M Sudo

T Hennino

E Tomasi-Gustafsson

Sensitivity to two photon exchange in time-like form factor measurements at PANDA

Keywords:

At PANDA(FAIR) proton time-like form factors measurements are foreseen. In this work, following detailed simulations, we study the sensitivity to possible two photon exchange contributions in the annihilation process p + p → e + + e -, through the induced asymmetry in the angular distribution. Asymmetries related to systematic errors are also investigated.

Let us consider the reaction p + p → e + + e -

The expression of the hadron electromagnetic current is usually derived assuming one photon exchange (OPE). The internal structure of the hadrons is parametrized in terms of two form factors (FFs), which are complex functions of q 2 , the four momentum squared of the virtual photon. Two photon exchange (TPE) is suppressed by a factor of α, the electromagnetic fine structure constant, α = 1/137. At large q 2 , however, two photon exchange could play a role, due, in particular, to a possible enhancement from a mechanism where the momentum is equally shared between the two photons. In this case, the simple formalism which allows the extraction of the electromagnetic form factors, would not hold anymore. The general analysis of experimental observables in the reaction p + p → e + + e -and in the time reversal channel, taking into account the TPE contribution, was done in Ref. [1], according to a model independent formalism developed for elastic electron proton scattering [2].

In particular, it has been shown that instead of two amplitudes, functions of q 2 , the general structure of the hadronic tensor contains five amplitudes, when one considers the TPE contribution.

The differential cross section of the reaction (1) for the case of unpolarized particles has the form:

dσ dΩ = α 2 4q 2 τ τ -1 D, τ = q 2 4m 2 , (2)
D = (1 + cos 2 θ)[|G M | 2 + 2(ReG M ∆G * M )] + 1 τ sin 2 θ[|G E | 2 + 2Re(G E ∆G * E )] + 2 τ (τ -1) cos θ sin 2 θRe( 1 τ G E -G M )F * 3 . (3)
where we have defined the following decomposition of the amplitudes: GM (q 2 , t) = G M (q 2 ) + ∆G M (q 2 , t), GE (q 2 , t) = G E (q 2 ) + ∆G E (q 2 , t).

where ∆G E,M corresponds to the TPE contribution to G E,M and F 3 is entirely due to TPE.

We neglect below the bilinear combinations of the terms ∆G M , ∆G E and F 3 since they are smaller (at least of the order of α), in comparison with the dominant ones.

Symmetry properties of the amplitudes with respect to the cos θ → -cos θ transformation can be derived from the C invariance of the considered 1γ ⊗ 2γ mechanism:

∆G M,E (cos θ) = -∆G M,E (-cos θ), F 3 (cos θ) = F 3 (-cos θ). (5)
One can see that in the Born approximation the expression (3) reduces to the result firstly obtained in Ref. [3]:

D = (1 + cos 2 θ)|G M | 2 + 1 τ sin 2 θ|G E | 2 . ( 6
)
The contribution of the OPE diagram leads to an even function of cos θ, whereas the TPE contribution leads to four new terms of the order of α compared to the dominant contribution. Different properties of the observables are described in detail in Ref. [1]. Here let us recall that, at the reaction threshold q 2 = 4m 2 , from which one gets G M = G E and the differential cross section becomes θ-independent in the Born approximation. This is not anymore true in presence of TPE terms. Note also that the contribution to the cross section due to TPE, being an odd function of the variable cos θ, does not contribute to the differential cross section for θ = 90 0 .

I. ANALYSIS OF THE DATA

Angular distributions are simulated according to Eqs. ( 2) and ( 6), assuming

|G E | = |G M |
for three values of q 2 =5.4, 8.2 and 13.6 GeV 2 , and a luminosity of L = 2 • 10 32 cm -2 s -1 , including the detector efficiency. Histograms are built as functions of cos θ, in 0.2 wide bins.

The experimental conditions are shown in Table I. The total number of counts is calculated assuming 120 days of measurement. In case of two photon exchange, the amplitudes (∆G E , ∆G M , F 3 ), are not known. Therefore we will use the presence of odd terms in cos θ as a (model independent) signature of TPE and approximate Eq. ( 3) by

D ≃ (1 + cos 2 θ)|G M | 2 + 1 τ sin 2 θ|G E | 2 + 2 τ (τ -1) cos θ sin 2 θ( 1 τ G E -G M )F 3 . (7)
Eq. ( 7) contains drastic approximations: we neglect the contributions ∆G E,M which, in any case, can not be inferred from unpolarized measurements. Then, in the last term, we consider only the real part of F 3 , G E , and G M , as their relative phases are not known. The purpose is to see up to which limit an odd cos θ contribution can be extracted from the data.

Eq. ( 7) contains odd cos θ terms of first and third order. To start with, we neglect the cos 3 θ term. In a second step, we give the results including the linear and the cubic terms.

For the three values of q 2 given above, we have simulated distributions according to Eq. ( 7), with |G E | = |G M | and F 3 /G M =0.02, 0.05 and 0.2. The asymmetry induced by these components can be seen in Fig. 1, where the angular distributions are shown as a function of cos θ. One can see that as one moves from no contribution (black circles), to 2% contribution (red squares), 5% contribution (green triangles) 20% contribution (blue triangle down), the vertex moves on the left side (it depends on the sign of the coefficient of the cos θ term) and the distribution becomes more and more asymmetric, with respect to cos θ = 0, being still a parabola. Including a cos 3 θ term makes a different distorsion of the distribution. As we keep the same coefficients for the linear and cubic terms, such distorsion cancels at cos θ=0 and ± 1, see Fig. 2.

In order to analyze the distributions, and extract the values of the two photon amplitude, we rewrite the angular distribution as a polynomial in cos θ. One can see that in case of OPE, Eqs. ( 2) and ( 6), can be rewritten as

dσ d(cos θ) = σ 0 1 + A cos 2 θ , (8)
where

σ 0 = α 2 4q 2 τ τ -1 |G M | 2 + 1 τ |G E | 2 (9)
is the value of the differential cross section at θ = π/2 and A can be written as a function of the FFs:

A = τ |G M | 2 -|G E | 2 τ |G M | 2 + |G E | 2 = τ -R 2 τ + R 2 . ( 10
)
where

R = |G E |/|G M |.
Therefore, at each value of q 2 , we can fit the angular distributions with a straight line in

cos 2 θ, y = a 0 + a 1 x with x = cos 2 θ, a 0 ≡ σ 0 , a 1 ≡ σ 0 A (11)
a 0 and a 1 are related to the physical FFs, through Eqs. ( 9) and (10).

Deviations from a straight line will be evidence of the presence of higher order terms, beyond Born approximation. Such C-odd terms appear as functions of cos θ. We can fit the distributions by a quadratic function (neglecting, as a first step, cos 3 θ):

y = a 0 + a 2 x ′ + a 1 x ′2 , x ′ = cos θ, (12)
or including cos 3 θ, with the same three parameters: with

y = a 0 + a 2 (x ′ -x ′3 ) + a 1 x ′2 , x ′ = cos θ, (13)
a 2 ≡ 2 τ (τ -1)(G E -τ G M )F 3 τ |G M | 2 + |G E | 2 . ( 14
)
These last fits are performed with MINUIT.

s [GeV 2 ] E [GeV] p σ [pb] Events 5.4 1. 1.

II. RESULTS

The results of the fit are reported in Table II, in the case where we neglect the cubic term.

We can see that, for the 1γ spectra, as expected, the coefficient a 2 is compatible with zero, for all q 2 values, its value being inferior to the other parameters. The odd cos θ contribution starts to be visible for F 3 /G M ≥ 5%. Note however that the extraction of R and A is not affected, in the limit of the error bars, by the presence of the C-odd term. The differential cross section is a quadratic function (parabola) in cos θ. The linear cos θ term is related to the position of the vertex of the parabola, a TPE contribution corresponds to a shift of this vertex. Therefore, the quadratic and constant coefficient are very stable. Only for the largest 2γ contribution, a systematic deviation of the reconstructed values of R and A from the input ones is observed. If one uses a two parameter fit, in presence of TPE, (Table III) the χ 2 becomes worse. Fig. 3 shows the angular distribution as a function of cos 2 θ. The lower (upper) branches correspond to backward (forward) emission for a negative lepton.

The solid (black) line is the result of the quadratic fit, the dashed (blue line) is the result of the linear fit.

Including the cos 3 θ term changes indeed the angular distributions, in particular canceling the distortion for collinear kinematics (where it is maximum in the previous case). However the previous conclusions do not change, as it is shown in Fig. 4 for the lowest q 2 value, in Fig. 5 for all considered cases, and in table IV, for the lowest value of q 2 : the parameter a 2

shows a sensitivity to TPE starting from F 3 /G M =0.05. The linear fit shows less deviation and a better χ 2 than in the case where only the linear cos θ term is kept: the distortion from a parabola cancels for collinear kinematics, and for cos θ = 0, see Table V.

III. EXPERIMENTAL ASYMMETRIES

Experimental bias, as possible misalignments of the detectors, approximations in the algorithms for reconstruction..., can appear as a distortion of the angular distribution, which translates into systematic errors. The experimental angular distribution can always be fitted by a function decomposed in a Fourier series of sines and cosines. In this section we look to the sensitivity to a sine term. Histograms are built according to a distribution obtained by adding a term C sin θ to Eq. ( 6), the expression which corresponds to one photon approximation. Numerical estimations have been done for different values of C sin θ (C=0,0.002,0.004,0.02) and the corresponding angular distributions are reported in Fig. 6.

The values of the coefficient C have been chosen in order to have a reasonable size of this term as compared to the two photon case. Similarly to the TPE case, the distributions have been fitted by the function:

y = a 0 + a 1 cos 2 θ + C sin θ. ( 15
)
The values of the parameters, as well as the extracted R and A are reported in table VI.

The quality of the fit can be seen in Fig. 6.

The results look very different from the previous cases, although the values of C are such to induce a quantitative comparable perturbation to the OPE distribution as in the previous section. Evidently the effect of the sine term is to induce a modulation in the distribution according to the angles: minimal for θ=0 and 180 degrees, maximal for θ=90 degrees. However, this distortion has the same sign for forward and backward angles as the contribution , 5% contribution, 20% contribution and for different q 2 values : from left to right q 2 =5.4, 8.2 and 13.6 GeV 2 . Backward scattering (green points), forward scattering (red points).

The solid (black) line correspond to the quadratic fit (12), the dashed (blue line) to the linear fit (11). main amplitudes (square of cosine and sine), therefore it is much more difficult to identify and extract it with a fitting procedure. One can see that the adding a sine term in the function, the error on the parameters becomes very large, even when the coefficient is zero.

Perturbations on the the values obtained in the linear fit (taking into account the error of the linear fit) can be seen even for no sine contribution. and for different q 2 values : from left to right q 2 =5.4, 8.2 and 13.6 GeV 2 .Notations as in Fig. 3.

The solid (black) line corresponds to the cubic fit (13), the dashed (blue line) to the linear fit (11).

IV. CONCLUSIONS

We have shown that the PANDA experiment will be sensitive to a contribution of TPE of 5% or more. TPE contribution is expected to increase at large q 2 , where the statistical error is also expected to be larger, making more difficult its extraction. Due to crossing symmetry properties, the reaction mechanism should be the same in SL and TL regions. If TPE is the reason of the discrepancy between the polarized and unpolarized FFs measurements in SL region, a contribution of 5% is necessary to bring the data in agreement, in the |q 2 | range between 1 and 6 GeV 2 [4]. The PANDA simulations show that such level of contribution will be detectable in the annihilation data. We have shown and discussed the stability of the extraction of R and A, from the data, even in presence of a relatively large contribution of TPE. This has to be taken with caution, as the relation of A with FFs, Eq. (10), holds only in frame of OPE. The signification of the extracted R as the ratio of moduli of the two electromagnetic form factors is not valid anymore. R has to be considered, in this case, as a kind of effective parameter.

However, one should note that if one does not detect the charge of the outgoing lepton, TPE contribution cancels in the angular distributions, and that the cross section at 90 • does not contain any odd contribution, by definition. TPE will cancel in the sum of the forwardbackward cross section, and will appear in the forward backward asymmetry. The moduli of the FFs ratio can be extracted knowing the cross section at at least two angles, even in presence of two photon exchange. This is the main advantage of such study in the time-like region: even in presence of TPE, the angular distribution of the scattered electron pair, for one setting of the antiproton beam, contains all the useful information. In space-like region, the extraction of FFs through the Rosenbluth fit requires measurements at different angles for the same Q 2 (i.e. it is necessary to change beam energy and detection angle at each point). In case of TPE, it is necessary to measure electron and positron scattering, in the same kinematical conditions.

One should note, however, that experimentally, no evidence of TPE (more exactly, of the real part of the interference between OPE and TPE) has been found in the experimental data on elastic scattering for spin zero [5], one half [6], and one [8]. An analysis of the BABAR data [7] does not show evidence of two photon contribution, in the limit of the uncertainty of the data [9].

We studied also the distortion induced by a sine term. In this case the results of the fit are very different, for the main parameters, even for small extra contributions.

Therefore we can conclude that a contribution of the order of 5% of two photon exchange, driven by odd terms in cos θ, will be visible. Care will have to be taken with the real data, in order to identify extra contributions which may induce distorsion in the angular distribution and make it different from the expectations given by the one photon exchange mechanism.

q 2 case a 0 a 2 a 1 χ 2 χ 2 /N f R A (GeV

FIG. 1 :

 1 FIG.1: Angular distributions for different contribution of TPE (including only the cos θ term at q 2 = 5.4 GeV 2 : no contribution (black circles), 2% contribution (red squares), 5% contribution (green triangles) 20% contribution (blue triangledown).

FIG. 2 :

 2 FIG.2: Same as Fig.1, including the cubic term.

FIG. 3 :

 3 FIG.3: Angular distributions as a function of cos 2 θ, built according to 7, but including only the linear cos θ term, for different contribution of TPE: from top to bottom: no contribution, 2%

FIG. 4 :

 4 FIG.4: Angular distributions as functions of cos 2 θ, according to Eq. (7), for q 2 =5.4 GeV 2 and for different contribution of TPE : no contribution (top left), 2% contribution (top right), 5% contribution (bottom left), and 20% contribution (bottom right). Notations as in Fig.3.The solid (black) line correspond to the cubic fit (13), the dashed (blue line) to the linear fit (11).

FIG. 5 :

 5 FIG.5: Angular distributions as functions of cos 2 θ, according to Eq. 7 for different contribution of TPE: from top to bottom : no contribution, 2% contribution , 5% contribution, 20% contribution

FIG. 6 :

 6 FIG.6: Angular distributions for different additional C sin θ contributions at q 2 = 8.4 GeV 2 : no contribution (black circles), C = 0.2% contribution (red squares), C = 0.4% contribution (green triangles), and C = 2% contribution (blue triangle down). Notations as in Fig.3.

FIG. 7 :

 7 FIG.7: cos 2 θ dependence for different contribution in sin θ term at q 2 = 8.4 GeV 2 : no contribution (top right),0.2% contribution (top left), 0.4% contribution (bottom right), and 2% contribution (bottom left). The solid (black) line correspond to the fit as in Eq. (15), the dashed (blue line) to the linear fit.

TABLE I :

 I Expected counting rates, for p + p → e + + e -.

TABLE II :

 II Results from the quadratic fit, for different values of q 2 and different TPE contributions, including only the cos θ term.

			q 2	case	a 0	a 1	χ 2 /N f
	2 )		(GeV 2 )
	5.4	1γ	46798 ± 182 -23 ± 239 9927 ± 485 1.94 0.11 1.004 ± 0.017 0.21 ± 0.01 5.4 1γ 46702 ± 182 9954 ± 485 0.10
		2γ • 0.02 46713 ± 182 119 ± 240 9926 ± 485 1.45 0.09 0.997 ± 0.017 0.21 ± 0.01 2γ • 0.02 46713 ± 182 9926 ± 485 0.09
		2γ • 0.05 46714 ± 182 662 ± 240 9924 ± 485 1.47 0.09 0.998 ± 0.017 0.21 ± 0.01 2γ • 0.05 46713 ± 182 9913 ± 485 0.5
		2γ • 0.20 46710 ± 182 3398 ± 240 9933 ± 485 1.13 0.07 0.997 ± 0.017 0.21 ± 0.01 2γ • 0.20 46700 ± 182 9611 ± 484 11.2
	8.2	1γ	2832 ± 30 -35 ± 42 1128 ± 85 3.66 0.22 1.001 ± 0.095 0.398 ± 0.030 8.2 1γ 2832 ± 30 1127 ± 85 0.22
		2γ • 0.02 2833 ± 29	45 ± 42 1130 ± 85 3.78 0.22 1.000 ± 0.095 0.399 ± 0.030 2γ • 0.02 2833 ± 29 1130 ± 85 0.27
		2γ • 0.05 2830 ± 30 163 ± 42 1136 ± 85 3.49 0.21 0.998 ± 0.096 0.401 ± 0.030 2γ • 0.05 2830 ± 30 1128 ± 85 1.03
		2γ • 0.20 2842 ± 30 805 ± 42 1106 ± 84 6.54 0.38 1.012 ± 0.092 0.389 ± 0.030 2γ • 0.20 2841 ± 30 899 ± 84 20.4
	13.84	1γ	85 ± 5 13.84	2 ± 9 1γ	39 ± 19 4.49 0.26 1.149 ± 1.09 0.469 ± 0.230 85 ± 5 40 ± 19 0.25
		2γ • 0.02	86 ± 5	9 ± 9 2γ • 0.02	41 ± 19 3.36 0.19 1.133 ± 1.116 0.481 ± 0.228 85 ± 5 41 ± 19 0.19
		2γ • 0.05	86 ± 5	16 ± 9 2γ • 0.05	41 ± 19 3.67 0.22 1.137 ± 1.107 0.478 ± 0.228 85 ± 5 39 ± 19 0.39
		2γ • 0.20	82 ± 5	59 ± 9 2γ • 0.20	55 ± 18 2.12 0.12 0.848 ± 2.121 0.672 ± 0.233 81 ± 5 26 ± 18 2.5

TABLE III :

 III Results from the linear fit, according to Eq. 11, on the histograms built for different values of q 2 and different TPE contributions, according to Eq. 7, but including only the cos θ term. 182 891 ± 442 9636 ± 485 1.82 0.11 1.004 ± 0.017 0.21 ± 0.01 2γ • 0.20 46789 ± 182 3537 ± 442 9657 ± 485 2.01 0.12 1.003 ± 0.017 0.21 ± 0.01 8.2 1γ 2832 ± 30 -45 ± 74 1127 ± 85 3.96 0.23 1.001 ± 0.095 0.398 ± 0.030 2γ • 0.02 2859 ± 30 45 ± 75 1057 ± 85 6.29 0.37 1.036 ± 0.087 0.369 ± 0.030 2γ • 0.05 2830 ± 30 163 ± 42 1136 ± 85 5.92 0.35 1.032 ± 0.088 0.373 ± 0.030 2γ • 0.20 2864 ± 30 769 ± 75 1043 ± 85 6.20 0.36 1.042 ± 0.086 0.364 ± 0.030

	q 2	case	a 0	a 2	a 1	χ 2 χ 2 /N f	R	A
	(GeV 2 )
	5.4	1γ	46798 ± 182 -4 ± 442 9927 ± 485 1.94 0.11 1.004 ± 0.017 0.21 ± 0.01
		2γ • 0.02 46795 ± 182 361 ± 442 9638 ± 485 1.96 0.11 1.004 ± 0.017 0.21 ± 0.01
	2γ • 0.05 46795 ± 13.84 1γ 85 ± 5	5 ± 14	40 ± 19 4.41 0.26 1.149 ± 1.09 0.469 ± 0.230
		2γ • 0.02	86 ± 5	13 ± 14	41 ± 19 3.17 0.19 1.136 ± 1.108 0.479 ± 0.228
		2γ • 0.05	86 ± 5	20 ± 14	41 ± 19 3.42 0.20 1.143 ± 1.095 0.474 ± 0.228
		2γ • 0.20	84 ± 5	65 ± 14	47 ± 19 3.69 0.22 0.998 ± 1.493 0.572 ± 0.231

TABLE IV :

 IV Same as table II, according to Eq. 13, i.e., including also the cubic term, for all considered q 2 values.

	q 2	case	a 0	a 1	χ 2 /N f
	(GeV 2 )
	5.4	1γ	46798 ± 182 9627 ± 485 0.11
		2γ • 0.02 46793 ± 182 9634 ± 485 0.33
		2γ • 0.05 46794 ± 182 9638 ± 485 0.15
		2γ • 0.20 46762 ± 182 9635 ± 485 3.66
	8.2	1γ	2832 ± 30 1127 ± 85 0.24
		2γ • 0.02 2859 ± 30 1057 ± 85 0.37
		2γ • 0.05 2830 ± 30 1065 ± 85 0.59
		2γ • 0.20 2849 ± 30 1035 ± 85 6.23
	13.84	1γ	85 ± 5	40 ± 19	0.25
		2γ • 0.02	85 ± 5	41 ± 19	0.22
		2γ • 0.05	85 ± 5	40 ± 19	0.31
		2γ • 0.20	81 ± 5	43 ± 18	1.37

TABLE V :

 V Results from the linear fit, for different values of q 2 and different TPE contributions, including both cos θ and cos 3 θ terms. 959 33 ± 943 1150 ± 636 4.3 0.26 0.986 ± 0.863 0.411 ± 0.267 C=0.002 2413 ± 947 483 ± 931 1251 ± 629 6.47 0.38 0.860 ± 1.440 0.518 ± 0.330 C=0.004 2204 ± 938 751 ± 922 1243 ± 624 8.2 0.48 0.806 ± 1.864 0.563 ± 0.371 C=0.02 1575 ± 905 1602 ± 888 1113 ± 607 8.2 0.49 0.633 ± 4.63 0.706 ± 0.559

	q 2	case	a 0	c	a 2	χ 2 χ 2 /N f	R	A
	(GeV 2 )
	8.2	C=0 2798 ±

TABLE VI :

 VI Same as table II,according to Eq. 15.


E Renaud
email: eddy.renaud@inrae.fr

A Husson
email: alain.husson@inrae.fr

P Eisenbeis
email: patrick.eisenbeis@smegreg.org

B De Grissac
email: bruno.degrissac@smegreg.org

Impacts of demography on drinking water supply networks in Gironde (France)

Keywords: Demographic and geographical data processing, Drinking water supply networks growth forecasting, Geographic information system, Water infrastructure asset management, Water losses, Water resources

To consider the long-term management of the water resources used to produce drinking water in Gironde (south-west of France), the SMEGREG (public watershed establishment in charge of deep aquifers in Gironde, France) and INRAE (French public research institute) are leading a collaboration which, in order to forecast changes in the level of water losses, focuses on the impact of demography and urbanization on the assets of drinking water supply networks (DWSN). Analyses are carried out at the scales of the Gironde county and the Bordeaux metropolitan area (BM), based on demographic and geographic databases, French observatory of water services database and the GIS of the BM's DWSN. The first results obtained show that an increase in the population served by a DWSN translates into a much smaller increase in the length of pipes and a comparable increase in the number of service connections. They also show that the higher the urban density of an area, the lower the impact of population growth on the increasing of pipes length.

Introduction

The Gironde county (GC), located in the south-west of France and whose main city is Bordeaux, is mainly supplied with drinking water produced from deep groundwater, some of which are overexploited. To consider the long-term management of these water resources, in addition to the drinking water consumption of users, it is necessary to take an interest in the needs of the system, most of which consists of water losses from the drinking water supply network (DWSN). Thus, estimating the evolution of these water losses requires taking an interest in the evolution of pipes networks, from a qualitative standpoint but also from a quantitative point of view.

To move forward on this question, the SMEGREG (public watershed establishment in charge of deep aquifers in Gironde, France) and INRAE (French public research institute) are leading a research collaboration to develop a prospective approach of water losses from the DWSNs in Gironde [START_REF] Renaud | Influence de la densité urbaine et de la démographie sur les pertes en eau des réseaux de distribution d'eau[END_REF]. While the first part of this study examines the links between urban density and the performance of services in terms of water losses, the second, which is the subject of this contribution, examines the influence of demography and town planning on the evolution of DWSNs assets.

Methods

The study is being carried out at two spatial scales, the territory of the GC (which includes the city of Bordeaux) and the territory of the Bordeaux metropolitan area (BM).

Approach at the GC scale

This first approach is based on the one hand on data from SISPEA, the French observatory of public drinking water services (SISPEA, 2021), and on the other hand on municipal demographic data from INSEE, the French institute of statistics (INSEE, 2021). The asset information available within SISPEA is, for each DWSN, the total length of pipes excluding service connections (L) and the number of subscribers (N) (generally close to the number of service connections). In addition to the population, the INSEE data includes, for each municipality, information on housing (type, occupation). After concatenation of demographic data at the DWSN level, the usable data cover the period 2010-2017. For each relevant variable X, an annual average change ratio RA-X is calculated. These ratios are then compared with each other.

Approach at the BM scale

This second approach mobilizes the GIS of BM's DWSN, the demographic data of the INSEE at the level of the IRIS which is an infra-municipal statistical subdivision (INSEE, 2021), as well as European land use mapping (Corine Land Cover, 2018) and IGN (French geographical institute) building mapping (IGN, 2021). After geoprocessing, the available data make it possible to study, at the IRIS scale and by land cover classes, the changes in the variables from 2006 to 2017.

Results

GC scale

In 2017, the GC had 1,584,000 inhabitants, or 703,000 subscribers served with drinking water by 19,000 km of pipes managed by around 100 DWSNs. For each service, the annual average changes in demographic indicators, network length (L) and number of subscribers (N) between 2010 and 2017 were calculated. For most services, the evolution of N is very close to that of the population (POP). On the other hand, it appears that, for most services, the length of the network grows much slower than the population. Figure 1 represents the annual average ratio of changes in length of pipes (RA-L) as a function of that of the population (RA-POP). It shows that most services with a growing population have a growth in the length of pipes lower than that of the population (groups G4 and G5 in green). The surface area of BM is 504 km 2 , more than half of which concerns predominantly agricultural or natural areas (Table 1). Logically, the population, which rose from 670,000 inhabitants in 2006 to 744,000 in 2017, resides mainly in the most urban areas. Table 2 summarizes the first results obtained. For the studied period (2006 -2017), BM's population (POP) increased by 11%. In number of inhabitants, this increase mainly concerns the "Discontinuous urban" and "Artificial" IRIS (classes CLC 2 and CLC 3), but in percentages, the less urbanized IRIS (CLC 3 to CLC 5) experience the strongest population growth.

If we look at the network lengths (L) we see that for all the classes their increase is in proportions very much lower than that of the population (which confirms the results obtained at the GC scale). Moreover, while more than half of the new pipes (nearly 30 km) were installed in "Agricultural" and "Natural" IRIS (classes CLC 4 and CLC 5), these only received a little more than 10% of the additional inhabitants. These observations are also reflected in the length of pipe required to serve a new inhabitant, which is increasing significantly from classes CLC 1 to CLC 5.

The variations in the number of service connections (Nsc) remain close to those of the population except for the "Artificial" IRIS (class CLC 3) for which the average increase in Nsc is much lower than that of the population (12% for 32%).

Conclusions

The study conducted by SMEGREG and INRAE at the scales of the Gironde county and the Bordeaux metropolitan area to assess the influences of demography and urbanization on the evolution of drinking water supply networks leads to the first following results:

-For the vast majority of DWSNs, an increase in the population translates into a much smaller increase in the length of pipes and a comparable increase in the number of service connections;

-The level of urbanization of the area concerned (assessed according to land use) has a strong impact on the influence of demography on the evolution of pipes length. This influence is zero for continues urban areas and increases when the level of urbanization decreases;

-Seen from another way, the additional pipes length per additional inhabitant is significantly lower than the average length per inhabitant of the service. For all land use classes average pipes length per inhabitant decreases as the population increases.

The study will now be extended to two other DWSNs within the GC (one on the outskirts of BM, the other on the coastline). Other geographic factors that can potentially impact the effects of demography on pipes length growth will be analysed (roads, structure of buildings, etc.). The impacts of demography on the characteristics of the network (materials, diameters, ages, etc.) will also be considered.

Ultimately, by combining these results with those obtained in the first part of the research collaboration (influence of urbanization on the performance of networks in terms of losses), orders of magnitude of water losses from drinking water supply networks associated with different demographic scenarios will be calculated.

Figure 1

 1 Figure 1 Comparative evolution of population and length of pipes for GC's DWSNs from 2010 to 2017 BM scale The types of land use in the Corine Land Cover inventory have been grouped into five classes: CLC 1 Continuous urban; CLC 2 Discontinuous urban; CLC 3 Artificial; CLC 4 Agricultural and CLC 5 Natural. Each IRIS of BM has been assigned the class which covers the largest part of its territory (Figure 2).

Figure 2

 2 Figure 2 Mapping of BM IRIS by land use class

Table 1

 1 BM. Population and surface areas by land use class

		IRIS	Surface area	2006 population Density
		u	km 2	%	u	%	u/km2
	CLC 1 -Continuous urban	65	21.7	4% 179 870	27%	8 304
	CLC 2 -Discontinuous urban 143 156.3	31% 384 688	57%	2 461
	CLC 3 -Artificial	32	70.0	14%	56 866	8%	812
	CLC 4 -Agricultural	12 100.5	20%	27 456	4%	273
	CLC 5 -Natural	10 155.1	31%	20 911	3%	135
	BM	262 503.6 100% 669 791 100%	1 330

Table 2

 2 BM. Comparative evolution of DWSNs and population between 2006 and 2017

		Variation between 2006 and	Evolution between	Pipe length per inhabitant
			2017 (%)		2006 and 2017 (units)	L/POP (m / capita)
		POP (%) L (%) Nsc (%)	POP (u)	L (m)	2006	New	2017
	CLC 1 -Continuous urban	4%	0%	3%	6 408	-797	2.31	-0.12	2.23
	CLC 2 -Discontinuous urban	11%	1%	9%	40 658	20 066	4.31	0.49	3.95
	CLC 3 -Artificial	32%	3%	12%	18 084	9 248	6.20	0.51	4.83
	CLC 4 -Agricultural	19%	4%	20%	5 170	10 103	9.94	1.95	8.67
	CLC 5 -Natural	20%	9%	22%	4 115	19 881	10.55	4.83	9.61
	BM	11%	2%	9%	74 435	58 501	4.36	0.79	4.00