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Frequency-dependent streaming potential coefficient measurements have been made upon Ottawa sand and glass bead packs using a new apparatus that is based on an electromagnetic drive. The apparatus operates in the range 1 Hz to 1 kHz with samples of 25.4 mm diameter up to 150 mm long. The results have been analysed using theoretical models that are either (i) based upon vibrational mechanics, (ii) treat the geological material as a bundle of capillary tubes, or (iii) treat the material as a porous medium. The best fit was provided by the Pride model and its simplification, which is satisfying as this model was conceived for porous media rather than capillary tube bundles. Values for the transition frequency were derived from each of the models for each sample and were found to be in good agreement with those expected from the independently measured effective pore radius of each material. The fit to the Pride model for all four samples was also found to be consistent with the independently measured steady-state permeability, while the value of the streaming potential coefficient in the low-frequency limit was found to be in good agreement with other steady-state streaming potential coefficient data.

There have only been 10 measurements of the frequency-dependent streaming potential coefficient of porous geological and engineering materials. A review of the existing measurements was carried out by Glover et al. [

While the first of these approaches mimics many of the possible applications more closely [

The second approach is capable of providing the streaming potential coupling coefficient at each frequency directly. Its disadvantage is that a high-quality harmonic driving pressure is required to create the time-varying flow. Various authors have shown that measurements on a range of materials are possible in the range 1 Hz to 600 Hz [

This paper reports research that uses the electromagnetic drive concept proposed by Glover et al. [

The steady-state streaming potential coefficient (the streaming potential per driving fluid pressure difference) has long been described by the Helmholtz-Smoluchowski (HS) equation, and is given in the form most convenient for application to rocks (e.g., [

In this equation

The importance of considering the surface conductance when applying the HS equation to geological materials has been discussed by a number of authors including [

There are several theoretical models for the frequency-dependent streaming potential coupling coefficient. The models fall into three categories: (i) models based only on vibrational mechanics [

If we apply the amplitude of the critically damped second-order vibrational behaviour [

for the frequency-dependent streaming potential coupling coefficient. In this equation

Both equations can be fitted to experimental data where

These vibrational mechanics models are purely formal and contain no underlying physics. They are interesting in that they can show that a system is behaving in a certain manner, but no inference can be made, for example, about what controls the damping coefficient. This lack of specificity often allows such models to apparently fit the data better than other models which include more of the underlying physics.

The capillary tube model was introduced by Packard [

where

where ^{3}) is the density of the bulk fluid,

Reppert et al. [

(their Equations 26 and 38). However, a recent study showed that the simplification is incorrect [

When (

An extremely important study by Pride [

where

Equation (

noting that the steady-state term in this model includes an additional factor

In (^{2}) is the steady-state fluid permeability, and

Recently, such a simplification of (

where

where^{−3 }mol/dm^{3} or more. If we take

which is dependent solely on the transition frequency.

If either the full Pride model (

Until recently only the Packard model [

Most of the theoretical models have a real and imaginary part. In this paper we have analysed these two contributions separately, comparing the measured data with the overall magnitude and each of the complex components of each model. We have taken an RMS measurement approach which provides the magnitude of the variation and maximises the precision with which it can be measured.

It is interesting to consider the physical meaning of the real and imaginary contributions to the streaming potential coupling coefficient. Currently we do not have sufficient information to answer this question with authority. However, there are some indications. The streaming potential coupling coefficient is defined as the ratio of the streaming potential to the pressure drop across the sample. However, it is the fluid velocity that separates the charge and causes the streaming potential. This implies that the frequency dependence of the streaming potential coupling coefficient depends on the frequency dependence of the dynamic fluid permeability. The dynamic fluid permeability at low frequencies is controlled by viscous flow that is represented by the real part of the dynamic permeability. However, when a critical frequency is reached, the inertial acceleration of the fluid begins to control the flow (e.g., [

Experimental tests were carried out on samples of Ottawa sand and on packs of glass beads using the apparatus described in the associated paper [

Physical properties of the sample material.

Property | Unit | Ottawa sand | Glass beads | Comment | ||

0. 5 mm | 1 mm | 2 mm | ||||

Modal grain radius (laser diffraction), | — | — | — | Using a Malvern Mastersizer 2000. | ||

Modal grain radius (image analysis), | Image analysis using Sigma Scan 4. | |||||

Modal grain radius (Hg injection), | — | — | — | Calculated from pressure data using Mayer-Stowe theory. | ||

D10 grain radius (sieving), | — | |||||

Effective pore radius, | Using the method of Glover and Walker [ | |||||

Effective pore throat radius, | Using Glover and Déry [ | |||||

Modal pore throat radius (Hg injection), | — | — | — | Using a Micromeritics AutoPore IV | ||

Porosity (gravimetry) | — | 0.314 | 0.383 | 0.380 | 0.382 | Please see text. |

Porosity (helium expansion) | — | 0.325 | 0.391 | 0.383 | 0.385 | Using a real gas expansion pycnometer. |

Porosity (mercury injection) | — | 0.304 | — | — | — | Using a Micromeritics AutoPore IV. |

Measured permeability, | m^{2} | The measured permeability at 5 Hz for the Ottawa sand and under steady-state conditions for the glass beads. | ||||

Predicted permeability using the RGPZ method, | m^{2} | Permeability predicted from electrical data and the grain diameter using the method of Glover et al. [ | ||||

Electrical conductivity, | S/m | For a 0.001 mol/L NaCl at 25°C. | ||||

Electrical conductivity of the fluid, | S/m | Measured on the recycled fluid at the sample outlet after equilibration. | ||||

Formation factor, | — | 4.676 | 4.13 | 4.18 | 4.16 | Calculated from the conductivities of the saturated rock and the saturating fluid. |

Connectedness, | — | 0.214 | 0.242 | 0.239 | 0.241 | Calculated from the formation factor [ |

Cementation exponent, | — | 1.372 | 1.48 | 1.48 | 1.48 | Calculated from the formation factor. |

Electrical tortuosity, | — | 1.519 | 1.583 | 1.589 | 1.588 | Calculated from the porosity and |

Theta factor, Θ | — | 3.705 | 3.53 | 3.57 | 3.55 | From the method of [ |

Predicted transition frequency, | Hz | 256.48 | 213.36 | 57.32 | 12.61 | At 24°C using the method of [ |

Ottawa sand was obtained from Fisher Scientific and washed repeatedly in distilled water in order to remove any rock powder before being dried in a vacuum oven prior to use. The pore throat and grain size distributions of the sample material were measured using mercury injection porosimetry and are shown as Figure

The quasi-steady-state permeability was obtained by calculating the volume of fluid flowing through the sample per second at 10 Hz using the measured piston displacement and also measuring the pressure required to move this fluid. The permeability at 10 Hz was _{10}^{−10 }m^{2}. We have taken this value to represent the steady-state permeability in the absence of steady-state permeability on the sample.

The measurements shown in this paper are the same as those reported in [

(a) The measured waveforms (streaming potential, fluid pressure and piston position (LVDT)) at 20 Hz. (b) The calculated streaming potential coupling coefficient as a function of piston amplitude at 10 Hz for Ottawa sand saturated with 10^{−3} mol/L NaCl solution at 24°C ^{2}). The errors in the frequency were calculated from the analysis of a train of approximately 500 cycles, while the errors in the streaming potential coupling coefficient were calculated from the errors in the RMS streaming potential and the measured RMS pressure difference (500 cycles).

Three sizes of soda lime glass bead (nominally 0.5, 1, and 2 mm in diameter) were obtained from Endecotts Ltd.. Samples of the beads were washed repeatedly in distilled water before being dried in a vacuum oven prior to use. No mercury porosimetry was carried out on the beads because they are too expensive to be disposed of after only one use. The grain size distribution was obtained by laser diffraction measurements and using the detailed calibration information provided by Endecotts Ltd., which is based on a sieve analysis. The porosity of the glass bead samples was measured using a helium pycnometer. The complex electrical properties of a saturated sample of the sand were measured at 60 frequencies between 1 Hz and 1 MHz using a Solartron 1260A Impedance Analyzer, and the results were used to calculate the mean pore size of each bead pack using the Glover and Walker method [

The steady-state permeability of each bead pack was measured using a gravitational pressure head. The results are shown in Table

The fluid used in the experiments was 10^{−3 }mol/L NaCl with a measured density of 997 kg/m^{3}, which agrees well with the equation of state of NaCl solutions (e.g., [^{−4 }Pa.s calculated using the model of [^{−2} S/m at 25°C before use, which is in good agreement with the model of Sen and Goode [^{−2} S/m. For the Ottawa sand, the conductivity of the fluid emerging from the apparatus during the measurement was extremely close to the original conductivity of the fluid ([^{−2} S/m at 25°C). In the case of the glass beads, the conductivity of the fluid slowly increased to 1.48 × 10^{−2} S/m, 1.38 × 10^{−2} S/m, and 1.33 × 10^{−2} S/m for 0.5, 1, and 2 mm diameter beads, respectively, while being circulated through the sample for 24 hours before the electrical and electrokinetic measurements were made. This amounts to an increase of concentration from 1 × 10^{−3 }mol/L to 1.23 × 10^{−3 }mol/L, 1.15 × 10^{−3 }mol/L, and 1.10 × 10^{−3 }mol/L, respectively, which we associate with dissolution of the grains during the attainment of physicochemical equilibrium between the grains and the fluid. For the Ottawa sand experiment, the pH of the fluid during the measurement was pH 6. In the case of the glass beads, the initial pH of the fluid was pH 6.9, which reduced during the recirculation of the fluids. The pH of the fluid was measured on samples of fluid emerging from the apparatus during the electrokinetic experiment. The stable values were pH 6.4, 6.6, and 6.7 for the 0.5, 1, and 2 mm diameter beads, respectively. We note again that the changes in the fluid conductivity and pH are not as great as some authors have experienced (e.g., Leroy et al. [

The cell was loaded with either a sample of sand or beads in layers of 1 cm with light tamping between the layers in the case of the sand, and with agitation after each layer in the case of the glass beads. The system was fully saturated with the process fluid, using back-pressure where necessary to remove all air bubbles. Once saturated, the steady-state permeability of the glass beads was measured using gravity-driven flow. The process fluid was then recycled through the sample for 24 hours to ensure full physicochemical equilibrium. During this time the permeability of the Ottawa sand was measured at a frequency of 10 Hz using the pressure transducers and calculating the flow by measuring the piston displacement with the LVDT.

Figure

In the case of Ottawa sand, tests were made up to 600 Hz, when the sample tube failed. It was observed that the seal between the piston and the tube let in air at frequencies higher than 200 Hz. Although the data for frequencies greater than 200 Hz seem to behave well, we have not reported them because the presence of air bubbles may make the measurements unreliable. We corrected the air leakage for the glass bead pack measurements simply by lubricating the piston seal.

This paper contains results for three diameters of glass bead (0.5, 1, and 2 mm). We also attempted to make measurements on glass bead packs with a 0.25 mm and 3.35 mm nominal diameter. Unfortunately we could not generate sufficient pressure to produce a streaming potential of sufficient size to measure the 3.35 mm beads with accuracy, and the experiment with the 0.25 mm beads did not provide data of sufficient quality to report.

The frequency-dependent streaming potential coefficients were calculated using the methods described in Reppert et al. [

Normalised electrical impedance data for (a) Ottawa sand, and glass beads with (b)

Ottawa sand

Glass beads,

Glass beads,

Glass beads,

One of the characteristics of an electromagnetic shaker is that the piston amplitude decreases with frequency for any set driving current [

We have carried out tests to examine the measured streaming potential as a function of the piston amplitude using a sample of the Ottawa sand and an arbitrary frequency of 10 Hz. Dynamic fluid pressure, dynamic streaming potential, and instantaneous piston position measurements were made while decreasing the shaker driving current in increments (and hence the piston amplitude) until the measured values were below the noise threshold. Measurements were then made while incrementally increasing the driving current until the maximum displacement was reached.

The results are shown in Figure

The measured streaming potential was affected at piston amplitude less than about 2 mm. Under these conditions the measured pressure difference is very small for our high permeability sample, and it is difficult to distinguish the measurements from the background noise. We believe that the observed increase in the streaming potential coupling coefficient for displacements less than 2 mm is due to the difficulty in measuring these small pressures. There was no evidence for turbulent fluid flow at large piston amplitudes.

Figures

The calculated streaming potential coupling coefficient (normalised to the value at 5 Hz, which was 0.518 V/MPa) as a function of frequency for Ottawa sand saturated with 10^{−3} mol/L NaCl solution at 24°C^{2}). (a) Magnitude with six models shown fitting the data [

The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.37 V/MPa) as a function of frequency for a pack of nominally 0.5 mm diameter glass beads saturated with 10^{−3} mol/L NaCl solution at 24°C (^{2}). (a) Magnitude with six models shown fitting the data [

The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.61 V/MPa) as a function of frequency for a pack of nominally 1 mm diameter glass beads saturated with 10^{−3} mol/L NaCl solution at 24°C (^{2}). (a) Magnitude with six models shown fitting the data [

The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.80 V/MPa) as a function of frequency for a pack of nominally 2 mm diameter glass beads saturated with 10^{−3} mol/L NaCl solution at 24°C (^{2}). (a) Magnitude with six models shown fitting the data [

It should be noted in these figures that the error bars become larger at the higher frequencies. This is due to the difficulty in measuring small streaming potentials at frequencies greater than the transition frequency.

The majority of the data analysis will concentrate on the frequency-dependent part of the streaming potential coupling coefficient. However, we should say a few words about the steady state streaming potential coupling coefficient. Although this was not measured in our apparatus, we can perhaps use the streaming potential coupling coefficient at the lowest frequency as a reasonable indication of that under true steady-state conditions considering that Figures _{o} = 5 sites/nm^{2}, pK_{me} = 7.5, pK_{-} = 8, while the formation factor, porosity, cementation exponent, grain diameter fluid concentration, and pH were set to the values related to each sample (Table

Summary of results.

Property | Unit | Ottawa sand | Glass beads | Comment | ||

0. 5 mm | 1 mm | 2 mm | ||||

Steady-state electrokinetic modelling | ||||||

Measured steady-state streaming potential coupling coefficient, | V/MPa | Value at lowest frequency measured. | ||||

Modelled steady-state streaming potential coupling coefficient, | V/MPa | 1.05 | 1.26 | 1.57 | 1.76 | Using [ |

Modelled zeta potential, | mV | −15.9 | −29.6 | −29.6 | −29.6 | Using [ |

pH for electrokinetic modelling | — | 6 | 6.7 | 6.7 | 6.7 | |

Transition frequencies | ||||||

Transition frequency, critically damped 2nd order vibrational model, | Hz | 230 | 234 | 54 | 13 | Using [ |

Transition frequency, 2nd order model with variable damping, | Hz | 748.8 | 636.9 | 176.4 | 41.7 | Using [ |

Damping factor, | — | 1.5 | 1.5 | 1.5 | 1.5 | Using [ |

Transition frequency from the Pride model, | Hz | 256.58 | 213 | 58.79 | 13.85 | Using [ |

Transition frequency from the Glover and Walker simplification, | Hz | 256.58 | 213.36 | 57.32 | 12.61 | Using [ |

Predicted effective pore radius | ||||||

Calculated effective pore radius from independent measurement, | From Table | |||||

From the Packard model, equivalent capillary tube radius, | 67.5 | 72 | 145 | 302 | Using [ | |

From the critically damped 2nd order vibrational model | 70.46 | 69.85 | 145.41 | 296.35 | Using ( | |

From the Pride model | 66.71 | 73.21 | 139.36 | 287.11 | Using ( | |

From the Glover and Walker simplification | 66.71 | 73.15 | 141.13 | 300.90 | Using ( | |

Characteristic length scale, | 62.40 | 67.76 | 131.45 | 280.24 | Using ( | |

Predicted permeabilities | ||||||

Measured permeability, | m^{2} | See text. | ||||

Predicted permeability using the RGPZ method, | m^{2} | Using [ | ||||

From the critically damped 2nd order vibrational model | m^{2} | Using ( | ||||

From the Pride model | m^{2} | Using ( | ||||

From the Glover and Walker simplification | m^{2} | Using ( |

(a) The steady-state streaming potential coupling coefficient measured in this work shown with a compilation of silica-based earth materials measured by (open symbols) or compiled by (solid symbols) Jaafar [^{2},

Figures

Figures

While not as effective as the Pride model and its simplification, the Packard model [

The critically damped second-order vibrational mechanics model also provides a reasonable fit to the data, giving transition frequencies and effective pore radii that are consistent with the independently obtained measurements (Table

The full Pride model calculates the transition frequency from the sample porosity, electrical tortuosity, and permeability as well as the density and viscosity of the pore fluid (

The calculated streaming potential coupling coefficient (normalised to the value at 2 Hz, which was 1.80 V/MPa) as a function of frequency using the Pride model [^{−3} mol/L NaCl solution at 24°C (^{2}). Each solid curve shows the results of the model for a different value of permeability. The dashed line is for the permeability of the sample that was measured independently.

The transition frequencies and capillary radii calculated from each of the 5 models are given in Table

The electrokinetic transition frequency as a function of the inverse square characteristic pore size (a) in full and (b) at expanded scale. Black symbols, previous data for capillary tubes, filters frits, and rocks, Figure 6 of [

The transition frequency can be used to predict the effective pore radius of each sample using (

The transition frequency can be used to predict the steady-state permeability of the sample using (

We have used the electromagnetic drive approach to create an experimental apparatus to measure the dynamic streaming potential coupling coefficient of disaggregated porous media between 1 Hz and 1 kHz. The apparatus has been used to measure samples of Ottawa sand and glass bead packs. Measurements were made on Ottawa sand between 5 Hz and 200 Hz, and on glass bead packs between 2 Hz and 500 Hz. In most cases the full variation either side of the transition frequency was captured. Measurements were possible up to 1 kHz, but in practice the streaming potential values became so small at high frequencies that they were unreliable; only those with reasonably small errors have been included in this paper.

Analysis of the steady-state part of the measured data, shows that the measured steady-state streaming potential coupling coefficient is compatible with the latest theoretical models of electrokinetics.

The dynamic experimental data, in the form of normalised streaming potential coupling coefficient, have been fitted with five theoretical models that were derived (i) from vibrational mechanics theory, (ii) for bundles of capillary tubes, and (iii) for porous media. The Pride model and its simplification, which were developed for porous media, fitted the data best and provided transition frequencies, characteristic length scales, and effective pore radii that were consistent with independently measured values for the samples. The Packard model and its simplification, which were developed for capillary tubes, also performed well. The second order vibrational mechanics model with variable damping only fitted the data when unreasonable transition frequencies were used, but the critically damped second order vibrational model performed reasonably well.

We found that the Pride model and its simplification models are extremely sensitive to the steady-state permeability which may cause difficulties in forward modelling given that this parameter is rarely known precisely and that there is often a large range of permeabilities even in isotropic, homogeneous, clean reservoir rock. However, the sensitivity is an advantage in reverse modelling as it should allow precise permeability determinations to be made by fitting this model to experimental dynamic streaming potential coupling coefficient data.

This work has been made possible thanks to funding by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant Programme. The authors would also like to thank Guillaume Lalande and the members of the mechanical engineering workshop for their help.