Comparative Study of Five Outcrop Chalks Flooded at Reservoir Conditions: Chemo-mechanical Behaviour and Profiles of Compositional Alteration

This study presents experimental results from a flooding test series performed at reservoir conditions for five high-porosity Cretaceous onshore chalks from Denmark, Belgium and the USA, analogous to North Sea reservoir chalk. The chalks are studied in regard to their chemo-mechanical behaviour when performing tri-axial compaction tests while injecting brines (0.219 mol/L \(\hbox {MgCl}_{2}\) or 0.657 mol/L NaCl) at reservoir conditions for 2–3 months (T = 130 \(^\circ \hbox {C}\); 1 PV/d). Each chalk type was examined in terms of its mineralogical and chemical composition before and after the mechanical flooding tests, using an extensive set of analysis methods, to evaluate the chalk- and brine-dependent chemical alterations. All \(\hbox {MgCl}_{2}\)-flooded cores showed precipitation of Mg-bearing minerals (mainly magnesite). The distribution of newly formed Mg-bearing minerals appears to be chalk-dependent with varying peaks of enrichment. The chalk samples from Aalborg originally contained abundant opal-CT, which was dissolved with both NaCl and \(\hbox {MgCl}_{2}\) and partly re-precipitated as Si-Mg-bearing minerals. The Aalborg core injected with \(\hbox {MgCl}_{2}\) indicated strongly increased specific surface area (from 4.9 \(\hbox {m}^{2}\hbox {/g}\) to within 7–9 \(\hbox {m}^{2}\hbox {/g}\)). Mineral precipitation effects were negligible in chalk samples flooded with NaCl compared to \(\hbox {MgCl}_{2}\). Silicates were the main mineralogical impurity in the studied chalk samples (0.3–6 wt%). The cores with higher \(\hbox {SiO}_{2}\) content showed less deformation when injecting NaCl brine, but more compaction when injecting \(\hbox {MgCl}_{2}\)-brine. The observations were successfully interpreted by mathematical geochemical modelling which suggests that the re-precipitation of Si-bearing minerals leads to enhanced calcite dissolution and mass loss (as seen experimentally) explaining the high compaction seen in \(\hbox {MgCl}_{2}\)-flooded Aalborg chalk. Our work demonstrates that the original mineralogy, together with the newly formed minerals, can control the chemo-mechanical interactions during flooding and should be taken into account when predicting reservoir behaviour from laboratory studies. This study improves the understanding of complex flow reaction mechanisms also relevant for field-scale dynamics seen during brine injection.     

Visual Investigation of Improvement in Extra-Heavy Oil Recovery by Borate-Assisted $$\hbox {CO}_{2}$$-Foam Injection

The significant reduction in heavy oil viscosity when mixed with \(\hbox {CO}_{2}\) is well documented. However, for \(\hbox {CO}_{2}\) injection to be an efficient method for improving heavy oil recovery, other mechanisms are required to improve the mobility ratio between the \(\hbox {CO}_{2}\) front and the resident heavy oil. In situ generation of \(\hbox {CO}_{2}\)-foam can improve \(\hbox {CO}_{2}\) injection performance by (a) increasing the effective viscosity of \(\hbox {CO}_{2}\) in the reservoir and (b) increasing the contact area between the heavy oil and injected \(\hbox {CO}_{2}\) and hence improving \(\hbox {CO}_{2}\) dissolution rate. However, in situ generation of stable \(\hbox {CO}_{2}\)-foam capable of travelling from the injection well to the production well is hard to achieve. We have previously published the results of a series of foam stability experiments using alkali and in the presence of heavy crude oil (Farzaneh and Sohrabi 2015). The results showed that stability of \(\hbox {CO}_{2}\)-foam decreased by addition of NaOH, while it increased by addition of \(\hbox {Na}_{2}\hbox {CO}_{3}\). However, the highest increase in \(\hbox {CO}_{2}\)-foam stability was achieved by adding borate to the surfactant solution. Borate is a mild alkaline with an excellent pH buffering ability. The previous study was performed in a foam column in the absence of a porous medium. In this paper, we present the results of a new series of experiments carried out in a high-pressure glass micromodel to visually investigate the performance of borate–surfactant \(\hbox {CO}_{2}\)-foam injection in an extra-heavy crude oil in a transparent porous medium. In the first part of the paper, the pore-scale interactions of \(\hbox {CO}_{2}\)-foam and extra-heavy oil and the mechanisms of oil displacement and hence oil recovery are presented through image analysis of micromodel images. The results show that very high oil recovery was achieved by co-injection of the borate–surfactant solution with \(\hbox {CO}_{2}\), due to in-situ formation of stable foam. Dissolution of \(\hbox {CO}_{2}\) in heavy oil resulted in significant reduction in its viscosity. \(\hbox {CO}_{2}\)-foam significantly increased the contact area between the oil and \(\hbox {CO}_{2}\) significantly and thus the efficiency of the process. The synergy effect between the borate and surfactant resulted in (1) alteration of the wettability of the porous medium towards water wet and (2) significant reduction of the oil–water IFT. As a result, a bank of oil-in-water (O/W) emulsion was formed in the porous medium and moved ahead of the \(\hbox {CO}_{2}\)-foam front. The in-situ generated O/W emulsion has a much lower viscosity than the original oil and plays a major role in the observed additional oil recovery in the range of performed experiments. Borate also made \(\hbox {CO}_{2}\)-foam more stable by changing the system to non-spreading oil and reducing coalescence of the foam bubbles. The results of these visual experiments suggest that borate can be a useful additive for improving heavy oil recovery in the range of the performed tests, by increasing \(\hbox {CO}_{2}\)-foam stability and producing O/W emulsions.     

Application of an Inverse Simulator of Single-Phase Flow Analysis for Leakage Pathway Estimation in a Multiphase System for Geologic $$\hbox {CO}_{2}$$ Storage

When \(\hbox {CO}_{2}\) is injected in a brine reservoir, brine or \(\hbox {CO}_{2}\) can be discharged into a permeable formation saturated with brine above the storage reservoir along a leakage pathway, if present. In most situations, the overlying formation can act as a single-phase aquifer with only brine leakage before \(\hbox {CO}_{2}\) leaks. This study examines the applicability of a developed inverse code for single-phase fluids to detect leakage pathway locations in view of the overlying formation using pressure anomalies induced by leaks. Before applying inverse analysis, forward modeling is performed using the TOUGH2 model to determine brine and \(\hbox {CO}_{2}\) leakage in a homogeneous conceptual model, and the simulated pressure profiles at monitoring wells are used as measurements in the inverse model. In the inverse code, an important consideration is that the vertical hydraulic conductivity and cross-sectional area of a leakage pathway that are inherent to a leakage term in the mass balance equation are integrated as a single parameter to estimate the leakage pathway locations. This method eliminates the impact of the uncertainty of the leakage pathway size on the accuracy of leakage pathway estimation. The inverse model examines the effect of the number of monitoring wells, monitoring periods and \(\hbox {CO}_{2}\) leakage into the overlying formation on the accuracy of leakage pathway estimation according to eleven application examples. The comparison between the results of the single-phase inverse code and iTOUGH2 code illustrates that the single-phase inverse model can be applicable to the leakage pathway estimation in a multiphase flow system.     

A new type of functional chemical sensitizer $$\hbox {MgH}_{2}$$ for improving pressure desensitization resistance of emulsion explosives

In millisecond-delay blasting and deep water blasting projects, traditional emulsion explosives sensitized by the chemical sensitizer \(\hbox {NaNO}_{2}\) often encounter incomplete explosion or misfire problems because of the “pressure desensitization” phenomenon, which seriously affects blasting safety and construction progress. A \(\hbox {MgH}_{2}\)-sensitized emulsion explosive was invented to solve these problems. Experimental results show that \(\hbox {MgH}_{2}\) can effectively reduce the problem of pressure desensitization. In this paper, the factors which influence the pressure desensitization of two types of emulsion explosives are studied, and resistance to this phenomenon of \(\hbox {MgH}_{2}\)-sensitized emulsion explosives is discussed.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

The role of angular momentum in the laminar motion of viscous fluids

In laminar flow, viscous fluids must exert appropriate elastic shear stresses normal to the flow direction. This is a direct consequence of the balance of angular momentum. There is a limit, however, to the maximum elastic shear stress that a fluid can exert. This is the ultimate shear stress, \(\tau _\mathrm{y}\), of the fluid. If this limit is exceeded, laminar flow becomes dynamically incompatible. The ultimate shear stress of a fluid can be determined from experiments on plane Couette flow. For water at \(20\,^{\circ }\hbox {C}\), the data available in the literature indicate a value of \(\tau _\mathrm{y}\) of about \(14.4\times 10^{-3}\, \hbox {Pa}\). This study applies this value to determine the Reynolds numbers at which flowing water reaches its ultimate shear stress in the case of Taylor–Couette flow and circular pipe flow. The Reynolds numbers thus obtained turn out to be reasonably close to those corresponding to the onset of turbulence in the considered flows. This suggests a connection between the limit to laminar flow, on the one hand, and the occurrence of turbulence, on the other.     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier

$$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$

with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Töplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Töplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

     

Approximation of Flat <Emphasis Type="Italic">W</Emphasis><Superscript>2,2</Superscript> Isometric Immersions by Smooth Ones

Let \({S\subset\mathbb{R}^2}\) be a bounded Lipschitz domain and denote by \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\) the set of mappings \({u\in W^{2,2}(S;\mathbb{R}^3)}\) which satisfy \({(\nabla u)^T(\nabla u) = Id}\) almost everywhere. Under an additional regularity condition on the boundary \({\partial S}\) (which is satisfied if \({\partial S}\) is piecewise continuously differentiable), we prove that the strong W ^2,2 closure of \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)\cap C^{\infty}(\overline{S};\mathbb{R}^3)}\) agrees with \({W^{2,2}_{\text{iso}}(S; \mathbb{R}^3)}\).     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x, y, t) that undergoes faceting is considered with periodic boundary conditions, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes \(u_1=h_{x}\) and \(u_2=h_y\) to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in \(\dot{H}^2_{per}\), we consider the solution operator \(S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}\), to gain our results. We prove the necessary continuity, dissipation and compactness properties.     

The Effects of Brine Species on the Formation of Residual Water in a \hbox {CO}_{2}–Brine System

Geological storage of \(\hbox {CO}_{2}\) in deep saline aquifers is achieved by injecting \(\hbox {CO}_{2}\) into the aquifers and displacing the brine. Although most of the brine is displaced, some residual groundwater remains in the rock pores. We conducted experiments to investigate factors that influence how much of this residual water remains after \(\hbox {CO}_{2}\) is injected. A rock sample was saturated with brines of two different salts. Supercritical \(\hbox {CO}_{2}\) was injected into the samples at aquifer temperature and pressure, and the displaced water and water–gas mixtures were collected and measured. The results show that deionized water drains more completely than either of the two brines, and NaCl brine drains more completely than \(\hbox {CaCl}_{2}\) brine. The ranking of the irreducible water saturation at the end of the experiment is deionized \(\hbox {water}<\hbox {NaCl brine } <\hbox {CaCl}_{2}\) brine. The process of drainage can be divided into three stages according to the drainage flow rates; the Pushing Drainage, Portable Drainage, and Dissolved Drainage stages. This paper proposed a capillary model which is used to interpret the mechanisms that characterize these three stages.     

Flow Dynamics of \hbox {CO}_{2}/brine at the Interface Between the Storage Formation and Sealing Units in a Multi-layered Model

Pressure distribution and \(\hbox {CO}_{2}\) plume migration are two major interests in \(\hbox {CO}_{2}\) geologic storage as they determine the injectivity and storage capacity. In this study, we adopted a three-layer model comprising a storage formation and the over- and underlying seals and determined three distinct flow regions based on the vertical flux exchange of \(\hbox {CO}_{2}\) and native brine. Regions 1 and 2 showed \(\hbox {CO}_{2}\) flowing from the storage formation to adjacent seals with counter-flowing brine. The characteristics of these fluxes in Region 1 were governed by permeability change due to salt precipitation whereas buoyancy force controlled the flux pattern in Region 2. Region 3 showed brine flowing from storage formation toward the over- and underlying seals, which enabled the displaced brine to escape from the storage formation and make room for \(\hbox {CO}_{2}\) to store as well as reduce the pressure build-up. In the multi-layered model, the counter-flowing brine in flow Region 1 resulted in localized salt precipitation at the upper and lower boundary of storage formation. We assessed the bottom-hole pressure and \(\hbox {CO}_{2}\) mass in caprock with respect to reservoir size. While the formation thickness influenced the bottom-hole pressure in the early stage of injection, the horizontal extension of the reservoir was more influential to pressure build-up during the injection period, and to the stabilized pressure during the post-injection period. The \(\hbox {CO}_{2}\) mass in caprock gently increased during the injection period as well as during the post-injection period and reached about 4–5 % of injected \(\hbox {CO}_{2}\) . The percentage of escaped brine from the storage formation ranged from 80–100 % of the \(\hbox {CO}_{2}\) mass stored in the storage formation depending on the reservoir scale.     

On the Damped Harmonic Oscillator with Time Dependent Damping Coefficient

Conditions guaranteeing asymptotic stability for the differential equation

$$\begin{aligned} x''+h(t)x'+\omega ^2x=0 \qquad (x\in \mathbb {R}) \end{aligned}$$

are studied, where the damping coefficient \(h:[0,\infty )\rightarrow [0,\infty )\) is a locally integrable function, and the frequency \(\omega >0\) is constant. Our conditions need neither the requirement \(h(t)\le \overline{h}<\infty \) (\(t\in [0,\infty )\); \(\overline{h}\) is constant) (“small damping”), nor \(0< \underline{h}\le h(t)\) (\(t\in [0,\infty )\); \(\underline{h}\) is constant) (“large damping”); in other words, they can be applied to the general case \(0\le h(t)<\infty \) (\(t\in [0,\infty \))). We establish a condition which combines weak integral positivity with Smith’s growth condition

$$\begin{aligned} \int ^\infty _0 \exp [-H(t)]\int _0^t \exp [H(s)]\,\mathrm{{d}}s\,\mathrm{{d}}t=\infty \qquad \left( H(t):=\int _0^t h(\tau )\,\mathrm{{d}}\tau \right) , \end{aligned}$$

so it is able to control both the small and the large values of the damping coefficient simultaneously.

     

A nonlocal species concentration theory for diffusion and phase changes in electrode particles of lithium ion batteries

A nonlocal species concentration theory for diffusion and phase changes is introduced from a nonlocal free energy density. It can be applied, say, to electrode materials of lithium ion batteries. This theory incorporates two second-order partial differential equations involving second-order spatial derivatives of species concentration and an additional variable called nonlocal species concentration. Nonlocal species concentration theory can be interpreted as an extension of the Cahn–Hilliard theory. In principle, nonlocal effects beyond an infinitesimal neighborhood are taken into account. In this theory, the nonlocal free energy density is split into the penalty energy density and the variance energy density. The thickness of the interface between two phases in phase segregated states of a material is controlled by a normalized penalty energy coefficient and a characteristic interface length scale. We implemented the theory in COMSOL Multiphysics\(^{\circledR }\) for a spherically symmetric boundary value problem of lithium insertion into a \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\) cathode material particle of a lithium ion battery. The two above-mentioned material parameters controlling the interface are determined for \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\), and the interface evolution is studied. Comparison to the Cahn–Hilliard theory shows that nonlocal species concentration theory is superior when simulating problems where the dimensions of the microstructure such as phase boundaries are of the same order of magnitude as the problem size. This is typically the case in nanosized particles of phase-separating electrode materials. For example, the nonlocality of nonlocal species concentration theory turns out to make the interface of the local concentration field thinner than in Cahn–Hilliard theory.     

Linearly Stable Quasi-Periodic Breathers in a Class of Random Hamiltonian Systems

In this paper, we construct linearly stable quasi-periodic breathers for the Hamiltonian systems in the form \({{\rm i} \dot{q}_n+v_n q_n+\delta|q_n|^2q_n+\varepsilon_n \left(q_{n+1}+q_{n-1} \right)=0,\quad n \in \mathbb{Z}}\) where \({\{v_n\}_{n \in \mathbb{Z}}}\) is a family of time independent identically distributed (i.i.d) random variables with common distribution \({g = dv_n, v_n \in [0,1]}\) and \({|\varepsilon_n| \leq \varepsilon e^{-\varrho |n|}}\) with \({\varepsilon,\varrho > 0}\) . We prove that for \({\varepsilon, \delta}\) sufficiently small, the equation admits a family of small-amplitude and linearly stable, time quasi-periodic solutions for most of the parameters \({\{v_n\}_{n \in \mathbb{Z}}}\) .     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.

     

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

In this paper we study the limit as \(\varepsilon \rightarrow 0\) of the singularly perturbed second order equation \(\varepsilon ^2 \ddot{u}_\varepsilon + \nabla _{\!x} V(t,u_\varepsilon (t))=0\), where V(t, x) is a potential. We assume that \(u_0(t)\) is one of its equilibrium points such that \(\nabla _{\!x}V(t,u_0(t))=0\) and \(\nabla _{\!x}^2V(t,u_0(t))>0\). We find that, under suitable initial data, the solutions \(u_\varepsilon \) converge uniformly to \(u_0\), by imposing mild hypotheses on V. A counterexample shows that they cannot be weakened.     

Editorial to the Special Issue on Reconstruction of Porous Media and Materials and Its Applications

The single-well chemical tracer test (SWCTT) has emerged in the past decades as a method for measuring oil saturation prior to and/or after EOR operations, to measure the recovery performance in-situ. To use this technology, the partition coefficients of the selected tracers are essential for estimating the level of residual oil at the targeted single well. Commonly, injection of short chain alcohols and ethyl acetate, a reactive tracer, is carried out for the tracer slug, mainly based on site-specific reservoir conditions, to accurately determine the level of oil saturation in-situ. However, injection of ethyl formate has been less common due to its fast hydrolysis rate under elevated temperature, which increases the challenges in data interpretation. Therefore, a systematic study for using ethyl formate under mid-range temperature \((<60\,^{\circ }\hbox {C})\), as commonly found in mature oil field in the USA, shows the potential to be applied for SWCTT. As part of the design effort for a series of EOR field tests to manage the project risk, we particularly assessed the relationships between the partition coefficients of reactive tracers and subsurface conditions such as salinity, temperatures, type of electrolytes, and the equivalent alkane carbon number (EACN) of the crude oil experiments was performed under various reservoir conditions as a function of actual site characteristics at the targeted high saline formations. In brief, our data clearly show that the (oil/water) partition coefficient of ethyl formate increases steadily with increasing NaCl concentrations, ranging from 10,000 (0.17 M) to 250,000 mg/L (4.28 M). A similar upward trend was observed for increasing temperature between 25 and \(52\,^{\circ }\hbox {C}\); however, the partition coefficient decreases inversely with increasing the crude oil EACN over the range from 8 to 12, which are common for domestic oil samples. It was also showed that brine with high NaCl concentration yielded higher partition coefficients. In contrast, brine with high \(\hbox {CaCl}_{2}\) and \(\hbox {BaCl}_{2}\) concentration yielded lower values. And \(\hbox {MgCl}_{2}\) performed somewhat unusual trend in our tests. These results further indicate that the partition coefficient of the reactive tracer, ethyl formate, is sensitive to change in salinity, temperatures, type of electrolytes and EACN, as observed for other chemical tracers. In addition, based on the hydrolysis rate of ethyl formate under various reservoir conditions, the appropriate window of shut-in time can be pre-determined before initiating the field test. We believe that the ability of better understanding the partition coefficients and predicting the shut-in time beforehand could drastically reduce the risks of SWCTT operations.