Homogenization of a multiscale model for water transport in vegetated soil
Abstract
In this paper we consider the multiscale modelling of water transport in vegetated soil. In the microscopic model we distinguish between subdomains of soil and plant tissue, and use the Richards equation to model the water transport through each. Water uptake is incorporated by means of a boundary condition on the surface between root tissue and soil. Assuming a simplified root system architecture, which gives a cylindrical microstructure to the domain, the two-scale convergence and periodic unfolding methods are applied to rigorously derive a macroscopic model for water transport in vegetated soil. The degeneracy of the Richards equation and the dependence of root tissue permeability on the small parameter introduce considerable challenges in deriving macroscopic equations, especially in proving strong convergence. The variable-doubling method is used to prove the uniqueness of solutions to the model, and also to show strong two-scale convergence in the non-linear terms of the equation for water transport through root tissue.
Keywords: Richards equation, degenerate parabolic equations, variable-doubling, dual-porosity, homogenization, two-scale convergence and periodic unfolding
MSC codes: 35B27, 35K51, 35K65
1 Introduction
Mathematical models for water transport in soil are used to inform practice within different environmental contexts such as calculating crop yields [22], monitoring flood risk [15], or assessing hill slope stability [16]. Moreover, predictions from the models are becoming even more important when developing strategies to deal with the increasing impact of climate change. The commonly employed models for soil water transport account for root-soil interactions only by adding a root water uptake sink term to Richards equation, whereas existing dual-porosity models [20, 42] do not relate the architecture of the macropores to the distribution and properties of roots within the soil.
Thus, the main aim of the paper is to derive a microscopic model that explicitly distinguishes between water flow through subdomains of soil and periodically arranged root branches and apply homogenization techniques to derive a macroscopic model for water flow through vegetated soil. For more general root systems, this simple geometry can be considered locally and, with a suitable transformation, a macroscopic model can be derived. In the microscopic model, the Richards equation, with different permeability and water retention functions, is used to model water flow through the soil and root subdomains and water uptake is described by the boundary conditions at the root surface. Pressure gradients within the root tissue are driven by transpiration, incorporated through a boundary condition at the root crown. The difference between the properties of the bulk soil and the rhizosphere is reflected through the explicit spatial dependence in the water retention and hydraulic conductivity functions.
The Richards equation, with its non-linearities in the time derivative and elliptic operator, belongs to a larger class of degenerate parabolic equations. Well-posedness results for these types of equations and systems have been obtained by many authors, see e.g. [4, 8, 14, 23, 27]. Existence of strong solutions to the Richards equation was addressed in [33], the hysteresis problem for the Richards equation was analysed in [40], and viscosity solutions for a general class of non-linear degenerate parabolic equations were studied in [28]. In [38, 29] the uniqueness of solutions for degenerate parabolic equations and systems was shown using the variable-doubling method [30], and proving the -contraction principle.
There are several results on multiscale analysis for the Richards equation and degenerate parabolic equations in general. The homogenization of non-linear parabolic equations with Dirichlet bounday conditions in a domain periodically perforated by small holes was studied in [35] and in [5, 7] the two-scale convergence method was applied to non-linear equations with oscilating coefficients modelling the two-phase flow in a porous medium. In [24] the homogenization of the Richards equation in a medium with deterministic almost periodic microstructure of coarse and fine impermeable soft inclusions is obtained using sigma-convergence method and an upscaling of the Richards equation to describe the flow in fractured porous media was considered in [31]. Parabolic and elliptic equations defined in domains with cylindrical microstructures, similar to the one considered here, were analysed in [13, 19]. Furthermore, in [34] homogenization techniques have been applied to dual-porosity models with the flow through porous particles and the inter-particle space described by the Richards equation. Similar to the model considered here, the permeability of the particles was dependent on the small parameter and, to show convergence of the non-linear terms, a specific form of test function was used in conjunction with the unfolding operator, also named as dilation operator and periodic modulation method [6, 10].
We consider a cylindrical microstructure representing the root branches where permeability of the root tissue is proportional to root branch radius and, hence, depends on the small parameter . The -dependence of the permeability tensor means that for the proof of the strong convergence, required to pass to the limit in the non-linear terms, a standard compactness argument cannot be used. Using the time-doubling method it is possible to show the equicontinuity of the unfolded sequence of solutions to the microscopic problem and prove the strong two-scale convergence. This is a novel application of the variable-doubling method in homogenization theory, which will allow the derivation of macroscopic equations for other degenerate parabolic problems. We would also like to remark that the ideas used to show the convergence results in [34] cannot be applied to our problem due to different scaling in the permeability function and different boundary conditions on the surfaces of microstructure. The paper is organised as follows. In section 2 we formulate the microscopic model for the water flow in vegetated soil. Section 3 is devoted to the analysis of the microscopic problem. The homogenization results and the derivation of the macroscopic model are then presented in section 4. Typical examples for the nonlinear hydraulic conductivity and water retention functions are given in Appendix.
2 Formulation of microscopic model
For the microscopic description of water flow in a vegetated soil we consider a simplified root system architecture where the root branches are periodically distributed throughout the domain. Consider representing a section of vegetated soil, with and . The regions occupied by root branches , the rhizosphere soil around each root branch , the bulk soil , and the combined total soil subdomain are defined as , ,
see Figure 1, where , the unit cell and with boundaries and respectively, , and . The lateral surfaces of the root branch microstructure and the boundaries between the rhizosphere and bulk soil are given by
and, with , the vertical boundaries are defined as
The water transport in unsaturated soil is modelled by the Richards equation
| (2.1) |
where . In root tissue, water is transported through the xylem, which can be regarded as a porous medium [21], and hence is also modelled by the Richards equation
| (2.2) |
In equations (2.1) and (2.2), and denote the hydraulic conductivity functions of soil and root tissue, and and are the soil and root water retention functions, respectively. The hydraulic properties of root tissue are anisotropic [18] and are defined in terms of root segment radius and the intrinsic axial conductance of the root tissue as
see e.g. [45, 44], and the function incorporates the dependence of hydraulic conductivity on xylem pressure head . To address the soil heterogeneity, we consider different hydraulic conductivity and retention functions for rhizosphere and bulk soil :
Here and are -periodic in such that for and
with denoting the characteristic function of the subdomain . Hence,
where and denote the pressure head in the rhizosphere and bulk soil, respectively. We have the continuity of the pressure head and flux on the interface between rhizosphere and bulk soil
and across the rhizosphere-root boundary , we set the flux of water as proportional to the difference between water pressure head in the soil and root tissue [45, 44].
At the soil-atmosphere interface , precipitation, evaporation and surface run-off are all accounted for by a function , and no-flux conditions are imposed on the lateral components of the external soil boundary . We also assume that the lower soil boundary is at the interface between the soil and the water table. At the root collars the outward flux is set as equal to the potential transpiration demand , often taken to be the product of a species-dependent crop coefficient and a climate-dependent reference evapotranspiration constant [3], and we prescribe a constant water pressure head on the lower boundary at the root cap . The full boundary conditions are therefore stated as
| (2.3) | ||||||
and
| (2.4) | ||||||
where the term is the product of the intrinsic radial conductance of the root tissue , the water density , and gravitational acceleration .
3 Existence and uniqueness results
In this section we prove the well-posedness of model (2.1), (2.3) and (2.2), (2.4). There are many well-posedness results for degenerate parabolic equations, but due to specific assumptions on the non-linear functions and boundary conditions considered here, we present the main steps of the proofs, using the methods as in [4, 23]. We consider the space
For and the dual product is denoted by and the inner product for and by , given a domain and . Similar notation is used for the inner product for and . We also denote and , where is a domain or boundary of a domain.
Assumption 3.1.
-
(A1)
The functions are strictly increasing and Lipschitz continuous, , and for all , where .
-
(A2)
The functions are continuous and for all , for .
-
(A3)
The function is continuous and for all .
-
(A4)
The initial conditions , are non-positive.
Remark 3.2.
The assumption , with , is not restrictive, since we can define shifted functions that will satisfy the same problem, where denotes the saturated water content for the soil and root xylem respectively.
Definition 3.3.
In the proofs we will use the functions , , , and , defined as
| (3.3) | ||||
The definition of and monotonicity assumptions on , for , imply
| (3.4) |
Remark 3.4.
For simplicity of presentation, in the proofs of existence, uniqueness, and non-positivity results, obtained for each fixed , we shall use the notation , , and .
Theorem 3.5.
Proof.
The existence of a solution can be shown using the Rothe-Galerkin method, see e.g. [26, 17]. Since is fixed, for the simplicity of notation we omit the superscript in and . With , where for some , and , we consider
where and are orthogonal bases for and , respectively, and are orthonormal in and , and the discrete-in-time equations
| (3.5) | |||
| (3.6) | |||
for all , , with and and and being the projections of and onto and respectively. The regularity of and implies in and in as .
Similar arguments as in [4, 39], imply that for given , under Assumption 3.1, there exists solution to (3.5) and (3.6), for all . To derive a priori estimates we consider and as test functions in (3.5) and (3.6). Summing over , for , and using Assumption 3.1 and inequality (3.4) yields
for some , where Cauchy’s inequality and properties of telescoping sum imply
To estimate the boundary integrals, we use the trace theorem, followed by the generalised Poincaré inequality, and the fact that on and on to obtain
| (3.7) | ||||
where may depend on . Combining the estimates from above and using assumptions on initial conditions, together with the Poincaré inequality, yield
| (3.8) |
for , where may depend on . Summing (3.5) over , with , and using the properties of the telescoping sum, then taking as a test function, and summing over , gives
Performing similar calculations with as test function in (3.6) and using estimates on and in (3.8) imply
| (3.9) |
To pass to the limit in (3.5) and (3.6) as we consider the piecewise constant interpolations in time
| (3.10) |
and , where for and . Then (3.8) and (3.9) yield
| (3.11) | |||
for , uniform with respect to and , where for some . Using a priori estimates (3.11) and equations (3.5) and (3.6) we obtain
| (3.12) |
where
Combining estimates (3.11) and (3.12) yields
| (3.13) | ||||||
The boundedness, Lipschitz continuity and monotonicity of imply
| (3.14) | ||||
for , where is independent of and and are the Lipschitz constants for . Using [41, Theorem 1], there exists , for , such that, upto a subsequence,
| (3.15) |
Since is strictly increasing and continuous, it admits a continuous inverse and
| (3.16) |
and, because in as , we have and , for . Applying the change of variables , with , in the last estimate in (3.14) and using the monotonicity and Lipschitz continuity of yield
| (3.17) | |||
for . Thus, using (3.16), we have
| (3.18) |
and a.e. in as , and hence
| (3.19) |
Boundedness of ensures and
| (3.20) |
as , for where and . Considering with and and using the last two convergence results in (3.13), together with the definition of the weak derivative, yields for . Continuity of implies a.e. in and, since , by the dominated convergence theorem we have
| (3.21) | ||||||
as , with and , and . Using the trace inequality applied to , along with the Lipschitz continuity of and boundedness of in , yields
Combining this with the convergence of in implies
| (3.22) |
and a.e. in , as . Then the continuity and boundedness of ensure the convergence of .
Remark 3.6.
In some models no water flux at root tips has been considered, i.e. on . Theorem 3.5 holds also for such boundary conditions and the only difference is in the estimation of the boundary integral. From (3.6), applying the trace theorem and Poincaré inequality, we obtain
Thus in the equation for the boundary integral involving can be estimated
Theorem 3.7.
The proof of Theorem 3.7 employs the same method as in [38]. For this we first prove two inequalities. Consider and , , for , given by
| (3.23) |
for , where if and if . Then for and for and we have and .
Similar as in the proof of Theorem 3.5, we omit the superscript in and .
Lemma 3.8.
Proof.
We shall prove (3.24) and the proof for , , , follows the same lines. Considering
| (3.25) |
for and , and using we can write
| (3.26) | ||||
where is extended by for . Since are Lipschitz continuous and and are monotone increasing, it follows that
| (3.27) | |||
Using (3.27) in (3.26), together with the non-negativity of , yields
| (3.28) | ||||
The same calculations hold for and , considering in the definition of in (3.25). Using the Cauchy-Schwarz inequality, change in the order of integration and that , we obtain , . The Lebesgue differentiation theorem and the regularity of and , imply in and, by applying the trace theorem, also in and , as . Hence, testing (3.1) and (3.2) with the corresponding and taking the limit as yield the inequality stated in the lemma. ∎
Proof of Theorem 3.7.
To prove the uniqueness result, assume there are two solutions and , consider a doubling of the time variable and define functions such that
Considering (3.24) with , , and , for non-negative , we obtain
| (3.29) | ||||
for a.e. , whereas considering , , and in the inequality with and , for , yields
| (3.30) | ||||
for a.e. . Integrating (3.29) with respect to and (3.30) with respect to , adding the resultant inequalities and using , we obtain
| (3.31) | ||||
where , , and . Using the definition of we have
| (3.32) | ||||
where is non-negative and singular as . The first term on the right-hand side of (3.32) is non-negative for and the second term is non-zero only if with
where and is the Lipschitz constant for , with . A similar estimate holds for . The definition of yields
where . When , , , and we directly have . If and , then and , resulting in
In the case where we have and hence . Analogous arguments imply when .
Using (3.31), the estimates above, and the fact that is non-decreasing, yields
| (3.33) |
The monotonicity of and and the non-negativity of ensure
for . A similar estimate is obtained for by using that . The boundedness of and implies and are uniformly bounded in and, since , we have
Additionally, if , and for , for , for . Then the Lipschitz continuity of implies
Using similar arguments for and and the fact that yields
| (3.34) | ||||
pointwise a.e. in as , for . The dominated convergence theorem implies the strong convergence of and in , as , for and . Taking in (3.33) the limits as we obtain
| (3.35) |
For any non-negative , there exists such that if or . For a non-negative of unit mass we have that , for and some , and the function
| (3.36) |
is admissible in (3.35), for . Applying the change of variables and denoting , yield
| (3.37) |
with the abbreviation for . To take the limit as , we first show strongly in , where . Assume and consider
The regularity of ensures that is an admissible test function in (3.1). Using an integration by parts and the regularity of , we obtain
where we used that for and for . An analogous arguments hold also for . Thus from equation (3.1) we have
| (3.38) | ||||
By the Lebesgue differentiation theorem, we have strongly in , and by the trace theorem, also in and . Hence the right hand side of (3.38) converges to as and from the Lipschitz continuity and monotonicity of we obtain
| (3.39) |
as . An identical argument is employed to show
| (3.40) |
Combining (3.39) and (3.40) with the continuity of and taking in (3.37) imply
| (3.41) |
Applying integration by parts and using that is compactly supported yield
and, since this holds for all non-negative , it follows that
for a.e. and a.e. in , for .
Using (3.24) with , and , and the corresponding inequality for and with and and performing the same calculations as above yield a.e. in , for . Thus, since and are strictly increasing, we obtain the uniqueness of and . ∎
Proposition 3.9.
Proof.
Using and as test functions in (3.1) and (3.2) and adding the resulting equations yield
| (3.42) | ||||
Notice that , since on . We first show
| (3.43) |
for all and , where is given by . Integrating by parts and using yield
For and , if , where for , we have
| (3.44) | ||||
Following a similar line of argument, result (3.44) is also obtained for . If , then since it follows
If , then since and we have
Combining estimates above yields (3.43). Multiplying each side of (3.43) by , using that and and the initial condition implies , and taking the limit as yields
| (3.45) |
The definition of and , implies
Combining the results above, together with and , it follows, from (3.42),
| (3.46) |
Using for and the boundary conditions on , gives
| (3.47) |
for . Thus from (3.46) and (3.47) we obtain
and that and are non-positive over and respectively. ∎
Remark 3.10.
Theorems 3.5 and 3.7 were proven for and . Physically realistic functions for water content , , and and hydraulic conductivity , , and are usually not defined for positive values of pressure head and, in the cases where they are, they often fail to satisfy Assumption 3.1, see [43, 25, 9]. In Appendix, we provide functions for , , and , which extend the expressions used in [43, 25, 9], to positive values of pressure head, in a way that the criteria of Assumption 3.1, Theorem 3.7 and Proposition 3.9 are satisfied. These extensions can therefore be assumed throughout the proofs of Theorems 3.5 and 3.7. Moreover, since Proposition 3.9 shows that and will remain non-positive, for any non-positive initial condition, the question of how realistic these extensions are for positive values of pressure head is not a concern.
4 Derivation of macroscopic model
To derive macroscopic equations from the microscopic model for the water transport in vegetated soil, we apply the two-scale convergence and periodic unfolding method, see e.g. [1, 11, 36, 37]. To pass to the limit as we first derive a priori estimates uniform in .
Lemma 4.1.
Proof.
Considering and as test functions in (3.1) and (3.2), respectively, adding the resulting equations, using Assumption 3.1 on and , for , and , yields
| (4.2) | ||||
for and . The definition of and , implies
| (4.3) | ||||
for . To estimate the third integral on the right hand-side of the second inequality in (4.3) we can use the continuity of and estimate
which follows from the definition of or, as in our case, using the boundedness of . The assumption on the microscopic structure of ensures existence of an extension of from to such that
| (4.4) |
where is independent of , see e.g. [12, Theorem 2.10]. By the trace theorem and (4.4), we have
| (4.5) | |||
Applying the trace theorem over the unit cell and the -scaling in -variables, together with the definition of the domain , yields
| (4.6) |
Thus, using (4.3), (4.5), and (4.6) in inequality (4.2) implies
| (4.7) | ||||
for and . Applying the generalised Poincaré inequality over and using the boundary condition on and the standard scaling argument, for the first term on the right hand side of (4.7) we have
| (4.8) |
where is independent of . Similarly for the second term on the right hand side of (4.7), the generalised Poincaré inequality, the extension properties, and at , imply
| (4.9) |
where are independent of . Using now (4.8) and (4.9) in (4.7), and considering an appropriate , yields the first and the last estimates stated in the lemma.
Estimates in (4.1), together with the properties of the extension (4.4) and of the two-scale convergence, see e.g. [1, 37, 36], imply the following convergence results
Lemma 4.2.
There exist and such that, upto a subsequence,
| (4.11) | ||||||
as , where .
The next lemma provides the convergence result for the sequence .
Lemma 4.3.
For an extension of by zero from into , there exists , with , such that
| (4.12) | ||||||
Proof.
From Lemma 4.1 we have that the sequence is bounded in and there exists a subsequence, denoted again by , which converges two-scale to with for , see e.g. [1]. From Lemma 4.1 we also have existence of a subsequence of that converges two-scale and weakly in , and
| (4.13) |
for . This, together with the two-scale convergence of and with
| (4.14) | ||||
as , implies for . Choosing , with , in the definition of the two-scale convergence of gives
and hence the third convergence in (4.12). The microscopic structure of implies . Then the estimate for , see Lemma 4.1, ensures existence of a subsequence of converging weakly in and, using the weak convergence of , we obtain the fourth convergence in (4.12). A priori estimates in (4.1) and properties of the two-scale convergence on oscillating boundaries, see e.g. [2, 36] ensure the second convergence result in (4.12). The uniform in estimate for , see (4.1), ensures weak convergence, up to a subsequence, of in , and hence the last convergence result in (4.12). ∎
To pass to the limit in the nonlinear terms in (3.1) and (3.2), we show the strong two-scale convergence of and , by using the Aubin-Lions-Simon compactness lemma [41] and showing the equicontinity of the corresponding sequences.
Lemma 4.4.
Proof.
Considering as a test function in (3.1) the following function
using an integration by parts, and applying the Cauchy-Schwarz inequality yield
A priori estimates in Lemma 4.1 and the uniform boundedness of and imply the first estimate in (4.15). Analogous calculations, with as a test function in (3.2), yield the second estimate in (4.15). ∎
To show the strong two-scale convergence of we use the unfolding operator given by
where , , , and is the unique integer combination, see e.g. [11]. Similar definition we have for .
Proof.
First we show the strong convergence of in and of in . The Lipschitz continuity of , with , properties of the unfolding operator, see e.g. [11], and Lemma 4.4 imply
for , , and , where and are independent of . Combining the estimates from above and using the compactness result in [41], yield the existence of and such that, up to a subsequence,
| (4.16) |
Since and are strictly increasing and continuous, we have
| (4.17) |
for . The two-scale convergence of implies in , see e.g. [11]. Using (4.16) and (4.17), together with , ensures strongly in and strongly two-scale, for . Then converges strongly two-scale to . ∎
Proof.
To prove strong convergence of we first show the equicontinuity of by using similar arguments as in the proof of the uniqueness result in Theorem 3.7. Consider and , with and , and define
and , , where . For , such that , and , where , we have , with . Using in the weak formulation of (2.2) and (2.4) over , integrating by parts in the time derivative and changing variables from to , for , yield
| (4.19) | ||||
Consider now the functions , , , and , with , as in (3.23). The arguments similar to those in the proof of Theorem 3.7 and Lemma 3.8 yield that and , for and , are admissible test functions in (4.19) and in the weak formulation of (2.2) over respectively, and we obtain the following inequalities
| (4.20) | |||
Considering a doubling of the time variable , with , , and in the first inequality, and , , and in the second inequality in (4.20), with non-negative , and adding the resulting inequalities yield
Taking in the above inequality and applying the arguments similar to the one used in the proof of (3.35), give
| (4.21) |
The function defined as in (3.36) is admissible in (4), in place of , and applying the change of variables and denoting we obtain
| (4.22) | ||||
Taking , applying the integration by parts, and using the compact support of , imply
| (4.23) |
Exchanging and in the calculations above yields (4.23) for . Applying the trace theorem over the unit cell , together with the standard scaling argument, and using estimates in Lemma 4.1, yield
| (4.24) |
for any non-negative . Using the regularity of initial conditions and Lipschitz continuity of , from (4.24) we obtain
Considering and using the definition of the unfolding operator imply
| (4.25) | ||||
where and for all , such that for some . For the finite number of estimate (4.25) follows from the continuity of the -norm. The estimates for and ensure, for and ,
| (4.26) | ||||
Using (4.25), (4.26), and the second estimate in (4.15), and applying the compactness theorem, see [41] and boundedness of , yields the strong convergence of in . This, together with the monotonicity and continuity of and two-scale convergence of to , implies a.e. in . Hence strongly in , which, applying the properties of the unfolding operator, implies the strong two-scale convergence of , stated in the lemma. ∎
Theorem 4.7.
Proof.
Considering , with and , and as test functions in (3.1) and (3.2) and applying the unfolding operator we have
| (4.29) | ||||
| (4.30) | ||||
Using the properties of the unfolding operator , i.e. for , with and or , for , and for , and the relations between the two-scale (strong two-scale) convergence of a sequence and weak (strong) convergence of the corresponding unfolded sequence, see e.g. [11], together with the convergence results in Lemmas 4.2, 4.3, 4.5, and 4.6, and taking in (4.29) and (4.30) the limit as we obtain
| (4.31) | |||
| (4.32) | ||||
Notice that uniform boundedness and continuity of , for , and convergence a.e. of and , together with the Lebesgue dominated convergence theorem, imply strong convergence in and in , for any . To show the convergence of we consider
| (4.33) | ||||
Here we used the trace theorem and the standard scaling argument to obtain
Then using in (4.33) the strong convergence of in , see Lemma 4.5, we obtain the strong convergence of in . The monotonicity and continuity of ensure a.e. in and then the continuity and boundedness of imply in .
Using the standard arguments, see e.g. [1, 12], from (4.31) by considering we obtain the unit cell problems (4.28) and the formula for . Then, considering (4.31) with and first and then and (4.32) first with and then yields the macroscopic equations (4.27). Considering and , with , as test functions in the weak formulation of (2.1), (2.3) and (2.2), (2.4) respectively, and using monotonicity of and implies that and satisfy the corresponding initial conditions.
Acknowledgements
AM was supported by the EPSRC Centre for Doctoral Training in Mathematical Modelling, Analysis & Computation (MAC-MIGS) funded by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/S023291/1, Heriot-Watt University, and the University of Edinburgh.
Appendix: Admissible functions for water content, hydraulic conductivity and water flux at the upper soil surface
The function for soil water content can be defined in accordance with the formulation of [43] and extended for positive values of soil water pressure head
| (4.34) |
Here the residual and saturated soil water contents are and respectively, and constants , and are shape parameters, where we have different sets of there parameters for bulk soil and rhizosphere. Water content for root tissue can be defined using the models in [25, 9] and extending for positive root tissue pressure heads in a similar way as (4.34):
| (4.35) |
Here is the root tissue porosity (or root xylem saturated water content), the air entry pressure head is , the root xylem elastic modulus is , is the Brooks and Corey exponent, and , and are as in (4.34). A suitable expression for can be defined by taking a regularisation of the Van Genuchten [43] formulation and extending it for positive pressure heads. The first step is to define the regularisation of the water content function
where . The regularised soil hydraulic conductivity satisfying the conditions in Theorems 3.5 and 3.7 is given as
In a similar way, we define a regularised expression for root tissue water content
where , and a regularised version of the function given as
An admissible function , for the water flux at the upper soil surface , is
where is the reference evapotranspiration , see [3], the function controls the amount of evaporation from the soil surface, and P is the precipitation. The function RO incorporates runoff, which occurs when precipitation lands on an already saturated soil surface and cannot infiltrate downwards. The evaporation function is defined in the same way as in [32] and the runoff function is defined as
where .
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