Numerical simulations of transport of non-ergodic solute plumes in heterogeneous aquifers

Transport of non-ergodic solute plumes by steady-state groundwater flow with a uniform mean velocity, μ, were simulated with Monte Carlo approach in a two-dimensional heterogeneous and statistically isotropic aquifer whose transmissivity, T, is log-normally distributed with an exponential covariance. The ensemble averages of the second spatial moments of the plume about its center of mass, <S _{i i} (t)>, and the plume centroid covariance, R _{i i} (t) (i=1,2), were simulated for the variance of Y=log T, σ _Y ²=0.1, 0.5 and 1.0 and line sources normal or parallel to μ of three dimensionless lengths, 1, 5, and 10. For σ _Y ²=0.1, all simulated <S _{i i} (t)>−S _{i i} (0) and R _{i i} (t) agree well with the first-order theoretical values, where S _{i i} (0) are the initial values of S _{i i} (t). For σ _Y ²=0.5 and 1.0 and the line sources normal to μ, the simulated longitudinal moments, <S ₁₁(t)>−S ₁₁(0) and R ₁₁(t), agree well with the first-order theoretical results but the simulated transverse moments <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) are significantly larger than the first-order values. For the same two larger values of σ _Y ² but the line sources parallel to μ, the simulated <S ₁₁(t)>−S ₁₁(0) are larger than but the simulated R ₁₁ are smaller than the first-order values, and both simulated <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) stay larger than the first-order values. For a fixed value of σ _Y ², the summations of <S _{i i} (t)>−S _{i i} (0) and R _{i i} , i.e., X _{i i} (i=1,2), remain almost the same no matter what kind of source simulated. The simulated X ₁₁ are in good agreement with the first-order theory but the simulated X ₂₂ are significantly larger than the first-order values. The simulated X ₂₂, however, are in excellent agreement with a previous modeling result and both of them are very close to the values derived using Corrsin's conjecture. It is found that the transverse moments may be significantly underestimated if less accurate hydraulic head solutions are used and that the decreasing of <S ₂₂(t)>−S ₂₂(0) with time or a negative effective dispersivity, defined as , may happen in the case of a line source parallel to μ where σ _Y ² is small.     

Numerical simulations of transport of non-ergodic solute plumes in heterogeneous aquifers

Transport of non-ergodic solute plumes by steady-state groundwater flow with a uniform mean velocity, μ, were simulated with Monte Carlo approach in a two-dimensional heterogeneous and statistically isotropic aquifer whose transmissivity, T, is log-normally distributed with an exponential covariance. The ensemble averages of the second spatial moments of the plume about its center of mass, <S _{i i} (t)>, and the plume centroid covariance, R _{i i} (t) (i=1,2), were simulated for the variance of Y=log T, σ _Y ²=0.1, 0.5 and 1.0 and line sources normal or parallel to μ of three dimensionless lengths, 1, 5, and 10. For σ _Y ²=0.1, all simulated <S _{i i} (t)>−S _{i i} (0) and R _{i i} (t) agree well with the first-order theoretical values, where S _{i i} (0) are the initial values of S _{i i} (t). For σ _Y ²=0.5 and 1.0 and the line sources normal to μ, the simulated longitudinal moments, <S ₁₁(t)>−S ₁₁(0) and R ₁₁(t), agree well with the first-order theoretical results but the simulated transverse moments <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) are significantly larger than the first-order values. For the same two larger values of σ _Y ² but the line sources parallel to μ, the simulated <S ₁₁(t)>−S ₁₁(0) are larger than but the simulated R ₁₁ are smaller than the first-order values, and both simulated <S ₂₂(t)>−S ₂₂(0) and R ₂₂(t) stay larger than the first-order values. For a fixed value of σ _Y ², the summations of <S _{i i} (t)>−S _{i i} (0) and R _{i i} , i.e., X _{i i} (i=1,2), remain almost the same no matter what kind of source simulated. The simulated X ₁₁ are in good agreement with the first-order theory but the simulated X ₂₂ are significantly larger than the first-order values. The simulated X ₂₂, however, are in excellent agreement with a previous modeling result and both of them are very close to the values derived using Corrsin's conjecture. It is found that the transverse moments may be significantly underestimated if less accurate hydraulic head solutions are used and that the decreasing of <S ₂₂(t)>−S ₂₂(0) with time or a negative effective dispersivity, defined as , may happen in the case of a line source parallel to μ where σ _Y ² is small.     

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.     

Low-salinity plumes in the oceanic region of the Basque Country

Low-salinity waters, within the upper layers of the water column, have been observed in the oceanic region of the southeastern limit of the Bay of Biscay (in March, 2007). This contribution assesses the potential role of large surface freshwater discharges from the Adour (France), Nervión, Oria, Deba, Urola, Urumea and Bidasoa (Basque Country) rivers, to explain the presence of these low-salinity waters. Such discharges, within the offshore waters, reached to at least 50 m in depth; likewise, extending over 15–20 km, in the horizontal. This pattern is confirmed by field data collected by offshore buoys, model results, and the analysis of satellite images. The presence of the low-salinity waters is corroborated by numerical simulations, performed by the ROMS (Regional Ocean Modeling System) hydrodynamic model, with the incorporation of river discharges. In order to simulate the freshwater movements and to identify their origin, particles were released at the river mouths and dispersed (by a Lagrangian particle-tracking model, LPTM), supplied by 3-hourly current fields derived from the ROMS model. The plumes of the Nervión, Oria, Urola and Urumea rivers appear to be the main factors contributing to the low-salinity field, at the offshore locations of the buoys. This pattern is related mainly to the prevailing oceano-meteorological conditions, over the period of analysis.     

Positive solution of two-dimensional solute transport in heterogeneous aquifers

The transport of contaminants in aquifers is usually represented by a convection-dispersion equation. There are several well-known problems of oscillation and artificial dispersion that affect the numerical solution of this equation. For example, several studies have shown that standard treatment of the cross-dispersion terms always leads to a negative concentration. It is also well known that the numerical solution of the convective term is affected by spurious oscillations or substantial numerical dispersion. These difficulties are especially significant for solute transport in nonuniform flow in heterogeneous aquifers. For the case of coupled reactive-transport models, even small negative concentration values can become amplified through nonlinear reaction source/sink terms and thus result in physically erroneous and unstable results. This paper includes a brief discussion about how nonpositive concentrations arise from numerical solution of the convection and cross-dispersion terms. We demonstrate the effectiveness of directional splitting with one-dimensional flux limiters for the convection term. Also, a new numerical scheme for the dispersion term that preserves positivity is presented. The results of the proposed convection scheme and the solution given by the new method to compute dispersion are compared with standard numerical methods as used in MT3DMS.     

Climate response to astronomical forcing

Power spectrum analysis of the oxygen isotopic records of deep-sea cores cannot be directly used for correlation studies with the astronomical forcing functions because of the rapidity of the major deglaciations. The ensuing step change in the isotopic curves, repeated at somewhat irregular intervals, produces, in fact, substantial spectral energy in a wide frequency band that includes the frequencies of the precessional parameters.The isotopic fluctuations within the ramp sections of a composite core curve have been found to be highly correlated with the precessional parameters, indicating a linear response of climate to these parameters and verifying, at the same time, the essential correctness of the time scale adopted.A direct correlation of eccentricity with the major glacial/interglacial cycle is doubtful because some of the eccentricity minima have absolute values of the same order as some of the maxima, while the isotopic amplitude remains essentially constant. In order to test if the major cycle could be related to a terrestrial parameter exhibiting an asymmetric relaxation fluctuation, we have added an asymmetric saw-tooth function to an artificial curve constructed from the precessional parameters on the basis of optimal response amplitude. The curve thus generated reproduces the original isotopic curve with a similarity sufficient to warrant a close search and analysis of the relaxation process.     

Laplace transform solutions for solute transport in fissured aquifers

The main processes affecting the migration of a solute in a fissured aquifer will be advection and dispersion in the fissures, diffusion into the porous matrix; and adsorption. This paper considers solute transport in an idealized fissured aquifer consisting of slabs of saturated rock-matrix separated by equally spaced, planar fissures. The solution of the transport equations is developed as far as Laplace transforms of the solute concentrations in the fissure and matrix water. Numerical inversion of the transforms is used to investigate characteristic behaviour of the model for a number of special cases.     

Stochastic modelling of groundwater flow and solute transport in aquifers

This paper presents an introductory overview of recently developed stochastic theories for tackling spatial variability problems in predicting groundwater flow and solute transport. Advantages and limitations of the theories are discussed. Lastly, strategies based on the stochastic approaches to predict solute transport in aquifers are recommended.     

Visualization of aquifer response to periodic forcing

Many studies of periodic forcing and response in aquifers have focused on describing the induced fluctuations in hydraulic head, without much consideration of the time-dependent flows. Visualization techniques presented in this paper can be applied to obtain a more physically intuitive impression of groundwater motion in aquifers that undergo periodic fluctuations of hydraulic head. The concepts of velocity ellipse and displacement ellipse are introduced as methods for visualizing oscillatory flows associated with individual forcing modes. Cyclical trajectories illustrate the potential complexity of flow paths that can arise due to superposition of modal responses. The full periodic motion that results due to superposition of the mean flow and modal flows is visualized using streaklines. An animated time series of streaklines provides an intuitive impression of the flow and affords insight into apparent dispersion phenomenon that can arise due to periodic fluctuations in both the strength and direction of groundwater flow. Electronic animations are available from the authors.     

Hillslope response to tectonic forcing in threshold landscapes

Hillslopes are thought to poorly record tectonic signals in threshold landscapes. Numerous previous studies of steep landscapes suggest that large changes in long‐term erosion rate lead to little change in mean hillslope angle, measured at coarse resolution. New LiDAR‐derived topography data enables a finer examination of threshold hillslopes. Here we quantify hillslope response to tectonic forcing in a threshold landscape. To do so, we use an extensive cosmogenic beryllium‐10 (¹⁰Be)‐based dataset of catchment‐averaged erosion rates combined with a 500 km² LiDAR‐derived 1 m digital elevation model to exploit a gradient of tectonic forcing and topographic relief in the San Gabriel Mountains, California. We also calibrate a new method of quantifying rock exposure from LiDAR‐derived slope measurements using high‐resolution panoramic photographs. Two distinct trends in hillslope behavior emerge: below catchment‐mean slopes of 30°, modal slopes increase with mean slopes, slope distribution skewness decreases with increasing mean slope, and bedrock exposure is limited; above mean slopes of 30°, our rock exposure index increases strongly with mean slope, and the prevalence of angle‐of‐repose debris wedges keeps modal slopes near 37°, resulting in a positive relationship between slope distribution skewness and mean slope. We find that both mean slopes and rock exposure increase with erosion rate up to 1 mm/a, in contrast to previous work based on coarser topographic data. We also find that as erosion rates increase, the extent of the fluvial network decreases, while colluvial channels extend downstream, keeping the total drainage density similar across the range. Our results reveal important textural details lost in 10 or 30 m resolution digital elevation models of steep landscapes, and highlight the need for process‐based studies of threshold hillslopes and colluvial channels. Copyright © 2012 John Wiley & Sons, Ltd.     

A test case for the simulation of three-dimensional variable-density flow and solute transport in discretely-fractured porous media

A test case has been developed for three-dimensional simulations of variable-density flow and solute transport in discretely-fractured porous media. The simulation domain is a low-permeability porous matrix cube containing a single non-planar fracture. The initial solute concentration is zero everywhere. A constant solute concentration is assigned to the top of the domain, which increases near-top fluid density and induces downward density-driven flow. The test case is therefore comparable to downwelling of a dense brine below a saline disposal basin or a waste repository. Numerous fingers and distinct convection cells develop early in the fracture but the fingers later coalesce and convection becomes less apparent. To help test other variable-density flow and transport models, results of the test case are presented both qualitatively (concentration contours and velocity fields) and quantitatively (penetration depth, mass flux, total mass stored, maximum fracture and matrix velocity).     

Sea surface temperature anomalies driven by oceanic local forcing in the Brazil-Malvinas Confluence

Sea surface temperature (SST) anomaly events in the Brazil-Malvinas Confluence (BMC) were investigated through wavelet analysis and numerical modeling. Wavelet analysis was applied to recognize the main spectral signals of SST anomaly events in the BMC and in the Drake Passage as a first attempt to link middle and high latitudes. The numerical modeling approach was used to clarify the local oceanic dynamics that drive these anomalies. Wavelet analysis pointed to the 8–12-year band as the most energetic band representing remote forcing between high to middle latitudes. Other frequencies observed in the BMC wavelet analysis indicate that part of its variability could also be forced by low-latitude events, such as El Niño. Numerical experiments carried out for the years of 1964 and 1992 (cold and warm El Niño-Southern Oscillation (ENSO) phases) revealed two distinct behaviors that produced negative and positive sea surface temperature anomalies on the BMC region. The first behavior is caused by northward cold flow, Río de la Plata runoff, and upwelling processes. The second behavior is driven by a southward excursion of the Brazil Current (BC) front, alterations in Río de la Plata discharge rates, and most likely by air-sea interactions. Both episodes are characterized by uncoupled behavior between the surface and deeper layers.     

Topographic forcing of supercritical convection in a porous medium such as the oceanic crust

We have performed laboratory experiments using a Hele-Shaw cell to model a saturated, porous layer with various sinusoidal upper boundaries. Our intent was to determine the range of conditions over which boundary topography can control the pattern of thermal convection within a porous layer, and thereby take the first step toward understanding why heat flow seems correlated with hypsography in many areas of the ocean floor.These experiments indicate that above the critical Rayleigh number, topography does not control the convection pattern, except when the topographic wavelength is comparable to the depth of water penetration. Scaled to the depth of the layer, the convective wavenumbers are restricted to values between 2.5 and 4.8—a range which brackets π, the natural wavenumber for convection in a porous slab with planar, isothermal, impermeable boundaries. Topographies within this range control the circulation pattern perfectly, with downwelling under valleys and upwelling aligned with topographic highs. Other topographies do not force the pattern, although in some cases, the convection wavenumber may be a harmonic of the topographic wavenumber. Unforced circulation cells wander and vary in size, because they are not locked to the topography.For these experiments we employed eight different topographies with non-dimensional wavenumbers between 1.43 and 8.17, and we studied the flow at Rayleigh numbers between zero and five times the critical Rayleigh number. The amplitude of each topography tapered linearly (over a factor of three to six) from one end of the apparatus to the other, and the mean topographic amplitude was 0.05 times the depth of the layer. Under these conditions, amplitude has only a minor effect on the structural form and vigor of supercritical convection.Our results may apply to submarine geothermal systems, sealed by a thin layer of impermeable sediment draped over the basement topography. In this case, the convection wavelength—as measured perhaps by the spatial periodicity of conductive heat flow—may be a good measure of the depth to which water penetrates the crust. Where the circulation correlates with the bottom topography, it may be because the topographic wavelength is comparable to the depth to which water penetrates the porous crust.     

Analytical solutions for the coefficient of variation of the volume-averaged solute concentration in heterogeneous aquifers

Under the assumption that local solute dispersion is negligible, a new general formula (in the form of a convolution integral) is found for the arbitrary k-point ensemble moment of the local concentration of a solute convected in arbitrary m spatial dimensions with general sure initial conditions. From this general formula new closed-form solutions in m=2 spatial dimensions are derived for 2-point ensemble moments of the local solute concentration for the impulse (Dirac delta) and Gaussian initial conditions. When integrated over an averaging window, these solutions lead to new closed-form expressions for the first two ensemble moments of thevolume-averaged solute concentration and to the corresponding concentration coefficients of variation (CV). Also, for the impulse (Dirac delta) solute concentration initial condition, the second ensemble moment of thesolute point concentration in two spatial dimensions and the corresponding CV are demonstrated to be unbound. For impulse initial conditions the CVs for volume-averaged concentrations axe compared with each other for a tracer from the Borden aquifer experiment. The point-concentration CV is unacceptably large in the whole domain, implying that the ensemble mean concentration is inappropriate for predicting the actual concentration values. The volume-averaged concentration CV decreases significantly with an increasing averaging volume. Since local dispersion is neglected, the new solutions should be interpreted as upper limits for the yet to be derived solutions that account for local dispersion; and so should the presented CVs for Borden tracers. The new analytical solutions may be used to test the accuracy of Monte Carlo simulations or other numerical algorithms that deal with the stochastic solute transport. They may also be used to determine the size of the averaging volume needed to make a quasi-sure statement about the solute mass contained in it.     

Influence of conduit network geometry on solute transport in karst aquifers with a permeable matrix

In karst aquifers with significant matrix permeability, water and solutes are exchanged between the conduits and carbonate matrix. Transport through the matrix increases the spread of solutes and increases travel times. This study numerically evaluates advective solute transport in synthetic karst systems that contain 3D branching conduit networks. Particle tracking is performed to analyze the spatial and temporal transport history of solute that arrives at the conduit outlet. Three measures of transport connectivity are used to quantify the solute migration behavior: the skewness of the particle arrival time distribution, the normalized fifth percentile of arrival times, and the fraction of the total travel time that occurs within conduits. All three of these metrics capture the influence of conduit network geometry on solute transport. A more tortuous network leads to enhanced conduit-matrix mixing, which reduces the transport connectivity and yields a broader distribution of solute arrival times. These results demonstrate that the conduit network geometry is an important control on solute transport in karst systems with a permeable matrix.     

On the effects of preferential or barrier flow features on solute plumes in permeable porous media

Despite that discrete flow features (DFFs, e.g. fractures and faults) are common features in the subsurface, few studies have explored the influence of DFFs on solute plumes in otherwise permeable rocks (e.g. sandstone, limestone), compared to low-permeability rock settings (e.g. granite and basalt). DFFs can provide preferential flow pathways (i.e. ‘preferential flow features’; PFFs), or can act to impede flow (i.e. ‘barrier flow features’; BFFs). This research uses a simple analytical expression and numerical modelling to explore how a single DFF influences the steady-state distributions of solute plumes in permeable aquifers. The analysis quantifies the displacement and widening (or narrowing) of a steady-state solute plume as it crosses a DFF in idealised, 1 × 1 m moderately permeable rock aquifers. Previous research is extended by accounting for DFFs as 2D flow features, and including BFF situations. A range of matrix-DFF permeability ratios (0.01 to 100) and DFF apertures (0.25 mm to 2 cm), typical of sedimentary aquifers containing medium-to-large fractures, are considered. The results indicate that for the conceptual models considered here, PFFs typically have a more significant influence on plume distributions than BFFs, and the impact of DFFs on solute plumes generally increases with increasing aperture. For example, displacement of peak solute concentration caused by DFFs exceeds 20 cm in some PFF cases, compared to a maximum of 0.64 cm in BFF cases. PFFs widen plumes up to 9.7 times, compared to a maximum plume widening of 2.0 times in BFF cases. Plumes crossing a PFF are less symmetrical, and peak solute concentrations beneath PFFs are up to two orders of magnitude lower than plumes in BFF cases. This study extends current knowledge of the attenuating influence of DFFs in otherwise permeable rocks on solute plume characteristics, through evaluation of 2D flow effects in DFFs for a variety of DFF apertures, and by considering BFF situations.     

Modelling of waves in the Irish Sea: effects of oceanic wave and wind forcing

This study uses a series of scenarios of wave (boundary) and wind (local) forcing to examine the sensitivity and to quantify the effects associated with nesting ProWAM and POLCOMS models for downscaling predictions of waves in the Irish Sea. The model results show that the response of the modelling system to the wave and wind forcing during the downscaling varies widely depending on wind conditions. Generally, the wave forcing has a greater effect on overall wave prediction in most of the Irish Sea, except for the eastern Irish Sea/Liverpool Bay. The study also suggests detailed look-up tables at specific locations to quantify the impacts of the different forcing scenarios over the Irish Sea, which can be readily extended to the location on any other sites.     

The unusual and large drawdown response of buried-valley aquifers to pumping

The buried-valley aquifers that are common in the glacial deposits of the northern hemisphere are a typical case of the strip aquifers that occur in many parts of the world. Pumping from a narrow strip aquifer leads to much greater drawdown and much more distant drawdown effects then would occur in a sheet aquifer with a similar transmissivity and storage coefficient. Widely used theories for radial flow to wells, such as the Theis equation, are not appropriate for narrow strip aquifers. Previously published theory for flow to wells in semiconfined strip aquifers is reviewed and a practical format of the type curves for pumping-test analysis is described. The drawdown response of strip aquifers to pumping tests is distinctive, especially for observation wells near the pumped well. A case study is presented, based on extensive pumping test experience for the Estevan Valley Aquifer in southern Saskatchewan, Canada. Evaluation of groundwater resources in such buried-valley aquifers needs to take into account the unusually large drawdowns in response to pumping.     

The behaviour of beach elevation contours in response to different wave energy environments

Within the context of a warming climate, there are wide and increasing concerns about the way beaches respond to different wave energy environments. However, behavioural differences in changes in beach elevation contours (including shorelines) in different wave energy environments remain unknown. Thus, it is unilateral to evaluate the changes in beaches based on a single elevation contour (e.g. shoreline) in coastal engineering and management applications. In this study, based on the collected shoreline and wave energy data of two international beaches, as well as the measured beach elevation contour data from Yintan Beach and the corresponding wave energy data simulated by Xbeach, our results show that frequency distributions of beach elevation contour changes exhibit distinct features under different wave energy environments. Under high wave energy environments, the frequency distributions of beach elevation contour changes show a Gaussian distribution. However, frequency distributions of beach elevation contour changes present a power law, intermediate between the logarithmic and Gaussian distributions under low and moderate wave energy environments, respectively. Furthermore, the conceptual model of beach elevation contour changes constructed by this study indicates that the relative importance of the wave energy and sediment resistance determines this phenomenon. © 2020 John Wiley & Sons Ltd