Predicting Collector Well Yields with MODFLOW

Groundwater flow models are commonly used to design new wells and wellfields. As the spatial scale of the problem is large and much local‐scale detail is not needed, modelers often utilize two‐dimensional (2D) or quasi three‐dimensional models based on the Dupuit‐Forchheimer assumption. Dupuit models offer a robust set of tools for simulating regional groundwater flow including interactions with surface waters, the potential for well interference, and varying aquifer properties and recharge rates. However, given an assumed operating water level or drawdown at a well screen, Dupuit models systematically overpredict well yields. For design purposes, this discrepancy is unacceptable, and a method for predicting accurate well yields is needed. While published methods exist for vertical wells, little guidance is available for predicting yields in horizontal screens or collector wells. In plan view, a horizontal screen has a linear geometry, and will likely extend over several neighboring cells that may not align with rows or columns in a numerical model. Furthermore, the model must account for the effects of converging three‐dimensional (3D) flow to the well screens and hydraulic interference among the well screens; these all depend on the design of a specific well. This paper presents a new method for simulating the yield of angled or horizontal well screens in numerical groundwater flow models, specifically using the USGS code MODFLOW. The new method is compared to a detailed, 3D analytic element model of a collector well in a field of uniform flow.     

Rossby wave resonance in the presence of a nonlinear critical layer

Abstract

The behavior of Rossby waves on a shear flow in the presence of a nonlinear critical layer is studied, with particular emphasis on the role played by the critical layer in a Rossby wave resonance mechanism. Previous steady analyses are extended to the resonant case and it is found that the forced wave dominates the solution, provided the flow configuration is not resonant for the higher harmonics induced by the critical layer. Numerical simulations for the forced initial value problem show that the solution evolves towards the analysed steady state when conditions are resonant for the forced wave, and demonstrate some of the complications that arise when they are resonant for higher harmonics. In relating the initial value and steady problems, it is argued that the time dependent solution does not require the large mean flow distortion that Haberman (1972) found to be necessary outside the critical layer in the steady case.     

Effect of the oceanic lithosphere velocity on free convection in the asthenosphere beneath mid-ocean ridges

The paper presents results obtained in experiments on a horizontal layer heated from below in its central part and cooled from above; the layer models the oceanic asthenosphere. Flow velocity and temperature profiles are measured and the flow structure under boundary layer conditions is determined (at Rayleigh numbers Ra > 5 × 10⁵). The flow in the core of a plane horizontal layer heated laterally and cooled from above develops under conditions of a constant temperature gradient averaged over the layer thickness. The flow core is modeled by a horizontal layer with a moving upper boundary and with adiabatic bounding surfaces under conditions of a constant horizontal gradient of temperature. Exact solutions of free convection equations are found for this model in the Boussinesq approximation. Model results are compared with experimental data. Temperature and flow velocity ranges are determined for the boundary layer regime. Based on the experimental flow velocity profiles, an expression is found for the flow velocity profile in a horizontal layer with a mobile upper boundary heated laterally and cooled from above. Free convection velocity profiles are obtained for the asthenosphere beneath a mid-ocean ridge (MOR) with a mobile lithosphere. An expression is obtained for the tangential stress at the top of the asthenosphere beneath an MOR and the total friction force produced by the asthenospheric flow at the asthenosphere-lithosphere boundary is determined.     

Hydraulics of a partially penetrating well with skin zone in a confined aquifer

A steady/quasi-steady model is developed for predicting flow into a partially penetrating well with skin zone in a confined aquifer overlying an impervious layer. The model takes into account flow through the bottom of the wellbore, finite skin thickness and finite horizontal and vertical extent of the aquifer. Moreover, the solution can be easily extended to include the mixed-type boundary condition at the well face, where a Dirichlet in the form of a specified hydraulic head and a Neumann in the form of zero flux coexist at the same time at different portions of the well face. The validity of the proposed solution is tested by comparing a few results obtained from the developed model with corresponding results obtained by analytical and numerical means. The study shows that, among other factors remaining constant, both the horizontal and vertical extent of an artesian aquifer, thickness of the skin zone, bottom flow and conductivity contrast of the skin and formation zones, play an important part in deciding flow to a well dug in the aquifer, and hence these factors must be considered while analyzing the problem. The model proposed here can be used to estimate skin thickness as well as hydraulic conductivities of the skin and formation zones of a well with skin zone in an artesian aquifer underlain by an impervious layer by utilizing pumping test data falling in the steady or quasi-steady state of a typical pumping test. As the proposed solution is of a general nature in the sense that it can handle, apart from partial penetration and bottom flow, the finite size skin zone and finite horizontal and vertical extent of an artesian aquifer together with the mixed-type boundary condition at the well face, it is hoped that the predictions coming out of the model will be more realistic than those obtained using solutions developed with more stringent assumptions.     

The existence of a thin low-viscosity layer beneath the lithosphere

The horizontal temperature gradient at the base of the lithosphere at an oceanic fracture zone, where plate of different ages is juxtaposed, is expected to drive a local circulation, the characteristics of which can be constrained by the amplitude, wavelength and age-dependence of the geoid. Two-dimensional numerical models of convection in a fluid layer overlain by a solid conducting lid have been used to generate theoretical geoid profiles at right angles to the fracture zone. Only a thin, low-viscosity layer provides a reasonable fit to the data. The best model so far obtained has a fluid layer 150 km thick with viscosity 1.5 × 10¹⁹ Pa s under a 75 km lid. Such a layer, which is incapable of transmitting strong horizontal shear stresses, could provide the decoupling mechanism between plate and deep mantle flow required to balance the forces on the plates.     

Heat transfer between a thermochemical plume channel and the surrounding mantle in the presence of horizontal mantle flow

Heat and mass transfer processes in the conduit of a thermochemical plume located beneath an oceanic plate far from a mid-ocean ridge (MOR) proceed under conditions of horizontal convective flows penetrating the plume conduit. In the region of a mantle flow approaching the plume conduit (in the frontal part of the conduit), the mantle material heats and melts. The melt moves through the plume conduit at the average velocity of flow v and is crystallized on the opposite side of the conduit (in the frontal part of the conduit). The heat and the chemical dope transferred by the conduit to the mantle flow are carried away by crystallized mantle material at the velocity v.The local coefficients of heat transfer at the boundary of the plume conduit are theoretically determined and the balance of heat fluxes through the side of the plume conduit per linear meter of the conduit height. The total heat generation rate, transmitted by the Hawaiian plume into the upper and lower mantle, is evaluated. With the use of regular patterns of heat transfer in the lower mantle, which is modeled on the horizontal layer, heated from below and cooled from above, the diameter of the plume source, the kinematic viscosity of the melt in the plume conduit, and the velocity of horizontal lower-mantle flows are evaluated and the dependences of the temperature drop, viscosity and Rayleigh number for the lower mantle on the diameter of the plume source are presented.     

Evidence for volatile-influenced differentiation in a layered alkali basalt flow, Penghu Islands, Taiwan

An approximately 20-m-thick alkali basalt flow on the Penghu Islands contains ∼20 cm thick, horizontally continuous (>50 m), vesicular layers separated by ∼1.5 m of massive basalt in its upper 8.5 m. The three layers contain ocelli-like "vesicles" filled with nepheline and igneous carbonate. They are coarse grained and enriched in incompatible elements relative to the massive basalt with which they form sharp contacts. These vesicular layers (segregation veins) formed when residual liquid in the underlying crystal mush was forced (gas filter pressing) or siphoned into three thermally induced horizontal cracks that opened successively in the advancing crystal mush of the flow's upper crust. Most vesicular layer trace elements can be modelled by residual melt extraction after 25–40% fractional crystallization of massive basalt underlying each layer. Sulphur, Cl, As, Zn, Pb, K, Na, Rb, and Sr show large concentration changes between the top, middle, and bottom layers, with each vesicular and underlying massive basalt forming a chemically distinct "pair." The large changes between layers are difficult to account for by crystal fractionation alone, because other incompatible elements (e.g., La, Sm, Yb, Zr, Nb) and the major elements change little. The association of these elements (S, Cl, etc.) with "fluids" in various geologic environments suggests that volatiles influenced differentiation, perhaps by moving alkali, alkaline earth, and chalcophile elements as magma-dissolved volatile complexes. Volatiles may have also led to large grain sizes in the segregation veins by lowering melt viscosities and raising diffusion rates. The chemical variability between layers indicates that a convection and concentration mechanism acted within the flow. The specific process cannot be determined, but different rates of vesicle plume rise (through the flow) and/or accumulation in the upper crust's crystal mush might account for the chemical pairing and extreme variations in Cl, S, As, and C. This study emphasizes the importance of sampling vesicular rocks in flows. It also suggests that volatiles play important physical and chemical roles in rapidly differentiating mafic magmas in processes decoupled from crystal fractionation. Received: 11 November 1996 / Accepted: 20 September 1998     

Dynamical structure and wind-driven upwelling in a summertime anticyclonic eddy within Funka Bay,Hokkaido, Japan

We conducted hydrographic observations in 2002 to investigate the anticyclonic eddy that emerges every summer in Funka Bay, Hokkaido, Japan, and elucidate dynamical structure and wind-driven upwelling within the eddy. The anticyclonic eddy has a vertical scale of 32 m and is characterized by a strong baroclinic flow and a sharp pycnocline with a concave isopycnal structure. The sharp pycnocline occurs below a warm and relatively low-salinity water termed summer Funka Bay water (FS), which is formed by heating from solar radiation and dilution from river discharge in summertime Funka Bay. Flow of the anticyclonic eddy rotates as a rigid body at each layer, and the horizontal scale and rotation period of the eddy in the surface layer are about 15 km and 2.2 days, respectively. The dynamical balance of the anticyclonic eddy is well explained by the gradient flow balance. The contribution of centrifugal force to the gradient flow balance is about 27%. Therefore, the effect of the nonlinear term associated with centrifugal force cannot be neglected in considering the dynamics of the anticyclonic eddy in summertime Funka Bay. In addition, upwelling of subsurface water was observed in the surface layer of the central part of the eddy. The formation mechanism of this upwelling is consistent with interaction between horizontal uniform wind and the eddy. This upwelling is driven by upward Ekman pumping velocity related to the horizontal divergence of Ekman transport. In summertime Funka Bay, there are two wind effects that affect the anticyclonic eddy: a decay effect of the upwelling of subsurface water resulting from horizontal uniform wind (mainly northwesterly wind), and a maintenance or spin-up effect of horizontal non-uniform wind (mainly southerly–southeasterly seasonal wind) with negative wind stress curl.     

Neutral planetary boundary layer above a nonhomogeneous surface

Abstract

Theoretical analyses of the atmospheric boundary layer are generally based on the assumptions of stationarity and horizontal homogeneity, which permit one to neglect the inertial terms in equations of motion. The importance of these terms is examined for the case where slight inhomogeneities may cause small but monotonic changes of surface shear in the direction of flow. Considering several models of the neutral barotropic Ekman layer, it is shown that horizontal gradient in the friction velocity of as small as one cm/sec per ten kilometers could make the wind hodograph deviate significantly from that for the homogeneous case. Significant deviations from the logarithmic profile in the surface layer, on the other hand, would occur for horizontal gradients of at least an order of magnitude greater. Thus, the practical criteria for assuming homogeneity may be far more strict and difficult to satisfy for the outer layer than these are for the surface layer.     

Heat and mass transfer in a thermochemical plume under an oceanic plate far from the mid-ocean ridge axis

Heat and mass transfer processes in the conduit of a thermochemical plume located beneath an oceanic plate far from a mid-ocean ridge (MOR) proceed under conditions of horizontal convective flows penetrating the plume conduit. In the region of a mantle flow approaching the plume conduit (in the frontal part of the conduit), the mantle material heats and melts. The melt moves through the plume conduit at the average velocity of flow v and is crystallized on the opposite side of the conduit. The heat and the chemical dope transferred by the conduit to the mantle flow are carried away by crystallized mantle material at the velocity v. The main equations of heat and mass transfer are obtained for a thermochemical plume interacting with a horizontal convective mantle flow. The joint multiparameter problem of heat and mass transfer is solved for a thermochemical plume located far from an MOR axis. The dope concentration at the base of the plume is found as a function of the Lewis number. The Lewis numbers and, accordingly, the diffusion coefficients of the chemical dope in the plume conduit far from the MOR axis are determined.     

The equatorial dynamics of a shallow,homogeneous ocean

Abstract

It is shown that the linear equatorial dynamics of a shallow ocean is characterized by two boundary layers of width γ^? L and γL (γ is the Ekman number of the flow, assumed small, and L is a horizontal dimension of the basin). In the γ^? layer stress in the bottom Ekman layer is comparable to that in the surface Ekman layer. In the γ layer vertical friction is important throughout the depth of the ocean. Should the Rossby number ? be so large as to invalidate a linear theory (? > γ^5/3), then inertial effects become important at a distance ?^2/5 L from the equator. The role played in the circulation of the basin by the non-linear equatorial current first studied by Charney (1960) is shown to be similar to that of the γ layer of the linear theory. Though lateral friction is unimportant in a linear model of the flow, shear layers at the equator are found to be a necessary feature of non-linear flow.     

Upper-ocean transport mechanisms from the Gulf of Maine to Georges Bank, with implications for Calanus supply

Potential upper-ocean pathways for the supply of biota from the Gulf of Maine to Georges Bank are investigated by numerically tracking particles in realistic 3-d seasonal-mean and tidal flow fields. The flow fields, obtained from a prognostic model forced by observed M₂ tides and seasonal-mean wind stress and density fields, include the major known observational features of the circulation regime in winter, spring and summer — a wind-driven surface layer (in winter and early spring) overlying seasonally-evolving baroclinic and tidally-rectified topographic gyres. The surface layer in winter and early spring, with generally southward drift for typical northwesterly wind stress, can act as a conveyor belt for the transport of biota to Georges Bank, provided that the biota can spend a substantial fraction of time in the surface Ekman layer. The numerical experiments indicate that the upper-ocean drift pathways for biota in the southern Gulf of Maine are strongly sensitive to biological and/or physical processes affecting vertical position in relation to the surface Ekman layer and horizontal position in relation to topographic gyres. The seasonality and location of the identified pathways are generally consistent with observed distributional patterns of Calanus finmarchicus based on the 11-year MARMAP surveys.     

Concerning the multiscale structure of solar convection

The discrete scale spectrum of the convective flows observed on the Sun has not yet received a convincing explanation. Here, an attempt is made to find conditions for the coexistence of convective flows on various scales in a horizontal fluid layer heated from below, where the thermal diffusivity varies with temperature in such a way that the static temperature difference across a thin sublayer near the upper surface of the layer is many times larger than the temperature variation across the remainder of the layer. The equations of two-dimensional thermal convection are solved numerically in an extended Boussinesq approximation, which admits thermal-diffusivity variations. The no-slip conditions are assumed at the lower boundary of the layer; either no-slip or free-slip conditions, at the upper boundary. In the former case, stable large-scale rolls develop, which experience small deformations under the action of small structures concentrated near the horizontal boundaries. In the latter case, the flow structure is highly variable, different flow scales dominate at different heights, the number of large rolls is not constant, and a sort of intermittency occurs: the enhancement of the small-scale flow component is frequently accompanied by the weakening of the large-scale one, and vice versa. The scale-splitting effects revealed here should manifest themselves in one way or another in the structure of solar convection.     

Convection in a rotating cylinder with its sidewall having a finite thermal conductance

Abstract

An investigation is made of steady thermal convection of a Boussinesq fluid confined in a vertically-mounted rotating cylinder. The top and bottom endwall disks are thermal conductors at temperatures T_t and T_b with δT = T_t ? T_b >0. The vertical sidewall has a finite thermal conductance. A Newtonian heat flux condition is adopted at the sidewall. The Rayleigh number of the fluid system is large to render a boundary layer-type flow. Finite-difference numerical solutions to the full Navier-Stokes equations are obtained. The vertical motions within the buoyancy layer along the sidewall induce weak meridional flows in the interior. Because of the Coriolis acceleration, the meridional flows give rise to azimuthal flows relative to the rotating container. Strong vertical gradients of azimuthal flows exist in the regions near the endwalls. As the stratification effect increases, concentration of flow gradients in thin endwall boundary layers becomes more pronounced. The azimuthal flow field exhibits considerable horizontal gradients. The temperature field develops horizontal variations superposed on the dominant vertical distribution. As either the sidewall thermal conductance or the stratification effect decreases, the temperature distribution tends to the profile varying linearly with height. Comparisons of the sizes of the dynamic effects demonstrate that, in the bulk of flow field, the vertical shear of azimuthal velocity is supported by the horizontal temperature gradient, resulting in a thermal-wind relation.     

Intermittent convection

Abstract

A theoretical analysis of pseudo two-dimensional, finite-amplitude, thermal convection is made for an infinite Prandtl number fluid which is subjected to a constant heat flux out of the top boundary and insulated at the bottom. For large Rayleigh numbers the convective flow becomes intermittent and the system is characterized by the following cyclic process: the formation of a thermal boundary layer by diffusion, the instability of this layer when it becomes sufficiently thick, the destruction of the layer by the convective flow, the dying down of the convection, and the reforming of the thermal boundary layer by diffusion. The periodicity and the horizontal wave number of the intermittent convective flow are found to be independent of the depth of the fluid layer but depend on the rate of cooling and the properties of the fluid.     

Boundary layer models of ocean floor spreading

Summary The equations of conservations of momentum and energy scaled with the characteristic values of the mantle indicate the presence of the upper boundary layer to produce the estimated rate of the ocean floor spreading by convection and the importance of the frictional heating. The depth of the upper boundary layer can be estimated from the balance of the viscous force with the horizontal pressure gradient at the sea floor. It is of the orders of 100 km and becomes deeper for the Pacific than for the Atlantic Ocean and also with frictional heating than without it. The frictional heating increases the surface heat flow of the heat conduction by ten to twenty percent for the Pacific Ocean but only by a few percent for the Atlantic Ocean. The similarity solutions are determined for the temperature and horizontal velocity in the upper boundary layer. These solutions are expressed in power series of the variabley x ^–n , wherex, y, andn are horizontal and vertical coordinates and numerical constant, respectively. Both temperature and horizontal velocity within the boundary layer are higher for the Pacific than for the Atlantic Ocean. When a larger viscosity is applied, it causes the increase of horizontal velocity below the surface because of the surface boundary conditions of the finite velocity and of vanishment of the velocity shear. The higher horizontal velocity generates higher temperature because it advects hotter material from the mid-ocean ridge site. The direct effect of frictional heating on the temperature distribution of the similarity solution is almost negligible, since the shear zone is deep and near the lower boundary of the upper boundary layer. In the similarity solution, the surface heat flow which is increased by the frictional heating is given as the boundary value. The effect of the frictional heating is important below the mid-ocean ridge.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

Numerical Study on Vortices in the Middle Layer of Flow around a Large Mountain under Rotating Stratified Conditions

Generation of cyclonic vortices in the middle layer of flow around a large mountain like Tibet and Rocky was investigated by means of a 3-D nonhydrostatic meteorological prognostic model. Special attention was paid to the effects of the earth’s rotation and stratification on the vortices detached successively from the slope of a high and large horizontal scale mountain. It was found the successive formation and detachment of such ‘von Karman-like vortices’ occurred in the flow regime at high Rossby numbers Ro and low Froude numbers Fr. It was successfully divided by the criterion of baroclinic instability. This means that if the condition is unstable baroclinically, a lee vortex is destabilized into a three-dimensional one, while under baroclinically stable conditions the lee vortex with vertical axis retains its standing structure and remains long lasting in the middle layer.     

On similarity solution of an unsteady laminar boundary layer along a flat plate

Summary A unified analysis has been made to obtain all possible similarity solutions of the steady and unsteady, forced flow, inside a boundary layer along a flat plate. Though previously, attempts were made to obtain similarity solutions of a steady boundary layer flow neglecting viscous dissipation term in the energy conservation equation but the treatments were not complete. Here we have taken account of the viscous dissipation term. In the steady case it has been shown that for a similarity solution of both velocity and temperature, there should be a relation between the undisturbed flow outside the boundary layer and the temperature of the plate. It has been shown that the similarity solution exists in the unsteady case if we neglet the viscous dissipation term in the energy equation.     

智利三联点南部含洋脊海洋板块俯冲过程的数值模拟

“三联点”是全球板块运动系统的重要组成部分.扩张的智利洋脊向南美板块俯冲形成了智利三联点,并造成了智利三联点以南数百千米范围内剧烈的地形变化.智利三联点区域的初始板块俯冲角度、洋脊扩张速率等因素的差异对南美板块岩石层热结构及地形起伏造成了显著影响.本文采用有限差分方法,构建了智利三联点区域洋脊俯冲的二维数值模型,模拟洋脊俯冲的动力学过程,探究扩张洋脊俯冲过程中的岩石层变形机制、地表热流的分布特征及其影响因素.结果表明,与俯冲速率相比,俯冲角度对地表热流的影响相对较大,较小俯冲角度（15°）情况下的计算热流分布与基于观测点的插值数据较为吻合,并且俯冲速率越大,上覆板块岩石层温度异常区的水平长度越大,其距海沟的距离最大可至190 km.我们的模拟结果支持智利三联点南部南极洲板块以低角度俯冲至南美大陆板块下方的观点.