3D geometric structures and biological activity: Application to microbial soil organic matter decomposition in pore space

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3D geometric structures and biological activity: Application to microbial soil organic matter decomposition in pore space Olivier Monga a,∗ , Mamadou Bousso a , Patricia Garnier b , Val´erie Pot b a b

IRD, Laboratoire MAT (UCAD2/IRD), UR GEODES, Centre IRD de Dakar, BP 1386, CP 18524, Dakar, Senegal INRA, Centre de Grignon, UMR environnement et grandes cultures, BP 01, 78850 Thiverval Grignon, France

a r t i c l e

i n f o

a b s t r a c t

Article history:

During the past 10 years, soil scientists have started to use 3D Computed Tomography in

Received 29 June 2007

order to gain a clearer understanding of the geometry of soil structure and its relation-

Received in revised form

ships with soil properties. We propose a geometric model for the 3D representation of pore

7 December 2007

space and a practical method for its computation. Our basic idea consists in representing

Accepted 17 April 2008

pore space using a minimal set of maximal balls (Delaunay spheres) recovering the shape

Published on line 12 June 2008

skeleton. In this representation, each ball could be considered as a maximal local cavity corresponding to the “intuitive” notion of a pore as described in the literature. The space

Keywords:

segmentation induced by the network of balls (pores) was then used to spatialize biologi-

Computational geometry

cal dynamics. Organic matter and microbial decomposers were distributed within the balls

Soil science

(pores). A valuated graph representing the pore network, organic matter and distribution of

Pore space modelling

micro-organisms was then defined. Microbial soil organic matter decomposition was simu-

Microbial decomposition simulation

lated by updating this valuated graph. The method was implemented and tested using real

3D Computer Vision

CT images. The model produced realistic simulated results when compared with data in the literature in terms of the water retention curve and carbon mineralization. A decrease in water pressure decreased carbon mineralization, which is also in accordance with findings in the literature. From our results we showed that the influence of water pressure on decomposition is a function of organic matter distribution in the pore space. As far as we know, this is the approach to have linked pore space geometry and biological dynamics in a formal way. Our next goal will be to compare the model with experimental data of decomposition using different soil structures, and to define geometric typologies of pore space shape that can be attached to specific biological and dynamic properties. © 2008 Elsevier B.V. All rights reserved.

1.

Introduction

Environmental disturbances to an ecosystem, such as changes to land use or climate, may affect the activity of microorganisms (Smith et al., 1998). In most current models for soil organic matter decomposition, micro-organisms are treated as a pool of organic matter. The predictive capacity of models should be improved by incorporating an explicit description

∗

Corresponding author. E-mail address: [email protected] (O. Monga). 0304-3800/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2008.04.015

of micro-organisms and their relationship with soil structure (Paustian, 1994). Soil structure determines the distribution of water and air in the soil pores which provide a habitat for micro-organisms. Unsaturated conditions have two principal effects. They reduce the diffusion of nutriments toward micro-organisms, and they may isolate the microbial habitat of particular species and inhibit competition to some extent (Long and Or, 2005). The influence of micro-organisms and

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spatial relationships affecting organic matter on biological dynamics is a recent area of study, targeted recently by Masse et al. (2008) in an unstructured 3D media. Until now, only a few studies have focused on the geometric modelling of 3D microscopic soil structures in order to gain a clearer understanding of the links between soil properties and soil structure geometry (Monga et al., 2007; Delerue and Perrier, 2002; Vogel and Roth, 2001). The first reason why soil scientists have been relatively little interested in 3D advanced geometric modelling has been the problem of obtaining 3D volumetric images of a soil sample with sufficient resolution to investigate microscopic biological activity. The second reason is that in the past, 3D Computer Vision (Ayache, 2003; Faugeras, 1993) and computational geometry (Boissonnat and Yvinec, 1998; Schmitt and Mattioli, 1994) were perhaps insufficiently advanced to cope with the complexity of soil geometric structures. Today, recent advances in image sensors and processing enables more precise investigation of the 3D geometry of soil structures (Gregory et al., 2003; Timmerman et al., 1999). Indeed, X-scanner Computed Tomography (CT) generates 3D images of soil samples with a resolution of 10 ␮m, which is at least sufficient to study biological dynamics. The most recent developments in 3D Computer Vision and computational geometry can also be adapted to the geometric modelling of soil structure (Monga et al., 2007). In this paper, we have applied this innovative paradigm to the geometric modelling of pore spaces in order to simulate microbial soil organic matter decomposition. The tools developed within this context could be of course applied to many other areas of soil science where the geometry of microscopic soil structures can be linked with soil properties. In Section 2, a representation of a pore space using an adjacency valuated graph of maximal balls (Delaunay spheres) which could be associated with the intuitive notion of “pores” widely used by soil scientists is presented. In Section 3, (i) water content distribution, (ii) initial repartition of organic matter and microbial decomposers and (ii) the rules of microbial soil organic matter decomposition are presented, thanks to the space description and the related geometrical constraints induced by the graph. Finally, Section 4 presents the results of microbial soil organic matter decomposition obtained using 3D Computed Tomography of real recomposed soil sample.

2.

Modelling 3D pore space

2.1.

The common notion of pore space

In the geoscience context, a pore is usually defined as a cavity within an empty space full of fluids (air or water) (Vogel and Roth, 2001). The pore space is commonly perceived as a set of pores, i.e. a set of cavities in the empty space. We have proposed a formal and rigorous geometric method to define the set of pores in poral space (Monga et al., 2007) that fits well with the intuitive notion of a pore as defined in the literature. The formal geometric method to define pores is described in Monga et al. (2007).

2.2.

Minimal recovery of a pore space skeleton

First of all, we define some standard notions of geometry (Boissonnat and Yvinec, 1998; Schmitt and Mattioli, 1994): • A “maximal” ball of a shape is a ball included in a shape but not included in any other ball included in the shape. • The skeleton of a shape is the set of all centres of the maximal balls. It can be computed using various algorithms relying on Delaunay triangulation, Voronoi diagrams or median axis (Samozino et al., 2006; Amenta and Bern, 1998; Attali et al., 2006). In the case of 3D shapes whose border is defined by a sparse number of points, the computation of the skeleton remains an open problem, mainly because of its inherent instability (Attali et al., 2006). Our basic premise was to look for a minimal set of “maximal balls” (Schmitt and Mattioli, 1994; Monga et al., 2007) that recovered the skeleton. Given that the skeleton is defined by the set of all centres of maximal balls, it follows that this set forms a recovery of open sets of the skeleton. If the skeleton is a compact set, then by definition, from each recovery of open sets we can extract a finite recovery. In most cases, the skeleton of a shape can be approximated using a compact set. This issue will be discussed more in more detail in a forthcoming paper. Therefore, it is possible to consider that the skeleton can always be recovered by a finite set of maximal balls. This is due to the well-known property that the skeleton of a shape is homotopic to that shape (Boissonnat and Yvinec, 1998). We chose to recover the skeleton using the minimal number of maximal balls. This provides a description of the shape while preserving the topology of the pore space. Moreover, it constitutes the most compact way to define the shape cavities that are called pores in the literature. We can generalize by considering the ␭-skeleton rather than the skeleton. The ␭-skeleton (Chazal and Lieuthier, 2005) can be considered as all centres of maximal balls whose radius is at least ␭. Therefore, previous developments can be extended, obtaining the notion of the minimal recovery of the ␭-skeleton of a shape using maximal balls with a radius of more than ␭. This is a meaningful way to achieve a simple definition of pore space at different scales. In the sections that follow, we will call MIRES(p) the graph associated to the minimal recovery of the skeleton of shape p (␭ MIRES(p) that is then associated to the ␭-skeleton of shape p.

2.3. Computation of a minimal set of maximal balls recovering the pore space skeleton It is possible to show that maximal balls can be considered as Delaunay spheres of the 3D Delaunay triangulation of shape boundaries (Boissonnat and Yvinec, 1998; Schmitt and Mattioli, 1994). In the specific case where sampling of the border is optimal (i.e. the situation considered here), the skeleton can be recovered from Delaunay triangulation using the algorithm described by Monga et al. (2007) and as follows: (i) computation of the shape volume border, (ii) computation of 3D Delaunay triangulation of the shape volume border, using an efficient implementation of Delaunay triangulation developed at INRIA by the GAMMA project (George, 2004; George and

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Borouchaki, 1998; Frey, 2001), (iii) computation of Delaunay spheres, (iv) pruning of Delaunay spheres, and (v) computing of the ␭-skeleton. We select all centres of Delaunay spheres included within the shape whose radius is at least ␭. This set of points provides the ␭-skeleton of the shape (Chazal and Lieuthier, 2005). The computation of minimal skeleton recovery using maximal balls is a complex optimization problem. Indeed, in terms of complexity, the existence of a reasonable algorithm to achieve this goal is not obvious. We thus proposed such an algorithm in the article by Monga et al. (2007), which practically constituted an efficient heuristic. Its basic principles were to situate the biggest ball iteratively by adding the constraint that it should be either tangent or disjoint to the other balls. If at the end of this process the skeleton was not completely recovered, then the constraint was relaxed.

2.4. Describing pore space (p) using the adjacency graph ( MIRES-graph(p)) The algorithm described in the previous section generates a set of balls that form a compact description of the cavities (pores) in the pore space. We propose using the adjacency graph of this set of balls to describe the pore space. We suggest defining this graph as follows. If p is the pore space defined by an indicative function (Monga et al., 2007) or a set of voxels, we define ␭ MIRES graph(p) = G(N,A), where G is a graph and N corresponds to a set of maximal balls bi forming a minimal recovery of the skeleton, i.e.: N = {b1 , b2 , . . . , bn } ∈  MIRES(p)

(1)

A defines the adjacency relationships between the balls, i.e.: (bi , bj ) ∈ A ⇔ bi ∩ bj = 

(2)

In the following, we will consider that: bi = (Ci , ri )

(3)

where bi is the ball whose centre is Ci and radius is ri . We can produce ␭ MIRES graph(p) after extracting a set of voxels corresponding to pore space (p) by thresholding soil 3D Computed Tomography (Delerue and Perrier, 2002; Monga et al., 2007). We compute ␭ MIRES(p) using the method described in Section 3 and then the ␭ MIRES-graph of p using the above definition. We also attach to each ball of the graph a set of features F which provide information about the pore: filling with water or air, organic matter biomass, microbial biomass, etc. For each ball bi , we note fi (k) the kth co-ordinate of feature vector Fi : F = {F1 , F2 , . . . , Fn },

where Fi = (fi (1), fi (2), . . .)

Therefore, ␭ MIRES graph(p) can be seen as a valuated graph representing biological exchanges in the 3D pore space at a given time.

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3. Using pore space modelling to simulate biological activity 3.1.

Principle

The aim of our method is to simulate microbial soil organic matter decomposition by (i) using the ␭ MIRES graph of pore space p to spatialize water content distribution, the mass of organic matter and microbial decomposers within the pore space, (ii) simulating the interactions between organic matter decomposition and microbial community dynamics (growth, respiration, mortality and dormant state) using an offer/demand modelling approach similar to that described by Cambier et al. (2007), and (iii) updating the ␭ MIRES graph using time discretization. The offer/demand approach differs from those that have normally been developed in the field of soil science (for example by Garnier et al., 2003) or in microbial ecology (for example by Fontaine and Barot, 2005). Under the former, the decomposition rate of organic matter is independent of the demand of micro-organisms; the supply of organic matter controls decomposition. In latter, microbial growth only controls organic matter decay. Under our present approach, the microbial OM decomposition is dependant of the demand of micro-organisms and supply of OM (following Cambier et al., 2007). We have defined the constraints of microbial soil organic matter decomposition using the “proximity” of organic matter and microbial decomposers within the graph. The decomposition of an organic matter through offer/demand approach is thus limited to the micro-organisms connected to the organic matter through a path water filled pores in order to simulate diffusion of extra cellular enzymatic decomposition process. Some assumptions have been made regarding this initial attempt. We hypothesize that oxygen was not limiting for decomposition. Microbial activity under anaerobic conditions is a highly complex problem with an abundant literature (for example, SchØnning et al., 2003) and specific models have been developed to address this process. Højberg et al. (1994) showed that oxygen can diffuse until 2–4 mm inside saturated aggregates. In this paper, we hypothesize that the size of our aggregate samples was small enough (a few mm3 ) to reasonably neglect anaerobic decomposition. The general conceptual diagram of our model is presented in Fig. 1.

3.2. Initial water content, organic matter and micro-organisms biomasses distributions within balls (pores) Water content distribution was simulated by a drainage iterative procedure at different water pressures j , following Vogel and Roth (2001) to take into account drainage hysteresis. The pore space system was initially saturated with water (fi (1) = 1 for all ball bi ) and all border balls bi whose radius ri was larger than the radius rj defined by the Young–Laplace law written for a totally wetting fluid (zero contact angle between water

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3.3.

Fig. 1 – Conceptual diagram of our model.

and soil) (4) was drained (fi (1) = 0): j

=

2 rj

(4)

where  is the surface tension water/air. Then an iterative algorithm drained the remaining balls bi if their radius ri was larger than rj and if they were in contact with a ball full of air. The iterative procedure was stopped when equilibrium was reached. Drainage was performed at different pressures, and the water retention curve relating water content to pressure, ( ) was obtained. We placed organic matter and microbial decomposers in the pores by attaching to each ball of the ␭ MIRES graph an organic matter mass and a microbial decomposer biomass. These biomasses represented masses of carbon. It is still difficult to visualize organic matter in tomography images (De Gryse et al., 2006) and techniques to explore microbial habitats in soil pores are still limited, despite recent advances (Nunan et al., 2003; Strong et al., 2004). We used two types of organic matter: Fresh Organic Matter (FOM) and Soil (humidified) Organic Matter (SOM), as used in current models of SOM decomposition (e.g., Garnier et al., 2003). FOM consisted of rapidly decomposable organic matter and SOM of slowly decomposable organic matter. We considered two types of micro-organisms (microbial decomposers): fast growing micro-organisms (type F) and slow growing micro-organisms (type S), as used in most current models of soil organic matter decomposition (Fontaine and Barot, 2005). Fast growing microorganisms (FM) decomposed FOM organic matter, while slow growing organisms (SM) decomposed SOM organic matter. We called the total mass of FOM and SOM MFOM and MSOM [M], respectively, and the total mass of FM and SM MFM and MSM [M], respectively. For each pore (ball) bi , we set fi (2) and fi (3), respectively, to the mass of FOM and SOM in (around) the pore [M] and fi (4) and fi (5), respectively, to the mass of FM and SM in (around) the pore [M] and fi (6) to the cumulative mineralized carbon due to FM and SM micro-organism respiration in the pore. We considered two types of initial repartition (Fig. 10). Under repartition 1, F(S)OM and F(S)M were equally distributed in N balls. Under repartition 2, F(S)OM and F(S)M were distributed as a function of ball i radius (ri ): mass being proportional to the radius.

Simulation of biological activity

Biological activity was simulated from the interactions between the decomposition of FOM and SOM, and the growth, respiration, mortality and dormant state of FM and SM. General data on FOM and SOM decomposition rates [T−1 ] were available, that were noted tFOM and tSOM , respectively. FM and SM growth rates [T−1 ], were noted tFM and tSM [T−1 ], respectively, FM and SM mortality rates [T−1 ], were noted dFM and dSM [T−1 ], respectively, and FM and SM respiration rates [T−1 ] were noted bFM and bSM [T−1 ], respectively. We stated that decomposition took place into water filled balls only. In each pore I filled with water, the FOM and SOM decomposition velocities were therefore fi (2)tFOM and fi (3)tSOM [M T−1 ], respectively, while the growing velocities of FM and SM were fi (4)tFM and fi (5)tSM , respectively. We further assumed that FOM and SOM within a pore i can be decomposed by micro-organisms of other pores l = (1, k) connected to pore i by a water-filled path, that is by k adjacent balls filled with water. This tried to simulate extra cellular enzymatic decomposition process through the connectivity of water filled pores. We attached to such balls the geodesic distance, ql , to ball i, according to the graph. To compute the geodesic distance we defined the distance between two balls as the distance between their two centers. The length of a path was set to the sum of the distances between adjacent balls. The geodesic distance is the minimal length of a path joining two balls. The geodesic distances were practically computed using the well-known Dijstra algorithm (Knuth, 2006). During a time step t, for a pore i, the maximal masses of FOM and SOM that could be decomposed were mFOM = min(fi (2), t fi (2))tFOM ) and mSOM = min(fi (3), t fi (3))tSOM ), respectively. These maximal decomposable masses represented the “organic matter offer” to microbial decomposers. Similarly, during a time step t, the total FOM and SOM demand of, respectively, FM and SM microorganisms included in pore i and in pores l = (1, k) connected to pore i are, respectively, DFM = t fi (4)tFM +

l=k

l=k

l=1

t fl (4)tFM and

DSM = t fi (5)tFM + t fl (5)tSM . These maximal decoml=1 posable masses represented the “organic matter demand” of microbial decomposers. When a micro-organism died, it became SOM. The quantity of mineralized carbon was calculated from respiration of micro-organisms. To prevent the disappearance of microorganisms from the pore space when demand from microbial decomposers could not be fulfilled, a dormant state of micro-organisms was introduced. For each pore (ball) bi , the biomasses fi (4) and fi (5) could not fall below a given threshold representing a percentage of the initial biomass (typically 5%). The decrease of decomposer mass due to mortality and respiration, fi (4), the increase of SOM due to mortality, fi (3) and cumulative mineralized carbon mass, fi (6) are presented in Fig. 11. The masses of FOM and SOM and decomposing FM and SM were calculated following an offer/demand approach (Fig. 12). In the case where “organic matter offer”, mFOM and mSOM of the organic matter are less than the “organic matter demand”, DFM and DSM , of the “water connected” microorganisms, the “organic matter offer” was shared according to the geodesic distances, ql , of each water connected pore l = (1,

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Fig. 2 – Top down: original soil sample; CT cross-sections; cross-sections of pore space obtained by thresholding the CT image.

k) using a polynomial repartition law (Fig. 12). In the opposite case, the micro-organisms satisfied completely their demand.

4.

Model application

4.1.

Soil samples

Soil was sampled from the surface layer (0–20 cm) of an agricultural field at the INRA experimental site of Feucherolles (FEU) (50 km west of Paris) in France, in January 2006. This was a typical hapludalf loamy soil (15% clay, 78% silt and 7% sand). The soil was sieved at its current volumetric water content of 0.15 g g−1 and only aggregates of a diameter between 3.15 and 2 mm were removed. The aggregates were compacted in cylin-

ders (3-cm high and 5-cm diameter) to obtain a soil dry density of 1.2 g cm−3 (Fig. 2). Three cylinders were thus prepared. The retention curve, relating the volumetric water content to the soil water pressure, of the three cylinders was measured using a pressure extractor method (Klute and Dirksen, 1986). The cylinder was saturated on a suction table by applying zero water suction at the base for 48 h. The volumetric water content of soil samples was measured at 0, −1, −3, −10, −20, −30, −55, −60, −75, −90, −100 and −130 cm of water pressure. The samples were kept for 3 days at each equilibrium stage and were then weighed. Tomography images were obtained from one of the cylinders using an X-scanner. Its scanner characteristics were 80 kVp and 500 ␮A. The X-scanner was made available by IRCAD (Soler, 1998). The initial image size was 512 × 100 × 100,

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Fig. 3 – Perspective views of the minimal set of maximal balls recovering the skeleton (top: only balls with a radius of more than 2 are displayed; bottom: only balls with a radius of more than 3 are displayed). We used Opendx (Delerue and Perrier, 2002) as visualization software.

Fig. 4 – Perspective views of the drained pore space at −20 cm water pressures. Top left is initially water saturated pore space, top right shows border balls drained and down shows equilibrium state. Grey balls are filled with air and white balls are filled with water.

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SOM was mostly in pores smaller than our resolution of 11 ␮m. In our model, the carbon pool of SOM is the result of mortality of micro-organisms. Since our model is not designed to perform aggregation and modification of pore sizes (Denef et al., 2002), we simply assumed that this SOM was located in the pores where micro-organisms died. We took similar rates to those used in the literature, with lower rates for SOM than for FOM: tFOM = 0.1 day−1 , tSOM = 0.05 day−1 , tFM = 0.7 day−1 , tSM = 0.1 day−1 , dFM = 0.01 day−1 , dSM = 0.001 day−1 , bFM = 0.05 day−1 , and bSM = 0.005 day−1 . The minimum dormant mass for SOM and FOM decomposers was 5%. The time step was 0.01 day. The total time of simulations was 120 days.

Fig. 5 – Water volume versus pressure: comparison of our model with experimental data.

resolution was 11 ␮m × 11 ␮m × 11 ␮m per voxel corresponding thus to a subsample of the core. The dimension of the modeled soil sample was 5.6 mm × 5.6 mm × 1.1 mm (Fig. 2).

4.2.

Modelling conditions for the biological process

4.2.1. Initial conditions and parameters for the biological process We adopted realistic initial conditions for the biomasses of FOM and micro-organisms (data from Angers and Recous, 1997). The initial mass of FOM was 1.8 g C/kg of dry soil for FOM and 125 mg C/kg of dry soil for micro-organisms were studied, split into 100 mg C/kg of dry soil for FOM decomposers and 25 mg C/kg dry soil for SOM decomposers. No account was taken of the initial SOM of soil because we assumed that this

4.2.2.

Geometric modelling of pore space

In X-ray scanner images, the gray levels were approximately proportional to material density. Thus the coarse pore space could be computed from 3D CT images of the soil using thresholding. Simple thresholding is employed in most papers (Delerue and Perrier, 2002; Vogel and Roth, 2001). This basic method performs reasonably well, although in some cases threshold determination is not entirely straightforward. We computed the pore space envelope from the 3D image using a simple algorithm to select boundary points. Opendx (Delerue and Perrier, 2002) was used as the visualization software. We computed 3D Delaunay triangulation of boundary points (Fig. 3) (George and Borouchaki, 1998). A simplified version of the Tetmesh-GHS3D code was employed to obtain (3D) Delaunay triangulation (George, 2004) and the free software MEDIT was used to visualize Delaunay tetrahedra (Frey, 2001). We should stress that because of the huge number of points forming the pore space envelope, it was crucial to apply efficient Delaunay Triangulation implementation. Delaunay triangulation of the pore space envelope was pruned by removing all

Fig. 6 – Initial distribution of FOM and (FM + SM) masses according to repartition 2 (Fig. 10) at −20 cm water pressures. Sizes of squares representing organic matter mass and micro-organisms biomasses are proportional to carbon mass values. For a better visualization, green squares representing organic matter mass are located at the ball surfaces while red squares representing micro-organisms mass are located inside balls. Down figure is the same as top figure but without balls.

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Fig. 7 – Organic matter (green color) and micro-organism (red color) masses at time 1.89 days (top) and 2.5 days (down).

Fig. 8 – Microbial organic matter decomposition at local scale of a macropore i and its “water connected” pores. Pore i is in blue, the “water connected” pores are in white. Top down, left right: time 0, time 1.89 day, time 2.5 days, time 3 days.

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Fig. 9 – Effect of water pressures (0, −50, −135 cm of water) on organic matter pools for the two distributions of organic matter in balls (Fig. 10).

tetrahedra that were not included in the pore space as defined by our initial data. We should stress that in this case, Delaunay triangulation pruning was straightforward using the 3D image defining pore space. Finally we used the algorithm described in Section 2 to obtain a minimal set of maximal balls to recover the skeleton (Fig. 3). We obtained a ␭ MIRES graph with 58755 balls. The maximal radius was 36.39 units (403.59 ␮m) and the minimal radius was 1 unit (11 ␮m). We observed that two balls represent macropores (diameter larger than 300 ␮m) while 54,000 balls

represent pores whose diameter is less than 30 ␮m (equivalent water pressure of −50 cm). We found 47,010 balls of diameter equal to 11 ␮m. The total porosity of the pore space of the ␭ MIRES GRAPH is 0.1556 cm3 /cm3 .

4.2.3. Drainage simulation results using pore space geometric modelling Drainage simulations were performed at different water pressures (from 0 to −135 cm) and water content was calculated after equilibrium was reached for each pressure.

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Fig. 10 – Initial repartition of FOM (fi (2)), SOM (fi (3)), FM (fi (4)) and SM (fi (5)) in ball i. Fig. 4 shows the drainage dynamics at −20 cm water pressure. Comparison between water retention curve obtained from experimental data and that obtained with our model is presented in Fig. 5. Curves were very close except for pressure near zero. The water retention curves near saturation are always very variable between samples even for a same soil and can explain the discrepancy between simulated results and experimental data. This result shows the pertinence of our pore space representation.

4.2.4. Biological activity simulation using pore space geometric modelling Simulations were performed at three different water pressures (0, −50 and −135 cm) and for two initial repartitions of organic matter and decomposers (Fig. 10). For the unsaturated conditions, the microbial organic matter decomposition started once the pore space was drained at the given pressures and the equilibrium state was reached. Fig. 6 shows the initial reparti-

tion of organic matter (FOM) and micro-organisms biomasses (FM and SM) according to repartition 2. Fig. 7 shows the biological activity at different time steps (1.89 and 2.5 days). Fig. 8 shows how simulation process works locally for a pore i of radius about 400 ␮m and its “water connected” balls at different time steps (0, 1.89, 2.5 and 3 days). Fig. 9 presents carbon mass evolution curves of organic matter pools using an homogeneous repartition (repartition 1) or an heterogeneous repartition (repartition 2) of organic matter and decomposers in all balls. Mineralized carbon mass increased with time while organic matter carbon mass decreased exponentially. Microorganisms mass increased due to organic matter decomposition, then decreased because of their mortality. FOM decomposers increased and then decreased rapidly, while SOM decomposers always increased due to their very low mortality rate. The simulated organic matter decomposition process followed similar trends found in typical experimental

Fig. 11 – Micro-organism mortality and respiration within a time step of t, fi (3), fi (4), fi (5), and fi (6) are the SOM, FM, SM, and cumulative mineralized carbon masses, respectively, of ball i.

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Fig. 12 – Decomposition of FOM and SOM according to an offer/demand approach within a time step of t, fi (2) and fi (3) are the masses of decomposed organic matter FOM and SOM in pore i, fpl (4) and fpl (5) are the masses of decomposing micro-organisms FM and SM in the adjacent water-connected balls pl to pi and ˛ is a fixed parameter defining the monotonic dependence of the decomposition process as a function of the geodesic distance. In practice, ˛ was set to 2. results of organic matter incubations like in Killham et al. (1993). From Fig. 9, we noticed that decomposition decreased as water pressure head decreased. For the homogeneous distribution, the higher decrease of mineralization was between −50 and −135 cm of water. We have drained a high amount of water balls (27,398 balls) during this pressure change. It corresponds to a higher amount of organic matter for homogeneous distribution than for heterogeneous one (where organic matter are mostly inside bigger balls). For pressure change from 0 to −50 cm, the decrease of decomposition was higher for repartition 2 than for repartition 1, because the 2079 drained balls were large and thus contains more organic matter in repartition 2. From our results we showed that the influence of water pressure on decomposition is a function of organic matter distribution in the pore space (Figs. 10–12).

5.

Conclusion and perspectives

As far as we know, this paper constitutes the first attempt to use a 3D intrinsic geometric representation of pore space to simulate the microbial decomposition of soil organic matter. We extracted the pore space by thresholding a 3D Computed Tomography image of a soil sample. The set of voxels (shape) representing the pore space was then approximated using a minimal set of Delaunay spheres (maximal balls) to recover the shape skeleton. This formal geometric representation fits well with the “intuitive” notion of a pore (“maximal” cavities) defined in the literature. Organic matter and microbial decomposers within pores were then specialized using the adjacency valuated graph of the balls (Delaunay spheres) describing the pore space. In this way, biological activity was modeled by updating the pore graph as a function of time sampling. The results of a simulated organic matter decomposition process

revealed trends similar to those observed during typical experiments of organic matter incubation, and clearly validated our method from a soil science point of view. In the future, we aim to focus on smaller pores using tomography images with a higher resolution. This type of approach can open new perspectives in terms of providing efficient simulators of biological activity that take account of soil structures. In the medium term, this method could be used to link the biological properties of soil to the specific geometric typology of soil structure.

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