Reservoir poroelastic properties from pore pressure oscillation method

Azar Hasanov a and Mike Batzle a

a Colorado School of Mines; 1500 Illinois str, Golden, CO 80401. ahasanov@mymail.mines.edu

Introduction

The oscillating pore pressure method for measurements of permeability and storage capacity has been previ-ously presented in details (Kranz et al., 1990; Fischer, 1992; Song and Renner, 2007). It is based on inducingtime-harmonic diffusive fluid flow through the porous sample and subsequent measurement of the upstream anddownstream pressure signals. The propagated through the specimen pressure sinusoid is always phase-lagged andattenuated. Once the amplitude ratio and phase has been calculated, we can solve the diffusivity equation withoscillatory boundary conditions and estimate the permeability and storage capacity of the sample.

Although it allows accurate determination of permeability, the estimation of the storage capacity yields large errors,sometimes on the order of several magnitudes (Bernabe et al., 2006). As mentioned by Bernabe et al. (2006) thevalues of the storage capacity are particularly uncertain if the experiment employs large downstream reservoirs.This is usually done in case of high permeability samples (100 mD and higher) in order to achieve measurablephase and amplitude ratio of the two pressure signals.

The storage capacity of the sample is a poroelastic constant, responsible for fluid storage in the sample and definedas (Wang, 2000; Kumpel, 1991):

βst = − 1

Vb

(∂Vfl∂Pp

)δPc=0

(1)

Song and Renner (2006) demonstrated that the knowledge of rock’s storage capacity, mineral bulk modulus, fluidcompressibility and porosity allows the estimation of drained bulk modulus, Biot-Willis and Skempton’s coeffi-cients. The diffusive pore fluid flow is a coupled poroelastic process, which gives rise to volumetric deformationof the sample. We measure this deformation directly by strain gauges and derive poroelastic parameters (includingstorage capacity) independently from the hydraulic pore pressure oscillations.

The poroelastic part of the oscillating pore pressure method inherits the advantages of the hydraulic experiment. Itis a continuous and harmonic measurement, which allows monitoring the changes in elastic properties of the sampleand relatively simple noise reduction. Since the measurement is frequency-dependent, the measured quantitiesreveal dispersion effect and effect of the attenuation (Hasanov, 2014). To our best knowledge, this experiment hasnever been carried out before.

Theoretical background

Poroelastic response of the rock is a coupled process, therefore it depends on both the confining and pore pressure.The so called pure compliance formulation is given according to Wang (2000):

[εζ

]=

[a11 a12a21 a22

]×

[δσδPp

](2)

Here ε is the volumetric strain, ζ is the change in fluid mass (or volume), δσ is an increment in confining pressure,δPp is an increment in pore pressure, [aij ] is the matrix of phenomenological coefficients. This matrix is definedas:

a11 =ε

δσ

∣∣∣δPp=0

=1

Kdr(3)

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3rd International Workshop on Rock Physics, Perth, 13th-17th April 2015

a12 = a21 =ε

δPp

∣∣∣∣δσ=0

=1

H(4)

a22 =ζ

δPp

∣∣∣∣δσ=0

=1

R= βst (5)

Coefficient a11 is the drained bulk compressibility (its inverse is drained bulk modulus Kdr, usually measuredin GPa units), defined as the amount of volumetric strain, caused by a unit change in confining pressure whilekeeping pore pressure constant. The condition of the constant pore pressure means that the compressibility of arock specimen in such an experiment is not affected by the saturant fluid, thus this modulus is sometimes referredas the dry bulk modulus, or the frame modulus - bulk modulus of the porous mineral frame of a rock sample.Diagonal coefficient a12 and a21 are always identical, following Onsager’s reciprocity theorem (Onsager, 1931),and are usually referred to as the poroelastic expansion coefficients. Coefficient a12 is defined as a change involumetric strain as a result of the pore pressure change at the constant confining stress. I measure this poroelasticmodulus directly during the oscillating pore pressure experiment.

The last coefficient a22 is of special interest to this work. It can be perceived as the amount of fluid, discharged oradded into the pore space by the unit change in pore pressure at constant confining stress condition. This definitionhas already been formulated by Equation 1, therefore a22 is the specific storage capacity of a rock sample. One ofthe main objectives of this work is to develop a robust method to determine specific storage independently fromthe purely hydraulic component of the oscillating pore pressure experiment.

Undrained condition is defined as the situation when any flow of the fluid in or out of the rock specimen is pro-hibited. In other words, the mass of the pore fluid is unchanged at all times, or ζ = 0. In a typical laboratoryexperiment undrained regime takes place if the tested rock sample is completely sealed (e.g., disconnected from apore pressure pump). The main parameter describing undrained condition is Skempton’s coefficient B (SkemptonA. W., 1954). Skempton’s coefficient may be intuitively defined as a change in pore pressure caused by a changein confining pressure during undrained experiment. It can be defined as following:

B = − δPpδσ

∣∣∣∣ζ=0

=R

H(6)

The Biot-Willis coefficient is a special case of the effective stress coefficient for the volumetric stress and com-monly defined as (Biot, 1941):

α = 1− Kdr

Kmin=Kdr

H(7)

Biot-Willis coefficient is close to unity for unconsolidated materials, and approaches zero for low-porosity rocks(e.g. granite). Any effective stress coefficient is stress-dependent itself, the bounds for Biot-Willis coefficient aregiven by Zimmerman et al. (1986) as φ < α < 1. The three constants Kdr, B and α are commonly accepted as thefundamental set of poroelastic properties (Wang, 2000; Pride, 2005).

Knowledge of a rock specimen’s porosity, mineral skeleton bulk modulus, pore fluid compressibility and one of theabove discussed poroelastic constants allows calculation of the other parameters. In this work, I indirectly deriveand compare these moduli from two independent measurements. Pore pressure oscillations method alone measuresa sample’s storage capacity βst. I utilize strain gauges to directly measure bulk rock deformation, caused by thepropagating pore fluid pulse at constant confining pressure steps. Equation 4 precisely depicts this experiment.Therefore the measured modulus is Biot’s poroelastic expansion coefficient H. The interrelationships betweenthese properties, specific storage and poroelastic expansion coefficient are summarized in Table 1.

Experimental setup and methodology

The experimental apparatus, built during the course of this thesis, consists of a pressure vessel, two pressure pumps,a function generator, three pressure transducers, strain gauge preconditioning modules, digitizing unit and a per-sonal computer. Our pressure vessel can sustain confining pressures up to 69 MPa (10000 psi). Two OMEGAPX612 pressure transducers, recording upstream and downstream pore pressures, are connected to the pressurevessel head through pore pressure plumbing. A third pressure transducer is placed at the bottom of the pressure

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3rd International Workshop on Rock Physics, Perth, 13th-17th April 2015

Table 1: Summary of poroelastic constants, expressed in terms of storage capacity and poroelastic expansioncoefficient.

Property from βst from H

Kdr

(βst +

(1 + φ)

Kmin− φ

Kfl

)−1 (1

H+

1

Kmin

)−1

α 1−

(βst +

(1+φ)Kmin

− φKfl

)−1

Kmin

Kmin

H +Kmin

B 1− φ

βst

(1

Kfl− 1

Kmin

)1

1 + φ(HKfl

− HKmin

)βst -

1

H+

φ

Kfl− φ

Kmin

H

(βst −

φ

Kfl+

φ

Kmin

)−1

-

vessel and records confining pressure. Electrical output signals from upstream and downstream pressure trans-ducers are recorded and digitized by MC USB-1608G data acquisition card. The upstream side of the plumbingis connected to a Teledyne ISCO 100DX syringe pressure pump, the downstream side is connected to the down-stream reservoir. An Agilent 33250A function generator supplies analog voltage input directly into the pressurepump.

Figure 1: Schematic of the oscillating pore pressureapparatus.

Figure 2: Schematic of a sample, instrumented withstrain gauges.

This allows the pump to generate pore pressure oscillations, amplitude, frequency and DC offset of which can bespecified with the function generator. Electrical voltage signals, generated by strain gauges during the ongoingsample deformation, are filtered and amplified by Validyne SG297A strain gauge preconditioning modules, andfurther digitized by the data acquisition card. Figure 1 presents a detailed schematic of the described appara-tus.

The typical sample is a slender cylinder of approximately 25 mm diameter and 100-140 mm length. We equipeach sample with six pairs of bi-axial strain gauges, evenly distributed along the length of the cylinder, as shownin Figure 2. Bi-axial strain gauges measure both longitudinal and transverse deformation.

After strain gauges are attached, the sample is placed between two aluminum endcaps and clamped. The sides of

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3rd International Workshop on Rock Physics, Perth, 13th-17th April 2015

endcaps attached to the sample were grooved for even fluid distribution and diffusion through a sample. We sub-sequently make an impermeable flexible epoxy jacket around the sample, which prevents leakage of the confiningpressure fluids into the sample.

Figure 3: Comparison of raw, drift-removed and fully-filtered data. The displayed data is the voltage record froma strain gauge, obtained at 100 mHz pulsing frequency.

As mentioned before, digital data acquisition of pressure oscillations allows effective noise suppression of thesignals. Strain gauge recording are usually sensitive and noise-prone. Figure 3 demonstrates two-stage filtering infrequency domain, applied to noisy data. The main source of noise is the plumbing resonance of the apparatus,temperature variations, as well as an interference from building’s electrical grid (60 Hz utility frequency). Onethe raw data has been filtered, we calculate the poroelastic expansion coefficient ”H” either by dividing peakamplitudes of the relevant signals in the frequency domain, or from the slope of the linear regression, fit to the porepressure as a function of strain data. The formula used for the calculation of H is:

H =Pup3 εup

≈ Pdown3 εdown

(8)

where Pup, Pdown are the amplitudes of the upstream and downstream pressure oscillations (measured in MPa),εup, εdown are the amplitudes of the upstream and downstream strain oscillation (in microstrain units) and H isthe poroelastic expansion coefficient (commonly expressed in GPa). Both linear regression and peak frequencymethods produce similar results. We calculate the values of H both from the upstream and downstream sensorsfor comparison. Once H is computed, the rest of the poroelastic parameters can be obtained according to Table1.

Results

Figures 4 and 5 summarize results, obtained for four analyzed samples at constant pulsing frequency of 100 mHz.As expected, drained bulk modulus steadily increases with increasing differential stress due to the closure of in-trinsic microcracks and compliant pores, and potentially pore collapse in case of San-Andres dolostone. Themeasured values for Berea sandstone are in good agreement with the published data (Hart and Wang, 1995), prov-ing the reliability of our poroelastic measurements. The values of Biot-Willis coefficients, derived from standalonehydraulic pore pressure oscillations experiment are not trustworthy due to errors in storage capacity estimations,generally resulting in non-physical numbers (higher than 0.95). On the contrary, poroelastically-measured Biot-Willis coefficients seem to be plausible, indicating a large stress-dependence of Berea and Navajo sandstones andleast stress-dependence in San-Andres dolostone. We conclude that the hydraulic oscillating pore pressure methodoverestimates storage capacity by an order of magnitude.

The frequency-dependence of the measured drained bulk modulus for various samples and differential stressesis demonstrated in Figures 6 and 7. The moduli steadily increase until the frequency of 0.2 Hz. AFter this fre-quency an unexpected, almost exponential decay of the drained bulk modulus is observed. Although this effectis not predicted by basic frequency-dispersion models, it is constantly observed in all of the samples. We suspectthat this deviation might be caused by the violation of the fundamental assumptions behind the pore pressure os-cillation method. Increasing oscillation frequencies probably result in either non-Darcy, or radial flow, causingthis unexpected response (Hasanov, 2014). We also suspect that this measurement can be improved by utiliz-ing a servo-controlled pressure intensifier as the oscillation source and by reducing the size of the downstreamreservoir.

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3rd International Workshop on Rock Physics, Perth, 13th-17th April 2015

Figure 4: Stress-dependent drained bulk moduli,measured for different samples. Data for Bereasandstone is in good agreement with the publisheddata by Hart and Wang (1995).

Figure 5: Stress-dependent Biot-Willis coefficients,measured at 100 mHz oscillation frequency. Thevalues derived from the hydraulic pore pressureoscillations method are not trustworthy, generallygrouping around 1.

Figure 6: Frequency-dependent real part of drainedbulk modulus of tested samples, measured at 34.47MPa differential stress.

Figure 7: Frequency-dependent real part of drainedbulk modulus of Fox Hills sandstone, measured atvarious differential stresses.

Conclusions

We have improved the conventional oscillating pore pressure method by instrumenting the samples with straingauges and measuring the elastic deformation, caused by the diffusing pore pressure front. This allowed us tosimultaneously measure hydraulic and poroelastic properties of the porous specimens. Measured values of thedrained bulk modulus and Biot-Willis coefficient are in good agreement with the literature. Poroelastic moduli,calculated from the hydraulically-derived storage capacity values are not trustworthy, indicating an error of a mag-nitude. Frequency-dependent drained bulk moduli exhibit a deviation from the predicted behavior after 0.2 Hzfrequency. We explain this deviation by non-Darcy or radial flow effects, which take place at higher frequencies,perhaps due to increased fluid flow velocity. We plan to perform more measurement on tighter samples and im-prove our setup by integrating a servo-controlled pressure intensifier and decreasing the volume of the downstreamreservoir.

Acknowledgements

We are thankful to Fluids & DHI consortium at Colorado School of Mines for providing financial support, as wellto Dr. Bob Kranz for valuable suggestions and discussion on the subject.

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3rd International Workshop on Rock Physics, Perth, 13th-17th April 2015

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