Buckley-Leverett Research Articles

First‐Order Empirical Interpolation Method for Real‐Time Solution of Parametric Time‐Dependent Nonlinear PDEs

ABSTRACTWe present a model reduction approach for the real‐time solution of time‐dependent nonlinear partial differential equations (PDEs) with parametric dependencies. A major challenge in constructing efficient and accurate reduced‐order models for nonlinear PDEs is the efficient treatment of nonlinear terms. We address this by unifying the implementation of hyperreduction methods to deal with nonlinear terms. Furthermore, we introduce a first‐order empirical interpolation method (EIM) to provide an efficient approximation of the nonlinear terms in time‐dependent PDEs. We demonstrate the effectiveness of our approach on the Allen–Cahn equation, which models phase separation, and the Buckley–Leverett equation, which describes two‐phase fluid flow in porous media. Numerical results highlight the accuracy, efficiency, and stability of the proposed method compared with both the Galerkin–Newton approach and hyper‐reduced models using the standard EIM.

Physics-Informed Generative Adversarial Network Solution to Buckley–Leverett Equation

Efficient and economical hydrocarbon extraction relies on a clear understanding of fluid flow dynamics in subsurface reservoirs, where multiphase flow in porous media poses complex modeling challenges. Traditional numerical methods for solving the governing partial differential equations (PDEs) provide effective solutions but struggle with the high computational demands required for accurately capturing fine-scale flow dynamics. In response, this study introduces a physics-informed generative adversarial network (GAN) framework for addressing the Buckley–Leverett (B-L) equation with non-convex flux functions. The proposed framework consists of two novel configurations: a Physics-Informed Generator GAN (PIG-GAN) and Dual-Informed GAN (DI-GAN), both of which are rigorously tested in forward and inverse problem settings for the B-L equation. We assess model performance under noisy data conditions to evaluate robustness. Our results demonstrate that these GAN-based models effectively capture the B-L shock front, enhancing predictive accuracy while embedding fluid flow equations to ensure model interpretability. This approach offers a significant advancement in modeling complex subsurface environments, providing an efficient alternative to traditional methods in fluid dynamics applications.

A third (fourth)-order computational scheme for 2D Burgers-type nonlinear parabolic PDEs on a nonuniformly spaced grid network

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

Stabilizing Discontinuous Galerkin Methods Using Dafermos’ Entropy Rate Criterion: II—Systems of Conservation Laws and Entropy Inequality Predictors

A novel approach for the stabilization of the Discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. First, estimates for the maximal possible entropy dissipation rate of a weak solution are derived. Second, families of conservative Hilbert–Schmidt operators are identified to dissipate entropy. Steering these operators using the bounds on the entropy dissipation results in high-order accurate shock-capturing DG schemes for the one-dimensional Euler equations, satisfying the entropy rate criterion and an entropy inequality. Other testcases include the one-dimensional Buckley–Leverett equation.

Performance study of variational quantum linear solver with an improved ansatz for reservoir flow equations

This paper studies the performance of the variational quantum linear solver (VQLS) with an improved ansatz for discretized reservoir flow equations for the first time. First, we introduce the two typical flow equations in reservoir simulation, namely, the diffusion equation for pressure and the convection-dominated Buckley–Leverett equation for water saturation, and their commonly used finite volume or finite difference-based discretized linear equations. Then, we propose an improved ansatz in VQLS to enhance the convergence and accuracy of VQLS and a strategy of adjusting grid order to reduce the complexity of the quantum circuit for preparing the quantum state corresponding to the coefficient vector of the discretized reservoir flow equations. Finally, we apply the modified VQLS to solve the discretized reservoir flow equations by employing the Xanadu's PennyLane open-source library. Four numerical examples are implemented, and the results show that VQLS can calculate reservoir flow equations with high accuracy, and the improved ansatz significantly outperforms the original one. Moreover, we study the effects of reservoir heterogeneity, the number of ansatz layers, the equation type, and the number of shots on the computational performance. Limited by the current computing capacity, the number of grids subject to the involved number of quantum bits in the implemented examples is small; we will further explore this quantum algorithm to practical examples that require a large number of quantum bits in the future.

A novel approach to solve hyperbolic Buckley-Leverett equation by using a transformer based physics informed neural network

A novel approach to solve hyperbolic Buckley-Leverett equation by using a transformer based physics informed neural network

A Novel Model for Forecasting Production Performance in Waterflood Oil Reservoirs

The importance of production performance forecasting in reservoir development and economic evaluation cannot be overstated. Previous models have shown deficiencies in accurately predicting production performance, necessitating the development of a new semianalytical model to enhance its application scope and prediction accuracy. This study proposes a novel semianalytical model based on the Buckley–Leverett (BL) equation and a newly proposed linear relationship between outlet water saturation and average water saturation, as well as Willhite’s formula of oil/water relative permeability. The results demonstrate the universality of this new model, as it can generate three equivalent log‐linear relations, including the previously proposed model. Sensitivity analysis confirms the applicability of the model in various reservoirs. In addition, both model and field case studies highlight the advantages of this technique in forecasting water cut and cumulative oil production, with an extensive application scope covering over 90% of the water cut range. A comparison of oil production prediction results from six different predictive methods reveals that the proposed semianalytical model exhibits the lowest error rate of −0.01%. Moreover, the semianalytical model can be utilized to directly solve for the approximate values of the exponents in Willhite’s oil/water relative permeability equations. In summary, this novel semianalytical forecasting model demonstrates a robust ability to forecast water cut, cumulative oil production, and recovery efficiency.

Numerical Solutions of the Classical and Modified Buckley-Leverett Equations Applied to Two-Phase Fluid Flow

Numerical Solutions of the Classical and Modified Buckley-Leverett Equations Applied to Two-Phase Fluid Flow

Development of oil extraction screening methodology taking into account innovative methods using the example of the Ukrainian field

The object of research in the paper is the process of fluid transfer through the pore space of the reservoir rock. The traditional method of estimating oil recovery by flooding has a large number of uncertainties. In this study, to address limitations of the current approach to determining oil production, let’s introduce a systematic algorithm aimed at enhancing result precision. The methodology for calculating the oil recovery coefficient for determining the amount of oil that can be extracted by flooding is presented. In this work, the step-by-step process of determining the oil recovery coefficient was analytically established, which achieves a certain degree of accuracy due to the inclusion of a number of methods of calculation of scientists from different countries of the world. In particular, the lithofacies distribution of the reservoir using the kriging method, the use of a representative elementary volume (REV) to increase the accuracy of determining the irreducible water saturation of each facies, and the use of the Buckley-Leverett equation in the calculation of the oil recovery coefficient are proposed. The number of facies (sandstone, argillaceous sandstone, siltstone) was determined on the example of the B-16n horizon of the «Ukrainian deposit» and the oil recovery coefficients were calculated for each separately (0.53, 0.47, 0.29). Further determination of the average oil recovery coefficient is described in the researched and requires close integration of the obtained data in three-dimensional space, as it allows to calculate the fraction of facies content in the reservoir volume. The use of the proposed action algorithm will help to build a more reliable three-dimensional hydrodynamic model, will lead to a much lower degree of uncertainty of reservoir properties, and in particular irreducible water saturation, as well as more accurate distribution of lithological properties using kriging. Also, this methodology for calculating the oil recovery coefficient involves the use of the Buckley-Leverett equation and fractional flow curves, the data of which are based on relative permeabilities and depend on the irreducible water saturation determined in the laboratory for each lithofacies. These techniques justify the collection of additional core material, the importance of lithofacies dismemberment of the formation and are closely integrated in the three-dimensional space, which makes it possible to simulate the existing processes, reproduce the proposed methodology and perform the forecast.

Analytical solution for chemical fluid injection into linear heterogeneous porous media based on the method of characteristics

Analytical solution for chemical fluid injection into linear heterogeneous porous media based on the method of characteristics

Physics-informed neural networks with adaptive localized artificial viscosity

Physics-informed neural networks with adaptive localized artificial viscosity

Physics-informed machine learning for solving partial differential equations in porous media

Physical phenomenon in nature is generally simulated by partial differential equations. Among different sorts of partial differential equations, the problem of two-phase flow in porous media has been paid intense attention. As a promising direction, physics-informed neural networks shed new light on the solution of partial differential equations. However, current physics-informed neural networks’ ability to learn partial differential equations relies on adding artificial diffusion or using prior knowledge to increase the number of training points along the shock trajectory, or adaptive activation functions. To address these issues, this study proposes a physics-informed neural network with long short-term memory and attention mechanism, an ingenious method to solve the Buckley-Leverett partial differential equations representing two-phase flow in porous media. The designed network structure overcomes the dependency on artificial diffusion terms and enhances the importance of shallow features. The experimental results show that the proposed method is in good agreement with analytical solutions. Accurate approximations are shown even when encountering shock points in saturated fields of porous media. Furthermore, experiments show our innovative method outperforms existing traditional physics-informed machine learning approaches. Cited as: Shan, L., Liu, C., Liu, Y., Tu, Y., Dong, L., Hei, X. Physics-informed machine learning for solving partial differential equations in porous media. Advances in Geo-Energy Research, 2023, 8(1): 37-44. https://doi.org/10.46690/ager.2023.04.04

Fractional-order integral generalization of the Rapoport–Leas equation

A one-dimensional time-fractional integral model for a two-phase flow through a porous medium with power-law memory in the presence of capillary forces is considered. Based on the time-fractional integral generalization of Darcy’s law and the classical mass balance equations for phases, neglecting gravity forces, a time-fractional integral generalization of the Rapoport–Leas equation is derived. It is shown that in the limiting case of constant capillary pressure, the obtained equation coincides with the classical Buckley–Leverett equation. An example of initial boundary value problem for the derived fractional integral equation is given for the case of a finite reservoir. Unlike the classical Rapoport–Leas equation, its time-fractional integral analogue does not have a traveling wave solution in general case. It is due to the fact that the fractional integral with a finite lower limit is not invariant with respect to translations in time. However, for the model problem of two-phase displacement in an unbounded reservoir, considered as a discontinuity breakup problem, the full memory approximation is valid. In this case the obtained equation has a traveling wave solution. In this approximation, relations for the main characteristics of the saturation jump on the shock are obtained. In particular, it is shown that the speed of the shock can be found from an equation that includes the time-fractional integral characteristic of the Leverett function. The behavior of the shock near its left and right boundaries is also asymptotically studied for different saturation levels at the boundaries. It has been proven that if the initial saturation exceeds the residual saturation of the displacing phase, then on the right boundary the saturation at the jump exponentially tends to the initial one. On the left boundary, the asymptotic behavior of the jump is also exponential, with the exception of one special case in which it becomes a fractional power law. For the equation in question, an approximate solution has been analytically constructed for the case of «weak» full memory, when the order of fractional integration is close to zero and can be considered as a small parameter.

Numerical Simulation Study on Temporary Well Shut-In Methods in the Development of Shale Oil Reservoirs

Field tests indicate that temporary well shut-ins may enhance oil recovery from a shale reservoir; however, there is currently no systematic research to specifically guide such detailed operations in the field, especially for the design of the shut-in scheme and multiple rounds of shut-ins. In this study, the applicability of well shut-in operations for shale oil reservoirs is studied, and a numerical model is built using the finite element method. In order to simulate the production in a shale oil reservoir, two separate modules (i.e., Darcy’s law and phase transport) were two-way coupled together. The established model was validated by comparing its results with the analytical Buckley–Leverett equation. In this paper, the geological background and parameters of a shale oil reservoir in Chang-7 Member (Chenghao, China) were used for the analyses. The simulation results show that temporary well shut-in during production can significantly affect well performance. Implementing well shut-in could decrease the initial oil rate while decreasing the oil decline rate, which is conducive to long-term production. After continuous production for 1000 days, the oil rate with 120 days shut-in was 9.85% larger than the case with no shut-in. Besides, an optimal shut-in time has been identified as 60 days under our modeling conditions. In addition, the potential of several rounds of well shut-in operations was also tested in this study; it is recommended that one or two rounds of shut-ins be performed during development. When two rounds of shut-ins are implemented, it is recommended that the second round shut-in be performed after 300 days of production. In summary, this study reveals the feasibility of temporary well shut-in operations in the development of a shale oil reservoir and provides quantitative guidance to optimize these development scenarios.

A Time-Continuous Embedding Method for Scalar Hyperbolic Conservation Laws on Manifolds

A time-continuous (tc-)embedding method is first proposed for solving nonlinear scalar hyperbolic conservation laws with discontinuous solutions (shocks and rarefaction waves) on codimension 1, connected, smooth, and closed manifolds (surface PDEs or SPDEs in {mathbb {R}}^2 and {mathbb {R}}^3). The new embedding method improves upon the classical closest point (cp-)embedding method, which requires re-establishments of the constant-along-normal (CAN-)property of the extension function at every time step, in terms of accuracy and efficiency, by incorporating the CAN-property analytically and explicitly in the embedding equation. The tc-embedding SPDEs are solved by the second-order nonlinear central finite volume scheme with a nonlinear minmod slope limiter in space, and the third-order total variation diminished Runge-Kutta scheme in time. An adaptive nonlinear essentially non-oscillatory polynomial interpolation is used to obtain the solution values at the ghost cells. Numerical results in solving the linear wave equation and the Burgers’ equation show that the proposed tc-embedding method has better accuracy, improved resolution, and reduced CPU times than the classical cp-embedding method. The Burgers’ equation, the traffic flow problem, and the Buckley-Leverett equation are solved to demonstrate the robust performance of the tc-embedding method in resolving fine-scale structures efficiently even in the presence of a shock and the essentially non-oscillatory capturing of shocks and rarefaction waves on simple and complex shaped one-dimensional manifolds. Burgers’ equation is also solved on the two-dimensional torus-shaped and spherical-shaped manifolds.

On one approach for the numerical solving of hyperbolic initial-boundary problems on an adaptive moving grids

On one approach for the numerical solving of hyperbolic initial-boundary problems on an adaptive moving grids

A neural network enhanced weighted essentially non-oscillatory method for nonlinear degenerate parabolic equations

In this paper, a new modification of the weighted essentially non-oscillatory (WENO) method for solving nonlinear degenerate parabolic equations is developed using deep learning techniques. To this end, the smoothing indicators of an existing WENO algorithm, which are responsible for measuring the discontinuity of a numerical solution, are modified. This is done in such a way that the consistency and convergence of our new WENO-DS (deep smoothness) method is preserved and can be theoretically proved. A convolutional neural network (CNN) is used and a novel and effective training procedure is presented. Furthermore, it is shown that the WENO-DS method can be easily applied to additional dimensions without the need to retrain the CNN. Our results are presented using benchmark examples of nonlinear degenerate parabolic equations, such as the equation of a porous medium with the Barenblatt solution, the Buckley–Leverett equation, and their extensions in two-dimensional space. It is shown that in our experiments, the new method outperforms the standard WENO method, reliably handles sharp interfaces, and provides good resolution of discontinuities.

Retracted] A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model

In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.

A fast method for solving time-dependent nonlinear convection diffusion problems

<abstract><p>In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study.</p></abstract>

Capillary hysteresis and gravity segregation in two phase flow through porous media

We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.