functional of time-ordered Wightman (i.e., Feynman) or Schwinger distributions, respectively, from the corresponding time-translation unitary group or contraction semigroup, respectively, as a path integral. For the unitary choice, that path integral can be rewritten in terms of the Lorentzian Einstein–Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantization and a reduction term that encodes the reduction of the full phase space to the phase space of observables. This path integral can then be treated with the methods of asymptotically safe quantum gravity in its Lorentzian version. We also exemplified the procedure using a concrete, minimalistic example, namely Einstein–Klein–Gordon theory, with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper, we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature, which has a strong impact on the convergence of “heat” kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation.

Optimization and Analysis of Flow Field Simulations Across Different Mach Regimes Using Discrete Numerical Methods

This article explores the development of flow field models in steady-state environments utilizing Euler equations and potential flow equations, with verification processes conducted using Python. The models demonstrate stability and high accuracy in low-velocity scenarios, capturing essential dynamics effectively. However, as the conditions transition to supersonic speeds, the models begin to exhibit increased errors. This discrepancy highlights the challenges faced in simulating high-speed aerodynamics accurately. The research underscores the importance of improving model fidelity in diverse Mach regimes, particularly in supersonic conditions where traditional methods struggle. Future research directions identified include the development of unsteady flow field models, which are crucial for dynamic analyses, optimization of grid structures for three-dimensional complex fields to improve computational efficiency, and the creation of extensive model libraries. These advancements aim to enhance the accuracy, reliability, and practicality of flow field simulations, extending their applicability in both academic studies and industry applications, particularly in aerospace engineering where precise flow modeling is critical.

Momentum flux coefficient effect on vertical two-phase flows: Numerical method comparison

The governing equations of two-phase flows have a safe operating range, and exceeding this range can negatively impact the results. To extend this safe range, one key strategy is to incorporate the momentum flux coefficient into the governing equations. This paper introduces the application of the momentum flux coefficient to no-pressure or free-pressure model equations for the first time, specifically evaluating its performance in downward co-current two-phase flows through numerical methods. A new and improved version of the no-pressure model is also presented. Findings suggest that the force approach method yields more accurate results compared to other methods, whereas the Richmeier method produces unrealistic outcomes at discontinuities. A comparison between the standard and the developed no-pressure models reveals that the latter does not prevent nonphysical mutations and even exacerbates their occurrence. However, the developed no-pressure model successfully reduces numerical distribution production.

Onset of global instability in a premixed annular V-flame

We investigate self-excited axisymmetric oscillations of a lean premixed methane–air V-flame in a laminar annular jet. The flame is anchored near the rim of the centrebody, forming an inverted cone, while the strongest vorticity is concentrated along the outer shear layer of the annular jet. Consequently, the reaction and vorticity dynamics are largely separated, except where they coalesce near the flame tip. The global eigenmodes corresponding to the linearised reacting flow equations around the steady base state are computed in an axisymmetric setting. We identify an arc branch of eigenmodes exhibiting strong oscillations at the flame tip. The associated eigenvalues are robust with respect to domain truncation and numerical discretisation, and they become destabilised as the Reynolds number increases. The frequency of the leading eigenmode is found to correspond to the Lagrangian disturbance advection time from the nozzle outlet to the flame tip. The essential role of this convective mechanism is also supported by resolvent analysis, which finds that the same flame-tip disturbance structure and frequency are optimally amplified when the flame is subjected to external white noise forcing. Strong non-modal effects in the form of pseudo-resonance are not found. Nonlinear time-resolved simulation further reveals notable hysteresis phenomena in the subcritical regime prior to instability. Hence, even when the flame is linearly stable, perturbations of sufficient amplitude can trigger limit-cycle oscillations and higher-dimensional dynamics sustained by nonlinear feedback. A Monte Carlo simulation of passive tracers in the unsteady flame suggests a nonlinear non-local instability mechanism. Notably, linear analysis of the subcritical time-averaged limit-cycle state yields eigenvalues that do not match the nonlinear periodic oscillation frequencies. This mismatch is attributed to the fundamentally nonlinear dynamics of the subcritical V-flame instability, where the dichromatic, non-local interaction between the heat release rate along the flame surface and the vortex dynamics in the jet shear layer cannot be approximated as a simple distortion of the mean flow.

A novel model for low-permeability reservoir numerical simulation considering dynamic capillary force

ABSTRACT Capillary force is not only affected by water saturation but also affected by fluid flow velocity, which is necessary to modify the prior numerical simulation methods. First, a dynamic capillary force testing device was developed, then, an experimental procedure was adopted to measure the dynamic capillary force curve. Three dynamic capillary force curves were obtained under different displacement velocity conditions. Finally, the dynamic capillary force was introduced into the flow equation of low-permeability reservoir numerical simulation. The oil-phase equation and water-phase equation were discretized in three-dimensional space, and the IMPES method was used to solve the water saturation and pressure difference. The calculation results between the traditional numerical simulation method and the novel method under heterogeneous and inhomogeneous conditions were analyzed. The results reveal that the difference in water cut is mainly reflected in the rapid rise of water cut. In the early stage of rapid increase, the water cut calculated by the new method is slightly higher than traditional method. When the water cut is more than 65%, the water cut calculated by the new method is slightly lower than the traditional method.

On the Scaling of Transport Phenomena at a Monotonously Changing Hydraulic Conductivity Field

Monotonously stratified porous medium, where the layered medium changes its hydraulic conductivity with depth, is present in various systems like tilled soil and peat formation. In this study, the flow pattern within a monotonously stratified porous medium is explored by deriving a non-dimensional number, Fhp, from the macroscopic Darcian-based flow equation. The derived Fhp theoretically classifies the flow equation to be hyperbolic or parabolic, according to the hydraulic head gradient length scale, and the hydraulic conductivity slope and mean. This flow classification is explored numerically, while its effect on the transport is explored by Lagrangian particle tracking (LPT). The numerical simulations show the transition from hyperbolic to parabolic flow, which manifests in the LPT transition from advective to dispersive transport. This classification is also applied to an interpolation of tilled soil from the literature, showing that, indeed, there is a transition in the transport. These results indicate that in a monotonously stratified porous medium, very low conducting (impervious) formations may still allow unexpected contamination leakage, specifically for the parabolic case. This classification of the Fhp to the flow and transport pattern provides additional insight without solving the flow or transport equation only by knowing the hydraulic conductivity distribution.

Metric flows with neural networks

Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow.

A Joint Estimation Method of Distribution Network Topology and Line Parameters Based on Power Flow Graph Convolutional Networks

Accurate identification of network topology and line parameters is essential for effective management of distribution systems. An innovative joint estimation method for distribution network topology and line parameters is presented, utilizing a power flow graph convolutional network (PFGCN). This approach addresses the limitations of traditional methods that rely on costly voltage phase angle measurements. The node correlation principle is applied to construct a node correlation matrix, and a minimum distance iteration algorithm is proposed to generate candidate topologies, which serve as graph inputs for the parameter estimation model. Based on the topological dependencies and convolutional properties of AC power flow equations, a PFGCN model is designed for line parameter estimation. Parameter refinement is achieved through an alternating iterative process of pseudo-trend calculation and neural network training. Training convergence and loss function values are used as feedback to filter and validate candidate topologies, enabling precise joint estimation of both topologies and parameters. The proposed method’s accuracy, transferability, and robustness are demonstrated through experiments on the IEEE-33 and modified IEEE-69 distribution systems. Multiple metrics, including MAPE, IAE, MAE, and R2, highlight the proposed method’s advantages over Adaptive Ridge Regression (ARR). In the C33 scenario, the proposed method achieves MAPEs of 4.6% for g and 5.7% for b, outperforming the ARR method with MAPEs of 7.1% and 7.9%, respectively. Similarly, in the IC69 scenario, the proposed method records MAPEs of 3.0% for g and 5.9% for b, surpassing the ARR method’s 5.1% and 8.3%.

An Improved Scheme for the Finite Difference Approximation of the Advective Term in the Heat or Solute Transport Equations

AbstractTransport equations are widely used to describe the evolution of scalar quantities subject to advection, dispersion and, possibly, reactions. Numerical methods are required to solve these equations in applications, adopting either the advective or conservative formulations. Conservative formulations are usually preferred in practice because they conserve mass. Advective formulations do not, but have received more mathematical attention and are required for Lagrangian solution methods. To obtain an advective formulation that conserves mass, we subtract the discretized fluid flow equation, multiplied by concentration, from the conservative form of the transport equation. The resulting scheme not only conserves mass, but is also elegant in that it can be interpreted as averaging the advective term at cell interfaces, instead of approximating it at cell centers as in traditional centered schemes. The two schemes are identical when fluid velocity is constant, and both have second-order convergence, but the truncation errors are slightly different. We argue that the error terms appearing in the proposed scheme actually imply an improved representation of subgrid spreading/contraction and acceleration/deceleration caused by variable velocity. We compare the proposed and traditional schemes on several problems with variable velocity caused by recharge, discharge or evaporation, including two newly developed analytical solutions. The proposed method yields results that are slightly, but consistently, better than the traditional scheme, while always conserving mass (i.e., mass at the end equals mass at the beginning plus inputs minus outputs), which the traditional centered finite differences scheme does not. We conclude that this scheme should be preferred in finite difference solutions of transport.

Modeling proppant transport and settlement in 3D fracture networks in geothermal reservoirs

In this paper, we develop an efficient proppant transport model using the Eulerian-Eulerian approach for simulating proppant transport in fractures and 3D fracture networks in geothermal reservoirs. The proposed model accounts for proppant settling, pack/bed formation, bridging/screenout, proppant concentration effect, fracture wall effect, and the transition from Poiseuille flow (fracture channel) to Darcy flow (proppant pack). Notably, the heat transfer process and its impact on proppant transport are also considered—a facet often overlooked in previous proppant transport models. A three-dimensional displacement discontinuity method (3D DDM) that incorporates the stress shadow effect is employed to generate the fracture geometry. The governing equations for slurry flow, proppant transport, and heat transfer are discretized and solved using the finite volume method (FVM). The model is verified against analytical solutions and published experimental data, demonstrating good agreement with these references. To demonstrate the proposed model, we applied it to both low-temperature (depleted hydrocarbon wells) and high-temperature (dry hot rocks) enhanced geothermal systems (EGS). The simulation results highlight the significant influence of reservoir temperature on proppant transport and settlement in a reservoir environment. Heating of the slurry by higher temperature reservoir rocks reduces fluid viscosity and accelerates proppant settling, thereby shortening the transport distance and reducing the coverage area of the proppant. Both ultra-light and micro-proppant are effective in mitigating proppant settlement in enhanced geothermal systems. However, proppant is susceptible to bridging at fracture intersections, where the fracture widths are narrower due to more pronounced stress shadow effects in these areas. Consequently, the use of micro-proppant could offer substantial benefits over ultra-light proppant in enhancing proppant coverage area in enhanced geothermal systems.

Transfer Learning-Based Physics-Informed Convolutional Neural Network for Simulating Flow in Porous Media with Time-Varying Controls

A physics-informed convolutional neural network (PICNN) is proposed to simulate two-phase flow in porous media with time-varying well controls. While most PICNNs in the existing literature worked on parameter-to-state mapping, our proposed network parameterizes the solutions with time-varying controls to establish a control-to-state regression. Firstly, a finite volume scheme is adopted to discretize flow equations and formulate a loss function that respects mass conservation laws. Neumann boundary conditions are seamlessly incorporated into the semi-discretized equations so no additional loss term is needed. The network architecture comprises two parallel U-Net structures, with network inputs being well controls and outputs being the system states (e.g., oil pressure and water saturation). To capture the time-dependent relationship between inputs and outputs, the network is well designed to mimic discretized state-space equations. We train the network progressively for every time step, enabling it to simultaneously predict oil pressure and water saturation at each timestep. After training the network for one timestep, we leverage transfer learning techniques to expedite the training process for a subsequent time step. The proposed model is used to simulate oil–water porous flow scenarios with varying reservoir model dimensionality, and aspects including computation efficiency and accuracy are compared against corresponding numerical approaches. The comparison with numerical methods demonstrates that a PICNN is highly efficient yet preserves decent accuracy.

Application of a New Mean Velocity Equation of Flow in a Natural Stream of 8.5m3/s, Its Verification and Extrapolation: Copper Creck River, Virginia (USA)

Application of a New Mean Velocity Equation of Flow in a Natural Stream of 8.5m3/s, Its Verification and Extrapolation: Copper Creck River, Virginia (USA)

Heat transfer enhancement using ternary hybrid nanofluid for cross-viscosity model with intelligent Levenberg-Marquardt neural networks approach incorporating entropy generation

Thermal heat transfer analysis of trihybrid nanofluids using an intelligent Levenberg-Marquardt neural network (ANN-LMA) approach, with a focus on entropy generation, has been conducted. The flow equations were modeled in Cartesian coordinates and simplified using dimensionless variables. Partial differential equations were converted into ordinary differential equations through appropriate similarity transformations. These ordinary differential equations were then solved using the finite element method applied to a data set evaluated from (ANN-LMA) approach. This dataset can be input into MATLAB to generate predicted solutions for flow patterns. The ANN-LMA technique was employed to evaluate the efficiency of heat transfer characteristics for nanofluids in various scenarios. Incorporating carbon nanotubes (both single-wall (SWCNT) and multi-wall (MWCNT)) along with iron oxide in water, the study demonstrates their effectiveness in enhancing heat transfer. These nanofluids have broad industrial applications, such as in coolant enhancement, cancer therapy, and solar radiation management, and show promising results. This study specifically examines the flow properties of water-based CNT cross-trihybrid nanofluids over a convectively heated surface, leveraging their unique characteristics. Improvements in heat transfer are achieved through the introduction of dissipative heat, thermal radiation, and external heat sources or sinks. The performance of the computational solver was assessed using error histograms, regression analyses, and Mean Squared Error (MSE) results. The physical significance of the designed factors is graphically depicted and discussed in detail. It was found that radiative heat increases surface heat energy through substantial accumulation, thereby enhancing heat transfer properties, while dissipative heat, due to Joule dissipation and other external sources, significantly raises the fluid temperature.

Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products

Abstract Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

Analysis of Capillary Water Absorption within Unsaturated Concrete Based on the Principle of Stationary Action

Capillary water absorption plays a critical role in the ingress of corrosive elements during the construction of concrete structures in corrosive environments. This study presented a novel approach for analyzing capillary water flow within unsaturated concrete based on the principle of stationary action. The flow of water within the concrete capillary pores can be regarded as a variational problem, while the principle of stationary action provides a method for determining the path solution. The evolution and distribution characteristics of water content and wetting front were explicitly determined using the exponential and power hydraulic functions. A simplistic yet effective approach for determining these hydraulic parameters was put forward based on the relationship between the position of the wetting front and the diffusivity parameters. The proposed approach exhibited enhanced theoretical robustness and entailed fewer hypotheses compared to existing methodologies. Furthermore, the material hydraulic parameters in the proposed approach can be determined explicitly. The governing equations for capillary water flow were derived in accordance with the principle of stationary action. Numerical simulations were carried out to verify the effectiveness of the proposed approach. The results demonstrated that the proposed approach can accurately predict capillary water flow and diffusivity parameters within unsaturated concrete. The findings of this study contribute to developing more effective strategies to mitigate moisture-related damage in concrete structures.

Using the total chemical potential to generalize the capillary pressure concept and therefrom derive a governing equation for two-phase flow in porous media

This work presents a governing equation (GE) for two-phase flow in porous media connecting capillary pressure to frictional pressure loss and external chemical potential supplied to a system in either stationary or diffusive equilibrium. It is based on the difference between non-wetting and wetting phase chemical potential (physically, pressure or energy density), which leads to a generalization of the capillary pressure concept. The difference in phase internal chemical potentials is characterized by changes in both interfacial areas and entropy densities due to variation in fluid saturations and is balanced by the system external chemical potential supplied. A definition of the capillary pressure concept is formulated based on the diffusive equilibrium criterion. The GE can explain the origin of the dynamic capillary pressure term, hysteresis, and connect the shift in the capillary pressure curve upon injecting water phases with varying salinities to all the other chemical potentials acting. It can connect, constrain, and potentially quantify all effects which can be formulated in terms of chemical potentials since it is based on a balance equation all two-phase flow systems must obey when either in stationary or diffusive equilibrium.

Learning a CNN with the finite volume method for modelling fluid seepage flow under closed boundary

Physics-informed neural networks (PINNs) are an efficient technique for incorporating physics law in models based on neural networks (NNs). In this paper, we propose a local refinement physics-informed convolutional neural network (LR-PICNN) to solve the seepage equation in closed boundary reservoirs. LR-PICNN employs a local refinement method to enhance focus on the region containing the well. First, the finite volume (FV) method is used to discretize the seepage equation on the global reservoir domain and on the local domain around the well to construct the loss functions respectively. Second, the boundary between the global and local domains are used as boundary conditions to couple the two systems. Finally, two loss functions are added to train the CNN to solve the seepage equation in autoregressive way and coupled without any labeled data. Experimental results show that LR-PICNN can more accurately solve seepage flow equation. Due to the coupling of solving equations and that LR-PICNN focuses on the behaviors of the well, compared to physics-informed Deep Convolutional neural network (PIDCNN) under the same well grid size, LR-PICNN has a faster solving speed that is at least 8 times faster, higher solution accuracy for BHP that is at least twice as high, and more stable solution, which could be used in numerical well testing.

Entropy generation of Williamson hydromagnetic non-linear radiative nanofluid flow near a stagnation point over a porous stretching sheet

Purpose This study aims to analyze the multi-slip effects of entropy generation in steady non-linear magnetohydrodynamics thermal radiation with Williamson nanofluid flow across a porous stretched sheet near a stagnation point. Also, the qualities of viscous dissipation, Cattaneo–Christove heat flux and Arrhenius activation energy are taken into account. Thermophoresis, Brownian motion and Joule heating are also considered. Design/methodology/approach The Navier–Stokes equation, the thermal energy equation and the Solutal concentration equations are the governing mathematical equations that describe the flow and heat and mass transfer phenomena for fluid domains. By using the proper similarity transformations, a set of ordinary differential equationss are retrieved from boundary flow equations. The classical Runge–Kutta fifth-order algorithm along with the shooting technique is implemented to solve the obtained first order differential equations. Findings The study concludes that the temperature distribution boosting for thermal radiation, magnetic field and Eckert number where as the velocity and entropy generation escalate for the Williamson parameter, diffusion parameter and Brinkman number. The skin-friction and heat and mass transfer rate increases with the fluid injection. In addition, tabulated values of friction drag and rate of heat and mass transfer for various values of constraints are provided. Originality/value The comparison of the present results is carried out with the published results and noted a good agreement.