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Journal ArticleDOI

Permeability measurements using oscillatory flows

01 Mar 2020-Experiments in Fluids (Springer Berlin Heidelberg)-Vol. 61, Iss: 3, pp 1-14
TL;DR: In this article, a spectral analysis of the pressure and fluid-displacement signals is performed for measuring the permeability of porous materials using oscillatory flows, and the measurements are shown to be in very good agreement with classical drainage experiments performed on the same device.
Abstract: We describe a versatile apparatus for measuring the permeability of porous materials using oscillatory flows. The permeabilities are measured by an original spectral analysis of the pressure and fluid-displacement signals. The measurements are shown to be in very good agreement with classical drainage experiments performed on the same device. Our apparatus and methodology will be useful if small fluid displacements are required, for example in reactive porous media.

Summary (3 min read)

1 Introduction

  • Porous media are ubiquitous in the natural environment and industrial applications.
  • Such as the Carmen-Kozeny equation [1], used to estimate the permeability of a porous medium from knowledge of its porosity, the permeability depends significantly on the internal morphology of the medium and must be measured in most practical applications.
  • The crystals can redissolve in response to flow of the pore liquid in the direction of any applied temperature field.
  • These observations indicate that the classical method to measure permeabilities with a sustained flow may not be sufficient for systems whose properties depend strongly on local gradients or on the mobilisation of the particles they contain.
  • The authors reverse the usual procedure by controlling the flow and measuring the consequent pressure drop across the sample.

2 Experimental apparatus and protocol

  • Sp whose displacement Dp(t) is measured over time.
  • This measurement determines the flow rate through the flow cell by volume conservation Q(t) = SpdDp(t)/dt.
  • The pressure drop across the porous medium ∆P (t) is measured with a Differential Pressure Sensor (DPS).
  • The pressure drop and flow rate signals are then used to compute the hydraulic resistance of the flow cell RH and the permeability of the porous medium.
  • The experimental apparatus comprises several elements, shown in figure 1 and detailed below.

2.1.1 Flow generation

  • Oscillating flows are generated by a rigid piston below the flow cell connected to a shaker (Data Physics SignalForce V20) supplied by a power amplifier (Data Physics SignalForce PA100E) and controlled by an acquisition card (Data Physics Quattro).
  • The piston is screwed to the moving part of the shaker through a thick (6mm) steel plate.
  • The authors experiments were performed at a single frequency f0 = 1Hz, chosen to be low enough to avoid vibration of the support frame and to keep the pore Reynolds number small.

2.1.2 Flow cell

  • The flow cell is a perspex cylinder with a 65mm internal diameter.
  • The porous medium is held between two metal plates drilled with regularly spaced 0.7mm diameter holes to ensure that the fluid enters the porous medium homogeneously.
  • The meshes are covered with a silk fabric to avoid the loss of the smallest beads and are supported by two inner perspex cylinders (A and B on Fig. 1), inserted in the flow cell.
  • Cylinder (A) keeps the porous medium away from the flow injection and ensures that the flow is unidirectional before entering the porous medium.
  • The cylinder (B) presses the upper mesh with a 5.65kg mass to compact the glass beads and avoid the fluidisation of the bed [8] when the flow velocity is directed upwards.

2.1.3 Sensors: calibration and sensitivity

  • The acquisition card records the piston stroke and the pressure drop between two points at the inlet and the outlet of the porous medium simultaneously.
  • This pressure drop ∆P is measured with a differential pressure sensor (DPS, Omega engineering, customized online: wet/wet differential, bidirectional, range 0.17bar, output ±10VDC), connected to the flow cell through meshed, rigid hoses.
  • The displacement transducer is a Linear Variable Differential Transformer (LVDT, Omega engineering LD620-10), which enables accurate measurements of the displacement of the piston in time Dp(t), proportional to the volume of fluid displaced in the flow cell V (t) = SpDp(t), with Sp the cross-sectional area of the piston.
  • The linearity of both sensors was checked with calibration experiments performed by measuring constant displacements or hydrostatic pressures over the whole range of the sensors (Fig. 2).
  • These calibration experiments determined the relative uncertainties of the measurements of the pressure and the volume of fluid displaced which are respectively 0.17% and 1.3%.

2.1.4 Porous media

  • To calibrate, test and characterize this new technique, the authors used porous media made of compacted glass beads placed in the flow cell between the two meshes.
  • 3 Signal acquisition A typical example of the signals acquired is shown in figure 3, which displays the shaker’s input voltage U(t) (Fig. 3a) with both measurements of the pressure drop ∆P (t) (Fig. 3b) and the volume of fluid displaced V (t) (Fig. 3c).
  • This behaviour does not impact the permeability measurement since both signals ∆P (t) and V (t) are recorded at the same time.
  • The flow regime is characterised by the Reynolds number, defined as Re = ρd〈v〉 µ (4) where 〈v〉 is the order of magnitude of the flow velocity during an experiment.
  • These observations indicate that the oscillation frequency f0 shall not exceed the order of fτ/100 for the quasi-static approximation to be valid.

3 Modelling and signal processing

  • The authors detail the experimental principle and how they compute the permeability K0 of a porous medium placed in the flow cell.
  • This allows the computation of the flow rate Q(t) by Fast Fourier Transform (FFT, see eq. 13).
  • The total hydraulic resistance RH is measured by processing the Fourier transforms of the signals Π(t) and Q(t) as shown in the following section.
  • This section details the processing applied to the signals to measure the permeability of the different porous media inserted in the flow cell.

3.2.1 Flow rate

  • The result is shown in figure 3d, which also indicates the magnitude of the filtration velocity v(t).
  • The shape of the computed signal is very similar to the pressure signal ∆P (t), although the pressure has a non-zero plateau when the flux is zero.

3.2.2 Hydraulic Resistance

  • The authors analyse the pressure and flux signals spectrally to measure the hydraulic resistance RH and deduce the permeability of the porous medium K0 from it (eq. 12).
  • The correlation between the two Fourier transforms is quantified by the normalized cross-correlation coefficient γ2 = 〈Q̃.Π̃∗〉2f 〈Π̃.Π̃∗〉f .〈Q̃.Q̃∗〉f . (15) A typical spectral analysis is shown in figure 5, where the authors compare the spectrum of Π with the computation of RHQ.
  • The measurement uncertainties associated with the measurement of K0 involve the relative uncertainty of the measurement of RH as shown in the next paragraph.
  • This could be improved further with more accurate measurements of RD and L0.

4 Results and observations

  • The measurements of the permeabilities of the porous media are first compared with classical measurements obtained by draining the fluid through the flow cell.
  • The impact of the flow regimes, characterised by the Reynolds number on the measurements is then assessed and provides further insights into the experimental method.
  • And the result of their comparison with oscillating measurements is shown in figure 6.
  • In both cases, the signal RHQ(t) slightly lags the pressure drop, and some deviations to the linearity between v(t) and Π(t) are observed.
  • This is not the case for the measurements performed with the biggest beads, as there is a slight decrease of the permeability values for Re ≥ 10, which indicates a deviation to Darcy’s law in this regime [1].

5 Conclusion

  • The authors have presented a new experimental technique and an original analysis for measuring the permeability of a porous medium using oscillating flows.
  • In particular, this technique differs from other methods using oscillating flows [12,13] by the spectral analysis presented in section 2 and the amplitude of the displacement of the fluid in the porous medium.
  • The setup is versatile and produces accurate permeability measurements with a relative precision of the order of 3.7%.
  • This could have also be done in the frequency domain.
  • This work was funded by grants from the British Antartic Survey Fundation and the Isaac Newton Trust.

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Noname manuscript No.
(will be inserted by the editor)
Permeability measurements using oscillatory flows
Baudouin eraud · Jerome A. Neufeld ·
Paul R. Holland · M. Grae Worster
Received: date / Accepted: date
Abstract We describe a versatile apparatus for measuring the permeability of
porous materials using oscillatory flows. The permeabilities are measured by an
original spectral analysis of the pressure and fluid-displacement signals. The mea-
surements are shown to be in very good agreement with classical drainage ex-
periments, performed on the same device. Our apparatus and methodology will
be useful if small fluid displacements are required, for example in reactive porous
media.
Include keywords, PACS and mathematical subject classification numbers as
needed.
Keywords Permeability · Oscillating flows · Spectral analysis
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Experimental apparatus and protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Modelling and signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Results and observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
A Drainage experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
B Analysis of the signal Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
B. eraud · M.G. Worster
Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge
CB3 0WA, UK
E-mail: mgw1@cam.ac.uk E-mail: bgeraud.pro@gmail.com
J.A. Neufeld
BP Institute, Bullard Laboratories, Madingley Road, Cambridge CB3 0EZ, UK
Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge
CB3 0WA, UK
Department of Earth Sciences, University of Cambridge
Tel.: +44-1-223-765709
E-mail: j.neufeld@bpi.cam.ac.uk
P.R. Holland
British Antarctic Survey, High Cross, Madingley Rd, Cambridge CB3 0ET, UK
E-mail: pahol@bas.ac.uk

2 Baudouin eraud et al.
1 Introduction
Porous media are ubiquitous in the natural environment and industrial applica-
tions. A fundamental characteristic of porous media is their permeability K, which
relates the volume flux per unit area v through a medium to the pressure gradient
applied across it and the body force, such as gravity, acting on the fluid within it,
according to Darcy’s law
v =
K
µ
(P ρg) , (1)
where µ is the dynamic viscosity of the pore fluid, ρ is its density, P is the local
pore pressure and g is the acceleration due to gravity. Therefore the inverse of
the permeability coefficient characterises the resistance to flow through a prorous
medium under a fixed pressure gradient. Although there are some semi-empirical
relationships, such as the Carmen-Kozeny equation [1], used to estimate the per-
meability of a porous medium from knowledge of its porosity, the permeability
depends significantly on the internal morphology of the medium and must be
measured in most practical applications.
Permeability is usually measured by applying a known pressure difference ∆P
across a sample of length L and cross-section S and measuring the volume flux
Q of a pore fluid of known viscosity [2,3]. The permeability is then given by the
formula
K =
µQ/S
∆P/L
. (2)
This method is simple and robust but relies on a sustained flow through the porous
medium being measured. However, if the porous medium is reactive then the pore
size and hence its permeability can evolve while a measurement is being made
in consequence of the sustained flow carrying heat and chemicals through the
medium. An example is a mushy layer such as sea ice, which is a reactive porous
medium made up of dendritic crystals formed during the freezing of alloys. The
crystals can redissolve in response to flow of the pore liquid in the direction of
any applied temperature field. The disturbance of the gradients in brine salinity
by a uni-directional flow can also lead to variations of the permeability of several
orders of magnitude [4–7]. Additionally in some systems containing fine particles,
a sustained flow can lead to pore clogging and result in an important decrease
of the permeability during the measurement [3]. These observations indicate that
the classical method to measure permeabilities with a sustained flow may not be
sufficient for systems whose properties depend strongly on local gradients or on
the mobilisation of the particles they contain.
Here, we describe our development of a new experimental apparatus to measure
permeability that uses an oscillating flow of small displacement and thus avoids
the imposition of a sustained uni-directional flow. We reverse the usual procedure
by controlling the flow and measuring the consequent pressure drop across the
sample. Spectral analysis of the measured signals allows for precise control of
measurement errors and has the added potential (by recording phase differences)
to measure characteristics of deformable poro-elastic media.
This paper is focused on detailing the experimental setup and the spectral
analysis used to determine permeabilities. As a preliminary test of the technique,
we report on experiments using packed beads of glass. We validate our results by
making supplementary measurements using the classical drainage technique.

Permeability measurements using oscillatory flows 3
2 Experimental apparatus and protocol
Central to our experimental design is the aim to produce an oscillating flow through
a porous medium in order to measure its permeability without the imposition
of a mean flow. The flow is driven by a rigid piston of cross-sectional area S
p
whose displacement D
p
(t) is measured over time. This measurement determines
the flow rate through the flow cell by volume conservation Q(t) = S
p
dD
p
(t)/dt.
The pressure drop across the porous medium ∆P (t) is measured with a Differential
Pressure Sensor (DPS). The pressure drop and flow rate signals are then used to
compute the hydraulic resistance of the flow cell R
H
and the permeability of the
porous medium.
2.1 Experimental setup
The experimental apparatus comprises several elements, shown in figure 1 and
detailed below.
2.1.1 Flow generation
Oscillating flows are generated by a rigid piston below the flow cell connected
to a shaker (Data Physics SignalForce V20) supplied by a power amplifier (Data
Physics SignalForce PA100E) and controlled by an acquisition card (Data Physics
Quattro). The piston is screwed to the moving part of the shaker through a thick
(6mm) steel plate. A displacement transducer is screwed to the steel plate to
measure the piston stroke D
p
(t) as a function of time. Our experiments were
performed at a single frequency f
0
= 1Hz, chosen to be low enough to avoid
vibration of the support frame and to keep the pore Reynolds number small. The
pore Reynolds number Re
p
= ρv
p
d/µ, where v
p
is a characteristic magnitude of
the pore velocity and d is a characteristic size of the pores, measures the ratio
of viscous stresses to inertia. It must be small in order for Darcy’s equation to
be valid, and this is achieved by ensuring that the pore velocity, of the order of
v
p
D
0
f
0
(where D
0
is a characteristic amplitude of D
p
(t) and φ the porosity)
is sufficiently small in our expriments.
2.1.2 Flow cell
The flow cell is a perspex cylinder with a 65mm internal diameter. The porous
medium is held between two metal plates drilled with regularly spaced 0.7mm
diameter holes to ensure that the fluid enters the porous medium homogeneously.
The meshes are covered with a silk fabric to avoid the loss of the smallest beads
and are supported by two inner perspex cylinders (A and B on Fig. 1), inserted in
the flow cell. Cylinder (A) keeps the porous medium away from the flow injection
and ensures that the flow is unidirectional before entering the porous medium.
The cylinder (B) presses the upper mesh with a 5.65kg mass to compact the glass
beads and avoid the fluidisation of the bed [8] when the flow velocity is directed
upwards. Another, cautionary mesh is set at the bottom of the flow cell to prevent
beads from dropping into the piston.

4 Baudouin eraud et al.
Fig. 1 Photos and sketch of the experimental setup. a): photo of the device. The metal frame
in background reduces any vibration of the device during the experiments. The Linear Variable
Differential Transformer (LVDT) and the Differential Pressure Sensor (DPS) are situated on
the right-hand side. b): photo of a mesh used to maintain the grains in the flow cell. The mean
diameter is 64.6mm and the typical hole radius is 0.7mm. Each mesh was covered with silk
fabric to retain the smallest grains. c): sketch of the device in cross-sectional view. The region
accessible to the fluid (water) is colored in light blue and the acquisition circuit is shown in red.
All the dimensions are indicated in mm, the diameters are indicated with double arrows and
the symbols Ø. The porous medium is compacted between two meshes borne by two perspex
cylinders (A and B), drilled for pressure measurements. The DPS measures the pressure drop
across the porous medium of length L
0
. The LVDT measures the piston stroke and therefore
its displacement D
p
(t), proportional to the displacement of the fluid in the flow cell D(t) by
volume conservation. The flux Q(t) is computed from the measurement of D
p
(t) (see section
2.3).

Permeability measurements using oscillatory flows 5
U
LVDT
(V)
-5 0 5
D
p
(cm)
0
1
2
3
U
DPS
(V)
-10 0 10
- P (× 10
4
Pa)
-1
0
1
D
p
(cm)
0 0.5 1 1.5 2 2.5
V (mL)
0
10
20
30
40
Fig. 2 Calibration curves of the sensors and the piston. The experimental points are
in blue, the red lines are linear fits. a) Displacement transducer calibration. Sensitivity:
2.0192 ± 0.0005mm/V . b) Differential Pressure Sensor calibration. Sensitivity: 1719 ± 3P a/V .
c) Measurement of the volume of liquid displaced in the flow cell for several piston positions.
The linear fit provides the section of the piston and so its diameter with an excellent precision:
45.4 ± 0.3mm.
2.1.3 Sensors: calibration and sensitivity
The acquisition card records the piston stroke and the pressure drop between
two points at the inlet and the outlet of the porous medium simultaneously. This
pressure drop ∆P is measured with a differential pressure sensor (DPS, Omega
engineering, customized online: wet/wet differential, bidirectional, range 0.17bar,
output ±10VDC), connected to the flow cell through meshed, rigid hoses. It is
important to note that there is no flow through the DPS so that the pressure
drops between the flow cell and each inlet of the sensor are hydrostatic.
The displacement transducer is a Linear Variable Differential Transformer (LVDT,
Omega engineering LD620-10), which enables accurate measurements of the dis-
placement of the piston in time D
p
(t), proportional to the volume of fluid displaced
in the flow cell V (t) = S
p
D
p
(t), with S
p
the cross-sectional area of the piston. The
linearity of both sensors was checked with calibration experiments performed by
measuring constant displacements or hydrostatic pressures over the whole range
of the sensors (Fig. 2). These calibration experiments determined the relative un-
certainties of the measurements of the pressure and the volume of fluid displaced
which are respectively 0.17% and 1.3%. We could then measure S
p
= 1.62×10
3
m
2
accurately by regression, as it is a crucial parameter for the measurements of the

Citations
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01 Dec 2013
TL;DR: In this article, the authors investigate the underlying mechanics by generating well- controlled, repeatable permeability enhancement in laboratory experiments, and they conclude that a flow-dependent mechanism associated with mobilization of fines controls both the magnitude of the pore enhancement and the recovery rate in their experiments.
Abstract: Article history: It is well-established that seismic waves can increase the permeability in natural systems, yet the mechanism remains poorly understood. We investigate the underlying mechanics by generating well- controlled, repeatable permeability enhancement in laboratory experiments. Pore pressure oscillations, simulating dynamic stresses, were applied to intact and fractured Berea sandstone samples under confining stresses of tens of MPa. Dynamic stressing produces an immediate permeability enhancement ranging from 1 to 60%, which scales with the amplitude of the dynamic strain (7 × 10 − 7 to 7 × 10 − 6 ) followed by a gradual permeability recovery. We investigated the mechanism by: (1) recording deformation of samples both before and after fracturing during the experiment, (2) varying the chemistry of the water and therefore particle mobility, (3) evaluating the dependence of permeability enhancement and recovery on dynamic stress amplitude, and (4) examining micro-scale pore textures of the rock samples before and after experiments. We find that dynamic stressing does not produce permanent deformation in our samples. Water chemistry has a pronounced effect on the sensitivity to dynamic stressing, with the magnitude of permeability enhancement and the rate of permeability recovery varying with ionic strength of the pore fluid. Permeability recovery rates generally correlate with the permeability enhancement sensitivity. Microstructural observations of our samples show clearing of clay particulates from fracture surfaces during the experiment. From these four lines of evidence, we conclude that a flow-dependent mechanism associated with mobilization of fines controls both the magnitude of the permeability enhancement and the recovery rate in our experiments. We also find that permeability sensitivity to dynamic stressing increases after fracturing, which is a process that generates abundant particulate matter in situ. Our results suggest that fluid permeability in many areas of the Earth's crust, particularly where pore fluids favor particle mobilization, should be sensitive to dynamic stressing.

83 citations

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Frequently Asked Questions (2)
Q1. What are the future works in "Permeability measurements using oscillatory flows" ?

This experimental technique will be used in further studies with porous media in equilibrium with their liquid phase. 

The authors describe a versatile apparatus for measuring the permeability of porous materials using oscillatory flows.