Time-dependent Injection Rates Research Articles

Optimal control of viscous fingering

The paper considers the problem of optimally filling a Hele-Shaw cell. The system is subject to viscous fingering effect. It is shown that, despite the threshold terms appearing on the right-hand side of the governing equations, the dynamics can be rewritten using several prime integrals. This allows reforming optimal control problems for the Fourier modes describing the fluid interface into smooth optimization problems, in the sense of Gâteaux derivative. Some numerical experiments illustrate the advantages of using the optimal solutions obtained using this reformulation instead of the currently known time-dependent injection rates.

Experimental and analytical approaches toward controlling viscous fingering at the interface of two immiscible fluids using the optimal constant piecewise injection rate

Viscous fingering (VF) at the interface of two immiscible fluids can have disruptive effects in many technological applications. Therefore finding methods for reducing it with an effective and simple approach plays a significant role. A proposed solution that restrains the expansion of such complex morphologies, is the time-dependent injection rate. This proposed solution is often done either analytically or by numerical simulations which need to be experimentally proven. In this study, we perform multiple experiments of low viscosity fluid injection into high viscosity fluid in order to study the formation of VF and find injection rates or protocols that reduce VF at the onset of the nonlinear regime and in the fully nonlinear regime by applying several constant piecewise injection rates. Moreover, analytically and numerically studies are done, in which we derive the optimal parameters in piecewise injection profiles to limit the VF and the tip-splitting effect. This optimal constant piecewise injection rate includes specified number of stages as well as the injection time and its rate in each stage. The obtained results revealed that depending on being at the beginning of the nonlinear regime or in the fully nonlinear regime, the optimal constant piecewise injection rate can considerably reduce the length of fingers and the interfacial fingering perturbations. Empirical results show similar trend in terms of instability reduction by applying the same optimal number of piecewise injection rate stages. In addition, evaluating the difference between this piecewise results and the results of linear injection rates, which we have done before, reveals that employing linear injection rates is more effective in controlling the instability in lower injection rates, while for higher injection rates, it is required to utilize a piecewise injection rate.

Electrically controlled self-similar evolution of viscous fingering patterns

A controlling protocol for the traditional viscous fingering instability is presented. By coupling time-dependent injection rates with time-varying electric currents, it is shown via boundary integral simulations that the strategy allows control of the self-similar regime (delay, promote, or suppress) and the relative finger instability size without altering the system's physical parameters.

Pseudo-three-dimensional simulations of a vertical hydraulic fracture with time-dependent injection rates

This study reports a hydraulic fracture pseudo-three-dimensional model with time-dependent injection rates in elastic layered reservoirs. The adopted solution method can capture the crack tip asymptotic behaviors. The model is validated against the PKN analytical solutions of storage and leak-off-dominated regimes. The results show good agreement between the P3D model and its corresponding PKN solutions. The validated model is then employed to conduct a series of simulations to gain insight into the variable injection rates, fracture toughness, and leak-off coefficient on the fracture propagation behavior. Results show that increasing the variable injection rate reduces the lateral growth and has slight effects on the width and vertical growth. The fracture toughness considerably increases the crack width and length, whereas the leak-off coefficient has the most impact on limiting the lateral extension.

Controlling fingering instabilities in Hele-Shaw flows in the presence of wetting film effects.

In this paper, the interfacial motion between two immiscible viscous fluids in the confined geometry of a Hele-Shaw cell is studied. We consider the influence of a thin wetting film trailing behind the displaced fluid, which dynamically affects the pressure drop at the fluid-fluid interface by introducing a nonlinear dependence on the interfacial velocity. In this framework, two cases of interest are analyzed: The injection-driven flow (expanding evolution), and the lifting plate flow (shrinking evolution). In particular, we investigate the possibility of controlling the development of fingering instabilities in these two different Hele-Shaw setups when wetting effects are taken into account. By employing linear stability theory, we find the proper time-dependent injection rate Q(t) and the time-dependent lifting speed b[over ̇](t) required to control the number of emerging fingers during the expanding and shrinking evolution, respectively. Our results indicate that the consideration of wetting leads to an increase in the magnitude of Q(t) [and b[over ̇](t)] in comparison to the nonwetting strategy. Moreover, a spectrally accurate boundary integral approach is utilized to examine the validity and effectiveness of the controlling protocols at the fully nonlinear regime of the dynamics and confirms that the proposed injection and lifting schemes are feasible strategies to prescribe the morphologies of the resulting patterns in the presence of the wetting film.

Control of viscous fingering through variable injection rates and time-dependent viscosity fluids: Beyond the linear regime.

During the past few years, researchers have been proposing time-dependent injection strategies for stabilizing or manipulating the development of viscous fingering instabilities in radial Hele-Shaw cells. Most of these studies investigate the displacement of Newtonian fluids and are entirely based on linear stability analyses. In this work, linear stability theory and variational calculus are used to determine closed-form expressions for the proper time-dependent injection rates Q(t) required to either minimize the interface disturbances or to control the number of emerging fingers. However, this is done by considering that the displacing fluid is non-Newtonian and has a time-varying viscosity. Moreover, a perturbative third-order mode-coupling approach is employed to examine the validity and effectiveness of the controlling protocols dictated by these Q(t) beyond the linear regime and at the onset of nonlinearities.

Optimal control of viscous fingers in radial Hele-Shaw cell

This work considers the displacement of a resident fluid in a radial Hele-Shaw cell by an invading low viscosity fluid driven by time-dependent injection rates. Finger formation in the circular interface through a sequence of bifurcations may be minimized by the optimal choice of a time-dependent injection rate. Approximate solutions of fluid equations predict how bifurcations can be suppressed or strongly reduced. Based on a computational fluid-dynamic approach, the magnitude of the fluctuations of invading interface is numerically evaluated, leading to the identification of the optimal parameter choice for any injection rate family. The combination of two time-dependent injection rates is investigated, where one decreases as a power-law and the second increases linearly. Results for a well tuned change between the two regimes reduce the injection time as compared to those based on a single rate whole process, with similar or reduced effects on interface fluctuations.

Chemistry of ion injection in supernova remnant shocks: hybrid simulations in the light of He/C/O data from AMS-02

The high precision spectrometry of galactic cosmic rays (CR), e.g. PAMELA experiment [1], accurately determined an approximately 0.1 difference between the rigidity spectral indices of protons and He ions. Similar deviations have been indicated earlier by other experiments [2] confirmed by the recent high-fidelity AMS-02 measurements [3]. These findings widen the window into a complicated selection process whereby collisionless shocks, such as supernova remnant (SNR) blast waves, extract different chemical elements from an interstellar matter mix. The similarity of He/p, C/p, and O/p rigidity spectra demonstrated by AMS-02 has provided new evidence that injection is a mass-to-charge (A/Z) dependent process. We investigate the particle injection into the diffusive shock acceleration using one- and two-dimensional self-consistent hybrid simulations. Our 1D simulations prove that an SNR shock can modify the chemical composition of accelerated CRs by preferentially extracting them from a homogeneous background plasma without additional, largely untestable assumptions. Our results confirm the earlier theoretical predictions of how the efficiency of injection depends on the shock Mach number MA. They show that selection rate of different ion species increases with A/Z, saturates, and peaks as a function of MA. The 2D simulations also evidence a deviation from the linear selection rate vs A/Z trend, pointing towards a saturation for higher A/Z. The integrated SNR rigidity spectrum for proton-to-helium flux ratio, obtained by the convolution of the time-dependent injection rates of protons and He ions with a decreasing shock strength over the active lives of SNRs, compares well with the AMS-02 and PAMELA data. The numerical results [4] correctly predict the decrease in p/He flux ratio with increasing rigidity at exactly the rate measured in the experiments for ℜ > 10 GV. Only at lower rigidities, ℜ < 10 GV, the difference between the data and our predictions becomes noticeable. Whether this deviation comes from the propagation through the interstellar medium or solar modulation, remains unclear. Except for this uncertainty, the suggested mechanism for A/Z-dependence of the injection fully explains the measured p/He ratio.

Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid–viscous fluid interface. Surprisingly, a converging or diverging taper of the plates leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Our simulations compare well with previous experimental results for various injection rates and geometric configurations. We demonstrate how the number of non-splitting fingers agrees with that predicted from linear stability theory up to some finger number; for larger values of our control parameter, the fully nonlinear dynamics of the problem leads to slightly fewer fingers than this linear prediction.

Acceleration of Cosmic Rays in Supernova Shocks: Elemental Selectivity of the Injection Mechanism

Abstract Precise measurements of galactic cosmic rays revealed a significant difference between the rigidity spectral indices of protons and helium ions. This finding is a notable contrast to the commonly accepted theoretical prediction that supernova remnant (SNR) shocks accelerate protons and helium ions with the same rigidity alike. Most of the earlier explanations for the “paradox” appealed to SNR environmental factors, such as inhomogeneous p/He mixes in the shock upstream medium, variable ionization states of He, or a multi-SNR origin of the observed spectra. The newest observations, however, are in tension with most past models. In this paper, we show by self-consistent hybrid simulations that such special conditions are not vital for explaining the cosmic-ray rigidity spectra. In particular, our simulations prove that an SNR shock can modify the chemical composition of accelerated cosmic rays by preferentially extracting them from a homogeneous background plasma without additional, largely untestable assumptions. Our results confirm the earlier theoretical predictions of how the efficiency of injection depends on the shock Mach number M. Its increase with the charge-to-mass ratio saturates at a level that grows with M. We have convolved the time-dependent injection rates of protons and helium ions, obtained from the simulations, with a decreasing shock strength over the active lives of SNRs. The integrated SNR rigidity spectrum for p/He ratio compares well with the AMS-02 and PAMELA data.

Control of viscous instability by variation of injection rate in a fluid with time-dependent rheology

Using variational calculus, we investigate the time-dependent injection rate that minimises the growth of the Saffman–Taylor instability when a finite volume of fluid is injected in a finite time, $t_{f}$, into a Hele-Shaw cell. We first consider a planar interface, and show that, with a constant viscosity ratio, the constant injection rate is optimal. When the viscosity of the displacing fluid, $\unicode[STIX]{x1D707}_{1}(t)$, gradually increases over time, as may occur with a slowly gelling polymer solution, the optimal injection rate, $U^{\ast }(t)$, involves a gradual increase in the flow rate with time. This leads to a smaller initial value of flow rate than the constant injection rate, finishing with a larger value. Such optimisation can lead to a substantial suppression of the instability as compared to the constant injection case if the characteristic gelling time is comparable to $t_{f}$. In contrast, for either relatively slow or fast gelling, there is much less benefit in selecting the optimal injection rate, $U^{\ast }(t)$, as compared to the constant injection rate. In the case of a constant injection rate from a point source, $Q$, then with a constant viscosity ratio the fastest-growing perturbation on the radially spreading front involves axisymmetric modes whose wavenumber increases with time. Approximating the discrete azimuthal modes by a continuous distribution, we find the injection rate that minimises growth, $Q^{\ast }(t)$. We find that there is a critical time for injection, $t_{f}^{\dagger }$, such that if $t_{f}&gt;t_{f}^{\dagger }$ then $Q^{\ast }(t)$ can be chosen so that the interface is always stable. This critical time emerges from the case with an injection rate given by $Q^{\ast }\sim t^{-1/3}$. As the total injection time is reduced to values $t_{f}&lt;t_{f}^{\dagger }$, the system becomes progressively more unstable, and the optimal injection rate for an idealised continuous distribution of azimuthal modes asymptotes to a flow rate that increases linearly with time. As for the one-dimensional case, if the viscosity of the injection fluid gradually increases over time, then the optimal injection rate has a smaller initial value but gradually increases to larger values than for the analogous constant viscosity problem. If the displacing fluid features shear-thinning rheology, then the optimal injection rate involves a smaller flow rate at early times, although not as large a reduction as in the Newtonian case, and a larger flow rate at late times, although not as large an increase as in the Newtonian case.

Hamiltonian formulation towards minimization of viscous fluid fingering.

A variational approach has been recently employed to determine the ideal time-dependent injection rate Q(t) that minimizes fingering formation when a fluid is injected in a Hele-Shaw cell filled with another fluid of much greater viscosity. However, such a calculation is approximate in nature, since it has been performed by assuming a high capillary number regime. In this work, we go one step further, and utilize a Hamiltonian formulation to obtain an analytical exact solution for Q(t), now valid for arbitrary values of the capillary number. Moreover, this Hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three-dimensional porous media. An analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided.

Adjoint based optimal control of partially miscible two-phase flow in porous media with applications to CO2 sequestration in underground reservoirs

With the target of optimizing CO2 sequestration in underground reservoirs, we investigate constrained optimal control problems with partially miscible two-phase flow in porous media. Our objective is to maximize the amount of trapped CO2 in an underground reservoir after a fixed period of CO2 injection, while time-dependent injection rates in multiple wells are used as control parameters. We describe the governing two-phase two-component Darcy flow PDE system, formulate the optimal control problem and derive the continuous adjoint equations. For the discretization we apply a variant of the so-called BOX method, a locally conservative control-volume FE method that we further stabilize by a periodic averaging feature to reduce oscillations. The timestep-wise Lagrange function of the control problem is implemented as a variational form in Sundance, a toolbox for rapid development of parallel FE simulations, which is part of the HPC software Trilinos. We discuss the BOX method and our implementation in Sundance. The MPI parallelized Sundance state and adjoint solvers are linked to the interior point optimization package IPOPT, using limited-memory BFGS updates for approximating second derivatives. Finally, we present and discuss different types of optimal control results.

Controlling fingering instabilities in nonflat Hele-Shaw geometries

In analogy to the viscous fingering problem between parallel flat plates, injection-driven flow in nonflat (conical and spherical) Hele-Shaw cells gives rise to ramified interfacial structures. The multiple occurrences of fingertip-splitting events makes the resulting pattern difficult to predict and to control. We calculate expressions for time-dependent injection rates that suppress the development of bifurcated shapes on curved confined environments. Implementation of these injection procedures leads to interfacial shapes with a robust n-fold symmetry. A weakly nonlinear approach is used to extract important analytical information about the influence of the geometric and topological features of the nonflat cells on the efficiency of the proposed fingering control processes.