TESTING A GEOMETRIC ROCK MODEL ON NATURALLY FRACTURED CARBONATE SAMPLES G. KORVIN *, KLAUDIA OLESCHKO **

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TESTING A GEOMETRIC ROCK MODEL ON NATURALLY
FRACTURED CARBONATE SAMPLESG. KORVIN , KLAUDIA
OLESCHKO A. ABDULRAHIM (KFUPM, UNAM,
MEXICO )
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Research Objective

• Given The measured porosity ?, permeability k,
  and cementation exponent m of a sedimentary rock.
• Problem Find an equivalent rock model
  characterized
• by 3 effective geometric 1 topological
  properties
• r (average pore radius)
• d (average distance between two nearest pores)
• ? (average throat radius)
• Z (average coordination number of a pore)
• Constraint The 4 parameters should be derivable
  from values of k, m, ? measured at atmospheric
  pressure, and they should exactly reproduce the
  measured values.

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Excursus Why four parameters?Because five are
too much!

• By Neumanns famous saying give me four
  parameters, and I will fit an elephant, give me a
  fifth, and I will make it wiggle its trunk.
  Actually (Wei, 1975) the contours of an elephant
  can be fit using 30 real coefficients, or at
  least 4 complex ones (Mayer et al. 2010) in the
  parametric Fourier representation

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Fitting an elephant LHS (Wei, 1975) sketch of
an elephant, fitting with 5, 10, 20 and 30 sine
coefficients RHS (Mayer et al., 2010) (a) four
complex coeficients, (b) five complex
coefficients make it wiggle the trunk
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4 to 5 parameters can describe very complex
rock-physics. Examples Three Thomeer MI
Capillary Pressure Parameters or Cole-Cole model
of IP (Induced Potential)

• Equivalent circuit model of the IP effect R0
  resistance of host rock, R1 resistance of the
  pore-filler liquid, Zm is complex impedance for
  the metallic grains. In the Cole-Cole model

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Some Definitions Pore
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Some Definitions Throat
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Some Definitions Coordination Number
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We simplified Doyens 1978 Max Entropy Model

• Doyens Assumptions
• The pores are fluid-filled ellipsoids with
  semiaxes (a, b, c), each pore is connected to Z
  nearby pores with throats of length l and
  elliptic cross-section with semi axes (r1, r2).
• One measures M bulk data B1, ,,BM for N
  pressure steps P1,,PN.

• Our Assumptions
• The pores are fluid-filled spheres with radius r,
  each pore is connected to Z nearby pores with
  throats of length d and circular cross-section
  with diameter ?
• We measure three bulk data ?, k, m for a
  single pressure step only.

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The parameters r, d, ?, Z that we use
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Theoretical Assumptions

• Z 2 m / (m-1)
• (From effective medium theory of granular
  materials, Yonezawa Cohen J. Appl. Phys.
  54 (1983) 2895)
• k (1/b) ?3 (1/S2) (1/t2)
• (Kozeny-Carman Eq., cf. Walsh Brace,
  1984 J. Geoph. Res.)
• t 1/ (?m-1)
• (Non standard assumption, follows from
  Peres-Rosales SPEJ, 1982 531-536 Archies
  Law)
• (Z coordination number, m cementation exponent, S
  
• specific surface, ? porosity, t tortuosity, k
  permeability.)

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Mathematical Solution (Exact Easily Computable)
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Results for a typical Saudi carbonate sample
(Khuff), red color pore
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Results of the KOA inversion on four carbonate
samples from a Mexican Well
Depth (m) Sample name () ?cplot (fraction) ?lab (fraction) K(1) (md) m r (µ) ? (µ) d (µ)
3772-3777 A 0.09 0.10 0.876(2) 2 9.36 0.0004 34
3772-3777 1 0.0723 91.3(3) 2 142 10 551
3772-3777 13 0.1351 7.49(3) 2 22 1.6 58
3850-3855 18 0.13 0.1243 2.06(3) 2.3 18 0.46 59
Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP Notes () in the crossplots we assumed sweet water based drilling liquid (1) m was estimated from the Lucia (1998) empirical equation (2) Klinkenberg corrected air permeability (3) Swanson permeability from MICP
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DETAILS FOR SAMPLE 18

• Sample 18 (3850-3855m) is limey dolomite, with
  vugs microfractures" according to the MIP
  Report. On the sonic-density crossplot it fairly
  well follows the time average rule, what excludes
  a large amount of vugs. Based on this, we assumed
  only 25 vug fraction, and Lucia's (1998)
  carbonate equation m2.14?vug1.76 yielded
  m 2.3. By Lab Rept., the sample has 0.1243
  porosity and from the log (taking the average
  crossplot porosity for this 5m depth range and
  assuming sweet water drilling fluid) we get 0.13
  porosity. For the computations we used the
  average between lab and cross-plot porosities,
  i.e. ?0.127. The MIP Report gives unimodal
  throat size distribution between 0.010 - 100 µ,
  the results of our inversion (0.46µ) fit into
  this range.

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Beauty of the model

• Mathematically simple (few degrees of freedom)
• We could (Ar.J.Geosci. 2014 ) derive other
  important rock physical properties from it (such
  as elastic moduli, P- and S wave velocities,
  resistivity, permeability, seismic attenuation,
  density, )
• Can be determined from three measured data k, F,
  m at atmospheric pressure
• Can be extended to multi-size pore systems (??)

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Select Bibliography

• Doyen, P.M. (1987) Crack geometry in igneous
  rocks A Maximum Entropy inversion of elastic and
  transport properties. Journal Geoph. Res. 92(B8)
  8169-8181
• Kirkpatrick, S. (1973) Percolation and
  conduction. Rev. Mod. Phys. 45(4)574-588
• Korvin, G., Klavdia Oleschko A. Abdulraheem
  (2014) A simple geometric model of sedimentary
  rock to connect transfer and acoustic properties.
  Arab. J. Geosci. 7 1127-1138
• Lucia, F.J., (1998) Carbonate Reservoir
  Characterization. Springer, Berlin.
• Mayer, J., Khairy, Kh., Howard, J. (2010)
  Drawing an elephant with four complex parameters.
  Am. J. Phys. 78(6) 648-649
• Perez-Rosales, C. (1982) On the relationship
  between formation resistivity factor and
  porosity. SPE J. (Aug. 1982) 531-536

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Thanks!