Crosswell tomography imaging of the permeability structure within a sandstone oilfield.
Tokuo Yamamoto(1), and Junichi Sakakibara(2)(1)

University of Miami and Yamamoto Engineering Corporation, (2) Kawasaki Steel Corporation.

ABSTRACT
Imaging the permeability structure of the earth is achallenge of geophysics. The first case of thepermeability structure image within a limestone aquifer
was obtained from a set of acoustic crosswell tomography
data through an analytical inversion based on a newporoelastic theory called the super-k model (Yamamoto,1998). In this paper we present the second exampleimage of permeability structure within a sandstone oilreservoir. High quality data of acoustic crosswell
tomography were obtained by a newly built system of thepseudo-random binary sequence (PRBS) based crosswell
tomography (U.S. Patent No. 5,406,530). The velocity
and attenuation images measured at a 500 Hz PRBS wereinverted analytically based on a squirt flow theory. The
acoustically measured permeability images agree with thepermeability information gathered through down holelogs and cores.
KEY WORDS: permeability, crosswell geophysics,acoustics, images, sandstone reservoir.
INTRODUCTION
Imaging the permeability structure of the earth
acoustically has been and is a challenge of geophysics. It
requires a high quality acoustic image measurement
method, physically consistent predictive models of
acoustic wave propagation through earth, and accurate
and stable inversion techniques.
Because the physics of acoustic wave propagation
through earth is very complicated, there is no single
theory that can accurately model the physics for all earthmaterials. It has been shown by Yamamoto et al (1994)
that the Biot (1956) theory predicts the acoustic wave
propagation through saturated, non-cemented sediments
at frequencies between 1 and 10 kHz. For sandstone, thesquirt flow theory by Mavko and Nur (1975) have beenshown to predict adequately for the wave propagation atcrosswell frequencies and ultrasonic frequencies.
Dvorkin and Nur (1993) combined the Biot theory andthe squirt flow theory. Yamamoto (1998) has developed
a new poroelastic theory for very permeable limestone

and accurately inverted the crosswell acoustic
tomography data for permeability. Some other theories
may be needed for fractured volcanic rock formations.
All these theories require the rock properties as giveninput. Empirical equations for bulk modulus and shear
modulus of skeletal frame are usually used for thispurpose.
From the point of view of inversion calculations, thesimpler the model the better. From this point of view, theBISQ (Biot and Squirt flow) model by Dvorkin and Nur(1993) is too complicated to find the correct permeability
inverse from measured acoustic data. It has been shownthat the analytical super-k model of limestone(Yamamoto, 1998) is accurate and stable for permeability
inversion for limestone. Similarly, it will be shown thatthe squirt flow model is suitable for inversion of acoustic
data for the permeability image within sandstone oilreservoirs.
SQUIRT FLOW THEORY
An analytical inverse theory for the permeability withinsandstone formation is built using the squirt flow theory.
Readers are refered to Mavko and Nur (1975); Dvorkinand Nur (1993); and Yamamoto (1998) for the physics
and mathematics behind the theory. The acoustic wave
slowness s is given by the squirt flow theory as
s2 = 
M + Fsq 2 / (1)where is the bulk density of the sediment, M is theuniaxial modulus of the frame, is a Biot elastic
coefficient, and is porosity. In this paper, vectors andcomplex variables are expressed by bold letters. Fsq isthe squirt flow modulus of pore fluid given by,
2 J1 ( R )Fsq = F [ 1 - --------------- ]RJ0 ( R )(2)

where F is the Biot elastic modulus of pore fluid stiffness,
R is the squirt flow length, and J0 and J1 are the Bessel

functions of the order zero and one. The squirt flow
wave number  is given through the following
nondimensionsal equation;
R2( R) = i------------- = i Fk2(3)where  is the angular frequency 2f, f is the frequency,is the viscosity of pore fluid, and k is the permeability.
The phase velocity VP and the intrinsic attenuation Q - 1 ofthe super-k medium are given as:
VP = 1/ Real ( s), and(4)Q - 1 = 2 Imag ( s ) / Real ( s)As mentioned earlier, the elastic moduli of theskeletal frame have to be estimated from the measured
sound speed VP. The empirical relation of Han et al.
(1986) is used to estimate the shear modulus and bulk
modulus of the skeletal frame. The values of clay content
c were assumed for sandstone and shale. The Biot
coefficient F and  are then calculated using the Biot
(1956) equations.
PERMEABILITY INVERSE
The permeability inverse is obtained through equations 1
through 4. First, the elastic moduli and porosity estimates
are obtained from the measured velocity VP through theHan et al. (1986) sandstone formula. Then thepermeability is inverted from the measured attenuation Q1. It should be noted that the attenuation and the velocity
dispersion are only the function of a dimension less, realpositive value parameter; Therefore, if one find theunique values of that satisfy the equations 4, one canfind the permeability through equation 3 as, R2k = --------------F

(5)

If the permeability is known from other measurements,
the viscosity of the pore fluid is inverted as,
Fk R2
This analytical inversion is simple and very stable as will
be shown in the following.

CROSSWELL ACOUSTIC TOMOGRAPHY
The crosswell tomography experiment using the newsystem YEC2111 was conducted in May and June of1998 using wells MD15 and Q1 at an oil field of TOL(TRINCAN OIL LTD), Trinidad. The resistivity logs
and gamma-ray logs of the two wells are reproduced inFigures 1 a and b. The two oil wells are separated by ahorizontal distance of 125 m. The cross-section between
the two wells and between the depths 350 m and 800 m
was imaged at 500 Hz PRBS of 4095 cycles, with asource and receiver depth interval of 6 m. This produces
a total of 60,000 source-receiver rays. The number ofstacking or averages varied from 100 to 400, making total
compression of up to 160,000 pulses. Due to thepowerful source out put of the new source ITC 6145 (180dB at 500 Hz, 198 dB at 1000 Hz, and 210 dB between
1500 and 5000 Hz), the test experiment of the newsystem YEC2111 went very smoothly and finished in 2days once the experiment became ready.
A 95 % of the 60,000 rays measured during thecrosswell experiment is used in inversion analysis
because of the extremely good quality of the wave data.
A typical source gather after correlation is plotted inFigure 2. Clear first arrival waves are easily recognizablefrom the source gather. Therefore, the inverted velocity
and attenuation images are of high quality and give a highconfidence in the inverted images. The least squares
SVD (Singular Value Decomposition) inversion codes forvelocity and attenuation by Bregman et al. (1989) wereused in the calculations. The pair of velocity andattenuation images are given in Figure 3 a and b. The
examination of images clearly show that the imaged
section can be roughly divided into four subsections: sandlayer 350 to 450 m, shale layer 450 to 530 m, sand layer
530 to 650 m, and shale layer 650 to 800 m. These roughsubdivisions of the section agree with the electric andgamma ray logs of the two wells. Actual geology of thesection is much more complicated than these simplified
divisions as will be seen in the permeability images.
POROSITY IMAGES
The proposed permeability inversions requires the valuesof porosity, shear modulus and bulk modulus of skeletalframe, viscosity of pore fluid, and squirt flow length. Theporosity, shear modulus and bulk modulus are estimatedfrom the velocity image Figure 3 a using the Han et al.
(1986) empirical formula. The extraction procedure ispresented in Yamamoto et al. (1995). The sandstone
porosity and the shale porosity images extracted by thisprocedure are shown in Figure 4 a and b. The clay
content of the sandstone is assumed at 25 % in thecalculation and 95 % for the shale. These porosity andelastic modulus values are used in the permeability
inversions.
MEASURED PERMEABILITY IMAGES
The permeability images for the sand layers (350-450 m,and 530-650 m) and the shale layers (450-530 m, and650-800 m) are inverted through the analytical inverse
procedure and shown in Figure 5 a and b. In theinversion calculations, a measured viscosity, 4.0 mPa s,of the oil from the two wells, and constant squirt flow
lengths 500 m for the sandstone and 20 m for the shale
are assumed. According to the analytical permeability
inversion, the top sandstone layer at 350-450 m haspermeability between 180 and 200 md and the bottom
sandstone layer at 530-650 m has a higher permeability
between 205 and 220 md. These values agree well withthe core permeability order of 200 md. The permeability
values of the two shale layers at 450-530 m and 650-800
m are very small and raging between 65 and 80 d. The
two oil baring sandstone reservoirs have good confining
layers according to the permeability images.

Biot, M. A., 1956, The theory of propagation of elastic
waves in a fluid-saturated porous solid, I lower
frequency range, J. Accoust. Soc. Am., 12, 155-164.
Mavko, G., and Nur, A., 1975, Melt squirt inasthenosphere, J. Geophys. Res., 80, 1444-1448.
Dvorkin, J., and Nur, A., 1993, Dynamic poroelasticity:a unified model with the squirt and the Biotmechanism, Geophysics, 58, 524-532.
Bregman, N. D., Chapman, C. H., and Baily, R. C., 1989,
Travel time and amplitude analysis in values of thesesandstone layers, which are of the seismic tomography,J. Geophys. Res., 54, 200-215.
Han, D., Nur, A., and Morgan, D., 1986, Effects ofporosity and clay content on wave velocities insandstones, Geohysics, 51, 2093-2107.
Yamamoto, T., Nye, T., and Kuru, M., 1995, Imaging thepermeability structure of a limestone aquifer bycrosswell tomography, Geophysics, 60, 1634-1645.

CONCLUSIONSAn analytical permeability inversion method and theexample permeability images within sandstone oil
reservoirs are presented. The velocity and attenuation
images measured by the patented PRBS based acoustic
crosswell tomography (US Patent No. 5,406,530). The
method is shown to be very stable and accurate according
to the core permeability results.
ACKNOLEGEMENTS
The analytical permeability inverse of acoustic velocity
and attenuation is partly a result of the senior author’s
research funded by US Office of Naval Research. ThePRBS crosswell tomography system was built by YEC
for KSC and used during the crosswell tomography
experiment by the two authors. All the expenses for theexperiment and publication were paid by KSC. TOL
provided the two wells for the experiment as well as thedown hole logs.
REFERENCES
Yamamoto, T., 1998, A poro-elastic model of extremely
permeable limestone for permeability imaging,
Geophysics (submitted).
Yamamoto, T., Nye, T., and Kuru, M., 1994, Porosity,
permeability, shear strength: cross- well tomography
below an iron foundry, Geophysics, 59, 1530-1541.

Figure 1a,Resistivity andGamma Ray Logs,Well Q1.

Figure 1b, Resistivity
and Gamma Ray Logs,Well MD-15.

Figure 2, Example 500 Hz PRBS source gather data, source at depth 431(1414 feet) in Well MD15.

Figure 3a, Velocity image
measured at 500 Hz PRBS.

Figure 3b, Attenuation (Q-1)
image measured at 500 Hz PRBS.

Figure 4a, Porosity image within sandstonelayers at 450 to 530 m and 650 to 800 m. Ignore
the images in the rest of depth intervals.

Figure 5a, Permeability image within sandstonelayers at 450 to 530 m and 650 to 800 m. Ignore
the images in the rest of depth intervals.

Figure 4b, Porosity image within shale layers
at 350 to 450 m and 530 to 650 m. Ignore theimages in the rest of depth intervals.

Figure 5b, Permeability image within shalelayers at 350 to 450 m and 530 to 650 m. Ignore
the images in the rest of depth intervals.