Microfluidic extensional rheometry using a hyperbolic contraction geometry

Microfluidic devices are ideally suited for the study of complex fluids undergoing large deformation rates in the absence of inertial complications. In particular, a microfluidic contraction geometry can be utilized to characterize the material response of complex fluids in an extensionally-dominated flow, but the mixed nature of the flow kinematics makes quantitative measurements of material functions such as the true extensional viscosity challenging. In this paper, we introduce the ‘extensional viscometer-rheometer-on-a-chip’ (EVROC), which is a hyperbolically-shaped contraction-expansion geometry fabricated using microfluidic technology for characterizing the importance of viscoelastic effects in an extensionally-dominated flow at large extension rates ( $\lambda \dot \varepsilon _a \gg 1$ , where $\lambda $ is the characteristic relaxation time, or for many industrial processes $\dot \varepsilon _a \gg 1$ s $^{-1}$ ). We combine measurements of the flow kinematics, the mechanical pressure drop across the contraction and spatially-resolved flow-induced birefringence to study a number of model rheological fluids, as well as several representative liquid consumer products, in order to assess the utility of EVROC as an extensional viscosity indexer.     

Semianalytical Solution for \text{ CO}_{2} Plume Shape and Pressure Evolution During \text{ CO}_{2} Injection in Deep Saline Formations

The injection of supercritical carbon dioxide ( $\text{ CO}_{2})$ in deep saline aquifers leads to the formation of a $\text{ CO}_{2}$ rich phase plume that tends to float over the resident brine. As pressure builds up, $\text{ CO}_{2}$ density will increase because of its high compressibility. Current analytical solutions do not account for $\text{ CO}_{2}$ compressibility and consider a volumetric injection rate that is uniformly distributed along the whole thickness of the aquifer, which is unrealistic. Furthermore, the slope of the $\text{ CO}_{2}$ pressure with respect to the logarithm of distance obtained from these solutions differs from that of numerical solutions. We develop a semianalytical solution for the $\text{ CO}_{2}$ plume geometry and fluid pressure evolution, accounting for $\text{ CO}_{2}$ compressibility and buoyancy effects in the injection well, so $\text{ CO}_{2}$ is not uniformly injected along the aquifer thickness. We formulate the problem in terms of a $\text{ CO}_{2}$ potential that facilitates solution in horizontal layers, with which we discretize the aquifer. Capillary pressure is considered at the interface between the $\text{ CO}_{2}$ rich phase and the aqueous phase. When a prescribed $\text{ CO}_{2}$ mass flow rate is injected, $\text{ CO}_{2}$ advances initially through the top portion of the aquifer. As $\text{ CO}_{2}$ is being injected, the $\text{ CO}_{2}$ plume advances not only laterally, but also vertically downwards. However, the $\text{ CO}_{2}$ plume does not necessarily occupy the whole thickness of the aquifer. We found that even in the cases in which the $\text{ CO}_{2}$ plume reaches the bottom of the aquifer, most of the injected $\text{ CO}_{2}$ enters the aquifer through the layers at the top. Both $\text{ CO}_{2}$ plume position and fluid pressure compare well with numerical simulations. This solution permits quick evaluations of the $\text{ CO}_{2}$ plume position and fluid pressure distribution when injecting supercritical $\text{ CO}_{2}$ in a deep saline aquifer.     

Numerical and experimental investigations of pseudo-shock systems in a planar nozzle: impact of bypass mass flow due to narrow gaps

During previous investigations on pseudo-shock systems, we have observed reproducible differences between measurement and simulations for the pressure distribution as well as for size and shape of the pseudo-shock system. A systematic analysis of the deviations leads to the conclusion that small gaps of $\Delta z=O(10^{-4})$  m between quartz glass side walls and metal contour of the test section are responsible for this mismatch. This paper describes a targeted experimental and numerical study of the bypass mass flow within these gaps and its interaction with the main flow. In detail, we analyze how the pressure distribution within the channel as well as the size, shape and oscillation of the pseudo-shock system are affected by the gap size. Numerical simulations are performed to display the flow inside the gaps and to reproduce and explain the experimental results. Numerical and experimental schlieren images of the pseudo-shock system are in good agreement and show that especially the structure of the primary shock is significantly altered by the presence of small gaps. Extensive unsteady flow simulations of the geometry with gaps reveal that the shear layer between subsonic gap flow and supersonic core flow is subject to a Kelvin–Helmholtz instability resulting in small pressure fluctuations. This leads to a shock oscillation with a frequency of $f= O(10^5) \hbox {s}^{-1}$ . The corresponding time scale $\tau $  (s) is 16 times higher than the characteristic time scale $\tau _\delta =\delta /U_\infty $ of the boundary layer given by the ratio of the boundary layer thickness $\delta $ directly ahead of the shock and the undisturbed free stream velocity $U_\infty $ . To assess the reliability of our numerical investigations, the paper includes a grid study as well as an extensive comparison of several RANS turbulence models and their impact on the predicted shape of pseudo-shock systems.     

Thin Shear Layer Structures in High Reynolds Number Turbulence

Three-dimensional tomographic time dependent PIV measurements of high Reynolds number (Re) laboratory turbulence are presented which show the existence of long-lived, highly sheared thin layer eddy structures with thickness of the order of the Taylor microscale and internal fluctuations. Highly sheared layer structures are also observed in direct numerical simulations of homogeneous turbulence at higher values of Re (Ishihara et al., Annu Rev Fluid Mech 41:165–180, 2009). But in the latter simulation, where the fluctuations are more intense, the layer thickness is greater. A rapid distortion model describes the structure and spectra for the velocity fluctuations outside and within ‘significant’ layers; their spectra are similar to the Kolmogorov (C R Acad Sci URSS 30:299–303, 1941) and Obukhov (Dokl Akad Nauk SSSR 32:22–24, 1941) statistical model (KO) for the whole flow. As larger-scale eddy motions are blocked by the shear layers, they distort smaller-scale eddies leading to local zones of down-scale and up-scale transfer of energy. Thence the energy spectrum for high wave number k is $E_X (k)\sim Bk^{-2p}$ . The exponent p depends on the forms of the large eddies. The non-linear interactions between the distorted inhomogeneous eddies produce a steady local structure, which implies that 2p?=?5/3 and a flux of energy into the thin-layers balancing the intense dissipation, which is much greater than the mean $\left<\epsilon\right>$ . Thence $B\sim\left<\epsilon\right>^{2/3}$ as in KO. Within the thin layers the inward flux energises extended vortices whose thickness and spacing are comparable with the viscous microscale. Although peak values of vorticity and velocity of these vortices greatly exceed those based on the KO scaling, the form of the viscous range spectrum is consistent with their model.     

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number $10^{-5}\le \hbox {Da}\le 10^{-3}$ , porous layer height ratio $0\le d/L\le 1$ , thermal conductivity ratio $1\le k_{1,3}\le 20$ , and dimensionless time $0\le \tau \le 1000$ on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.     

Double-Diffusive Natural Convection in Anisotropic Porous Medium Bounded by Finite Thickness Walls: Validity of Local Thermal Equilibrium Assumption

Double-diffusive natural convection in fluid-saturated porous medium inside a vertical enclosure bounded by finite thickness walls with opposing temperature, concentration gradients on vertical walls as well as adiabatic and impermeable horizontal ones has been performed numerically. The Darcy model was used to predict fluid flow inside the porous material, while thermal fields are simulated based on two-energy equations for fluid and solid phases on the basis of a local thermal non-equilibrium model. Computations have been performed for different controlling parameters such as the buoyancy ratio $N$ , the Lewis number Le, the anisotropic permeability ratio $R_\mathrm{p}$ , the fluid-to-solid thermal conductivity ratio $R_\mathrm{c}$ , the interphase heat transfer coefficient $\mathcal{H}$ , the ratio of the wall thickness to its height $D$ , the wall-to-porous medium thermal diffusivity ratio $R_\mathrm{w}$ , and the solid-to-fluid heat capacity ratio $\gamma $ . Thus, the effects of the controlling parameters on heat and mass transfer characteristics are discussed in detail. Moreover, the validity domain of the local thermal equilibrium (LTE) assumption has been delimited for different set of the governing parameters. It has been shown that Le has a noticeable significant effect on fluid temperature profiles and that higher $N$ values lead to a significant enhancement in heat and mass transfer rates. Moreover, for higher $\mathcal{H}, R_\mathrm{c}$ , $R_\mathrm{p}, R_\mathrm{w}$ , or $D$ values and/or lower $\gamma $ values, the solid and fluid phases tend toward LTE.     

A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs

The flow of oil and water in naturally fractured reservoirs (NFR) can be highly complex and a simplified model is presented to illustrate some main features of this flow system. NFRs typically consist of low-permeable matrix rock containing a high-permeable fracture network. The effect of this network is that the advective flow bypasses the main portions of the reservoir where the oil is contained. Instead capillary forces and gravity forces are important for recovering the oil from these sections. We consider a linear fracture which is symmetrically surrounded by porous matrix. Advective flow occurs only along the fracture, while capillary driven flow occurs only along the axis of the matrix normal to the fracture. For a given set of relative permeability and capillary pressure curves, the behavior of the system is completely determined by the choice of two dimensionless parameters: (i) the ratio of time scales for advective flow in fracture to capillary flow in matrix $\alpha =\tau ^f/\tau ^m$ ; (ii) the ratio of pore volumes in matrix and fracture $\beta =V^m/V^f$ . A characteristic property of the flow in the coupled fracture–matrix medium is the linear recovery curve (before water breakthrough) which has been referred to as the “filling fracture” regime Rangel-German and Kovscek (J Pet Sci Eng 36:45–60, 2002), followed by a nonlinear period, referred to as the “instantly filled” regime, where the rate is approximately linear with the square root of time. We derive an analytical solution for the limiting case where the time scale $\tau ^{m}$ of the matrix imbibition becomes small relative to the time scale $\tau ^{f}$ of the fracture flow (i.e., $\alpha \rightarrow \infty $ ), and verify by numerical experiments that the model will converge to this limit as $\alpha $ becomes large. The model provides insight into the role played by parameters like saturation functions, injection rate, volume of fractures versus volume of matrix, different viscosity relations, and strength of capillary forces versus injection rate. Especially, a scaling number $\omega $ is suggested that seems to incorporate variations in these parameters. An interesting observation is that at $\omega =1$ there is little to gain in efficiency by reducing the injection rate. The model can be used as a tool for interpretation of laboratory experiments involving fracture–matrix flow as well as a tool for testing different transfer functions that have been suggested to use in reservoir simulators.     

Generalized Resolvent Estimates of the Stokes Equations with First Order Boundary Condition in a General Domain

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain ${\Omega}$ of the N-dimensional Eulidean space ${\mathbb{R}^N, N \geq 2}$ . This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter ${\lambda}$ varying in a sector ${\Sigma_{\sigma, \lambda_0} = \{\lambda \in \mathbb{C} \mid |\arg \lambda| < \pi-\sigma, \enskip |\lambda| \geq \lambda_0\}}$ , where ${0 < \sigma < \pi/2}$ and ${\lambda_0 \geq 1}$ . The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution ${p \in \hat{W}^1_{q, \Gamma}(\Omega)}$ to the variational problem: ${(\nabla p, \nabla \varphi) = (f, \nabla \varphi)}$ for any ${\varphi \in \hat W^1_{q', \Gamma}(\Omega)}$ . Here, ${1 < q < \infty, q' = q/(q-1), \hat W^1_{q, \Gamma}(\Omega)}$ is the closure of ${W^1_{q, \Gamma}(\Omega) = \{ p \in W^1_q(\Omega) \mid p|_\Gamma = 0\}}$ by the semi-norm ${\|\nabla \cdot \|_{L_q(\Omega)}}$ , and ${\Gamma}$ is the boundary of ${\Omega}$ . In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in ${(\lambda_0, \infty)}$ . Our assumption is satisfied for any ${q \in (1, \infty)}$ by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q =  2.     

Unsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink

This paper presents an analytical study of the unsteady MHD free convective heat and mass transfer flow of a viscous, incompressible, gray, absorbing-emitting but non-scattering, optically-thick and electrically conducting fluid occupying a semi-infinite porous regime adjacent to an infinite moving hot vertical plate with constant velocity. We employ a Darcian viscous flow model for the porous medium the Rosseland diffusion approximation is used to describe the radiative heat flux in the energy equation. The homogeneous chemical reaction of first order is accounted in mass diffusion equation. The governing equations are solved in closed form by Laplace-transform technique. A parametric study of all involved parameters is conducted and representative set of numerical results for the velocity, temperature, concentration, shear stress function $\frac{\partial u}{\partial y} \vert_{y=0}$ , temperature gradient $\frac{\partial \theta }{ \partial y}\vert_{y=0}$ , and concentration gradient $\frac{ \partial \phi }{\partial y}\vert_{y=0}$ is illustrated graphically and physical aspects of the problem are discussed.     

Mixed Convection Boundary-Layer Flow Over a Vertical Surface Embedded in a Porous Material Subject to a Convective Boundary Condition

The mixed convection boundary-layer flow on one face of a semi-infinite vertical surface embedded in a fluid-saturated porous medium is considered when the other face is taken to be in contact with a hot or cooled fluid maintaining that surface at a constant temperature $T_\mathrm{{f}}$ . The governing system of partial differential equations is transformed into a system of ordinary differential equations through an appropriate similarity transformation. These equations are solved numerically in terms of a dimensionless mixed convection parameter $\epsilon $ and a surface heat transfer parameter $\gamma $ . The results indicate that dual solutions exist for opposing flow, $\epsilon <0$ , with the dependence of the critical values $\epsilon _\mathrm{{c}}$ on $\gamma $ being determined, whereas for the assisting flow $\epsilon >0$ , the solution is unique. Limiting asymptotic forms for both $\gamma $ small and large and $\epsilon $ large are also discussed.     

Dynamics of a two-dimensional flow subject to steady electromagnetic forces

A novel experimental setup is presented to study the dynamics of a two-dimensional (2D) flow formed of an electrolyte subject to steady electromagnetic forcing. A thin layer of potassium hydroxide is poured into a square-base container with a strong magnetic field ( $\vec{B}$ ) achieved by permanent neodymium magnets inserted underneath the base. The set of electrodes of alternating polarity distributed along the perimeter of the container generates currents ( $\vec{j}$ ) in opposite directions. Coherent primary vortices of scales about 2 cm are thus generated by the $\vec{j} \times \vec{B}$ force. We also show, and for the first time, that fluid motion is caused by the magnetic field gradient where the amplitude of $\vec{B}$ is equal to zero. It leads to the generation of jets with size about that of the container, that is, 25 cm. The interaction between these gradB jets and the edge vortices leads to a final flow dominated by large-scale vortices resulting from the inverse cascade process that destroys the small-scale coherent structures on one hand and on the other modifies the initial scale and direction of the gradB jets.     

A Combined Theoretical/Experimental Approach for Reducing Ringing Artifacts in Low Dynamic Testing with Servo-hydraulic Load Frames

One of the most challenging tasks facing computer-aided engineering (CAE) analysis is the acquisition of accurate tensile test data that spans quasi-static to low dynamic (10^?5/s?≤ $ \overset{.}{\varepsilon } $ ≤5?×?10²/s) strain rates ( $ \overset{.}{\varepsilon } $ ). Critical to the accuracy of data acquired over the low dynamic range is the reduction of ringing artifacts in flow data. Ringing artifacts, which are a consequence of the inertial response of the load frame, are spurious oscillations that can obscure the desired material response (i.e. load vs. time or load vs. displacement) from which flow data are derived. These oscillations tend to grow with increasing strain rate and peak at the high end of the low dynamic range on servo-hydraulic tensile test frames. Common practices for addressing ringing are data filtering, which is often problematic since filtering introduces distortion in smoothed material data, or trial-and-error design of test specimen geometries. This renders techniques for reducing ringing based upon the mechanics of the load frame and optimization of tensile specimen geometry quite attractive. In the present paper, relationships between load, stress wave propagation, and specimen geometries are addressed, to both quantify ringing and to develop specimen designs that will reduce ringing. A combined theoretical/experimental approach for tensile specimen design was developed for reducing ringing in flow data over the low dynamic range of strain rates (10^?5/s≤ $ \overset{.}{\varepsilon } $ ≤5?×?10²/s). The single camera digital image correlation (DIC) method was used to measure the displacement fields and strain rates with specimens resulting from the combined theoretical/experimental approach. While the approach was developed on a specific commercial load frame with a TRIP steel subject to a two-step quenching and partitioning heat treatment (Q&P980), it is readily adaptable to other servo-hydraulic load frames and metallic alloys. The developed approach results in a 90 % reduction in ringing artifact (with no filtering) in a tensile flow curve for Q&P980 at $ \overset{.}{\varepsilon}\kern-4pt $ = 5?×?10²/s. Results from split Hopkinson bar tests of Q&P980 were performed at $ \overset{.}{\varepsilon } $ = 500/s and compare favorably with the test data generated by the developed testing approach. Since the Q&P980 steel represents a new generation of advanced high strength steels, we also evaluated its strain rate sensitivity over the low dynamic range.     

A KAM Theorem for Higher Dimensional Nonlinear Schrödinger Equations

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional nonlinear Schrödinger equation $$\begin{aligned} iu_t-\triangle u +M_\xi u+f(|u|^2)u=0, \quad t\in \mathbb{R }, x\in \mathbb{T }^d \end{aligned}$$ with periodic boundary conditions, where $M_\xi $ is a real Fourier multiplier and $f(|u|^2)$ is a real analytic function near $u=0$ with $f(0)=0$ . We obtain for the equation a Whitney smooth family of real-analytic small-amplitude linearly-stable quasi-periodic solutions with a nice linear normal form.     

Inertia effect in conical converging flow of viscoelastic fluids

The convergent flow of viscoelastic fluids in conical nozzles has been examined. The experimentally determined streamlines agreed with those obtained from calculations with an approximation up to \(\dot V^2 \) Because of the elasticity of the fluid a rotationally symmetric eddy arises. It clings to the wall whereas the output flow proceeds in the middle. When $$\frac{\varrho }{{\eta _0 }}\left( {\frac{{\dot V^2 }}{{t_0 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} > 40$$ > 40 another eddy can arise in front of the first one. This second eddy is situated in the middle and causes the output to proceed as a flow near the wall. The higher the afore-mentioned dimensionless value is, the higher is the inclination of the flow to instability.     

Effect of Conduction in Bottom Wall on B��nard Convection in a Porous Enclosure with Localized Heating and Lateral Cooling

Darcy-Bénard convection in a square porous enclosure with a localized heating from below and lateral cooling is studied numerically in the present paper. A finite-thickness bottom wall is locally heated, the top wall is kept at a lower temperature than the bottom wall temperature, and the lateral walls are cooled. The finite difference method has been used to solve the dimensionless governing equations. The analysis in the undergoing numerical investigation is performed in the following ranges of the associated dimensionless groups: the heat source length?? ${0.2\leq H \leq 0.9}$ , the wall thickness?? ${0.05\leq D \leq 0.4}$ , the thermal conductivity ratio?? ${0.8\leq K_{\rm r} \leq 9.8}$ , and the Biot number?? ${0.1\leq Bi \leq 1.1}$ . It is observed that the heat transfer rate could increase with increasing heat source lengths, thermal conductivity ratio, and cooling intensity. There exists a critical wall thickness for a high wall conductivity below which the increasing wall thickness increases the heat transfer rate and above which the increasing wall thickness decreases the heat transfer rate.     

Suspended Particles Transport and Deposition in Saturated Granular Porous Medium: Particle Size Effects

An experimental study on the transport and deposition of suspended particles (SP) in a saturated porous medium (calibrated sand) was undertaken. The influence of the size distribution of the SP under different flow rates is explored. To achieve this objective, three populations with different particles size distributions were selected. The median diameter $d_{50}$ of these populations was 3.5, 9.5, and $18.3~\upmu \hbox {m}$ . To study the effect of polydispersivity, a fourth population noted “Mixture” ( $d_{50} = 17.4\; \upmu \hbox {m}$ ) obtained by mixing in equal proportion (volume) the populations 3.5 and $18.3\;\upmu \hbox {m}$ was also used. The SP transfer was compared to the dissolved tracer (DT) one. Short pulse was the technique used to perform the SP and the DT injection in a column filled with the porous medium. The breakthrough curves were competently described with the analytical solution of a convection–dispersion equation with first-order deposition kinetics. The results showed that the transport of the SP was less rapid than the transport of the DT whatever the flow velocity and the size distribution of the injected SP. The mean diameter of the recovered particles increases with flow rate. The longitudinal dispersion increases, respectively, with the increasing of the flow rates and the SP size distribution. The SP were more dispersive in the porous medium than the DT. The results further showed that the deposition kinetics depends strongly on the size of the particle transported and their distribution.     

A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range

Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.     

Conjugate Natural Convection in a Porous Enclosure Sandwiched by Finite Walls Under the Influence of Non-uniform Heat Generation and Radiation

Conjugate natural convection in a square porous enclosure sandwiched by finite walls under the influence of non-uniform heat generation and radiation is studied numerically in the present article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures, while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the ratio of wall thickness to its width $(0.02 \le D \le 0.3)$ ( 0.02 ≤ D ≤ 0.3 ) , the wall to porous thermal conductivity ratio $(0.1 \le k_\mathrm{r} \le 10.0)$ ( 0.1 ≤ k r ≤ 10.0 ) , the internal heating $(0 \le \gamma \le 5)$ ( 0 ≤ γ ≤ 5 ) , and the local heating exponent parameters $(1 \le \lambda \le 20)$ ( 1 ≤ λ ≤ 20 ) . It is found that the average Nusselt number on the hot and cold interfaces increases with increasing the radiation intensity. Very high non-uniformity heating does not affect the average Nusselt number at very thick walls.     

Drainage of Capillary-Trapped Oil by an Immiscible Gas: Impact of Transient and Steady-State Water Displacement on Three-Phase Oil Permeability

In a previous paper (Dehghanpour et al., Phys Rev E 83:065302, 2011a), we showed that relative permeability of mobilized oil, $k_\mathrm{ro}$ , measured during tertiary gravity drainage, is significantly higher than that of the same oil saturation in other tests where oil is initially a continuous phase. We also showed that tertiary $k_\mathrm{ro}$ strongly correlates to both water saturation, $S_\mathrm{w}$ , water flux (water relative permeability), $k_\mathrm{rw}$ , and the change in water saturation with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . To develop a model and understanding of the enhanced oil transport, identifying which of these parameters ( $S_\mathrm{w},\,k_{\mathrm{rw}}$ , or $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ ) plays the controlling role is necessary, but in the previous experiments these could not be deconvolved. To answer the remaining question, we conduct specific three-phase displacement experiments in which $k_{\mathrm{rw}}$ is controlled by applying a fixed water influx, and $S_\mathrm{w}$ develops naturally. We obtain $k_{\mathrm{ro}}$ by using the saturation data measured in time and space. The results suggest that steady-state water influx, in contrast to transient water displacement, does not enhance $k_{\mathrm{ro}}$ . Instead, reducing water influx rate results in excess oil flow. Furthermore, according to our pore scale hydraulic conductivity calculations, viscous coupling and fluid positioning do not sufficiently explain the observed correlation between $k_{\mathrm{ro}}$ and $S_{\mathrm{w}}$ . We conclude that tertiary $k_{\mathrm{ro}}$ is controlled by the oil mobilization rate, which in turn is linked to the rate of water saturation decrease with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . Finally, we develop a simple model which relates tertiary $k_{\mathrm{ro}}$ to transient two-phase gas/water relative permeability.