Michael G. Trefry

Surface water‐groundwater interaction near shallow circular lakes: Flow geometry in three dimensions

Steady flow regimes for three-dimensional lake-aquifer systems are studied via idealized mathematical models that are extensions of earlier simplified vertical section models of interaction between shallow lakes and underlying aquifers. The present models apply to a shallow circular lake at the surface of a rectangular aquifer of finite depth, yielding a truly three-dimensional representation of the resulting flow system. Flux boundary conditions are applied at the ends of the aquifer, with net vertical recharge or evapotranspiration at the water table. The lake is defined by a region with constant head. By determining and visualizing solutions to the discretized saturated flow equations, a range of possible flow regimes is identified, and their topological properties are studied. Tools for analyzing flow regimes are described, including a method for locating and mapping three-dimensional dividing surfaces within steady flow fields. Results show strong similarities between two- and three-dimensional systems, including a large number of flow-through, recharge, and discharge regimes and reverse flow cells. Flow lines calculated on a vertical plane through the middle of a lake resemble but are not identical to two-dimensional streamlines for a range of aquifer flow and recharge conditions. Estimates of the widths and depths of capture and release zones for various lake-aquifer geometries are asymptotic to earlier results for two-dimensional systems. Numerical predictions are compared with analytical results for certain limiting flow regimes.

Periodic forcing in composite aquifers

Observations of periodic components of measured heads have long been used to estimate aquifer diffusivities. The estimations are often made using well-known solutions of linear differential equations for the propagation of sinusoidal boundary fluctuations through homogeneous one-dimensional aquifers. Recent field data has indicated several instances where the homogeneous aquifer solutions give inconsistent estimates of aquifer diffusivity from measurements of tidal lag and attenuation. This paper presents new algebraic solutions for tidal propagation in spatially heterogeneous one-dimensional aquifers. By building on existing solutions for homogeneous aquifers, comprehensive solutions are presented for composite aquifers comprising of arbitrary (finite) numbers of contiguous homogeneous sub-aquifers and subject to sinusoidal linear boundary conditions. Both Cartesian and radial coordinate systems are considered. Properties of the solutions, including rapid phase shifting and attenuation effects, are discussed and their practical relevance noted. Consequent modal dispersive effects on tidal waveforms are also examined via tidal constituent analysis. It is demonstrated that, for multi-constituent tidal forcings, measured peak heights of head oscillations can seem to increase, and phase lags seem to decrease, with distance from the forcing boundary unless constituents are separated and considered in isolation.

The potential for convection and implications for geothermal energy in the Perth Basin, Western Australia

Convection of groundwater in aquifers can create areas of anomalously high temperature at shallow depths which could be exploited for geothermal energy. Temperature measurements in the Perth Basin (Western Australia) reveal thermal patterns that are consistent with convection in the Yarragadee Aquifer. This observation is supported by Rayleigh number calculations, which show that convection is possible within the range of aquifer thickness, geothermal gradient, salinity gradient and permeability encountered in the Yarragadee Aquifer, assuming that the aquifer can be treated as a homogeneous anisotropic layer. Numerical simulations of convection in a simplified model of the Yarragadee Aquifer show that: (1) the spacing of convective upwellings can be predicted from aquifer thickness and permeability anisotropy; (2) convective upwellings may be circular or elongate in plan view; (3) convective upwellings create significant temperature enhancements relative to the conductive profile; (4) convective flow rates are similar to regional groundwater flow rates; and (5) convection homogenises salinity within the aquifer. Further work is required to constrain the average horizontal and vertical permeability of the Yarragadee Aquifer, to assess the validity of treating the aquifer as a homogeneous anisotropic layer, and to determine the impact of realistic aquifer geometry and advection on convection.RésuméLa convection des eaux souterraines dans les aquifères peut créer des zones de températures anormalement élevées à faible profondeur qui pourraient être exploitées pour l’énergie géothermique. Les mesures de température dans le bassin de Perth (Australie occidentale) révèlent des caractéristiques thermales cohérentes avec une convection dans l’aquifère de Yarragadee. Cette observation est étayée par des calculs du nombre de Rayleigh, qui montrent que la convection est possible pour l’ordre de grandeur d’épaisseur de l’aquifère, le gradient géothermal, le gradient de salinité, et la perméabilité rencontrées dans l’aquifère de Yarragadee, en supposant que l’aquifère puisse être traité comme une couche homogène et anisotrope. Des simulations numériques de la convection dans un modèle simplifié de l’aquifère de Yarragadee montrent que : (1) l’espacement des flux convectifs ascendants peut être prédit d’après l’épaisseur de l’aquifère et l’anisotropie de la perméabilité; (2) les flux convectifs ascendants peuvent être circulaires ou allongés dans une vue en plan ; (3) les flux convectifs ascendants créent des accroissements de température significatifs par rapport au profil du conduit ; (4) les débits des flux convectifs sont similaires aux débits des écoulements régionaux des eaux souterraines; et (5) la convection homogénéise la salinité dans l’aquifère. D’autres travaux sont nécessaires pour tenir compte de la perméabilité moyenne horizontale et verticale de l’aquifère de Yarragadee, pour juger de la validité d’un traitement de l’aquifère comme couche homogène anisotrope, et pour déterminer l’impact d’une géométrie réaliste de l’aquifère et de l’advection sur la convection.ResumenLa convección del agua subterránea en acuíferos puede crear áreas de temperaturas anómalamente altas en profundidades someras que podrían ser explotadas para energía geotérmica. Las medidas de temperatura en la cuenca de Perth (Australia Occidental) revelaron esquemas termales que son consistentes con la convección en el acuífero Yarragadee. Esta observación está apoyada por cálculos del número de Rayleigh, que muestra que la convección es posible dentro del rango de los espesores del acuífero, gradiente geotérmico, gradiente de salinidad y permeabilidad encontradas en el acuífero Yarragadee, suponiendo que el acuífero puede ser tratado como una capa homogénea anisotrópica. Las simulaciones numéricas de convección en un modelo simplificado del acuífero Yarragadee muestra que: (1) El espaciado de surgencias convectivas puede ser predicho a partir del espesor del acuífero y de la anisotropía de la permeabilidad; (2) Las surgencias convectivas pueden ser circulares o alongados en vista en planta; (3) Las surgencias convectivas crean un acrecentamiento significativo de la temperatura en relación al perfil conductivo; (4) Los ritmos de flujos convectivos son similares a los ritmos de flujo regional del agua subterránea; y (5) La convección homogeiniza la salinidad dentro del acuífero. Se requiere trabajo adicional para restringir la permeabilidad promedio vertical y horizontal del acuífero Yarragadee, para evaluar la validez de tratar al acuífero como una capa anisotrópica homogénea y determinar el impacto de la geometría realística del acuífero y de la advección sobre la convección.摘要含水层中地下水的对流可在浅部造成局部高温异常,并用于开采地热能。澳大利亚西部珀斯盆地的测温量热揭示出与Yarragadee含水层对流相一致的热模式。雷利数的计算支持这一观测结果,并表明在Yarragadee含水层中对流发生在一定含水层厚度、地温梯度、盐分梯度以及渗透系数范围内,假定含水层为均质各向异性的。对Yarragadee含水层简化模型的对流数值模拟表明:(1)对流上涌的间距可通过含水层厚度及渗透性各向异性预测;(2)在平面上对流上涌是可循环或拉长的;(3)相对于传导剖面对流上涌可产生更大的温度增强;(4)对流速度与区域上的地下水流速相似;(5)对流使得含水层盐分均一化。进一步工作需要限制Yarragadee含水层的水平和垂直平均渗透系数,用于评价视含水层为均质各向异性的有效性,确定真实含水层尺度及对流中平流的影响。ResumoA conveção em aquíferos pode criar áreas anormais de alta temperatura a profundidades baixas, as quais podem ser exploradas para energia geotérmica. Medições de temperatura na Bacia de Perth (Austrália Ocidental) mostram padrões térmicos consistentes com conveção no Aquífero Yarragadee. Esta observação é apoiada pelos cálculos do número de Rayleigh, os quais mostram que a conveção é possível dentro da gama de espessuras, gradientes geotérmicos, gradientes de salinidade e permeabilidades encontradas no Aquífero Yarragadee, assumindo que o aquífero pode ser tratado como uma camada anisotrópica homogénea. Um modelo simplificado de simulações numéricas de conveção do Aquífero Yarragadee mostra que: (1) O espaçamento de upwellings convetivos pode ser previsto a partir da espessura do aquífero e da anisotropia da permeabilidade, (2) Upwellings convetivos podem ser circulares ou alongados no plano horizontal, (3) Upwellings convetivos criam condições significativamente melhores na temperatura em relação ao perfil condutivo, (4) As taxas de fluxo convetivas são similares às taxas de fluxo regional da água subterrânea, e (5) A conveção homogeneiza a salinidade dentro do aquífero. Torna-se necessário realizar algum trabalho adicional para restringir os parâmetros da permeabilidade média horizontal e vertical do Aquífero Yarragadee, para avaliar a validade de considerar o aquífero como uma camada anisotrópica homogénea, e para determinar o impacte da geometria real do aquífero e da adveção na conveção.

Analytical series expressions for Hantush's M and S functions

M. S. Hantush established relationships between the dynamics of groundwater mounding beneath recharge zones and two integral functions, M and S. Exact algebraic expressions for these functions are developed in terms of a formal power series expansion. This expansion may be reordered to provide two independent analytical partial summations involving elementary functions. The convergence characteristics of these two formulae are discussed and compared with numerical quadratures of M and, hence, S. The algebraic expressions are used to generate identities for related integrals. Compact algebraic approximations to M and S can be deduced from the series expansions with essentially arbitrary accuracy, while retaining valuable functional information. For example, a simple two-term truncated sum yields a reasonable approximation to M over a useful range of arguments. The results are amenable for use in further theoretical studies of groundwater percolation and mounding where numerical quadratures may be undesirable.

The use of mass depletion-mass flux reduction relationships during pumping to determine source zone mass of a reactive brominated-solvent DNAPL

Mass depletion-mass flux relationships usually applied to a groundwater plume were established at field scale for groundwater pumped from within the source zone of a dense non-aqueous phase liquid (DNAPL). These were used as part of multiple lines of evidence in establishing the DNAPL source mass and architecture. Simplified source mass-dissolved concentration models including those described by exponential, power, and error functions as well as a rational mass equation based on the equilibrium stream tube approach were fitted to data from 285 days of source zone pumping (SZP) from a single well which removed 152 kg of dissolved organics from a multi-component, reactive brominated solvent DNAPL. The total molar concentration of the source compound, tetrabromoethane and its daughter products was used as a single measure of contaminant concentration to relate to source mass. A partitioning inter-well tracer test (PITT) conducted prior to the SZP provided estimates of groundwater travel times, enabling parameterisation of the models. After accounting for capture of the down-gradient dissolved plume, all models provided a good fit to the observed data. It was shown that differentiation between models would only emerge after appreciably more pumping from the source zone. The model fits were not particularly sensitive to the exponent parameters and variance of groundwater travel time. In addition, the multi-component nature of the DNAPL did not seem to affect the utility of the models for the period examined. Estimates of the DNAPL mass prior to the start of SZP from the models were greatest where the log of the variance of travel time was used explicitly in the source depletion models (mean 295kg) compared to where the associated power exponent and variance was fitted freely (mean 258 kg). The estimates of source mass were close to that of 220kg determined from the PITT. In addition to the PITT, multi-level groundwater sampling from within the source zone provided important supporting information for developing the conceptual model of the source zone. It is concluded that SZP may be an effective and relatively simple means for characterising DNAPL source zones.

Analytical Solutions for Partitioned Diffusion in Laminates: I. Initial Value Problem with Steady Cauchy Conditions

Studies of the transport of contaminants and nutrients in industrial and environmental systems are complicated by the heterogeneous nature of the supporting porous or permeable media, and by the numerical problems associated with high Peclet number advection and sharp interface models. In order to provide independent theoretical checks of numerical transport theories, this set of papers presents analytical solutions to diffusive transport equations in simplified (one-dimensional) laminate systems subject to partitioning interactions. Here, in Part I, a standard separation of variables technique is used to develop analytical eigenfunction expansions of the concentration solution in an N-laminate system subject to steady Cauchy (third-type) nonhomogeneous boundary conditions. Both Cartesian and radial (axisymmetric) coordinate systems are considered. The solutions are developed for two different interface partitioning formulations, allowing the partitioning processes to be described by instantaneous equilibration mechanisms, or in terms of gradual equilibration mediated by mass transfer coefficients. Worked examples are presented and limitations of the approach discussed.

On oscillating flows in randomly heterogeneous porous media

The emergence of structure in reactive geofluid systems is of current interest. In geofluid systems, the fluids are supported by a porous medium whose physical and chemical properties may vary in space and time, sometimes sharply, and which may also evolve in reaction with the local fluids. Geofluids may also experience pressure and temperature conditions within the porous medium that drive their momentum relations beyond the normal Darcy regime. Furthermore, natural geofluid systems may experience forcings that are periodic in nature, or at least episodic. The combination of transient forcing, near-critical fluid dynamics and heterogeneous porous media yields a rich array of emergent geofluid phenomena that are only now beginning to be understood. One of the barriers to forward analysis in these geofluid systems is the problem of data scarcity. It is most often the case that fluid properties are reasonably well known, but that data on porous medium properties are measured with much less precision and spatial density. It is common to seek to perform an estimation of the porous medium properties by an inverse approach, that is, by expressing porous medium properties in terms of observed fluid characteristics. In this paper, we move toward such an inversion for the case of a generalized geofluid momentum equation in the context of time-periodic boundary conditions. We show that the generalized momentum equation results in frequency-domain responses that are governed by a second-order equation which is amenable to numerical solution. A stochastic perturbation approach demonstrates that frequency-domain responses of the fluids migrating in heterogeneous domains have spatial spectral densities that can be expressed in terms of the spectral densities of porous media properties.

Chaotic advection at the pore scale: Mechanisms, upscaling and implications for macroscopic transport

Abstract The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the pore scale generate chaotic advection—involving exponential stretching and folding of fluid elements—the mechanisms by which it arises and implications of microscopic chaos for macroscopic dispersion and mixing. Prohibited in steady 2D flow due to topological constraints, these phenomena are ubiquitous due to the topological complexity inherent to all 3D porous media. Consequently 3D porous media flows generate profoundly different fluid deformation and mixing processes to those of 2D flow. The interplay of chaotic advection and broad transit time distributions can be incorporated into a continuous-time random walk (CTRW) framework to predict macroscopic solute mixing and spreading. We show how these results may be generalised to real porous architectures via a CTRW model of fluid deformation, leading to stochastic models of macroscopic dispersion and mixing which both honour the pore-scale kinematics and are directly conditioned on the pore-scale architecture.

Impact of Rock Heterogeneity on Interactions of Microbial-Enhanced Oil Recovery Processes

Residual oil saturation reduction and microbial plugging are two crucial factors in microbial-enhanced oil recovery (MEOR) processes. In our previous study, the residual saturation was defined as a nonlinear function of the trapping number, and an explicit relation between the residual oil saturation and the trapping number was incorporated into a fully coupled biological (B) and hydrological (H) finite element model. In this study, the BH model is extended to consider the impact of rock heterogeneity on microbial-enhanced oil recovery phenomena. Numerical simulations of core flooding experiments are performed to demonstrate the influences of different parameters controlling the onset of oil mobilization. X-ray CT core scans are used to construct numerical porosity-permeability distributions for input to the simulations. Results show clear fine-scale fingering processing, and that trapping phenomena have significant effects on residual oil saturation and oil recovery in heterogeneous porous media. Water contents and bacterial distributions for heterogeneous porous media are compared with those for homogenous porous media. The evolution of the trapping number distribution is directly simulated and visualized. It is shown that the oil recovery efficiency of EOR/MEOR will be lower in heterogeneous media than in homogeneous media, largely due to the difficulty in supplying surfactant to unswept low-permeability zones. However, MEOR also provides efficient plugging along high-permeability zones which acts to increase sweep efficiency in heterogeneous media. Thus, MEOR may potentially be more suited for highly heterogeneous media than conventional EOR.

Analytical Solutions for Partitioned Diffusion in Laminates: II. Harmonic Forcing Conditions

The migration of organic compounds in stratified media is of fundamental concern in environmental and chemical engineering research. The diffusive transport of volatile organic compounds through laminate systems is characterized by partitioning, i.e., the development of concentration discontinuities at the interfaces between the individual laminae. If the transport is governed by cyclic transients, the relevant equations can be written in terms of coupled systems of diffusion equations subject to sinusoidal boundary conditions. This paper solves these systems of equations to present new algebraic solutions for propagation of the sinusoidal modes through arbitrary (finite) numbers of contiguous one-dimensional laminae. Both Cartesian and radial coordinate systems are considered. Two independent formulations of the lamina interface matching conditions are considered, corresponding to (1) an instantaneous partitioning model and to (2) a mass-limited partitioning model. It is shown that sinusoidal components of the concentration solutions propagate dispersively throughout the laminates. This is manifested, for example, in changes in shape of concentration pulses as measured at different points in the laminate system. Algorithms for generating exact dispersion relations for partitioning laminates are given, and an experimental technique for studying interfacial dynamics via frequency domain measurements is proposed.