Duhamel's Principle Research Articles

Pressure transient behavior of layered commingled reservoir with vertical inhomogeneous closed boundary

The multilayered deposition is the main feature of the tidal flat reservoir in the western Sichuan Basin. The boundary distance among layers of the tidal flat reservoir is different. To understand the influence of vertical inhomogeneous closed boundary (VICB) on pressure transient behavior, an extended n-layer VICB reservoir model is presented.Firstly, the VICB is discretized into n boundary segments representing the characteristics of the individual layer, and the reservoir is divided into n horizontal layers accordingly. The pressure transient model of the n-layer VICB reservoir is established by the seepage equations and boundary conditions of the individual layer. Secondly, solving Bessel equations consisting of 2n boundary conditions by the Cramer's rule to obtain the bottom-hole pressure solution in Laplace space. The wellbore storage and skin effect are taken into account by Duhamel's principle in the Laplace domain. Lastly, the bottom-hole pressure solution in the real-time domain is obtained by the Stehfest numerical inversion algorithm. The comparison result with the bottom-hole pressure of the two-layer reservoir proves the reliability of the n-layer VICB reservoir model.In the n-layer VICB reservoir model, a new VICB radial flow regime is identified which easily mistaken as the outer radial flow of the composite reservoir model. The result of sensitivity analysis shows that the pressure transient behavior of the n-layer VICB reservoir has the “equivalent compound zone effect” and “equivalent boundary radii effect”. There is a quantitative relation between the distribution of VICB and the characteristics value of pressure derivative during the VICB radial flow regime stage.The “equivalent compound zone effect”, “equivalent boundary radii effect” and the quantitative relation provide a simple and fast analytical method for identifying the vertical distribution and distance range of VICB. Finally, a field case from the western Sichuan Basin multilayered gas reservoir is discussed in detail to validate the capabilities of our model.

Pointwise structure of a radiation hydrodynamic model in one‐dimension

In this paper, we study the nonlinear stability and the pointwise structure around a constant equilibrium for a radiation hydrodynamic model in 1‐dimension, in which the behavior of the fluid is described by a full Euler equation with certain radiation effect. It is well‐known that the classical solutions of the Euler equation in 1‐D may blow up in finite time for general initial data. The global existence of the solution in this paper means that the radiation effect stabilizes the system and prevents the formation of singularity when the initial data is small. To study the precise effect of the radiation in this model, we also treat the pointwise estimates of the solution for the original nonlinear problem by combining the Green's function for the linearized radiation hydrodynamic equations with the Duhamel's principle. The result in this paper shows that the pointwise structure of this model is similar to that of full Navier‐Stokes equations in 1‐D.

Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling.

We consider a formulation for the Hopf functional differential equation which governs statistical solutions of the Navier-Stokes equations. By introducing an exponential operator with a functional derivative, we recast the Hopf equation as an integro-differential functional equation by the Duhamel principle. On this basis we introduce a successive approximation to the Hopf equation. As an illustration we take the Burgers equation and carry out the approximations to the leading order. Scale invariance of the statistical Navier-Stokes equations in d dimensions is formulated and contrasted with that of the deterministic Navier-Stokes equations. For the statistical Navier-Stokes equations, critical scale invariance is achieved for the characteristic functional of the dth derivative of the vector potential in d dimensions. The deterministic equations corresponding to this choice of the dependent variable acquire the linear Fokker-Planck operator under dynamic scaling. In three dimensions it is the vorticity gradient that behaves like a fundamental solution (more precisely, source-type solution) of deterministic Navier-Stokes equations in the long-time limit. Physical applications of these ideas include study of a self-similar decaying profile of fluid flows. Moreover, we reveal typical physical properties in the late-stage evolution by combining statistical scale invariance and the source-type solution. This yields an asymptotic form of the Hopf functional in the long-time limit, improving the well-known Hopf-Titt solution. In particular, we present analyses for the Burgers equations to illustrate the main ideas and indicate a similar analysis for the Navier-Stokes equations.

Flowback fracture closure of multi-fractured horizontal wells in shale gas reservoirs

In multi-fractured horizontal wells (MFHW), fracture properties such as permeability and fracture half-length significantly deteriorate during production, which negatively affects gas production from shale reservoirs. Therefore, it is crucial to evaluate the temporal changes in fracture properties using production data. This paper presents two multiphase flowback models for gas and water phase under boundary dominated flow (BDF) condition by considering gas influx from matrix. In addition, a workflow is proposed to quantitatively evaluate hydraulic fracture closure and fracture properties using both water-phase and gas-phase flowback data. A material balance equation (MBE) is derived to obtain average pressure in the fracture by calculating gas influx from matrix under variable bottomhole pressure (BHP) condition based on Duhamel's principle. The flowback models, average pressure calculation method, and workflow are validated against numerical simulations and applied to an MFHW drilled in Marcellus Shale. The results show that the developed models are capable of predicting fracture properties, and MBE could accurately calculate average pressure in the fracture. Furthermore, the proposed workflow provides quantitative insights into the performance of stimulation jobs by accurately predicting fracture half-length, fracture permeability, and permeability modulus using flowback data. By reconciling flowback analysis with long-term production data analysis, a significant decrease in fracture properties due to fracture closure is observed in the Marcellus MFHW.

Global existence and optimal time-decay estimates of solutions to the generalized double dispersion equation on the framework of Besov spaces

We investigate the initial value problem for the generalized double dispersion equation in any dimensions. Inspired by [28] for the hyperbolic system of first order PDEs, we develop Littlewood-Paley pointwise energy estimates for the dissipative wave equation of high-order. Furthermore, with aid of the frequency-localization Duhamel principle, we establish the global existence and optimal decay estimates of solutions in spatially critical Besov spaces. Our results could hold true for any dimensions (n≥1). Indeed, the proofs are different in case of high dimensions and low dimensions owing to interpolation tricks.

The optimal time‐decay estimate of solutions to two‐fluid Euler‐Maxwell equations in the critical Besov space

AbstractWe obtain the optimal time‐decay rate of classical solutions to two‐fluid Euler‐Maxwell equations in , which is a remaining question in the framework of critical Besov space (see [1]). The system is of regularity‐loss, so it is difficult to get decay rates in the solution space. In this paper, the new estimate of ‐‐ type and something like “square formula of Duhamel principle” are mainly used. It is shown that in the critical Besov space, the solution decays to constant equilibrium at the rate with if and if .

Production Performance Evaluation of Multifractured Horizontal Wells in Shale Oil Reservoirs: An Analytical Method

Hydraulic fracturing stimulation has become a routine for the development of shale oil and gas reservoirs, which creates large volumes of fracturing networks by helping the hydrocarbon to transport quickly into the wellbore. However, the optimal fracture spacing distance and fracture conductivity are still unclear for the field practice, even though the technique has improved significantly over the last several years. In this work, an analytical method is proposed to address it. First, the analytical production rate for a single fracture is proposed, and then, we apply Duhamel principle to obtain the production rate of a horizontal well with multifractures. Based on this model, the flow regimes and essential affecting factors including fracture spacing, fracture conductivity, and skin factor are analyzed in this work. The optimal values and suggestion are provided based on the simulation results.

A fast compact exponential time differencing method for semilinear parabolic equations with Neumann boundary conditions

In this paper we propose a fast compact exponential time differencing method for solving a class of semilinear parabolic equations with Neumann boundary conditions. The model equation is first discretized in space by a fourth-order compact finite difference scheme with an appropriate treatment of the boundary condition, the resulting semi-discretized system is then diagonalized with fast Fourier transforms, and further expressed in a temporal integral formulation by the use of exponential integrators according to the Duhamel principle. The fully discrete scheme is finally obtained by using multistep interpolations for the nonlinear terms and exact evaluations of the underlying integrals. Some numerical experiments are performed to demonstrate the accuracy and efficiency of the proposed method.

A symplectic approximation with nonlinear stability and convergence analysis for efficiently solving semi-linear Klein–Gordon equations

It is noted that the geometric integration for nonlinear Hamiltonian PDEs has led to the development of numerical schemes which systematically incorporate qualitative features of the underlying problem into their structure. The symplectic approximation to nonlinear Hamiltonian PDEs should be useful when studying the geometric integration. However, it is also an important aspect to analyse the nonlinear stability and convergence when a fully discrete symplectic scheme is designed for nonlinear Hamiltonian PDEs. In this paper, we develop a symplectic approximation for efficiently solving semi-linear Klein–Gordon equations, which can be formulated as an abstract Hamiltonian system of second-order ordinary differential equation (ODE). To this end, we first analyse an extended Runge–Kutta–Nyström-type (RKN-type) approximation based on the operator-variation-of-constants formula (also known as the Duhamel Principle) for the abstract Hamiltonian system of second-order ODE. We then present the symplectic conditions for the approximation, and derive some practical symplectic approximation schemes for semi-linear Klein–Gordon equations. The most important is that we commence the nonlinear stability and convergence analysis for the symplectic approximation to semi-linear Klein–Gordon equations. The results of various numerical experiments, including Klein–Gordon equations in the nonrelativistic limit regime where the solution is highly oscillating in time, demonstrate the remarkable advantage and efficiency of the symplectic approximation schemes in comparison with existing numerical schemes in the literature.

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method.

An approximate semianalytical method for two-phase flow analysis of liquid-rich shale gas and tight light-oil wells

Two-phase flow of oil and gas is quite common in the reservoir during the depletion of liquid-rich shale gas and tight light-oil. Many analytical models proposed for multistage fractured horizontal wells completed in unconventional reservoirs are restricted to single phase flow problem. In this paper, an approximate semianalytical method is proposed to handle the complexities of phase change, pressure dependent PVT properties, two-phase flow behavior and complex fracture networks. The black oil model is used to model phase change and two-phase flow of oil and gas. The secondary fractures in the stimulated reservoir volume and the reservoir beyond the main fractures are considered in the physical model to make long-term production forecasts. The forecasted producing GOR is used to decrease the equations of the two-phase flow model, so only the flow equations of primary phase are used to develop the mathematic model. Then an approximate method is used to linearize the model and the analytical solution is obtained in Laplace space for constant pressure inner boundary problem. The Duhamel's principle is applied to correct the production rate under variable flowing pressure inner boundary, and a procedure is proposed for history matching of field production data and making forecasts. Finally, the semianalytical model is validated with two simulation examples, the average relative error is shown to be less than 10%. After that, two field examples are used to show its utility.

Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}_{+}^{n} $\end{document}</tex-math></inline-formula>

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in \begin{document}$ \mathbb{R}^n_+ $\end{document} is investigated. We describe the double mechanism of the hyperbolic effect and the parabolic effect using the explicit functions. With the absorbing and radiative boundary condition, we show that the Green's function for the half space linear problem can be described in terms of the fundamental solution for the Cauchy problem and the reflected fundamental solution coupled with a boundary operator. Using the Duhamel's principle, we see that due to the fast decay property of the Green's function and the high nonlinearity, the pointwise decaying rate for the nonlinear solution and extra time decaying rate for its first order derivative are obtained.

Investigation on the transient pressure response of water injector coupling the dynamic flow behaviors in the wellbore, waterflood-induced fracture and reservoir: Semi-analytical modeling and a field case

There is growing evidence showing that water injection may induce formation fracturing around injectors in tight reservoirs. Because waterflood-induced fractures (WIFs) are not strengthed by proppants, they close gradually during the field-testing period, which results in “fracture-closure-induced” flow rate, shrinking fracture length (SFL) and decreasing fracture conductivity (DFC). In this paper, we propose a novel semi-analytical model to characterize the bottom-hole pressure (BHP) behavior of water injectors by coupling the dynamic flow in the wellbore, WIF, and reservoir. Flows between reservoir and WIF are linked through a fracture-storage coefficient and fracture-face skin factor, while flows between WIF and wellbore are coupled via wellbore-storage coefficient and choked-fracture skin factor. Perturbation theory method is deployed to include the DFC effect, and Duhamel principle is invoked to characterize flow rate changes caused by wellbore and fracture storage effects. Results show that bi-storage effects can be identified as two unit slopes in the pressure derivative curve. In the absence of extra pressure drop between wellbore and WIF, i.e., choked-fracture skin equals to zero, a prolonged storage period with a considerably large storage coefficient can be obtained. In addition, we find that SFL could cause the variable fracture storage effect while DFC may lead to the upward of pressure derivative curve at late times. Finally, the model is successfully applied in the Changqing Oilfield to validate its reliability.

Decline Curve Analysis of Fractured Horizontal Wells Through Segmented Fracture Model

Nowadays, production performance evaluation of a multifractured horizontal well (MFHW) has attracted great attention. This paper presents a mathematical model of an MFHW with considering segmented fracture (SF) for better evaluation of fracture and reservoir properties. Each SF consists of two parts: fracture segment far from wellbore (FSFW) and fracture segment near to wellbore (FSNW) in segmented fracture model (SFM), which is different from fractures consists of only one segment in common fracture model (CFM). Employing the source function and Green's function, Newman's product method, Duhamel principle, Stehfest inversion algorithm, and Laplace transform, production solution of an MFHW can be obtained using SFM. Total production rate is mostly contributed from FSNW rather than FSFW in many cases; ignoring this phenomenon may lead to obvious erroneous in parameter interpretation. Thus, clear distinctions can be found between CFM and SFM on the compound type curves. By using decline curve analysis (DCA), the influences of sensitive parameters (e.g., dimensionless half-length, dimensionless production rate, conductivity, and distance between SF) on compound type curves are analyzed. The results of sensitivity analysis are benefit of parameter estimation during history matching.

The formulation and analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations

Abstract In this paper we focus on the analysis of energy-preserving schemes for solving high-dimensional nonlinear Klein–Gordon equations. A novel energy-preserving scheme is developed based on the discrete gradient method and the Duhamel principle. The local error, global convergence and nonlinear stability of the new scheme are analysed in detail. Numerical experiments are implemented to compare with existing numerical methods in the literature, and the numerical results show the remarkable efficiency of the new energy-preserving scheme presented in this paper.

A new method for production data analysis in shale gas reservoirs

For most shale gas wells, frequent shut-ins and nozzle size changing often occur, leading to significant discontinuities in production data. This study proposes a new production data analysis (PDA) method to address the abrupt change issue based on the virtual equivalent time. With the virtual equivalent time, the qualities of type curve matching and production data analysis can be improved when there are abrupt changes in production data. The mathematical model with a variable flow rate provides new definitions of normalized pseudopressure and pseudotime with consideration of the pressure dependent permeability for multi-fractured horizontal wells in shale gas reservoirs. Subsequently, the virtual equivalent time is calculated based on the average formation pressure to consider the abrupt changes of the production data, which also can reduce the time of superposition principle calculations. Duhamel's principle, Laplace transform and inversion, and Newman's method are employed to solve the PDA model, which is validated by analytical and numerical solutions. Then sensitivity analysis are performed on the dimensionless constrained axial modulus, which shows that the larger the dimensionless constrained axial modulus, the lower and later the curves. Finally, a field case study is carried out using the proposed method. The results show that by using the virtual equivalent time, the matching qualities are good, and the issue caused by abrupt changes is satisfactorily addressed. Therefore, it has great potential for estimating the formation parameters and predicting the well production performance more effectively and practically.

Green's function and pointwise estimate for a generalized Poisson‐Nernst‐Planck‐Navier‐Stokes model in dimension three

AbstractThe Cauchy's problem of the generalized Poisson‐Nernst‐Planck‐Navier‐Stokes model in dimension three is considered. First, after dividing the physical domain into two parts: finite Mach number region and outside finite Mach number region, we give pointwise estimates of the Green's function by using long‐wave short‐wave decomposition and weighted energy estimate method in each region separately. Then from Duhamel's principle and some estimates on the nonlinear interactions between different wave patterns, we give the pointwise estimates of the solution for the nonlinear problem –, which exhibit generalized Huygens' principle for mass density, macroscopic momentum, negative charge distribution and positive charge distribution. As a byproduct, we find that the decay rates in () for both negative charge distribution and positive charge distribution are faster than those for the mass density ρ and the macroscopic momentum. More importantly, we obtained both electrostatic potential and the difference between negative charge distribution and positive charge distribution have the exponential decay rate in ‐norm when .

A stability-improved efficient deconvolution algorithm based on B-splines by appending a nonlinear regularization

Previous deconvolution algorithms based on B-splines are much easier to be understood and programmed for academic researchers and engineers. However, due to the use of a linear regularization, their stability is weaker than that of the commonly used von Schroeter et al.’s deconvolution algorithm in which a nonlinear regularization is used; the linear regularization can make the deconvolution algorithms less tolerant to data errors. Good stability for the deconvolution algorithms is very important in order to make deconvolution as a viable tool for well-test analysis. In the paper, in order to improve the stability of the deconvolution algorithms based on B-splines, a nonlinear regularization by minimizing the curvature of pressure derivative response, as used in von Schroeter et al.’s algorithm, is appended instead of the linear regularization. And the corresponding nonlinear regularization equations are appropriately deduced. In particular, the improved algorithm is based on the Duhamel principle directly, and the complex transformation by the nonlinear z function, as used in von Schroeter et al.’s algorithm, is avoided; it does simplify the whole deconvolution process; moreover, the sensitivity matrix of an involved basic linear system from the measured pressure and rate data can also be solved directly by the piecewise analytical integration method, which can largely improve the deconvolution computation speed. Ultimately, in combination with the nonlinear regularization equations, a nonlinear least-squares problem is formulated for the stability-improved deconvolution algorithm based on B-splines. Besides, a constraint condition for tuning the parameter values of the B-spline base and an involved smooth factor is presented for restricting the nonlinear regularization process. Through a simulated case study, it is found that the nonlinear least-squares problem can be solved stably by the advanced Powell's Dog Leg method due to its great convergence ability and numerical stability; and the solution accuracy is also verified. Then the effects of the two parameters on the type curves of the deconvolution results are analyzed. And the effect of the error in the initial formation pressure on the type curves of the deconvolution results is also analyzed. Then a statement on how to perform the nonlinear regularization is presented specifically.Furthermore, through the study on two simulated cases with added data errors and an actual case, it is demonstrated that when the nonlinear regularization is appended, the stability of the deconvolution algorithm based on B-splines can be largely improved for mitigating the effect of data errors; besides, the stability-improved algorithm based on B-splines even exhibits higher stability than von Schroeter et al.’s algorithm that takes the same nonlinear regularization method, and the reason can be attributed to the superior properties of the representation of the wellbore pressure derivative (to be deconvolved) by B-spline functions in the numerical stability of computations and the inherent smoothness. Through the test of some simulated cases, it is also concluded that the stability-improved algorithm based on B-splines by appending the nonlinear regularization still has a high-level computation speed, which is nearly twenty times more than that of von Schroeter et al.’s algorithm. It can be attributed to the more undetermined coefficients and the computational complexity resulted from the z-function transformation in the formulation of von Schroeter et al.’s algorithm.

Long time behavior for the visco-elastic damped wave equation in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^n_+$\end{document}</tex-math></inline-formula> and the boundary effect

In this paper, we investigate the existence and long time behavior of the solution for the nonlinear visco-elastic damped wave equation in \begin{document}$\mathbb{R}^n_+$\end{document} , provided that the initial data is sufficiently small. It is shown that for the long time, one can use the convected heat kernel to describe the hyperbolic wave transport structure and damped diffusive mechanism. The Green's function for the linear initial boundary value problem can be described in terms of the fundamental solution (for the full space problem) and reflected fundamental solution coupled with the boundary operator. Using the Duhamel's principle, we get the \begin{document}$ L^p $\end{document} decaying rate for the nonlinear solution \begin{document}$\partial_{{\bf x}}^{\alpha}u$\end{document} for \begin{document}$|\alpha|\le 1$\end{document} .

Control for well-posedness about a class of non-Newtonian incompressible porous medium fluid equations

Considering the non-Newtonian fluid equation of incompressible porous media, using the properties of operator semigroup and measure space and the principle of squeezed image, Fourier analysis and a priori estimate in the measurement space are used to discuss the non-compressible porous media, the properness of the solution of the equation, its gradual behavior and its topological properties. Through the diffusion regularization method and the compressed limit compact method, we study the overall decay rate of the solution of the equation in a certain space when the initial value is sufficient. The decay estimation of the solution of the incompressible seepage equation is obtained, and the asymptotic behavior of the solution is obtained by using the double regularization model and the Duhamel principle.