On the characterization of non-linear diffusion equations. An application in soil mechanics

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1 On the characterization of non-linear diffusion equations. An application in soil echanics GARCÍA-ROS, G., ALHAMA, I., CÁNOVAS, M *. Civil Engineering Departent Universidad Politécnica de Cartagena Paseo Alfonso XIII 52, Cartagena (Murcia) SPAIN gonzalo.garcia@upct.es; ivan.alhaa@upct.es * Metallurgical and Mining Engineering Departent Universidad Católica del Norte Av. Angaos 060, Antofagasta (Región de Antofagasta) CHILE anuel.canovas@ucn.cl Abstract: - The search of diensionless groups in engineering probles ruled by partial differential equations presents any probles whereby it is an untreated topic in the scientific literature. The ain difficulties arise in the suitable choice of the reference quantities needed to define the diensionless variables which ust also be noralized, i.e., extended to the range of values (0,). After setting the steps for a correct nondiensionalization protocol in this kind of probles, its application is illustrated by studying the soil consolidation proble, a process in which the constitutive dependences between the physical paraeters and the dependent variables are strongly non-linear. Results are verified by nuerical siulations. Key-Words: - Nondiensionalization, diensionless groups, soil consolidation, non-linear Introduction The nondiensionalization of the atheatical odel (governing equations and boundary conditions) is a known ethod currently used for extracting the diensionless groups that influence the solution of a large variety of coplex probles in physics or engineering. This process rests on the properties of hoogeneous functions, long tie ago set by Buckingha in his paper of 94 []. His faous pi-theore, applied in the sae anuscript to probles of different fields of physics, can be stated as follows: any equation or syste of equations that contains atheatical forulation of laws that deterine a physical phenoenon can be represented as a r elation between diensionless quantities. Fro that date, a lot of papers have been published in the search for those relations or diensionless nubers, a basic inforation for odelling [2]. Aong the clearly differentiated protocols for the search of these nubers is, firstly, the diensional analysis. This procedure, very extended in the literature despite its poor results in coplex probles, does not part of the governing equations but a list of relevant variables fro which a nuber of independent onoials of zero diension can be set [3]. The difficulty of obtaining accurate results with this technique is the cause of its strong rejection by the ajority of researchers. Secondly, the diensionless groups ay be obtained by writing, whether for theoretical reasoning for which a deep understanding of the phenoena involved in the proble is required or laboratory tests, the ratios between agnitudes that balance each other in the proble. Many faous nubers, such as R eynolds [4], were deduced in this way. Finally, the sought diensionless nubers ay be derived by relatively siple atheatical anipulation fro the diensionless-noralized for of the governing equations, for which a correct deduction of the sae is required. This last procedure, called nondiensionalization process, leads to the ost precise solution and is the one we will follow in this work. However, the use of this technique in the scientific literature both journals and specialized books to obtain the diensionless groups has been reduced to linear processes. Thus, lots of works devote to apply the ethod in various engineering probles ruled by partial differential equations such as heat transfer [5], fluid dynaics [6], fluid E-ISSN: Volue 2, 207

2 flow and solute transport [7] and geotheric [8] or by ordinary differential equations such as echanical engineering [9]. In all of the, the authors do not distinguish the potentially anisotropic character of the ediu which causes a l ack of discriination and leads inexorably to a less precise solution. Discriination [0] assues as different quantities those of the sae nature but with an inherent vector character for exaple, two perpendicular velocities in 2D probles are assued to be diensionally different quantities with discriination and joins classical nondiscriinated diensionless nubers in the new, ore precise, independent groups that really rule the solution of the proble. As regards non-linear probles, very few works can be found in which the nondiensionalization procedure is applied to derive diensionless groups, whether in isotropic or anisotropic doains [7,8]. The reasons for this absence ay be several but, undoubtedly, the ost iportant is that non-linear probles generally involve non pr oportional dependences between one or ore paraeters and the dependent or, eventually, independent variables. The result is that the non-linear ters within the governing equations cannot turn into diensionless ters in a direct way, doing so that this equation is very difficult to anage in order to derive the sought diensionless groups. Fro the point of view of its perforance, nondiensionalization is a siple ethod that ay indeed soeties should precede or even supplant rigorous atheatical analysis. To set an equation in its diensionless for we ust firstly select the reference quantities, explicit or iplicit in the stateent, that noralize the diensionless dependent and independent variables, for which a deep understanding of the physical processes involved is required. This noralization allows to average to an order of agnitude unity the changes of these variables and their derivatives. Once the governing equation is siplified the diensionless nubers are the independent ratios between two addends of the equation. After suarize the steps of the procedure in the next section, nondiensionalization is applied to the non-linear soil consolidation proble. The derived groups provided by the nondiensionalization are verified by a set of nuerical siulations using the network ethod []. 2 Procedure of nondiensionalization The steps to apply the nondiensionalization procedure are suarized as follows: i) To choice the references (explicit or not in the proble stateent) to ake diensionless and noralized both the dependent and independent variables. References do not given in the stateent are treated as unknowns of the proble. ii) To define the diensionless and noralized dependent and independent variables. According to an appropriate choice of references, the range of values of these variables should be extended to the interval (0,) or very close to it. iii) To yield the diensionless governing equations. To do this, each ter of the equation is averaged so that the factors of such ter related to the diensionless noralized variables and their changes are assue to be of order of agnitude unity in first approxiation. Very sharp nonlinearities in the proble ay eventually ove away fro this assuption leading to diensionless nubers of an order of agnitude larger or saller than unity, accordingly. Also, the existence of coplex atheatical expressions of the dependent variables that coe fro the constitutive relations should be studied in a particular way. iv) To get the diensionless nubers as the quotients fored by the groupings of the paraeters of the proble. There are as any diensionless groups as ters of the equation inus one. However, the nuber of ters ay be less than expected since soe nubers would appear in ore than one equation. Again, non-linearities could ake it advisable to separate a coplex group into two for easier anageent for the engineer. In addition, the set of independent final diensionless groups can be established in different fors by siple atheatical anipulation. For convenience, the final set can be chosen in such a way that each unknown appear in a single group. v) If there are +n diensionless groups fro which each of π contains a different unknown and the rest (π n ) does not contain unknowns, the solution for π is an arbitrary function (ψ) of the π n groups: π i, i = ψ(π j,+ j +n ) () If all the groups are of the order of agnitude unity, it is evident that the arbitrary function also has this property. Fro the -relations (), the order of agnitude of each unknown can be obtained. 3 Noenclature c v, e e o H initial coefficient of consolidation ( 2 /s) void ratio (diensionless) initial void ratio (diensionless) initial thickness () E-ISSN: Volue 2, 207

3 H 2 final thickness () k hydraulic conductivity (/s) k initial hydraulic conductivity (/s) v coefficient of voluetric copressibility ( 2 /N) v, initial coefficient of voluetric copressibility ( 2 /N) q o surface applied load (N/ 2 ) t tie (s) u excess pore water pressure (N/ 2 ) Ū σ average degree of pressure dissipation (diensionless) V volue ( 3 ) z vertical spatial coordinate () γ k nonlinear coefficient of change of pereability with effective pressure (diensionless) γ v non-linear coefficient of copressibility (diensionless) γ w specific weight of water (N/ 3 ) π i diensionless group or nuber (diensionless) σ effective pressure (N/ 2 ) σ initial effective pressure (N/ 2 ) σ 2 final effective pressure (N/ 2 ) σ ean value of the effective pressure (N/ 2 ) τ o,σ characteristic tie which takes the excess pore pressure to dissipate until approxiately the value of zero (s) Ψ arbitrary atheatical function Subscripts I, II denote different nubers Superscripts denote diensionless agnitude Figure. Physical schee of D consolidation The non-linearities of this civil engineering proble [2,3] relate to the constitutive dependences of the paraeters void ratio (e) and hydraulic conductivity (k) on the effective pressure, generally of potential or logarithic type. Since the foral procedure of nondiensionalization does not depend on the constitutive dependences, we will assue those of the odel of Juárez-Badillo [4]. 4. Matheatical odel Under the conditions iposed by the oedoeter test u z = σ z, u = σ t t the constitutive dependences dv V = γ v dσ σ, dk k = γ k dσ σ (2) and soe atheatical anipulation, the governing equation dσ dt z = H z = 0 t = 0 = γ w v z qo H u k (3) z can be written in ters of an only dependent variable, the effective pressure σ t = h qo z = H 2 z = 0 4 The soil consolidation proble A physical schee of this proble is depicted in Fig.. When a layer of clay soil saturated with water suddenly subits to loads on its surface, the initial excess pore water pressure gradually dissipates until all the load is supported by the soil skeleton. It is a typically asyptotic diffusion process whose duration (or characteristic tie) depends on t he hydrological and echanical soil properties. σ t where = k σ σ k γ γ γ w γ v σ k σ σ z σ z 2 (4) k σ γ w γ v = k γ w v, = c v, (5) is the initial coefficient of consolidation. 4.2 Diensionless groups To ake diensionless equation (4), the diensionless variables (σ ) = σ σ σ 2 σ, t = t τ o,σ and z = z H are defined, with τ o,σ the characteristic E-ISSN: Volue 2, 207

4 tie an unknown reference. Thus, equation (4) writes in its diensionless for as σ 2 σ σ σ τ o,σ t + c v, σ H2 σ σ 2 H2 σ = c v,γ k σ γ k (σ2 σ ) σ γ k (σ2 σ ) 2 σ z 2 σ 2 + z 2 (6) with σ a ean value for the effective pressure. By averaging the above equation and deleting the ters whose order of agnitude is unity diensionless variables and their derivative expressions, the three eergent coefficients τ o,σ, k σ γ w γ v γ k k σ γ w γ v H2 σ 2 σ H2 σ σ γ k σ, σ γ k (7) give rise to two diensionless independent groups: π I = γ w γ v H 2 τ o,σ k σ σ π II = γ k σ 2 σ σ σ γ k (8) The coplex expression of π II, dependent on σ, akes understandable the partition of this group into two ore siple groups, γ k and σ 2 σ, giving to σ the value of any of the ends of its range since they are of the sae order of agnitude. Thus, the groups π I and π II disclose in three groups π = τ o,σ k σ γ w γ v H 2 π 2 = γ k π 3 = σ 2 σ (9) Doing this, fro pi theore π = ψ(π 2, π 3 ), the order of agnitude of τ o,σ is given by τ o,σ = γ w γ v H 2 k σ Ψ τ γ k, σ 2 σ (0) As regards Ū σ, since it also depends on tie, the solution is given by Ū σ = Ψ U t, γ τ o,σ k, σ 2 σ () with Ψ τ and Ψ U unknown functions of their arguents. 5 Verification of the results This section is devoted to check the solutions found in the above section, i.e., the dependences for τ o,σ and Ū σ, expressions (0) and () respectively. Eight sets of siulations have been run; in each one, soe of the particular paraeters or initial values of the proble have been changed to give, as appropriate, the sae or different values to the diensionless groups in the search of the sae or different solutions. Changes in the values of the individual paraeters are enough to cover uch of the real scenarios. First, a reference set is established with all the rest referred to it or copared to each other. The physical and geoetrical characteristics that change are: γ v, γ k, e o, H (), σ (N/ 2 ), σ 2 (N/ 2 ) and k (/year), Table. The values of π 2 and π 3 are derived fro the while π is obtained once the characteristic tie τ o,σ is read fro the siulation, Table 2. The criterion for the choice of τ o,σ is the tie required by the soil to reach 90% of the total excess pore pressure dissipation. set γ v γ k e o H σ σ 2 k Table. Physical and geoetrical paraeters set τ o,σ π π 2 π Table 2. Characteristic tie and diensionless groups In view of the results, Tables and 2, the characteristic tie (τ o,σ ) is different for each proble, depending on the paraeters and geoetric characteristics of the sae. However, the diensionless for of the characteristic tie (π ) reains invariant as long as the values of π 2 and π 3 are kept constant, onoials on which we have already seen it depends (0). As for Ū σ (), two representations of this variable are given, Fig. 2 and Fig. 3. In the first, Ū σ is represented as a function of tie, so that for each E-ISSN: Volue 2, 207

5 of the 8 sets studied a different curve of Ū σ is obtained. In the second graph, when we ake its representation as a function of the diensionless tie, t/τ o,σ, we see that the curves are arranged very close to each other. In addition, all cases with equal values of π 2 and π 3, and therefore of π, are represented by a single curve, as in our case for sets to 5. This is a sign that the nondiensionalization process has been applied successfully, providing the ost precise solutions to the non-linear consolidation proble. Average degree of pressure dissipation Average degree of pressure dissipation 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, 0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, 0 Set Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8 0 0,5,5 2 2,5 3 3,5 4 4,5 5 tie (years) Figure 2. Ū σ as a function of tie Sets -5 Set 6 Set 7 Set 8 0 0,25 0,5 0,75,25,5,75 2 2,25 2,5 diensionless tie Figure 3. Ū σ as a function of diensionless tie 6 Conclusion For the first tie, as far as we know, the nondiensionalization procedure has been applied to a non-linear diffusion proble in order to deterine the independent diensionless groups that rule its solution. It has been deonstrated that the difficulties that eerge in the treatent of the nondiensionalization protocol can be surpassed what justifies the application of the ethod to such an iportant objective. The non-linearities inherent to soil consolidation are ainly related to the constitutive dependences between the paraeters or coefficients of the governing equations and the dependent variable (they are potential type functions). These non-linearities are, perhaps, not very sharp, but undoubtedly coplex in relation with the nondiensionalization protocol. In contrast to linear probles, attention should be paid to the choice of the references to noralize the dependent and independent variables, as well as the data averaging needed to ake unity the order of agnitude of the variables and their derivatives thus deleting the in the diensionless governing equation, allowing the coefficients of the averaged equation to be of the sae order of agnitude. Once the diensionless nuber are derived, they can be organized in the best way for the anageent of the engineer and, as o ccurs in consolidation, separated in new groups (although it supposes a greater nuber), ore failiar to the field in which the proble is involved. To check the solutions provided by the nondiensionalization process applied to soil consolidation a set of siulations has been carried out by using a reliable nuerical ethod. In these siulations, when the individual paraeters are odified in such a way that the diensionless groups do not change their value, the solutions reains unchanged. References: [] Buckingha, E., On physically siilar systes: illustrations of the use of diensional equations, Phys. Rev., 4 (4), , 94 [2] Gibbings, J.C., On diensional analysis, J. Phys A: Math. Gen, 3, pp 75-89, 980 [3] Holzbecher, E., Modelling Density-Driven Flow in Porous Media, Springer, Berlin, Gerany, 998 [4] Reynolds, O., An experiental investigation of the circustances which deterine whether the otion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Proceedings of the royal society of London, 35 ( ), 84-99, 883 [5] Madrid, C.N. and Alhaa, F., Discriinated diensional analysis of the energy equation: application to lainar forced convection along a flat plate, International Journal of Theral Sciences, 44 (4), , 2005 [6] Madrid, C.N. and Alhaa, F., Study of the lainar natural convection proble along an isotheral vertical plate based on discriinated diensional analysis, Cheical Engineering Counications, 95 (2), , 2008 E-ISSN: Volue 2, 207

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