Measurements of streaming potential versus applied pressure for porous rocks

e

Phys. Chem. Earth, Vol. 22, No. 1-2, pp. 81-86, 1997 © 1997 Published by Elsevier Science Ltd. Ali rights reserved

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Measurements of Streaming Potential Versus Applied Pressure for Porous Rocks M. F. Middleton Department of Geology, Chalmers University of Technology, S-412 96, Goteborg, Sweden

Received JO May 1996; accepted 20 November 1996

Abstract. Current hypotheses of streaming potentiaI behaviour suggest a simple proportionaIity between the streaming potentiaI and applied pressure in porous rocks. However, it is known that the behaviour departs from this simple linear re1ationship with increasing applied pressure. It is commonly assumed that the reason for this departure is due to the fluid flow being "not fu11y developed". This study bas investigated the behaviour of streaming potential versus applied pressure for sandstone and fractured granite. The results indicate that a11 the samples studied disp1ayed the simple re1ationship AV = A AP for AP 1ess than 0.075 MPa, where AV is the streaming potential, A is ca11ed the "coupling coefficient", and AP is the applied pressure. Some rocks exhibited a departure from this 1inear behaviour for applied pressures in the range 0.008 to 0.03 MPa. This non-linear behaviour can be described by the relationship: AV = A [l - B Apm-1 ]AP, where B and m are constants. From the observed data, it is found that the constant m is typica11y in the range 2 to 3, when non-linearity exists. An hypothesis, based on flow separation, is proposed to explain the observed behaviour. As 1inear re1ationships between pressure gradient and flow rate were aIso observed in aIl experiments, it is concluded that the flow separation within the pore space is due to irregu1arities in pore geometry rather than non-Darcy flow. The exponent "m" in the non-linear equation is proposed to reflect pore roughness. Values of "m" in the range 2 to 3 suggest high pore roughness, and values greater than 5 indicate decreased roughness. © 1997 Published by Elsevier Science Ltd

Mizutani, 1981; Sill, 1983; Morgan et aI., 1989; Sprunt et aI., 1994; Jouniaux and Pozzi, 1995). Streaming potential bas also recieved recent attention as a means of monitoring fluid flow in oit field systems (Wurmstich and Morgan, 1994; Middleton, 1994). The aim of this paper is to present the observed behaviour of streaming potential versus applied pressure in varlous porous rocks, and to propose a simple model to explain this behaviour. The classical behaviour of streaming potential is described by the Helmboltz-Smoluchowski equation (Levine et al., 1975; Sprunt et aI., 1994): AV 1 AP =

[s

ç

Rw] 1 [4 7t

~] ,

(1)

where AV is the streaming potential (a potential difference), AP is the applied pressure difference, s is the dielectric constant, ç is the zeta potentiaI, Rw is the pore fluid resistivity and ~ is the fluid viscosity. As aIl the parameters on the right-hand side of Eq.(I) are considered constant physical properties for each rockfluid system, the relation between streaming potential and applied pressure is predicted to be linear. The ratio AV/AP is termed the "streaming potential coupling coefficient" . The behaviour in Eq.(l) assumes that (a) the fluid flow is laminar, (b) the surface conductance of the matrix grains is negligible, and (c) the pore (capillary) radius is very much larger than the electrical double layer thickness (5) at the solid/fluid interface. These conditions are valid for many porous rocks.

1 Introduction Streaming potential in porous rocks, as a consequence of fluid pressure gradient, has been studied for many years ( Lynch, 1962; Ahmed, 1964; Ogitvy et al., 1969; Corwin and Hoover, 1979; Fitterman, 1979; Ishido and

2 Experimental Streaming potential can be measured in the laboratory using a simple apparatus (Fig. 1), where a pressure

81

82

M. F. Middleton

difference across a rock sample is applied by connecting a water reservoir at constant head to the inlet of the measurement apparatus. The experimental procedure used in these experiments has been described by Middleton (1996).

fluid used for the sandstones was water of resistivity Il nm.

'1

El ~

! ! 1

SAMPLE

INLET

• TrlHlle IInllstanl 'larnlMll,,) D n .. S.adst... (AilUraUIl

; = ... ,

r-------~a'r_------~

-.=TL--JI

11

Iu=....

e

. 2 0.05 8.11 tlPPLIEI NlESSURE

OUTLET

là". 1.1 lll'a1.15

I.ZI

ZI~------~-------r-------'------~

frKtlll'H arenltes 'SMldI.,

=>

"'

~ IS Fig. 1. Schematic diagram of the experimental apparatus used to detennine streaming potential.

!I ~

1[DUmOI H-------+. li

~

li

The rock samples measured were (1) a sandstone from the Flag Sandstone of the Australian North West Shelf with porosity of 20 %, (2) a sandstone of Triassic age from Bornholm with porosity of 5 %, and (3 & 4) two samples of fractured granite from the Zinkgruvan Mine, Sweden. The fractured granites had porosities less than 1 %. The Flag sandstone sample had a permeability of 10- 13 m2 , the sandstone from Bornholm had a permeability of 10- 14 m2 , and the fractured granites had a permeability of approximately 10- 17 m2. The maximum applied pressure gradient for these experiments was 4 MPa/m, and a linear relationship was observed between pressure gradient and flow rate for ail samples. It was thus concluded that a Darcy flow regime existed in ail samples. The results of these experiments are plotted as observed electrical potential, which is the streaming potential, versus applied pressure difference (Fig.2). Middleton (1996) has discussed the results from Fig.2, and concluded that an empirical curve of the type: AV =

A {l - B[AP]m-1} AP ,

(2)

could be matched to the data for applied differential pressures of less that 0.03 MPa. The constant A has been termed the "coupling coefficient" (AV/AP for the initiallinear part of the curve). For the sandstones (Fig. 2a) , we find that the Triassic sandstone from Bornholm exhibits linear behaviour, and the Flag Sandstone exhibits non-linear behaviour. For the Bornholm sandstone the coupling coefficient, A, is 272 mVIMPa. For the Flag Sandstone, A = 435, B = 635 and m = 2.95, measuring AV in millivolts and AP in MPa. When linear behaviour is observed, we May consider that the constant B=O or mis large (>5). The

i

~

5

'" I.DS •• 11 0.15 1.21 .PPLIED PRESSURE Fig. 1. Plots of Slreaming pOiential venul applied prelsure difference for the samplel Flag Sandstone and Triasaic sandstone from Bornholm, (b) fractured granite rocks from Sweden. Eqn.l, linear, and Eqn.2, non-linear, are matched to the data al appropriate.

1à".'.l "'1

In the case of the fractured granites (Fig.2b), we again find one that exhibits linear behaviour and another that has non-linear behaviour. For the linear behaviour, A = 1012 mV/MPa, and for the non-linear case, A = 598, B = 132 and m = 2.5 (AV in millivolts and AP in MPa). Water of resistivity 25 nm was used with the granites.

Reflected-light photographs (50x magnification) were taken of fractures on the surface of the granites in order to investigate the possibility that the different streaming potential behaviour, seen in Fig.2b, May be caused by texturaI differences. Examination of a typical fracture in the granite exhibiting Iinear behaviour (Fig.3a) shows that the fracture filling is calcite with strongly developed secondary porosity; a vug of dimensions 0.75 mm by 0.25 mm is clearly visible in the centre of the photograph. The granite with non-linear behaviour, and almost half the "A" (coupling coefficient) value, shows healed calcite fractures with very little secondary porosity (Fig.3b). Thus, the fractures in the two samples appear to show quite clear texturaI differences. Therefore, it is tempting to suggest that a greater coupling coefficient, and linear behaviour, are both associated with greater porosity, or perhaps greater pore throats.

Measurements of Streaming Potential Versus Applied Pressure Indeed, it has been long known that capillaries of small radius (approaching the size of the double layer thickness, B) give rise to lower coupling coefficients than those with greater radius (I~vine et al., 1975), due to effects such as electroviscous retardation and surface conductivity. Such behaviour has been recently confirroed by louoiaux and Pozzi (1995), who show that the coupling coefficient decreases dramatically (nearly three orders of magnitude) for saodstones if their perroeability is decreased from 10- 12 m2 to 10- 16 m2 . They do not, however, observe the non-linear hehaviour in their samples, and the non-linear hehaviour observed here (and by others, such as Morgan et al., 1989) may he due to an effect other than pore-throat size.

"j

that sorne samples exhibit non-linear hehaviour at higher pressures (Eq.2). Morgan et al. (1989) reported that the streaming potential versus pressure departed from the expected linear hehaviour for pressure differences greater than 0.01 MPa. They suggested that the departure was due to the lack of a "fully established flow regime" at these greater applied pressure differences, and thereafter confined their studies to the linear-hehaviour regime. Il would he convenient to have an explanation of the departure of the streaming potential from the linear relationship to applied pressure difference. Such a deviation may he related to non-Darcy, non-turbulent flow within the rock with Reynolds numhers in the range 1 to 10 (1 10 (Bear, 1972, p. 177, 182). This type of flow is possibly what Morgan et al. (1989) referred to as a not "fully established flow regime". Bear (1972) suggests that this non-Darcy flow region (before the onset of turbulence) is probably caused by flow separation, although the precise cause is still in debate. The observations in this study, however, preclude this interpretation for pressure gradients up to 4 MPa/m, as a Darcy flow regime was observed in ail samples.

.

, ..t; ' .. ::,.

83

~

a

An hypothesis is now presented that explains the nonlinearity of the streaming potential versus applied pressure curve in terms of flow separation, due to pore irregularities, or pore roughness. Thus, parts of the pore walls become isolated due to this flow separation, but the main fluid flow path is not restricted and a Darcy flow regime is maintained.

3 Theory

b Fig. 3. Retleeted light photographs (50x magnifieation) of typieal fractures in the granites shown in Fig.2b; (a) granite with linear behaviour, and (b) granite with non-linear behaviour. The width of field in eaeh photograph is approximately 1.5 mm.

Sprunt et al. (1994) investigated the streaming potential for pressure differences in the range 0.14 to 0.41 MPa. They found that the "coupling coefficient" was typically half that found by Morgan et al. (1989), for the same fluid resistivities. This conflict in reported values for the "coupling coefficient" may he explained the observation

A quantitative description of the departure of streaming potential versus pressure difference may he derived by considering the theory of streaming potential for thin capillaries proposed by Levine et al. (1975), but with consideration of flow separation in part of the capillary. We consider the model shown in Fig.4, which shows three stages of flow development: (a) full laminar flow (Fig.4a), (b) flow with a small degree of flow separation (Fig.4b) and (c) flow with a relatively large degree of flow separation (Fig.4c).

We first consider the theory presented by Levine et al. (1975) for the type of flow in Fig.4a. They propose (assuming axial symmetry) that the axial electrostatic potential, V(r,x), in the pore capillary is V(r,x) = Vo - Ex x

+ ",(r) ,

(3)

where x is the axial direction, r is the radial direction, Vois the potential due to the axial electrical field at the

84

M. F. Middleton

beginning of the capillary (x=O), Ex (=-ôV/âx) is the axial electric field strength, and \jI(r) is the radial potential due to the double layer effect. Equation (3) cao be generalized to V(r,x) =

Vo - 1Ex dx

+ \jI(r) .

(4)

with the boundary conditions: ux(a) = 0, \jI(a-a) = ç and a> >li ,

(5)

where V2 is the Laplacian operator, U x is the velocity of the fluid in the x-direction, P is pressure, Pe(r) is the radial distribution of charge density, ltotal is the total electrical current flowing, lcond is the electrical current due to conduction alone, lconv is the electrical current due to convection alone, r de dr is an element of the cross-sectionaI area of the pore capillary, a is the radius of the pore, ç is the zeta potential as presented in Eq.(I), and li is the electrical double layer thickness. When the streaming potential effect occurs, ltotal =0. Levine et al. (1975) show that with this condition, the axial electrical field cao be written as: Ex

= -[ôV/âx] = {l& ç Rw] 1 [41t I.I.]}

{ôPlâx} ,

(6)

which is the same as the Helmboltz-Smoluchowski equation, as in Eq.(I). Equation (6) shows that the potential difference, dV(x,r), along the capillary bas no dependence on the radius of the capillary provided the conditions (5) are met. We now consider the condition of flow separation in Fig.4b. In the region where the flow separation occurs, the boundary conditions (5) are not met, especiaUy to note is that the radial potential \jI(r) at the boundary of the laminar flow, fJam' is considerably less than ç: \jI(fJam) < < ç. Thus, the electrical field , Ese~' in the region of separation will be very mucb less than the electrical field, Ex, in the rest of the capillary : Esep < < Ex' Hence, if we calculate the streaming potential , AV, using Eq.(3), we find: Fig. 4. Schematic sketch of the proposed simple pore model: (a) with laminar tlow through the complete pore &pace, (b) with a 8ma11 lenglh, SI' of tlow separation, (c) with a relatively long lenglh, S2' of tlow separation.

Levine et al. (1975) solve the equations for fluid flow momentum, Gauss' law and electrical current flow: 1.1.

V2[ux(r)] = [ôP/âx] - Ex Pe(r) , (fluid momentum) -[4

ltotal = lcond

lconv

1f

1t

Pe(r)] 1 &

+ lconv'

Pe

Ux

r de dr,

,

(Gauss' Law)

AV =

- Ex [L -

Sil '

(7)

where AV is V(x,r) - V0 ' L is the length of the complete pore and SI is the lengtb of the zone of flow separation (see Fig.4b). In the case in Fig.4c, the flow separation occurs over a longer zone S2' Similarly, we cao write: AV = - Ex [L - SV,

We cao see from these cases that the length of flow separation is a function of the applied pressure difference AP: S = S(AP). Thus, we cao generalize this bebaviour to the following relationsbip:

(electric current flow) AV = - Ex [L - S(AP)] = - {l& ç Rw] 1 [41t I.I.]} {L-S(AP)} {ôP/âx} .

Measurements of Streaming Potential Versus Applied Pressure

Then, if L=dx, the length of the pore path, we can write this as: AV =

{[s ç Rw] / [41t

~]}

{1 - S(AP)/L} AP . (8)

Thus, equation (8) describes the departure of the streaming potential, AV, from the classical linear relationship to the applied pressure difference, AP.

4 Discussion The theoretical expression based on our simple model of flow separation in the pore capillary (Bq.8) is similar in form to the empirically based Eq.(2). In comparing the two equations, it is seen that AB[AP]m-1 from Eq.(2) is equal to {[s ç Rw ] / [4 1t ~]} {S(AP)/L} from Eq.(8). Thus, it is concluded that the length of flow separation S is related to pressure by: S(AP)

=

constant [AP]m-l .

(9)

A further consequence of this hypothesis is that the departure of the streaming potential versus applied pressure curve from the linear relationship is an indication of how difficult one might find moving liquid hydrocarbons from irregular pores. Furthermore, a worthy conjecture is that the exponent "m" in Eq.(2) and Eq.(9) May provide some index of pore roughness. It is heyond the scope of the present study to draw conclusions on the type or degree of pore roughness that a particular "m" value might represent, but it appears that the lower the "m" value: the greater the roughness. This conjecture is supported by the texturai differences seen in the granites (Fig.3a and Fig.3b), which suggest that the more restricted pore volume (non-linear behaviour) is presenting a less regular, perhaps more tortuous, path to the flowing fluid. 15r---------~----------~--------~

85

A further consideration of the present analysis is that the conflict between the "coupling coefficients" reported by Morgan et al. (1989) and Sprunt et al. (1994) May he resolved by consideration of the non-linear hehaviour discussed above. Figure 5 shows observed data for the Flag Sandstone up to applied differential pressures of approximately 0.04 MPa. The Flag Sandstone data are consistent with the Morgan et al. (1989) linear hehaviour for 1:1 electrolytes (AV liIj 4 Rw AP) for applied pressure differences of less than 0.01 MPa. For pressure differences greater than 0.01 MPa, the Flag Sandstone data exhibit non-linear hehaviour, with a trajectory towards the Sprunt et al., (1994) data. Figure 5, therefore, suggests that the Morgan et al. "coupling coefficient" May he reconciled, at least partially, with the Sprunt et al. "coupling coefficient", due to this observed non-linear hehaviour. This argument is, of course, contingent upon the assumption that the samples analysed by Morgan et al. (1989) have a similar zeta potential to those analysed by Sprunt at al. (1994). This may not, in fact, he completely the case, as the former researchers used crushed granite (of unknown permeability), and the latter used Berea Sandstone with permeability of approximately 2 x 10- 13 m2 . Nevertheless, the Flag Sandstone with permeability of 10- 13 m2 is consistent with the Morgan et al. hehaviour for low differential pressures.

5 Conclusion This study presents an hypothesis to explain the nonlinear hehaviour of streaming potential versus applied pressure at high pressure gradients, which is observed in some rocks. This non-linear hehaviour is proposed to he caused by flow separation due to pore roughness. An exponent "m" is proposed to characterize this nonlinearity for applied pressure gradients up to 4 MPalm (Eq.9). Values of m in the range of 2 to 3 appear to indicate rough, or irregular, pore geometry, and values greater than 5 suggest low pore roughness, or perhaps straighter capillary-like pores.

References

.~--------~~------__~~.~.h~se~r~~d~d~l~tl 1.5 1.' 1.5 "LIE' PRfSSIIIE IAPI, 1.1 IIPI Fia. 5. Experimencal data showing streaming potential versus applied pre88Ure difference for Flag Sandstone samples. The linear behaviour propoaed by Morgan et al. (1989) for a rockcontaing similar water resistivity, and the data region of Sprunt et al. (1994) are also shown. The curve shown (Eqn.2) may offer a reconciliation to the different data seIa.

Ahmad, M.U., A laboratory study of streaming potentials, Geophys. Prospect., 12,49-64, 1964. Bear, J., Dynamics of Fliuds in Porous Media, Elsevier, 1972. Corwin, R.F. and Hoover, D.B., The self-potential method in geothermal exploration, Geophysics, 44, 226-245, 1979. Fitterman, D.V., Calculations of self-potential anomalies near vertical contacts, Geophysics, 44, 195205, 1979.

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M. F. Middleton

Ishido, T. and Mizutani, H., Experimental and theoretica1 basis of electrokinetic phenomena in rock-water systems and its application to geophysics, J. Geophys. Res., 86,1763-1775, 1981. Jauniaux, L. and Pozzi, J-P., Streaming potential and permeability of saturated sandstones under triaxial stress: Consequences for electroteUuric anomalies prior to earthquakes, J. Geophys. Res., 100, 10197-10209, 1995. Levine, S., Marriott, J.R., Neale, G. and Epstein, N., Theory of electrokinetic flow in fine cylindrica1 capillaries at high zeta-potentials, J. Colloid. Interface Sei., 52, 136-149, 1975. Lynch, E.J., FOmuJlion Evaluatioll, Harper & Row, New York, 1962. Middleton, M.F., S.P. measurements in porous rocks, Forskningsseminarium, 12 Oecember 1994, Sammanfattningar av Anroranden, Chalmers Tekniska HogskoIa, Geologiska Institutionen Publ. B 407, 3639,1994. Middleton, M.F., The influence of pressure on streaming potential for porous rocks, Nordic symposium on PETROPHYSICS AND RESERVOIR MOOELLING - Fractured Reservoirs, 25-26 January, 1996, Gothenburg, EXTENOEO ABSTRACTS, Chalmers Tekniska HogskoIa, Geologiska Institutionen Publ. B 425,37-44, 1996.

Morgan, F.O., Williams, E.R. & Madden, T.R., Streaming potential properties ofWesterly Granite with applications, J. Geophys. Res., 94, 12449-12461, 1989. Ogilvy, A.A., Ayed, M.A. & Bogoslovsky, V.A., Geophysica1 studies of water leakages trom reservoirs,Geophys. Prospect., 17, 36-62, 1969. Sill, W.R., Self-potential modeling from primary flows, Geophysics, 48, 76-86, 1983. Sprunt, E.S., Mercer, T.B. & Ojabbarah, N.F., Streaming potential from multiphase flow, Geophysics, 59,707-711, 1994. Wurmstich, B. & Morgan, F.O., Modeling of streaming potential responses caused by oil weil pumping, Geophysics, 59, 46-56, 1994.