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54-512: The soil moisture velocity equation or SMVE is a Lagrangian reinterpretation of the Eulerian Richards' equation wherein the dependent variable is the position z of a wetting front of a particular moisture content θ {\displaystyle \theta } with time. where: The first term on the right-hand side of the SMVE is called the "advection-like" term, while the second term

108-399: A fluid relates temperature , pressure , and volume in this manner. The triple product rule for such interrelated variables x , y , and z comes from using a reciprocity relation on the result of the implicit function theorem , and is given by where each factor is a partial derivative of the variable in the numerator, considered to be a function of the other two. The advantage of

162-434: A fine water-content discretization and solution method. This equation was converted into a set of three ordinary differential equations (ODEs) using the method of lines to convert the partial derivatives on the right-hand side of the equation into appropriate finite difference forms. These three ODEs represent the dynamics of infiltrating water, falling slugs, and capillary groundwater, respectively. This derivation of

216-568: A function Z R ( θ , t ) {\displaystyle Z_{R}(\theta ,t)} that describes the position of a particular moisture content within the soil using a finite moisture-content discretization. Employing the Implicit function theorem , which by the cyclic rule required dividing both sides of this equation by − ∂ θ / ∂ z {\displaystyle {-\partial \theta }/{\partial z}} to perform

270-1375: A function f ( x , y , z ) = 0 , where x , y , and z are functions of each other. Write the total differentials of the variables d x = ( ∂ x ∂ y ) d y + ( ∂ x ∂ z ) d z {\displaystyle dx=\left({\frac {\partial x}{\partial y}}\right)dy+\left({\frac {\partial x}{\partial z}}\right)dz} d y = ( ∂ y ∂ x ) d x + ( ∂ y ∂ z ) d z {\displaystyle dy=\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz} Substitute dy into dx d x = ( ∂ x ∂ y ) [ ( ∂ y ∂ x ) d x + ( ∂ y ∂ z ) d z ] + ( ∂ x ∂ z ) d z {\displaystyle dx=\left({\frac {\partial x}{\partial y}}\right)\left[\left({\frac {\partial y}{\partial x}}\right)dx+\left({\frac {\partial y}{\partial z}}\right)dz\right]+\left({\frac {\partial x}{\partial z}}\right)dz} By using

324-478: A given porous medium 's properties, we are going to have to measure samples of the porous medium. If the sample is too small, the readings tend to oscillate. As we increase the sample size, the oscillations begin to dampen out. Eventually the sample size will become large enough that we begin to get consistent readings. This sample size is referred to as the representative elementary volume. If we continue to increase our sample size, measurement will remain stable until

378-427: A macroscopically isotropic material). On the other hand, the sample should be small enough to be analyzed analytically or numerically. In continuum mechanics generally for a heterogeneous material, RVE can be considered as a volume V that represents a composite statistically, i.e., volume that effectively includes a sampling of all microstructural heterogeneities (grains, inclusions, voids, fibers, etc.) that occur in

432-413: A short time later t +Δ t (dashed). The wave maintains its shape as it propagates, so that a point at position x at time t will correspond to a point at position x +Δ x at time t +Δ t , This equation can only be satisfied for all x and t if k  Δ x − ω  Δ t = 0 , resulting in the formula for the phase velocity To elucidate the connection with the triple product rule, consider

486-433: Is Where D ( θ ) [L/T] is 'the soil water diffusivity' as previously defined. Note that with θ {\displaystyle \theta } as the dependent variable, physical interpretation is difficult because all the factors that affect the divergence of the flux are wrapped up in the soil moisture diffusivity term D ( θ ) {\displaystyle D(\theta )} . However, in

540-522: Is also referred to as stochastic volume element in finite element analysis, takes into account the variability in the microstructure. Unlike RVE in which average value is assumed for all realizations, SVE can have a different value from one realization to another. SVE models have been developed to study polycrystalline microstructures. Grain features, including orientation, misorientation, grain size, grain shape, grain aspect ratio are considered in SVE model. SVE model

594-450: Is being done relative to the microscale (d). As L/d goes to infinity, the RVE is obtained, while any finite mesoscale involves statistical scatter and, therefore, describes an SVE. With these considerations one obtains bounds on effective (macroscopic) response of elastic (non)linear and inelastic random microstructures. In general, the stronger the mismatch in material properties, or the stronger

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648-523: Is called the "diffusion-like" term. The advection-like term of the Soil Moisture Velocity Equation is particularly useful for calculating the advance of wetting fronts for a liquid invading an unsaturated porous medium under the combined action of gravity and capillarity because it is convertible to an ordinary differential equation by neglecting the diffusion-like term. and it avoids the problem of representative elementary volume by use of

702-531: Is known to not produce fluxes. Another instance when the diffusion-like term will be nearly zero is in the case of sharp wetting fronts, where the denominator of the diffusion-like term ∂ ψ / ∂ z → ∞ {\displaystyle \partial \psi /\partial z\to \infty } , causing the term to vanish. Notably, sharp wetting fronts are notoriously difficult to resolve and accurately solve with traditional numerical Richards' equation solvers. Finally, in

756-444: Is noteworthy that the SMVE advection-like term solved using the finite moisture-content method completely avoids the need to estimate the specific yield . Calculating the specific yield as the water table nears the land surface is made cumbersome my non-linearities. However, the SMVE solved using a finite moisture-content discretization essentially does this automatically in the case of a dynamic near-surface water table. The paper on

810-428: Is very similar to the kinematic wave approximation. In this case, the flux of water to the j th {\displaystyle j^{\text{th}}} bin occurs between bin j and i . Therefore, in the context of the method of lines : and which yields: Note the "-1" in parentheses, representing the fact that gravity and capillarity are acting in opposite directions. The performance of this equation

864-541: The chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero ( ∂ x ∂ y ) ( ∂ y ∂ z ) + ( ∂ x ∂ z ) = 0 {\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)+\left({\frac {\partial x}{\partial z}}\right)=0} Subtracting

918-434: The cyclic chain rule , cyclic relation , cyclical rule or Euler's chain rule , is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics , where frequently three variables can be related by a function of the form f ( x , y , z ) = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for

972-411: The state variables of pressure (P), volume (V), and temperature (T) via which can be written as so each state variable can be written as an implicit function of the other state variables: From the above expressions, we have A geometric realization of the triple product rule can be found in its close ties to the velocity of a traveling wave shown on the right at time t (solid blue line) and at

1026-562: The 1-D soil moisture velocity equation for calculating vertical flux q {\displaystyle q} of water in the vadose zone starts with conservation of mass for an unsaturated porous medium without sources or sinks: We next insert the unsaturated Buckingham–Darcy flux: yielding Richards' equation in mixed form because it includes both the water content θ {\displaystyle \theta } and capillary head ψ ( θ ) {\displaystyle \psi (\theta )} : Applying

1080-469: The Green and Ampt (1911) assumption is employed, represents the capillary head gradient that is driving the flow in the j t h {\displaystyle j^{th}} discretization or "bin". Therefore, the finite water-content equation in the case of infiltration fronts is: After rainfall stops and all surface water infiltrates, water in bins that contains infiltration fronts detaches from

1134-430: The RVE as a sample of a heterogeneous material that: In essence, statement (1) is about the material's statistics (i.e. spatially homogeneous and ergodic ), while statement (2) is a pronouncement on the independence of effective constitutive response with respect to the applied boundary conditions . Both of these are issues of mesoscale (L) of the domain of random microstructure over which smoothing (or homogenization)

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1188-405: The RVE was proposed by Drugan and Willis: The choice of RVE can be quite a complicated process. The existence of a RVE assumes that it is possible to replace a heterogeneous material with an equivalent homogeneous material. This assumption implies that the volume should be large enough to represent the microstructure without introducing non-existing macroscopic properties (such as anisotropy in

1242-566: The SMVE, the three factors that drive flow are in separate terms that have physical significance. The primary assumptions used in the derivation of the Soil Moisture Velocity Equation are that K = K ( θ ) {\displaystyle K=K(\theta )} and ψ = ψ ( θ ) {\displaystyle \psi =\psi (\theta )} are not overly restrictive. Analytical and experimental results show that these assumptions are acceptable under most conditions in natural soils. In this case,

1296-415: The Soil Moisture Velocity Equation is equivalent to the 1-D Richards' equation, albeit with a change in dependent variable. This change of dependent variable is convenient because it reduces the complexity of the problem because compared to Richards' equation , which requires the calculation of the divergence of the flux, the SMVE represents a flux calculation, not a divergence calculation. The first term on

1350-420: The Soil Moisture Velocity Equation was highlighted by the editor in the issue of J. Adv. Modeling of Earth Systems when the paper was first published, and is in the public domain. The paper may be freely downloaded here by anyone. The paper describing the finite moisture-content solution of the advection-like term of the Soil Moisture Velocity Equation was selected to receive the 2015 Coolest Paper Award by

1404-402: The adjacent figure, the RVE consists of a split-ring resonator and its surrounding backing material. There does not exist one RVE size and depending on the studied mechanical properties, the RVE size can vary significantly. The concept of statistical volume element (SVE) and uncorrelated volume element (UVE) have been introduced as alternatives for RVE. Statistical volume element (SVE), which

1458-437: The advection-like SMVE solution against the numerical solution of Richards' equation. The advection-like term of the SMVE can be solved using the method of lines and a finite moisture content discretization . This solution of the SMVE advection-like term replaces the 1-D Richards' equation PDE with a set of three ordinary differential equations (ODEs). These three ODEs are: With reference to Figure 1, water infiltrating

1512-531: The case of dry soils, K ( θ ) {\displaystyle K(\theta )} tends towards 0 {\displaystyle 0} , making the soil water diffusivity D ( θ ) {\displaystyle D(\theta )} tend towards zero as well. In this case, the diffusion-like term would produce no flux. Comparing against exact solutions of Richards' equation for infiltration into idealized soils developed by Ross & Parlange (1994) revealed that indeed, neglecting

1566-623: The case of periodic materials, one simply chooses a periodic unit cell (which, however, may be non-unique), but in random media, the situation is much more complicated. For volumes smaller than the RVE, a representative property cannot be defined and the continuum description of the material involves Statistical Volume Element (SVE) and random fields . The property of interest can include mechanical properties such as elastic moduli , hydrogeological properties, electromagnetic properties, thermal properties, and other averaged quantities that are used to describe physical systems. Rodney Hill defined

1620-482: The chain rule of differentiation to the right-hand side of Richards' equation: Assuming that the constitutive relations for unsaturated hydraulic conductivity and soil capillarity are solely functions of the water content, K = K ( θ ) {\displaystyle K=K(\theta )} and ψ = ψ ( θ ) {\displaystyle \psi =\psi (\theta )} , respectively: This equation implicitly defines

1674-2023: The change in variable, resulting in: ∂ Z R ∂ t = − K ′ ( θ ) ψ ′ ( θ ) ∂ θ ∂ z − K ( θ ) ψ ″ ( θ ) ∂ θ ∂ z − K ( θ ) ψ ′ ( θ ) ∂ 2 θ / ∂ z 2 ∂ θ / ∂ z + K ′ ( θ ) {\displaystyle {\frac {\partial Z_{R}}{\partial t}}=-K'(\theta )\psi '(\theta ){\frac {\partial \theta }{\partial z}}-K(\theta )\psi ''(\theta ){\frac {\partial \theta }{\partial z}}-K(\theta )\psi '(\theta ){\frac {\partial ^{2}\theta /\partial z^{2}}{\partial \theta /\partial z}}+K'(\theta )} , which can be written as: ∂ Z R ∂ t = − K ′ ( θ ) [ ∂ ψ ( θ ) ∂ z − 1 ] − K ( θ ) [ ψ ″ ( θ ) ∂ θ ∂ z + ψ ′ ( θ ) ∂ 2 θ / ∂ z 2 ∂ θ / ∂ z ] {\displaystyle {\frac {\partial Z_{R}}{\partial t}}=-K'(\theta )\left[{\frac {\partial \psi (\theta )}{\partial z}}-1\right]-K(\theta )\left[\psi ''(\theta ){\frac {\partial \theta }{\partial z}}+\psi '(\theta ){\frac {\partial ^{2}\theta /\partial z^{2}}{\partial \theta /\partial z}}\right]} . Inserting

Soil moisture velocity equation - Misplaced Pages Continue

1728-471: The composite. It must however remain small enough to be considered as a volume element of continuum mechanics. Several types of boundary conditions can be prescribed on V to impose a given mean strain or mean stress to the material element. One of the tools available to calculate the elastic properties of an RVE is the use of the open-source EasyPBC ABAQUS plugin tool. Analytical or numerical micromechanical analysis of fiber reinforced composites involves

1782-768: The definition of the soil water diffusivity: into the previous equation produces: ∂ Z R ∂ t = − K ′ ( θ ) [ ∂ ψ ( θ ) ∂ z − 1 ] − D ( θ ) ∂ 2 ψ / ∂ z 2 ∂ ψ / ∂ z {\displaystyle {\frac {\partial Z_{R}}{\partial t}}=-K'(\theta )\left[{\frac {\partial \psi (\theta )}{\partial z}}-1\right]-D(\theta ){\frac {\partial ^{2}\psi /\partial z^{2}}{\partial \psi /\partial z}}} If we consider

1836-416: The departure from elastic behavior, the larger is the RVE. The finite-size scaling of elastic material properties from SVE to RVE can be grasped in compact forms with the help of scaling functions universally based on stretched exponentials. Considering that the SVE may be placed anywhere in the material domain, one arrives at a technique for characterization of continuum random fields. Another definition of

1890-472: The derivatives on the right hand side gives the triple product rule Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the exact differential dz , the ability to construct a curve in some neighborhood with dz  = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on mathematical analysis would eliminate these potential ambiguities. Suppose

1944-527: The diffusion-like term resulted in accuracy >99% in calculated cumulative infiltration. This result indicates that the advection-like term of the SMVE, converted into an ordinary differential equation using the method of lines, is an accurate ODE solution of the infiltration problem. This is consistent with the result published by Ogden et al. who found errors in simulated cumulative infiltration of 0.3% using 263 cm of tropical rainfall over an 8-month simulation to drive infiltration simulations that compared

1998-469: The early career members of the International Association of Hydrogeologists . Representative elementary volume In the theory of composite materials , the representative elementary volume (REV) (also called the representative volume element (RVE) or the unit cell ) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. In

2052-416: The flux and the remaining conductivity term K ′ ( θ ) {\displaystyle K'(\theta )} represents the ability of gravity to conduct flux through the soil. This term is responsible for the true advection of water through the soil under the combined influences of gravity and capillarity. As such, it is called the "advection-like" term. Neglecting gravity and

2106-423: The land surface can flow through the pore space between θ d {\displaystyle \theta _{d}} and θ i {\displaystyle \theta _{i}} . Using the method of lines to convert the SMVE advection-like term into an ODE: Given that any ponded depth of water on the land surface is h p {\displaystyle h_{p}} ,

2160-399: The land surface. Assuming that the capillarity at leading and trailing edges of this 'falling slug' of water is balanced, then the water falls through the media at the incremental conductivity associated with the j th   Δ θ {\displaystyle j^{\text{th}}\ \Delta \theta } bin: This approach to solving the capillary-free solution

2214-456: The point p 1 at time t and its corresponding point (with the same height) p̄ 1 at t +Δ t . Define p 2 as the point at time t whose x-coordinate matches that of p̄ 1 , and define p̄ 2 to be the corresponding point of p 2 as shown in the figure on the right. The distance Δ x between p 1 and p̄ 1 is the same as the distance between p 2 and p̄ 2 (green lines), and dividing this distance by Δ t yields

Soil moisture velocity equation - Misplaced Pages Continue

2268-399: The right-hand side of the SMVE represents the two scalar drivers of flow, gravity and the integrated capillarity of the wetting front. Considering just that term, the SMVE becomes: where ∂ ψ ( θ ) / ∂ z {\displaystyle {\partial \psi (\theta )}/{\partial z}} is the capillary head gradient that is driving

2322-412: The sample size gets large enough that we begin to include other hydrostratigraphic layers. This is referred to as the maximum elementary volume (MEV). Groundwater flow equation has to be defined in an REV. While RVEs for electromagnetic media can have the same form as those for elastic or porous media, the fact that mechanical strength and stability are not concerns allow for a wide range of RVEs. In

2376-609: The scalar wetting front capillarity, we can consider only the second term on the right-hand side of the SMVE. In this case the Soil Moisture Velocity Equation becomes: This term is strikingly similar to Fick's second law of diffusion . For this reason, this term is called the "diffusion-like" term of the SMVE. This term represents the flux due to the shape of the wetting front − D ( θ ) ∂ 2 ψ / ∂ z 2 {\displaystyle -D(\theta ){\partial ^{2}\psi /\partial z^{2}}} , divided by

2430-407: The second derivative will equal zero. One example where this occurs is in the case of an equilibrium hydrostatic moisture profile, when ∂ ψ / ∂ z = − 1 {\displaystyle \partial \psi /\partial z=-1} with z defined as positive upward. This is a physically realistic result because an equilibrium hydrostatic moisture profile

2484-506: The second term and multiplying by its inverse gives the triple product rule ( ∂ x ∂ y ) ( ∂ y ∂ z ) ( ∂ z ∂ x ) = − 1. {\displaystyle \left({\frac {\partial x}{\partial y}}\right)\left({\frac {\partial y}{\partial z}}\right)\left({\frac {\partial z}{\partial x}}\right)=-1.} The ideal gas law relates

2538-492: The spatial gradient of the capillary head ∂ ψ / ∂ z {\displaystyle {\partial \psi /\partial z}} . Looking at this diffusion-like term, it is reasonable to ask when might this term be negligible? The first answer is that this term will be zero when the first derivative < ∂ ψ / ∂ z = C {\displaystyle <\partial \psi /\partial z=C} , because

2592-665: The speed of the wave. To compute Δ x , consider the two partial derivatives computed at p 2 , Dividing these two partial derivatives and using the definition of the slope (rise divided by run) gives us the desired formula for where the negative sign accounts for the fact that p 1 lies behind p 2 relative to the wave's motion. Thus, the wave's velocity is given by For infinitesimal Δ t , Δ x Δ t = ( ∂ x ∂ t ) {\displaystyle {\frac {\Delta x}{\Delta t}}=\left({\frac {\partial x}{\partial t}}\right)} and we recover

2646-443: The study of a representative volume element (RVE). Although fibers are distributed randomly in real composites, many micromechanical models assume periodic arrangement of fibers from which RVE can be isolated in a straightforward manner. The RVE has the same elastic constants and fiber volume fraction as the composite. In general RVE can be considered same as a differential element with a large number of crystals. In order to establish

2700-416: The triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example, Various other forms of the rule are present in the literature; these can be derived by permuting

2754-459: The variables { x , y , z }. An informal derivation follows. Suppose that f ( x , y , z ) = 0. Write z as a function of x and y . Thus the total differential dz is Suppose that we move along a curve with dz = 0, where the curve is parameterized by x . Thus y can be written in terms of x , so on this curve Therefore, the equation for dz = 0 becomes Since this must be true for all dx , rearranging terms gives Dividing by

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2808-1089: The velocity of a particular water content θ {\displaystyle \theta } , then we can write the equation in the form of the Soil Moisture Velocity Equation : d z d t | θ = ∂ K ( θ ) ∂ θ [ 1 − ( ∂ ψ ( θ ) ∂ z ) ] − D ( θ ) ∂ 2 ψ / ∂ z 2 ∂ ψ / ∂ z {\displaystyle \left.{\frac {dz}{dt}}\right\vert _{\theta }={\frac {\partial K(\theta )}{\partial \theta }}\left[1-\left({\frac {\partial \psi (\theta )}{\partial z}}\right)\right]-D(\theta ){\frac {\partial ^{2}\psi /\partial z^{2}}{\partial \psi /\partial z}}} Written in moisture content form, 1-D Richards' equation

2862-448: Was applied in the material characterization and damage prediction in microscale. Compared with RVE, SVE can provide a comprehensive representation of the microstructure of materials. Uncorrelated volume element (UVE) is an extension of SVE which also considers the co-variance of adjacent microstructure to present an accurate length scale for stochastic modelling. Triple product rule The triple product rule , known variously as

2916-467: Was verified, using a column experiment fashioned after that by Childs and Poulovassilis (1962). Results of that validation showed that the finite water-content vadose zone flux calculation method performed comparably to the numerical solution of Richards' equation. The photo shows apparatus. Data from this column experiment are available by clicking on this hot-linked DOI . These data are useful for evaluating models of near-surface water table dynamics. It

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