Buckley-Leverett Problem Research Articles

Analytical model for CO2-water displacement with rate-dependent phase permeability for geological storage

Analytical modelling is an effective tool for predicting reservoir behaviour under high uncertainties, like CO2 storage in aquifers, where multiple simulation runs are necessary for stochastic modelling and risk assessment. This paper formulates the Buckley-Leverett problem with x-dependent fractional flow. We derive a new analytical model for displacement of brine by CO2 accounting for (i) rate-dependent phase permeability during radial flows and (ii) radial flows of Forchheimer's high-rate gas injection; this analytical model is also valid for (iii) linear flows during flooding in micro heterogeneous or composite cores and (iv) CO2-water flow upscaling in reservoirs where layering is perpendicular to flux direction. An exact solution of the flow equation is based on the observation that the flux of each phase conserves along the characteristic trajectories. We discuss S-shape fractional flow function, which is typical for reservoir rocks. The solution includes formulae for phase saturations and fluxes and trajectories of the displacement CO2-water front and of the forward and rear mixture zone boundaries. The fast analytical model can be used for multivariate sensitivity study and sweep efficiency prediction.

Numerical method to solution of generalized model Buckley-Leverett in a class of discontinuous functions

A new numerical method is proposed for solving the generalized Buckley-Leverett problem, which describes the movement of two-phase mixtures of Bazhenov bed sediments in a class of discontinuous functions. To this end, we introduce an auxiliary problem that has advantages over the main problem, and using these advantages, an original finite difference method to solve of the auxiliary problem is developed. Using the suggested auxiliary problem, a solution which expresses exactly all physical characteristics of the problem is obtained.

Data-Free and Data-Efficient Physics-Informed Neural Network Approaches to Solve the Buckley–Leverett Problem

Physics-informed neural networks (PINNs) are an emerging technology in the scientific computing domain. Contrary to data-driven methods, PINNs have been shown to be able to approximate and generalize well a wide range of partial differential equations (PDEs) by imbedding the underlying physical laws describing the PDE. PINNs, however, can struggle with the modeling of hyperbolic conservation laws that develop shocks, and a classic example of this is the Buckley–Leverett problem for fluid flow in porous media. In this work, we explore specialized neural network architectures for modeling the Buckley–Leverett shock front. We present extensions of the standard multilayer perceptron (MLP) that are inspired by the attention mechanism. The attention-based model was, compared to the multilayer perceptron model, and the results show that the attention-based architecture is more robust in solving the hyperbolic Buckley–Leverett problem, more data-efficient, and more accurate. Moreover, by utilizing distance functions, we can obtain truly data-free solutions to the Buckley–Leverett problem. In this approach, the initial and boundary conditions (I/BCs) are imposed in a hard manner as opposed to a soft manner, where labeled data are provided on the I/BCs. This allows us to use a substantially smaller NN to approximate the solution to the PDE.

Physics-Informed Neural Networks Help Predict Fluid Flow in Porous Media

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 203033, “Prediction of Fluid Flow in Porous Media Using Physics-Informed Neural Networks,” by Muhammad M. Almajid, SPE, and Moataz O. Abu-Alsaud, SPE, Saudi Aramco. The paper has not been peer reviewed. The authors of the complete paper write that the realm of reservoir engineering is considered to belong to the small-data regime because of the complex physical systems involved and the inherent uncertainty of the problems. The objective of the paper is to present a physics-informed neural network (PINN) technique that is able to use information from the fluid-flow physics and observed data to model the Buckley-Leverett problem. Fractional-Flow Theory Through their study of fluid displacement in sands, Buckley and Leverett introduced the fractional-flow theory. The theory estimates the rate at which one fluid displaces another and, consequently, the change in fluid saturations. The theory has been applied to many fluid-flow processes in porous media such as waterflooding, carbonated waterflooding, alcohol flooding, miscible flooding, steam flooding, foam flooding, low-salinity waterflooding, and carbon sequestration. The authors use the fractional-flow theory to simulate the displacement of water-filled porous media by gas, a process detailed in the complete paper. Once the Buckley-Leverett solution is obtained, the saturation profile of gas or water can be plotted at different times. It is noteworthy that a gas viscosity of 0.2 cp in this case is used so that the shock is visually apparent in the Buckley-Leverett solution. Additionally, having this large of a gas viscosity is justified. For instance, foamed gas can develop apparent viscosities orders of magnitude larger than unfoamed gas. Fig. 1 plots the gas-saturation profile along the dimensionless distance. Obtaining the corresponding water-saturation profiles is trivial. The authors compare the analytical solution and the PINN prediction with respect to water saturation.

Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem

Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set.

Prediction of porous media fluid flow using physics informed neural networks

Due to the explosion of the digital age of data, deep learning applications for different physical sciences have gained momentum. In this paper, we implement a physics informed neural network (PINN) technique that incorporates information from the fluid flow physics as well as observed data to model the Buckley-Leverett problem. The classical problem of drainage of gas into a water-filled porous medium is used to validate our implementation. Several cases are tested that signify the importance of the coupling between observed data and physics-informed neural networks for different parameter space. Our results indicate that PINNs are capable of capturing the overall trend of the solution even without observed data but the resolution and accuracy of the solution are improved tremendously with observed data. Adding a small amount of diffusion to the PDE-constrained loss function improved the solution slightly only when observed data were used. Moreover, the PINN is used to solve the inverse problem and infer the most optimal multiphase flow parameters. The performance of the PINN is compared to that of an artificial neural network (ANN) without any physics. We show that the ANN performs comparably well to the PINN when the observed data used to train the ANN include times that span the early- and late-time behavior. As opposed to the PINN, the ANN is not able to predict the solution when only early-time saturation profiles are provided as observed data and extrapolation are needed.

A Class of Lagrangian–Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms

In this work, we develop an improved shock-capturing and high-resolution Lagrangian–Eulerian method for hyperbolic systems and balance laws. This is a new method to deal with discontinuous flux and complicated source terms having concentrations for a wide range of applications science and engineering, namely, 1D shallow-water equations, sedimentation processes, Geophysical flows in 2D, N-Wave models, and Riccati-type problems with forcing terms. We also include numerical simulations of a 1D two-phase flow model in porous media, with gravity and a nontrivial singular $$\delta $$ -source term representing an injection point. Moreover, we present approximate solutions for 2D nonlinear systems (Compressible Euler Flows and Shallow-Water Equations) for distinct benchmark configurations available in the literature aiming to present convincing and robust numerical results. In addition, for the linear advection model in 1D and for a smooth solution of the nonlinear Burgers’ problem, second order approximations were obtained. We also present a high-resolution approximation of the nonlinear non-convex Buckley–Leverett problem. Based on the work of A. Harten, we derive a convergent Lagrangian–Eulerian scheme that is total variation diminishing and second-order accurate, away from local extrema and discontinuous data. Additionally, using a suitable Kružkov’s entropy definition, introduced by K. H. Karlsen and J. D. Towers, we can verify that our improved Lagrangian–Eulerian scheme converges to the unique entropy solution for conservation laws with a discontinuous space-time dependent flux. A key hallmark of our method is the dynamic tracking forward of the no-flow curves, which are locally conservative and preserve the natural setting of weak entropic solutions related to hyperbolic problems that are not reversible systems in general. In the end, we have a general procedure to construct a class of Lagrangian–Eulerian schemes to deal with hyperbolic problems with or without forcing terms. The proposed scheme is free of Riemann problem solutions and no adaptive space-time discretizations are needed. The numerical experiments verify the efficiency and accuracy of our new Lagrangian–Eulerian method.

Fluid Displacement in a Dual-Permeability Medium with Local Capillary Equilibrium

The solution to the Buckley–Leverett problem in fractured porous media is investigated by applying the dual-porosity dual-permeability model. It is shown that the dual-permeability medium is equivalent to an upscaled homogenized medium at long injection times when local capillary equilibrium is established. Simple analytical relationships for the petrophysical properties and saturation functions of the equivalent medium are derived. The upscaled relative permeability and the fractional flow function can exhibit kinks, resulting in more complicated solutions to the Buckley–Leverett problem as compared to the classical case. It is found that immiscible displacement in fractured porous media leads to the appearance of saturation profiles with several displacement fronts and rarefaction waves. Up to four solution types are possible, which are constrained in a solution map constructed in the space of the dimensionless parameters of the problem. It is shown that, for particular parameters, the displacing fluid moves faster through the matrix than through the fractures.

An engineering approach to study the effect of saturation-dependent capillary diffusion on radial Buckley-Leverett flow

1D water oil displacement in porous media is usually described by the Buckley-Leverett equation or the Rapoport-Leas equation when capillary diffusion is included. The rectilinear geometry is not representative for near well oil displacement problems. It is therefore of interest to describe the radially symmetric Buckley-Leverett or Rapoport-Leas equation in cylindrical geometry (radial Buckley-Leverett problem). We can show that under appropriate conditions, one can apply a similarity transformation (r,t) rightarrow eta = r^{2}/(2t) that reduces the PDE in radial geometry to an ODE, even when capillary diffusion is included (as opposed to the situation in the rectilinear geometry (Yortsos, Y.C. (Phys. Fluids 30(10),2928–2935 1987)). We consider two cases (1) where the capillary diffusion is independent of the saturation and (2) where the capillary diffusion is dependent on the saturation. It turns out that the solution with a constant capillary diffusion coefficient is fundamentally different from the solution with saturation-dependent capillary diffusion. Our analytical approach allows us to observe the following conspicuous difference in the behavior of the dispersed front, where we obtain a smoothly dispersed front in the constant diffusion case and a power-law behavior around the front for a saturation-dependent capillary diffusion. We compare the numerical solution of the initial value problem for the case of saturation-dependent capillary diffusion obtained with a finite element software package to a partially analytical solution of the problem in terms of the similarity variable η.

Effective Asymptotic Model of Two-Phase Flow through Fractured-Porous Media

The solution of the Buckley-Leverett problem classical for the theory of flow through a porous medium, generalized to the case of two-phase flows in fractured-porous media, is considered. In this case, immiscible displacement of fluid in the porous medium is complicated by the absence of local capillary equilibrium between pore spaces of different scales and in the generic case the solution of problem is not self-similar. The flow through a porous medium is considered in the limiting case of large time scales when capillary equilibrium is established and the flow parameter distributions, as shown in the study, tend to self-similar asymptotics. For the effective ordinary porous medium the average equations of equilibrium flow through the porous medium which describe these asymptotics are obtained

INSIM-FT in three-dimensions with gravity

For situations where reservoir management based on developing and running a full-scale reservoir simulation model is not plausible, data-driven models may provide an attractive alternative. Recently, we developed a simple data-driven model, INSIM-FT for reservoir flow and transport as a replacement for a reservoir simulator. However, INSIM-FT only considers two-dimensional flow in a reservoir that contains only vertical wells. In this work, we develop a new data-driven model that extends the interwell numerical simulation model with front-tracking (INSIM-FT) from single-layer reservoirs to full three-dimensional (3D) multi-layer reservoirs. The new model, which is referred to as INSIM-FT-3D, can be used for history matching and reservoir performance predictions for a three-dimensional reservoir under waterflooding. The novelty of the new approach includes: (1) INSIM-FT-3D replaces the original Riemann solver in INSIM-FT by a new Riemann solver based on a convex-hull method that enables the solution of the Buckley–Leverett problem with gravity; (2) unlike the original INSIM-FT model, which assumes all wells are vertical, the INSIM-FT-3D model allows for the inclusion of wells with arbitrary trajectories with multiple perforations; (3) INSIM-FT-3D applies Mitchell's best-candidate algorithm to automatically generate the imaginary wells that are evenly distributed in the reservoir given a set of prefixed actual well nodes and (4) INSIM-FT-3D utilizes our own modification of Delaunay triangulation to build the 3D connection map necessary to use the general INSIM-FT-3D formulation. History matching of the INSIM-FT-3D model parameters is performed using the ensemble-smoother with multiple data assimilations. The model parameters for history matching include the connection-based parameters, the parameters defining the power-law relative permeabilities and the parameters defining the well indices for the perforated zones of a well with multiple perforated well segments.The reliability of the analytical water-saturation solution obtained with INSIM-FT-3D is validated by comparison with corresponding results from an Eclipse [47] model with very fine grids and small time steps. After illustrating the utility and reliability of the method for a three-dimensional synthetic problem, the method is shown to give reasonable history-matching and prediction results for a field example. At last, the proposed method is tested for Brugge field, a field-scale synthetic reservoir simulation model.

Application of carbon dioxide as working fluid in geothermal development considering a complex fractured system

The advantages of using supercritical carbon dioxide as working fluid in geothermal development (potential carbon dioxide geological storage and relatively high mobility) have attracted considerable interest in the simulation of a carbon dioxide-based enhanced geothermal system. This study proposed a numerical model based on the three-dimensional unified pipe-network method to simulate the two-phase flow and thermal-hydraulic coupled process in a carbon dioxide-based enhanced geothermal system considering complex fracture networks. The van Genuchten capillary model and relative permeability model are adopted to characterize the displacement process in both the fractures and the rock matrix. A pipe equivalent technique is employed to discretize the governing equations for the two-phase flow and the heat transmission with the local thermal non-equilibrium concept. Two-phase fluid transfer at material interfaces is solved based on a pipe superposition principle in a pipe-network system. Discretized forms are solved by a sequential implicit time scheme alleviating the Courant-Friedrichs-Lewy condition in fracture networks. The proposed model is verified against analytical results for the Buckley-Leverett problem. Saturation evolution curves at different time steps converge to the analytical solutions as the node number increases. Sensitivity analyses using a doublet system horizontally embedded with one large fracture indicate that the capillary contrast between the rock matrix and the fractures and the injection flow rate pose impacts on the produced carbon dioxide saturation and the outlet fluid temperature drawdown. The simulation of circulating processes considering randomly generated fracture networks demonstrates that the more promising potential of heat extraction can be achieved using the pure carbon dioxide as the working fluid instead of cooled water. Analyzing the effects of the initial reservoir carbon dioxide saturation shows that increasing the reservoir carbon dioxide saturation before heat excavation can improve the heat production. Both increase the matrix permeability and the fracture aperture enhances the performance of the heat extraction and the carbon dioxide sequestration.

Flow and displacement of waxy crude oils in a homogenous porous medium: A numerical study

In the present study, the displacement of waxy crude oils is numerically/theoretically investigated in the water-flooding operation. The oil was assumed to obey the Houska model―a robust thixotropic fluid model which is often realized to well describe the rheology of waxy oils in different parts of the world. Based on the concept of effective viscosity, a modified version of the Darcy's law was developed for this particular fluid model in order to describe its flow through a homogenous porous medium. Use was made of numerical and theoretical methods to study the displacement of Houska fluid by water in two benchmark problems: (i) the Buckley–Leverett problem, and (ii) the five-spot problem. It was found that the yield stress of the Houska fluid being variable (i.e., shear- and time-dependent) has a retarding effect on the water breakthrough phenomenon. The breakdown-to-rebuild ratio in the Houska model was shown to play a key role in the water breakthrough phenomenon provided that it is very large. At this extreme, however, the effect was attributed mostly to the shear-thinning behavior of the Houska fluid rather than its thixotropic behavior. In fact, at sufficiently low breakdown-to-rebuild ratios (i.e., when fluid's thixotropy becomes progressively more important) it had no significant effect on the water breakthrough phenomenon. Therefore, it is concluded that in competition with shear-thinning, the thixotropic behavior of Houska fluid plays a secondary role, if any, in the water-flooding operation.

An efficient discontinuous Galerkin method for two-phase flow modeling by conservative velocity projection

Purpose – The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection. Design/methodology/approach – The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations. Findings – The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements. Research limitations/implications – The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible. Practical implications – The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media. Originality/value – The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

Characterization of CO2 Injection into Deep Saline Aquifers Using Two-Phase Darcy-Forchheimer Flow

AbstractIn this study the generalized Darcy-Forchheimer model is used to characterize two-phase fluid flow where spatial flow characteristics may transition between Darcy and Forchheimer flow behavior. The local transition between the two flow regimes is characterized using the Forchheimer number as the criterion. A three-dimensional numerical model is developed that utilizes a control-volume method to simulate two-phase inertial, immiscible, and incompressible flow in a nondeformable homogeneous porous medium. The numerical model is validated by comparing its results with those obtained using a semianalytical solution of the Buckley-Leverett problem. The critical Forchheimer number is characterized using experimental data and is used to transition the local flow domain between Darcy and Forchheimer flow regions for both single-phase and multiphase analysis. The saturation-dependent distribution of the critical Forchheimer number is then used to analyze the Darcy and Forchheimer flow regions to provide a ...

A control volume finite element method for adaptive mesh simulation of flow in heap leaching

Accurate determination of fluid flow within heap leaching is crucial for understanding and improving performance of the process. Numerical methods have the potential to assist by modelling the process and studying the transport phenomena within the porous medium. This paper presents an adaptive mesh numerical scheme to solve for unsaturated incompressible flow in porous media with applications to heap leaching. The governing equations are Darcy’s law and the conservation of mass. An implicit pressure explicit saturation method is used to decouple the pressure and saturation equations. Pressure is discretized using a control volume (CV) finite element method, while for saturation a node-centred CV method is employed. The scheme is equipped with dynamic anisotropic mesh adaptivity to update the mesh resolution as the leaching solution infiltrates through the heap. This allows for high-fidelity modelling of multiscale features within the flow. The method is verified against the Buckley–Leverett problem where a quasi-analytical solution is available. It is applied for two-phase flow of air and leaching solution in a simplified two-dimensional heap geometry. We compare the accuracy and CPU efficiency of an adaptive mesh against a static mesh and demonstrate the potential to achieve high spatial accuracy at low computational cost through the use of anisotropic mesh adaptivity.

Gravity Effect on Two-Phase Immiscible Flows in Communicating Layered Reservoirs

An upscaling method is developed for two-phase immiscible incompressible flows in layered reservoirs with good communication between the layers. It takes the effect of gravity into consideration. Waterflooding of petroleum reservoirs is used as a basic example for application of this method. An asymptotic analysis is applied to a system of 2D flow equations for incompressible fluids at high-anisotropy ratios, but low to moderate gravity ratios, which corresponds to the most often found reservoir conditions. The 2D Buckley–Leverett problem is reduced to a system of 1D parabolic equations in a layered reservoir. For low-gravity ratios, it can further be reduced to a system of hyperbolic equations. The number of the 1D equations in the system is equal to the number of layers in the reservoir. The method is tested on different examples of displacement in a layer-cake reservoir. Different combinations of gravity-viscous and anisotropy ratios are tested. Solutions by our method are compared with the results of 2D simulations carried out by the COMSOL solver. The results are comparable, especially if the layers of the reservoirs are further subdivided into sublayers, in order to account better for gravity segregation. The effects of gravity are analyzed.

Numerical study on two-phase flow through fractured porous media

Based on the flux equivalent principle of a single fracture, the discrete fracture concept was developed, in which the macroscopic fractures are explicitly described as (n−1) dimensional geometry element. On the fundamental of this simplification, the discrete-fractured model was developed which is suitable for all types of fractured porous media. The principle of discrete-fractured model was introduced in detail, and the general mathematical model was expressed subsequently. The fully coupling discrete-fractured mathematical model was implemented using Galerkin finite element method. The validity and accuracy of the model were shown through the Buckley-Leverett problem in a single fracture. Then the discrete-fractured model was applied to the two different type fractured porous media. The effect of fractures on the water flooding in fractured reservoirs was investigated. The numerical results showed that the fractures made the porous media more heterogeneous and anisotropic, and that the orientation, size, type of fracture and the connectivity of fractures network have important impacts on the two-phase flow.

Generalized Buckley-Leverett problem

Analysis of the effect produced by thermal and gas treatment on oilfield reservoirs of the Bazhenov formation involves the dynamics of a porous saturated medium with a multiphase multicomponent fluid, taking into account the chemical and physicochemical transformations under thermal and gas injection. This work is a qualitative investigation into such processes based on a simplified partial model allowing analytical study. The process of hot water injection into hydrocarbon-saturated reservoirs is considered. The physical statement of the problem is the generalization of the BuckleyLeverett problem additionally taking into account mass and heat sources. The purpose is to study the mechanism of heat and mass transfer in a porous medium with allowance for chemical and physicochemical transformations. By way of example, an analytical solution of the problem is considered using a specific algebraic expression for the Buckley-Leverett function. Our research makes it possible to identify some qualitative features of the general model and to draw significant practical conclusions. The results can be applied to the investigation into a wide scope of processes of liquid-liquid displacement in a porous medium with two-phase and multicomponent fluid in the presence of heat and mass sources.

Upscaling of Two-Phase Immiscible Flows in Communicating Stratified Reservoirs

A semi-analytical method for upscaling two-phase immiscible flows in heterogeneous porous media is described. This method is developed for stratified reservoirs with perfect communication between layers (the case of vertical equilibrium), in a viscous dominant regime, where the effects of capillary forces and gravity may be neglected. The method is discussed on the example of its basic application: waterflooding in petroleum reservoirs. We apply asymptotic analysis to a system of two-dimensional (2D) mass conservation equations for incompressible fluids. For high anisotropy ratios, the pressure gradient in vertical direction may be set zero, which is the only assumption of our derivation. In this way, the 2D Buckley–Leverett problem may be reduced to a one-dimensional problem for a system of quasi-linear hyperbolic equations, of a number equal to the number of layers in the reservoir. They are solved numerically, based on an upstream finite difference algorithm. Self-similarity of the solution makes it possible to compute pseudofractional flow functions depending on the average saturation. The computer partial differential equation solver COMSOL is used for comparison of the complete 2D solutions with averaged 1D simulations. Cases of both discrete and continuous (log-normal) permeability distribution are studied. Generally, saturation profiles of the 1D model are only slightly different from the 2D simulation results. Recovery curves and fractional flow curves fit well. Calculations show that at a favorable mobility ratio (displaced to displacing phase) crossflow increases the recovery, while at an unfavorable mobility ratio, the effect is the opposite. Compared with the classical Hearn method, our method is more general and more precise, since it does not assume universal relative permeabilities and piston-like displacement, and it presumes non-zero exchange between layers. The method generalizes also the study of Yortsos (Transp Porous Media 18:107–129, 1995), taking into account in a more consistent way the interactions between the layers.