HAL Id: jpa-00230371
https://hal.archives-ouvertes.fr/jpa-00230371
Submitted on 1 Jan 1990
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
ULTRASONIC DIAGNOSIS IN MULTIPHASE FLOWS
J.-C. Bacri, D. Salin
To cite this version:
J.-C. Bacri, D. Salin. ULTRASONIC DIAGNOSIS IN MULTIPHASE FLOWS. Journal de Physique
Colloques, 1990, 51 (C2), pp.C2-13-C2-16. �10.1051/jphyscol:1990203�. �jpa-00230371�
COLLOQUE DE PHYSIQUE
Colloque C2, supplement au n°2. Tome 51, Fevrier 1990 ler Congres Frangais d'Acoustique 1990
C2-13
ULTRASONIC DIRGNOSIS IN MULTIPHASE FLOWS
J.-C. BACRI and D. SALIN
Laboratolre d'Ultrasons, VRR CNRS n°789. University Pierre et Marie Curie, Tour 13, 4 Place Jussieu, F-75252 Paris Cedex, France
Résumé. - Nous présentons ici une nouvelle méthode de diagnostic ultrasonore dans les milieux poreux naturels. Cette méthode, fondée sur les variations de vitesse acoustique, nous a permis de mesurer la répartition spaciale de deux liquides (eau et huile) dans un grès des Vosges.
Abstract. - We present here a new ultrasonic diagnosis technique in natural, porous rocks. This method, based on the ultrasonic velocity changes, allows us to measure the transient spacial distribution of two liquids (water and oil) in a sandstone.
1 - INTRODUCTION
The characterization of immiscible fluid displacements in porous media requires the knowedge of the transient spatial distribution of fluids. This requirement cannot be met by large-volume average measurement techniques.
We present here a ultrasonic diagnosis technique based on the ultrasonic velocity changes with fluid saturation in porous media. We first recall the basics of ultrasonic waves propagation in porous media and show how ultrasonic data are linked to the static properties of the medium such as density and elastic constants but also to hydrodynamic properties, namely tortuosity and permeability. We apply this method to immiscible flows in porous media.
2 - ACOUSTICS OF POROUS MEDIA
Although dating back to 1956, Biot's theory [1] remains the best way to describe the acoustic wave propagation in porous media. Numerous consequences of this theory have been experimentally tested [2]. In this short review, we shall adopt a much simpler approach which provide the main physical features of the theory. We shall focus in particular on bulk longitudinal waves, transverse waves and extension to surface waves having been already discussed elsewhere [3].
We consider a multiconnected solid porous medium saturated with a fluid. At each point r , we define displacement vectors for both fluid UpCr.t) and solid u"s(r,f), averaged over volumes large enough in comparison with the medium heterogeneities. As our approach is a continuous medium one, the wave length X of the acoustic wave is larger than the averaging volume and hence larger than any detail of the pore structure. Since we deal with two media, we can write down two wave equations :
(1)
where pp , ps > E F and Es are the densities and elastic constants for respectively fluid and solid. Dots are time derivatives and r superscripts space derivatives. For plane longitudinal wave vector q = 2rc/X and frequency CO/2TC . The displacements are proportional to the propagating factor exp [i(q r - cot)] . Inserted in Eq. (1), we obtain the sound velocities V = co/q of the fluid (Vp = (EF/pF)1/2) and of the solid (Vs = (Es/ps)1/2) • A first natural way to couple solid and fluid displacements is through viscous friction : this friction involves the relative
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990203
c2-14 COLLOQUE DE PHYSIQUE
velocity iF
- us
and the permeability k of the porous medium through a friction coefficient f = q (?/k where q is the viscosity of the fluid and @ the porosity. Equation (1) becomes :where Es is now the elastic constant of the porous medium. Seeking for plane wave solutions of this set of equations, we get the dispersion equation q(o) which has two wave solutions. The prediction of two modes is indeed the main feature of the theory but their simultaneous experimental observation is quite recent 141. A quick glance at Eq. (2) shows that the friction term is of the order of fW/02pF compared with 1 for the other terms : at low frequency, friction will be dominant while at high frequency the friction coupling vanishes. We can define a characteristic frequency Q which separates these two regimes :
The physical meaning of oc is simple : in a viscous fluid, shear surface oscillations decay over the viscous skin depth 6 = 2 p q / p F o
.
At low frequency 6 is larger than the typical pore radius (which is of the order of kin) and the whole fluid is involved in the solid oscillation. At high frequency, only the fluid over 6 is involved. The cross-over between these two behaviours occurs when 6 is of the order of kln, which is equivalent to Eq. (3).In the low frequency regime (o c o c ) , Eq. (2) is valid and leads to two modes : a propagating mode for which the fluid and solid displacements are in phase and another mode for which the fluid and the solid oscillate out of phase. For this second mode there is a large viscous dissipation and this mode is viscoelastic : the amplitude of the displacements vanishes within a few wavelengths. The velocity of the propagating mode is :
with p = @ p ~ + ( l - @ ) p ~ and H = K b + i N
+
(Ks-
Kb)2 KF (1-
@)KsKF + @K:-
KbKS In this expression, resulting from Biot's theory, Kb and N are the bulk and shear moduli of the dry solid skeleton of the porous medium, Ks and KF bulk moduli of solid medium and fluid. In this regime the attenuation is proportional to c$ and k.In the high frequency regime (w B we), the friction coupling vanishes. Other couplings such as density and elastic couplings become predominant. For density coupling, eq. (2) transforms into :
where ~ F S = (1
-
a ) @ p ~ describes the inertial drag that the fluid exerts on the solid. The tortuosity a is a purely geomemcal factor (a = (1+
@-')I2 for a suspension of spheres) ; a governs also the dispersion at zero flow rate [5]. The two modes propagate but, except in devoted experiments [4], only the fast one is observed. In the simple case of an unconsolidated porous medium (Kb = N = 0), we get :where K: = @ ~ - ; + ( l - a ) ~ :
and p e = a p F (1 -
$IPS
+ (1-
a - ' ) $ p ~-
+ (a-
2$ + $2)pFI<e
and pe are here the effective compressibility and density of the system.Such an expression shows the physical parameters involved in ultrasonic velocity measurements : densities, solid and fluid velocities, porosity and tortuosity. The permeability does not appear in these expressions but is involved in attenuation measurements. Expression (6) is valid for a suspension of concentration c = 1 - @
.
3
-
IMMISCIBLE FLUIDS IN POROUS MEDIAOne of the basic experimental problem of oil recovery and multiphase flow is to determine the local saturations in oil (So) and water (S,). It has been generally accepted that acoustic methods are not appropriate for the determination of saturations in sandstones. This acceptance is related to the fact that the measured sound velocity in rock saturated with either oil and water are close to one another (figure 1 at S, = 0 % and S, = 100 %). In this figure we plot the sound velocity versus the water saturation Sw (the oil saturation is So = 1
-
S,). Water (in fact a brine) is the totally wetting fluid. These data [6] show clearly the nonlinear extrapolation currently accepted.Figure 1. Sound velocity variations in a sandstone saturated with both brine
(Sw) and oil (So = 1
-
Sw). The circles correspond to imbibition injection, the crosses to drainage. The curve through the data is our extension of Biot's theory.The right hand side branch (+) corresponds to the injection of a non-wetting fluid (oil) in a fully water saturated and water-wet sample. The dashed-dotted line through these data is our theoretical expression of Biot's theory to immiscible fluids. In our experiment, the ultrasonic frequency is 400 kHz and we are ih the regime o<<w, : sandsone permeability K = 110 mDarcy (1 mDarcy = 10-11 cm2), fluid viscosity q = 10-2 cp. Equation (4) holds with a single fluid consisting of an effective mixture of oil and brine with p~ = Swpw + Sopo and K;,' = S,K;
+
SoK; (where p,, po, Kw and K,, are water and oil densities and compressibilities).pw = 1015 kgm-3 , po = 755 kgm-3 , K, = 2.4 x 109 Pa , KO = 1,2 x 109 Pa
.
The porosity of the sample is@ = 21.5 % , Ks = 3.7 x 101° P a , ps = 2650 kgm-3 , Kb = 3.2 x 109 P a .
The left hand brand ( 0 ) corresponds to the injection of water in a fully oil saturated water-wet medium. The decrease of the sound velocity with S, results from the continuous decrease of the frame modulus due to the modification of the grain contacts by the injection of the totally wetting fluid. The accuracy of the method is around 1 % in S,.
COLLOQUE DE PHYSIQUE
4
-
PROFILE MEASUREMENTSIn immiscible displacements [7] (imbibition and drainage), the sample (sandstone) is typically 4 x 4 x 30 cm in size and is coated with epoxy. The fluids flow vertically along the largest dimension. Saturation measurements are performed using ten transmitter-receiver pairs of transducers laid every 2.5 cm along the sample. As the transducers are at fixed position, we get a very good accuracy AV/V = 10-4. The ten positions are generally scanned in less than 2 minutes. Recent improvements in the technique decrease this time doyn to 10 seconds. Our automatic procedure records the time dependence of the saturation c(x,t) for each transducer pairs.
This system allows us to analyse the following experiment. When a totally wetting fluid (here water) injection is performed in a prewet sample saturated in oil (i.e. a medium at irreducible water saturation), the capillary diffusion presents symptoms of hyperdiffusion, which means that the saturation profile extends more widely than in classical diffusions [7, 81.
CONCLUSION
Ultrasonic diagnosis is an accurate way to study local oil and water saturation in a natural rock. This technique is a devoted one for studying transient special distribution of fluids in multiphase flow experiments.
REFERENCES
[I] BIOT, M. A., J. Acous. Soc. Am. 28 (1956) 168.
[2] JOHNSON, D. L., PLONA, T. J., J . Acous. Soc. Am. 7 2 (1982) 556.
[3] FENG, S. and JOHNSON, D. L., J. Acous. Soc. Am. 74 (1983) 906.
[4] PLONA, T. J., Appl. Phys. Lett. 36 (1980) 259.
[5] BACRI, J.-C., LEYGNAC, C . and SALIN, D., J. Physique Lett. (France) 45 (1984) L-767.
[6] BACRI, J.-C. and SALIN, D., Geophys. Res. Lett. 13 (1986) 326.
[7] BACRI, J.-C., LEYGNAC, C. and SALIN, D., J. Physique Lett. (France) 46 (1985) L-467.
[8] BACRI, J.-C., ROSEN, M. and SALIN, D., Europhys. Lett. ?? (1990)
