Maderas. Ciencia y tecnología. 4(1):50-68, 2002

ARTICULO

FLUID MIGRATION IN TWO SPECIES OF BEECH (FAGUS SILVATICA AND FAGUS ORIENTALIS) : A PERCOLATION MODEL ABLE TO ACCOUNT FOR MACROSCOPIC MEASUREMENTS AND ANATOMICAL OBSERVATIONS.

Patrick Perré¹ and Ali Karimi^2
1 LERMAB (Laboratory of Wood Science), UMR INRA-ENGREF-UHP 1093
ENGREF, 14 rue Girardet, F-54042 Nancy Cedex, France.
²Faculty of Natural Resources, University of Teheran, Iran

Corresponding author: perre@engref.fr

---

ABSTRACT
The aim of this study is to visualise, measure and understand the pathway of liquid and gas at both macroscopic and microscopic levels on specimens of beech (Fagus silvatica and Fagus orientalis). The permeability to air and to water has been measured using devices developed in our laboratory. The extension of the area coloured by dyed water is available as well. At the microscopic level, the permeability has been calculated using Poiseuille's equation from the vessel diameters determined on cross sections by image processing. Using dyed water, the proportion of active vessels is also reported at different distances from the injection surface.

Our data confirm that the permeability decreases significantly when the sample distance increases. Moreover, the value extrapolated for a zero-length sample is similar to the value predicted from the vessel diameters. This observation stands for both for sapwood and heartwood, in spite of the great permeability difference noticed between these zones. At the microscopic level, the percentage of active vessels decreases with the increase of sample total length and the increase of the distance from the injection point. Several simulations performed using a bond percolation model confirmed our experimental results obtained at the macroscopic and microscopic levels.

KeyWords: Air, Beech, Length effect, Liquid, Percolation model, Permeability, anatomy.

---

INTRODUCTION

Mass migration in wood has been the subject of numerous works, including liquid and gaseous flow as well as migration in the hygroscopic range of moisture content. A good overview of these works can been found in Siau (1984). Subsequently, the corresponding physical formulation and the parameters as determined by various experimental systems can be used in comprehensive computational models to account for these mechanisms and provide simulation facilities (Perré and Turner, 1996, 1999).

Nevertheless, some particular phenomena need further investigation: determination of relative permeability curves (Dullien, 1992), non-steady state liquid migration (Banks, 1981), decreasing of permeability values with sample length (Bramhall, 1971).

Not only macroscopic measurements but also anatomical observations are required to support an efficient modelling approach capable of prediction. In an attempt to advance in this direction, the present work proposes the analysis of different experimental data gathered on beech (Fagus silvatica and Fagus orientalis):

• gaseous permeability measurement at different sample lengths,
• liquid injection, with macroscopic and microscopic measurements,
• image analysis on cross sections to predict the permeability value by Poiseuille's equation,

Finally, a percolation model is proposed to analyse the effect of sample length, both at macroscopic and microscopic scale.

MATERIAL AND METHODS

Wood samples

Samples of Fagus orientalis L. used in this study comes from an experimental forest located 7 km east of Nowshahr (North of Iran) belonging to the Faculty of natural resources, University of Teheran. Five trees of good quality have been selected and felt from plot 116 in the Patom stand (Table 1). Samples of Fagus silvatica L. comes from an experimental forest located 5 km South-West of Nancy (North-East of France) belonging to the ENGREF. Four trees of good quality have been selected and felt from plot 113 in the Sivrite stand (Table 2)

One log of about 1.5 meter long has been cut from each tree. In order to ensure that the sample length is along the longitudinal direction, each log has been split into four sectors.

Table 1: Some characteristics of Fagus orientalis logs.

Tree number	Age (years)	Breast height diameter (cm)
1	153	45
2	125	42
3	86	47
4	155	50
5	150	51

Table 2: Some characteristics of Fagus silvatica logs.

Tree number	Age (years)	Breast height diameter (cm)
1	83	46
2	70	40
3	75	43
4	74	42

Liquid injection device

The device used to determine liquid permeability is described elsewhere (Karimi 1995, Karimi and Perré 1997). It comprises different elements (Fig. 1):

a tank containing dyed water,

compressed air with a pressure regulator,

a series of tubes at different diameters equipped with valves,

a injector designed to supply the dyed water to the wood sample without leakage,

an electronic balance connected to a computer for data storage.

Fig. 1: The experimental device developed and used to inject liquid and measure its flux in wood samples

Thank to the air pressure regulator, the dyed water is injected at one end of the sample at a controlled pressure. In order to reduce any hindrance to the liquid flux, the water is filtered before injection (the pore diameter of the filter is less than 0.3 µm) and the injection face of the sample is carefully prepared with a scalpel.

The series of tubes allows the input flux to be measured. Thanks to the compressed air line, the pressure level is the same in the tank and in the tubes. In order to determine the flux, one calibrated tube is selected by opening the Vt valve of this tube (all other Vt valves remain closed). The flux measurement starts when the tank valve (VT) is closed. From this instant, the liquid flux is removed from the tube without any pressure change in the sample. The liquid flow rate of the injected water is determined by the level variation over a certain time. Some centimetres of level variation are sufficient to obtain a good accuracy, so that pressure variation due to the level variation of the liquid column remains negligible compared to the total pressure. The existence of several tubes with a wide range of sections allows the operator to match any value of the actual flux.

The output flux flowing out of the sample is continuously weighted using a precision electronic balance. These values are different during the transient period, but becomes identical in steady-state conditions (after less than one or two dozens of minutes for one-meter long samples).

The use of dyed water allows the liquid pathway in the sample to be analysed just after the injection experiment. The macroscopic stained area is determined each 10 centimetres (usually, the samples are 100 cm long) simply by drawing the stain pattern of a tracing paper. The area is then measured using a digital camera and an image processing software. Just after, cross sections of the same sample parts are cut using a sledge microtome and observed through an optical microscope without any preparation. This procedure allows the coloured vessels to be recognised easily. The percentage is determined by counting the total number of vessels and the number of active vessels on the same part of the sample.

Gaseous permeability measurement

The device developed to measure the permeability to gas uses the surrounding air as a fluid. The sample geometry is a parallelepiped cut along the material directions of wood (Fig. 2). The two main faces of the sample are insulated with a double coat of epoxy resin. Air at controlled pressure is injected in the central hole, whereas the airflow is free on all lateral faces of the sample. The air pressure, the volumetric air flux and the pressure level at certain positions of the sample (Fig. 2) build up the set of experimental data. With the aid of an approached analytical solution of the pressure field, this configuration allows the longitudinal permeability and the anisotropy ratio to be determined (Perré 1992). Samples have to be cut as close as possible along the material axes: they are planed from split logs.

Fig. 2: Sample geometry and boundary conditions chosen to determine the longitudinal gaseous permeability and the anisotropy ratio at the same time.

Anatomical characterisation

Cross sections of beech samples were sectioned using a sledge microtome. The sections of around 15 µm were then prepared and mounted being first immersed in sodium hypochlorite and then stained with safranin to enhance contrast. The samples were observed using a Zeiss optical microscope and photographed using a colour CCD camera. An image processing software, Visilog 5.0, allowed us to extract objective information from these images (image improvement, thresholding, labelling and determination of Feret’s diameters each 10 degrees).

Assuming that wood in longitudinal direction is made up of capillary tubes (vessels) placed in parallel, with a tortuosity factor equal to the unit, the permeability value can be calculated from microphotographs of cross sections of wood with the following expression (Dullien 1992):

(1)

S is the total cross section of the observed part and R_i is the radius of vessel number i. Each vessel radius is deduced from the vessel diameter as determined by image processing. Because the vessels are usually not isotropic, the representative diameter has been calculated as the quadratic average of the maximum Feret's diameter and the Feret’s diameter along the direction orthogonal to the first one.

EXPERIMENTAL PERMEABILITY DATA

Permeability measurements

Figure 3 depicts a typical example of injection test. The input flux is plotted versus time for two samples: one in sapwood and the other in heartwood. As it was expected, heartwood is less permeable than sapwood (by a factor 3 to 4). Moreover, it is important to notice that, for both samples, the permeability value decreases as the injection time increases. This effect is usually explained by entrapped air, which creates meniscus along the flow pathway and partly blocks the flux. In addition, in spite of the filter placed at the entrance of the liquid, the existence of particulate matter that can exist within the wood sample is able to clog up the smallest anatomical openings (Comstock 1967, Choong et al. 1974, Bolton and Petty 1978). For all these reasons, gaseous permeability values are much easier to analyse. Consequently, rather than measure permeability, we usually kept the liquid injection device to analyse the macroscopic and microscopic path in wood.

Fig. 3: Typical example of permeability values calculated from the liquid flux as a function of injection time (Fagus silvatica)

As previously stated, our gaseous experiment allows the permeability to be determined along two material directions simultaneously. Table 3 gives some typical values reported for beech. The anisotropy ratio is spectacular, particularly the tangential over longitudinal ratio. However, these huge ratios are in agreement with results obtained by different authors for diffuse hardwood species (Choong et al., 1974, Siau, 1984). As expected, the heartwood part of beech is less permeable than sapwood, but the ratio of about 3 found here remains small compared to other species. In the case of softwood for example, the permeability of heartwood can be only a small percentage (1 to 10 %) of the value measured in sapwood (Comstock and Côté, 1968, Meyer, 1971, Bolton and Petty, 1977). This dramatic reduction is due to the aspiration of bordered pits. Some hardwood species like some European oaks, prone to tyloses development, present even a much higher difference of permeability between heartwood and sapwood.

Table 3: Example of gaseous permeability measured on boards with central hole (Fagus orientalis, sample half-length = 40 cm).

Zone	Sawing pattern	KL (m².1012)		KT (m².1016)	KR (m².1016)
sapwood	flatsawn	4.4	65000	0.7	-
	quatersawn	3.2	3 000	-	10
heartwood	quatersawn	1.4	3 000	-	4

Figure 4 summarises all permeability measurements performed in this study. The averaged permeability values measured using gas and liquid for sapwood and heartwood parts are reported for the two species. The flow length was always equal to 10 cm, which means a 10-cm long sample for liquid injection and a 20 cm long sample for the gaseous device. From these values, it becomes evident than Fagus orientalis is slightly less permeable than Fagus silvatica. As expected, for both species, the heartwood part is less permeable than sapwood. Note however that the ratio is less than two. Finally, although we calculated the liquid permeability from the liquid flux measured for short injection times, the permeability values are smaller when measured with liquids. This observation is consistent with our previous remark.

Fig. 4: Averaged permeability values measured using gas and liquid for sapwood and heartwood parts of both beech species (sample length = 10 cm).

Permeability calculated from cross-sections

It is interesting to compare the actual permeability values with the theoretical value calculated from the vessel characteristics (equation 1 applied to the data collected by image processing on cross sections of wood). In figure 5, a complete set of data is depicted for Fagus orientalis. The number of vessels on the image (430 x 430 µm²), the averaged vessel diameter and the calculated permeability is exhibited for different trees and different zones within the trees (sapwood and heartwood). All anatomical views have been sorted according to their position within the annual ring (early wood, middle wood and late wood). The values collected in figure 5 are the average over all images allocated to the same position within the annual ring. Due to some technical problems, some positions are missing for certain trees. The calculated values exhibit a dramatic reduction of the permeability from early wood to late wood. By analysing carefully the data, it becomes obvious that the vessel diameter, rather than the vessel number, is responsible for this drastic reduction. This is a direct effect of Poiseuille's regime, with turns into a power four of the radius in equation 1. The same reason explain the very high value obtained for tree number 1, as a result of the highest average vessel diameter (close to 70 µm). Obviously, no fundamental difference has to be expected between sapwood and heartwood: heartwood formation can affect extractives, chemical composition and eventually tyloses formation, but does not change the pore morphology. Possible difference may however exist between sapwood and heartwood due to the age of tree, which can affect the forest stand (growth rate) and the physiological behaviour.

Fig. 5: The theoretical permeability value calculated from anatomical croos sections of wood using the Poiseuille’formula (Fagus orientalis, average over several cross sections, S stands for sapwood and H for heartwood)

Considering the problem encountered with liquid measurement, these theoretical values have been compared to the actual values determined with the gas device. The theoretical values of table 4 have been calculated as a weighted mean of values obtained in the different parts of the annual ring. Because these parts are placed in parallel in the case of longitudinal permeability, the weight factor must to be proportional to the section. Considering that the late wood part of the annual ring is usually small, we used 2, 2 and 1 as factors for early, middle and late wood parts respectively. In the case of missing data, we assumed that the permeability variation within the annual ring follows a geometrical progression.

The ratio of these values (actual over predicted) is also reported in figure 6. Keeping in mind that the calculation assumes capillary tubes of even diameter placed in parallel, the theoretical value is in fact an upper bound. So, it is comforting to notice that the measured value is always smaller than the predicted one. Furthermore, except for tree number 1, the percentage is always smaller for the heartwood samples. From the anatomical structure, heartwood has the same potential as sapwood for fluid flow, but the possibility of connection between vessels elements is reduced by the presence of tyloses or extractives.

Table 4: Comparison of he permeability value calculated on anatomical cross sections (zero length) and the actual gaseous permeability value measured on 10 cm long samples (Fagus orientalis).

	Longitudinal permeability (m².1012)
Tree number	Sapwood		Heartwood
	Calculated from anatomy	Measured with air (L=10 cm)	Calculated from anatomy	Measured with air (L=10 cm)
1	30.93	3.40	-	3.17
2	15.48	4.83	-	4.53
3	17.36	4.77	17.11	2.17
4	20.61	8.47	18.54	3.10
5	14.11	5.68	13.22	3.65

Fig. 6: Comparison of he permeability value calculated on anatomical cross sections (zero length) and the actual gaseous permeability value measured on 10 cm long samples.The ratio of these values is expressed in percent (same trees and position within logs as in figure 5)

Effect of sample length

Numerous studies proved that the permeability value, which is supposed to be an intrinsic property of the porous medium, indeed depends on the sample length. This observation is generally explained by a certain probability of connection of the pores within the porous medium (Bramhall, 1971, Meyer, 1971, Trénard, 1980, Siau, 1984, Kauman et al. 1994). Consequently, the value determined using formula (1), named K₀, can be considered as the permeability obtained as the sample length tends towards zero.

In order to investigate the effect of sample length on permeability, the sample is shortened by successive cuts during the experiment. This procedure has been applied to liquid and gas measurements.

Figure 7 depicts a typical result obtained on a heartwood sample of Fagus sylvatica with liquid injection. One can notice that the permeability starts to increase when the sample length is reduced, but this trend is reversed for very low values of the length. An entrance effect can be responsible for this surprising decrease for short samples (Kauman et al. 1994, Karimi, 1995). However, the effect of the time injection is another possible explanation for this final decreasing. Actually, the remaining part of the shortened sample underwent the longer injection time, which can explain a lower permeability value (see figure 3). Due to this uncertainty, no further analyse is proposed for liquid measurements.

Fig. 7: Effect of sample length on liquid permeability (Fagus silvatica, one representative sample).

Figure 8 depicts the gaseous permeability measured at different sample lengths for sapwood and heartwood. On average, the permeability value increases by a factor 2 for sapwood and by a factor 5 for heartwood when the length is reduced from 40 to 10 centimetres. The experimental data have been fitted using a classical exponential dependence of permeability with length. The exponential factor is equal to –3.2 m^-1 for sapwood and –6.5 m^-1 for heartwood. This observation is consistent with other experimental works: the decrease in the logarithm of the permeability with length is higher for samples with low permeability values (Siau 1984). Thanks to this difference in decreasing rate and in spite of the great permeability difference measured on samples, the intercept with the y-axis, which represents the extrapolated value for a zero-length sample, K₀, is similar for both zones: 12.2.10^-12 m² for sapwood and 9.5.10^-12 m² for heartwood. It is also important to notice that these values are reasonably close to the values predicted using formula (1) (see table 4).

Fig. 8: Effect of sample length on gaseous permeability (Fagus silvatica, average over 3 samples of sapwood and heartwood).

All these data tend to prove that, among all possible pathways in beech samples, only a small percentage is actually used. In addition, this percentage decreases with sample length. The use of dyed water allowed us to confirm this analysis. This is the object of the next paragraph.

The use of dyed water to observe the path in wood

Macroscopic injection stains

At the macroscopic level, the stains are easily observed just after the experiment. In order to avoid the diffusion process that takes place after injection, the sample is cut in several sections just after the experiment and the shape of each stain is drawn using tracing paper, for later analysis. Figure 9 depicts the evolution of the dimensionless macroscopic stained area along the sample for two injection times for representative samples of Fagus silvatica. The effect of the injection time becomes obvious for distances greater than 40 centimetres. After 10 minutes, the area decreases significantly with the distance, whereas it is almost constant throughout the sample after 300 minutes. Note that the reference area is much larger after 300 minutes (see table 5). This can be due to the effect of transverse migration. Also, the time required for the peripheral zone, undergoing a lower flux, to be coloured could explain this observation.

Table 5: Evolution of macroscopic stained area along the sample for two injection times (Fagus silvatica, representative samples).

Distance from injection point (cm)	Stained area (mm²) for a short injection duration (10 minutes)		Stained area (mm²) for a long injection duration (300 minutes)
	sapwood	heartwood	sapwood	heartwood
10	105	112	238	302
20	103	93	237	322
30	91	75	258	340
40	81	53	250	330
50	56	38	240	329
60	41	16	213	319
70	33	3	200	290
80	30	1	171	263
90	22	-	-	-
100	18	-	-	-

Fig. 9: Evolution of the dimensionless macroscopic stained area along the sample for two injection durations : a = 10 minutes and b = 300 minutes (Fagus silvatica, representative samples). The effect of the injection duration becomes obvious for distances greater than 40 centimeters. Note also that the reference area is much larger after 300 minutes (see table 5).

Percentage of active vessels

Cross-sections of wood are prepared just after injection. By microscopic observation, we discovered that only a small percentage of vessels were active during liquid flow (Fig. 10). Moreover, the diffusion of dyed water in the transverse direction is very limited. Because the macroscopic stains seem very uniform, these observations are quite surprising. Table 6 proves that the number of active vessels decreases significantly as the distance from the injection point increases. However, contrary to what was observed on macroscopic stains, the effect of injection time on the number of coloured vessels seems negligible here. It is probably easy to detect active vessels, whereas a certain accumulation time is required for the macroscopic stain to be observable with the naked eye. Several observations told us that not only this percentage depends on the total length of the sample but also depends on the distance from the injection face. Figure 11 points out representative results. A long sample has been cut in shorter samples during injection. The effect of position from the injection point can be observed by the decrease of each continuous curve along the x-axis. The gap between two successive curves (at each cutting place) is due to the effect of total sample length.

Fig. 10: Microphotograph of a sample penetrated by dyed water: only a small percentage of vessels were active during injection (a = active vessel).

Table 6: Number of coloured vessels on the image (430 x 430 µm2) versus the distance from the injection point for different injection times (Fagus silvatica, representative samples of sapwood and heartwood).

	Number of coloured vessels
	sapwood			heartwood
Distance from injection point (cm)	Injection time (minutes)			Injection time (minutes)
	5	20	40	5	20	40
10	60	62	47	36	37	30
40	25	38	24	6	18	16
70	8	13	8	1	7	2
100	1	5	4	0	0	0

Fig. 11:The effect of both the distance from the injection point and the sample length on the percentage of active vessels.

PERCOLATION MODEL

In the previous parts, important results have been gained :

• the actual sample permeability is much smaller than the value calculated assuming that the wood sample is a bundle of capillary tubes (the vessels),
• the sample permeability decreases as the total sample length increases,
• the number of active vessels decreases both when the sample length increases and when the distance from the injection point increases.

Only statistical models can account for these outcomes. In such models (Adler 1992, Dullien 1992), the porous medium is assumed to be made up of bonds having a certain conduction probability (Fig. 12a). In order to account for wood anisotropy, our 2-D model considers two different conduction probabilities: one along the longitudinal direction and the other in the transverse direction. When injecting the fluid in one part of the external surface, only the connected pores act for the liquid migration (Fig. 12b). But, among these connected bonds, the flow is only effective in a few of them. This is the concept of active pores, those for which the flow rate exceeds a threshold value, which requires the pressure field within the sample to be solved (Fig. 12c).

(a) Conducting bonds	(b) Connected bonds	(c) Active bonds
Fig. 12: The random networks to explain the liquid flow in wood. The probability for a conducting bond is 0.8 in the longitudinal direction and 0.2 in the transverse one. (The thick line is the zone of fluid injection)

Once the pressure field is solved in the stochastic network, the dimensionless volumetric flux can be calculated. The theoretical permeability value is deduced from this dimensionless flux together with the zero-length permeability value, as calculated from anatomical cross-sections. Figure 13 depicts some simulations obtained with different sets of probability that allow the experimental variation of permeability with sample length to be fitted. Note that the difference between sapwood and heartwood only lies in the probability values, not in the zero-length permeability. In this simulation, the vessels, composed of many vessel cells, are supposed to have an averaged length of two centimetres (Zimmerman, 1983).

Fig. 13: Prediction of permeability value as a function of sample length Three different sets of connection probability (in percent, longitudinal and transverse direction respectively) that allows the experimental data to be approached.

In this model, the concept of active vessel has been tied to the pressure gradient within each bond. In order to obtain a dimensionless gradient, the pressure gap through each bond is divided by the reference gap (pressure gap over the sample divided by the number of bonds in the longitudinal direction). In the example depicted in figure 14, a threshold value equal to 0.7 allows the percentage of active bonds to be close to the measured percentage of active vessels.

Fig. 14: Percentage of active vessels predicted as a function of sample length for one set of connection probability (in longitudinal and transverse direction) that allows the experimental data to be approached

Although this concept has been developed for a 2-D network here, it allowed all the reported steady-state observations to be predicted:

• the effect of sample length on permeability,
• the variation of the active pore fraction within the sample and with the sample length,
• the differences between gaseous and liquid permeability or between sapwood and heartwood,
• the apparent deviation of the permeability calculated on cross-sections from the actual value measured on real samples.

CONCLUSION

In this work, the permeability to air and to water has been measured on specimens of beech (Fagus silvatica and Fagus orientalis) using devices developed in our laboratory. At the microscopic level, the permeability was calculated using Poiseuille's equation from the vessel diameters determined on cross sections by image processing. Using dyed water, the extension of the macroscopic active area and the proportion of active vessels have been described at different distances from the injection surface.

Based on the analysis of this comprehensive set of data, a percolation model has been proposed and used to describe fluid flow in beech in steady-state configurations. This model allowed all the observed trends to be simulated:

• the permeability decreases with an increase of the sample length,
• the percentage of active pores depends on both on the location within the sample and on the total sample length,
• differences between gaseous and liquid permeability or between sapwood and heartwood can be explained by different values of conduction probability,
• the permeability value predicted for very short samples is in agreement with the values predicted from Poiseuille's law on cross-sections of wood.

Based on the percolation concept, a double scale interpretation of the mechanisms that govern transient moisture migration in beech will be derived in the future. Such a model could explain the surprising behaviour observed during unsteady state flow in hardwood, namely during soaking or drying experiments (Perré, 1996).

ACKNOWLEGMENTS

One of the authors, Patrick Perré, would like to thank Walter Kauman for the many fruitful discussions concerning physics, thermodynamics and wood anatomy conducted over several years on this interesting and fascinating scientific field of heat and mass transfer in wood. I first met Walter Kauman during my PhD work in 1985. His broad knowledge, together with his clear understanding in this field, were a significant help and a great source of motivation for the young scientist I was then. It was a great honour to have him as a member of the committees for both my PhD and my DSc theses and it is with great pleasure that I dedicate this work to him.

REFERENCES

Adler, P.M. 1992. Porous Media: Geometry and Transports, Butterworth-Heinemann Series in Chemical Engineering.

Banks, W. B. 1981. Addressing the problem of non-steady state liquid flow in wood, Wood Sci. Technol. 15: 171-177.

Bolton, A.J.; Petty, J.A. 1977. Influence of critical point and solvent exchange drying on the gas permeability of conifer sapwood, Wood Science 9: 187-193.

Bolton, A.J.; Petty, J.A. 1978. The relationship between the axial permeability of wood to dry air and to a non polar solvent, Wood Sci. Technol. 12: 111-126.

Bramhall, G. 1971. The validity of Darcy’s law in the axial penetration of wood, Wood Sci. Technol. 5: 121-134.

Choong, E.T.; Tesoro, F.O.; and Manwiller, F.G. 1974. Permeability of twenty-two small diameter hardwoods growing on southern pine sites, Wood and Fiber 6(1) 91-101.

Comstock, G.L. 1967. Longitudinal permeability of wood to gases and nonswelling liquids, Forest Products J. 17(10) 41-46.

Comstock, G. L. and W. E, Côté. 1968. Factors affecting permeability and pit aspiration in coniferous sapwood, Wood Science and Technology 2, 279-291.

Dullien, F.A.L. 1992. Porous media - Fluid transport and pore structure, Second Edition, Academic Press, London.

Jacquiot, C.; Trénard, Y.; Dirol, D. 1973. Atlas d’anatomie des bois des Angiospermes, CTBA, Paris.

Karimi, A. 1995. Caractérisation microscopique et macroscopique de l’écoulement dans le hêtre (Fagus silvatica et Fagus orientalis): interprétation à l’aide d’un modèle de percolation, PhD dissertation, ENGREF, Nancy.

Karimi, A. and Perré, P. 1997. Un dispositif de mesure du débit et de la progression du liquide dans la direction longitudinale du bois, Les Cahiers Scientifiques du Bois, volume 1, 149-158, numéro spécial Instrumentation, ARBOLOR, Nancy.

Kauman, W.G.; Ananias, R.A.; Gutiérrez, M. and Valenzuela, H. 1994. Non-Darcian permeability in Chileau Tepa (Laurelia philippiana), Holzforschung, 48: 77-81.

Meyer, R. W. 1971. Influence of pit aspiration on earlywood permeability of Douglas-Fir, Wood and Fiber, 2, 328-339.

Perré, P. 1992. Transferts couplés en milieux poreux non-saturés. Possibilités et limitations de la formulation macroscopique, Habilitation à Diriger des Recherches, INPL, Nancy.

Perré, P. 1996. Mesure de l’humidité du bois au cours d.“essais d’imbibition par détection directe de rayons-X” Colloque S.I.B., 243-250, Arbolor, Nancy.

Perré, P and Turner, I. 1996. Using a set of macroscopic equations to simulate heat and mass transfer in porous media: some possibilities illustrated by a wide range of configurations that emphasize the role of internal pressure, Numerical methods and Mathematical modelling of the Drying Process, 83 - 156, edited by I.W.Turner and A. Mujumdar, Marcel Dekker.

Perré, P and Turner, I. 1999. A 3D version of TransPore: A Comprehensive Heat and Mass Transfer Computational Model for Simulating the Drying of Porous Media, Int. J. Heat Mass Transfer, 42(24): 4501-4521.

Siau, J.F. 1984. Transport Processes in Wood, Springer-Verlag, Berlin.

Trénard, Y. 1980.Comparaison et interprétation de courbes obtenues par porosimétrie au mercure sur diverses essences de bois. Holzforschung 34(4): 139-146.

Zimmerman, M. H. 1983. Xylem structure and the ascent of sap, Springer-Verlag, Berlin.