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Testing the New Gassmann solid-fluid substitution on the Triassic sandstone reservoirs in the Southern North Sea basin.


In sandstone reservoirs, it is common to perform a fluid substitution in the pore space between different fluids like gas, water and oil. This substitution is usually calculated by using the Gassmann equations (Gassmann, 1951). The prediction of seismic properties for pores filled with different fluids is one of the most important problems in the rock physics analysis of logs, cores and seismic data (Mavko et al., 1998).

Brown and Korringa (1975) generalised the Gassmann equation for anisotropic porous media. Ciz and Shapiro, 2007 and Ciz et al. 2008 published literatures on the extension of the Gassmann equation for solid substitution in the pore space. One of the most common problems in the North Sea is the occurrence of salt in the pores of Triassic sandstones. Here, gas-bearing good-quality reservoir rock looks almost identical to salt-plugged rock. Many wells failed for this reason; hence, solution is to be found by modelling and characterizing the salt-plugging scenarios. For this purpose, this new theory of the extended Gassmann model is used perform solid-fluid substitution in order to distinguish between salt-plugged and potentially gas-bearing reservoirs the result of which will crucial for the Oil and Gas industry.

The Aim of this study is to validate the extended Gassmann model on real data of a potentially salt-plugged reservoir in the Southern North Sea. To this end, modelling codes were generated using Matlab (scientific research and modelling programme) and RokDoc[TM] (an industry-standard software for reservoir characterization and modelling, provided and supported by Ikon Science). RokDoc[TM] is a powerful tool for performing rock physics modelling on wells, displaying the results and integration of different wells in one project. These codes were for the respective substituent-gas, water and salt on the bases of which predictions were made for the elastic moduli and the AVO behaviours. AVO (Amplitude Variation with Offset) responses, elastic and acoustic parameters are used to indicate the differences of the fluid- and solid-filled reservoir scenarios. For the AVO analysis here, the solution of Shuey's (1985) approximated Zoeppriz (1919) equation which assumes small layer contrasts was used. The equation describes amplitude of reflections as a function of the angle of incidence of a seismic wave. Three groups of wells (water-, Gas- and Salt-bearing) were subjected to modelling. Figure 1 shows location map of the North Sea Basin (Off shore, Netherlands).


Backgrounds of the Gassmann Equations

Gassmann equation is commonly applied to predict the elastic moduli of rocks saturated with different fluids. It assumes that all pores are interconnected and that the pore pressure is in equilibrium in the pore space. Furthermore, the porous frame is macroscopically and microscopically homogeneous and isotropic. Shapiro and Kaselow (2005) extended the theory to the case where the pore-filling material is an anisotropic elastic solid.

Classical Gassmann Equation for Fluid Substitution

This is one of the best known prediction methods for seismic properties in rocks. This predicts changes from one fluid to another by calculating the dry rock velocities and bulk modulus and then substitutes with a second fluid. Commonly, this fluid substitution is between water, gas and oil. Here, when a pore pressure change is induced by a wave such as a passing seismic wave, the stiffness of the rock, and hence the bulk modulus K, changes. It is important, to note that the Gassmann equation is only valid for sufficiently low frequencies < 100Hz (sufficient time for the pore fluid to flow and eliminate wave-induced pore pressure gradients). Further assumptions are

* That the rock is isotropic;

* All minerals making up the rock have the same bulk and shear moduli;

* The fluid-bearing rock is completely saturated.

Equations 1 and 2 are the classical Gassmann's equation for fluid substitutions. The procedure is to first transform the moduli from the initial fluid saturation to the dry state (equation 1; forward substitution)

[K.sub.dry] = [K.sub.sat](([phi][K.sub.0]/[K.sub.fl] + 1 - [phi]) - [K.sub.0])/[phi][K.sub.0]/[K.sub.fl] + [K.sub.sat]/[K.sub.0] - 1 - [phi] (1)

and then transform from the dry state into the new fluid-saturated state (equation 2; backward substitution).

[K.sub.dry] = [K.sub.dry] + [(1 -[K.sub.dry]/[K.sub.0]).sup.2]/[phi]/[K.sub.fl] + 1 - [phi]/[K.sub.0]- [K.sub.dry]/[K.sup.2.sub.0] (2)


[K.sub.0] = bulk modulus of mineral material making up the rock

[K.sub.dry] = effective bulk modulus of dry rock

[K.sub.sat] = effective bulk modulus of the rock with pore fluid

[K.sub.fl] = effective bulk modulus of the pore fluid

[phi] = Porosity

For more details, see Mavko et al.,1998.

The Extended New Gassmann Model

For this theory, a porous rock of porosity, [phi] is considered and the pore space is interconnected, representing Biot's medium. Biot (1962) defined an isotropic rock where all minerals making up the rock have the same bulk and shear moduli. According to Brown and Korringa (1975) and Shapiro and Kaselow (2005) a deformation of a rock sample is described by symmetric tensors. The deformation is described (after appropriate mathematical derivations) by:

[delta][[eta].sub.ij]/V = [S.sup.dry.sub.ijkl] [partial derivative][[sigma].sup.d.sub.kl][delta][[sigma].sup.f.sub.kl] (3)

where [[sigma].sup.d.sub.kl] = [[sigma].sup.c.sub.kl]-[[sigma].sup.f.sub.kl] is the effective compliance tensor of the composite porous rock with a solid infill [S.sup.*.sub.ijkl] defined as

[S.sup.*.sub.ijkl] = 1/V([partial derivative][[eta].sub.ij]/[partial derivative][[sigma].sup.c.sub.kl])con. (4)

Ciz and Shapiro (2007) generalized Brown and Korringa's equation for a solid infill of the pore space for an isotropic material. The compliance tensor is then expressed by bulk (K) and shear (u) moduli, and the following isotropic Gassmann equations for a solid-saturated porous rock are the main results:

[K.sup.*-1.sub.sat] = [K.sup.-1.sub.dry]-[([K.sup.-1.sub.dry]-[]).sup.2]/[phi]([K.sup.-1.sub.if]- [K.sup.-1.sub[phi]])+([K.sup.-1.sub.dry]-[])


[[mu].sup.-1.sub.sat] = [[mu].sup.-1.sub.dry]-[([[mu].sup.-1.sub.dry]-[[mu]]).sup.2]/[phi]([[mu].sup.- 1.sub.if]-[[mu].sup.-1.sub.[phi]]+([[mu].sup.-1.sub.dry]-[[mu]]) (6)


[K.sup.*-1.sub.sat] and [[mu].sup.-1.sub.sat] are solid saturated bulk and shear moduli;

[K.sub.dry] and [[mu].sub.dry] denote drained bulk and shear moduli of the porous frame;

[] and [[mu]] represent bulk and shear moduli of the grain material of the frame;

[K.sub.if] and [[mu].sub.if] are bulk and shear moduli related to the solid body of the pore infill;

[K.sub.[phi]] and [[mu].sub.[phi]] are the bulk and shear moduli related to the pore space of the frame.


* The workflow is shown in Fig. 2 and consists of the following steps

* Well Log Analysis and Quality Control

* Initial well Tie

Wavelet Estimation Procedure

For a reliable result of the modelling exercise, good data quality was ensured by way of cross-plot analysis and calibration of measured density with core data. In calibrating data sets, Gamma ray, porosity, compressional wave velocity ([V.sub.P]), Shear wave velocity ([V.sub.S]) and density logs were displayed and analysed in the well-viewer of the RokDoc project (figure 3). The Volpriehausen formation, which is a water bearing well (WTR) is at a depth of 3903 m-3942m (figure3) where


The green lines and red rectangle indicate the interval. The first column to the right of the display are the density logs: The light green (DENC) signature is the log measurement with the density


Tool, while the black curve represents density readings from laboratory measurements on core data. The measured log density seems to be systematically higher than expected based on the core data (black). To circumvent this problem, a new density was calculated based on the porosity log using a modified Wyllie's equation (equation 7).

[[rho].sub.bulk] = (1-[PHI]) * [[rho].sub.matrix] + [PHI] * [[rho].sub.pore] (7)

Where [[rho].sub.matrix] is an average value of 2.63 g/[cm.sup.3] and water was been taken to be pore fill with 1.1 g/[cm.sup.3]. The newly calculated density (blue, first column right) corresponds better with the core measurements and was therefore used for further calculations. Therefore, the newly calculated densities are used and are calculated the same way for a. o. wells.

Looking at the whole Volpriehausen sandstone interval (fig. 3) reveals a rather clean sandstone, except for a small interval between 3912 m and 3918 m which is indicated by two green lines (salt-plugging top and base) with a red rectangle. This interval marks the evidence of salt-plugging scenario in the Volpriehausen reservoir by a decreasing porosity, increasing Vp, Vs and [rho]. While density increases to 2.64 g/[cm.sup.3], velocity goes up to 5000 m/s. This observation is similar for a. o. wells where the salt-plugging scenarios occur.

With the cross-plotting tool, spikes were removed from data, which allowed for data review and adjustment for every well. Within the well viewer odd data were also removed and splined on selected intervals. All cross plots represent the Volpriehausen sandstone interval. RokDoc uses default constants data from the Gulf of Mexico; therefore, specific parameters like Greenberg-Castagna and fluid properties were checked and updated according to real data in the Southern North Sea.

Conceptional and Operational Modelling

The Typical gas, water and salt parameters (Tapan Mukerji et al.,1998) were used to determine their effective elastic moduli (table1). Conceptional input parameters (Vp and Vs), as the average sets of each type of well (see table 2) were modelled with the Matlab[R] (Matrix laboratory) programming, using empirical relation based on the locally derived Greenberg-Castagna parameters (Castagna et al.,1998).

Then, with the RokDoc programming, the bulk (K) and shear moduli (u) were calculated for all types of pore fillings the using following relations respectively (equations 8 and 9):

[mu] = [V.sup.2.sub.s] [rho] (8)


K = [V.sup.2.sub.p] - 4/3 [mu] (9)

Based on these elastic properties and characteristic velocities, the reflectivity between an average shale and the sandstone with different pore-filling was calculated, again with code generation using Matlab. This was performed to check the sensitivity of parameter input in the Shuey's equation and to understand the steps behind RokDoc blocky AVO analysis. With these modelling exercises, the rock physics model has been fully constrained and calibrated. Table 2 shows the average sets for both of the elastic and wave parameters.

Solid and Fluid Substitutions

Further, using Birch's (1961) principle, as applied in Brown and Korringa (1975), the elastic properties like Young's modulus E, Poisson's ratio Act, Acoustic Impedance AI, Elastic Impedance EI were calculated with average initial velocities using Matlab and the imported into RokDoc as plug-in. Using the RokDoc modeller, codes were generated based on the new Gassmann model for solid substitution which were used to calculate the bulk and shear moduli. To this end:

* Gas-bearing reservoir was substituted into salt- and water-bearing reservoir intervals

* Salt-plugged well was substituted into water and gas well

* Water-bearing well was substituted into gas and salt pore fillings

Table 3 shows the calculated average values for the elastic moduli for initial and substituted wells. In the diagonal (red) are the initial wells. From left to right (first to third row respectively in each column): second column represents real gas bearing and the substituted water and salt wells/zones into gas. Third column represents real and substituted water moduli, while the fourth column represents the real and substituted salt moduli.

A similar modelling codes were generated and calculations were made for velocities and densities. Tables 4 shows the average velocity (m/s) and density (g/[cm.sup.3]) for initial and substituted wells, where bold letters represent the real gas, water and salt-plugged averages. The averages are derived from calculations performed on log data within one well, namely the salt substitution within GAS2 and WTR.'

For the AVO analysis, sets with average elastic properties were created for each example well. The near stack angle is defined as 10[degrees] and the far stack angel as 50[degrees].

Table 5 shows the Intercept (I) and the gradient (G) values for different interfaces. These values are used to calculate the weighted stack.

Results and Discussion

Figure 4 shows the result of the AVO analysis for the classical Gassmann model while figures 5 show the AVO response of a gas bearing well, the water to gas substitution and the solid substitution from salt to gas. All the curves show a very small positive gradient and a negative intercept. Figure 6 shows the AVO of a real water bearing well, a substituted gas-bearing well to water pore filling and a salt-plugged to gas substitution. The intercept of all lines is around -0.06 and the gradient is slightly positive especially in the far offset. The AVO response of solid substitution from gas to salt and water to salt is visible in figure 7. All curves show a polarity change from positive to negative, starting with a positive intercept of 0.03-0.06. The gradient of the curves is between -0.1 and -0.2. These responses and values in the salt-plugged scenario and clearly different from those of the normal reservoir (fluid-filled) conditions.

Also, looking at the values in tables 3,we see that salt-plugged wells show clearly different elastic parameters with respect to gas- and water-bearing wells. Interpreting from the diagonals (red), salt-plugged zones have elastic moduli (K= 27.5 and [mu] = 20.8) compared to that of water-filled reservoir pores (K= 22.1 and u=11.7) and gas-filled reservoir pores (K= 17.7 and [mu]= = 13.4). Similarly from table 4, salt-plugged reservoir interval have P-wave velocity, S-wave velocity and density (Vp = 4763,Vs = 2926 and p = 2.528) respectively compared to that of water (Vp = 4084, Vs = 2287 and [rho] =2.371) and gas (Vp = 4053, Vs = 2352 and [rho] = 2.29).





In order to check if the new Gassmann equation is applicable for this type of data, cross-plot analysis for substituted wells was done. For a classic Gassmann substitution between gas and water, the Vp and [rho] logs plot (figure 8) shows that the velocity prediction overlays nicely with real water and real gas wells. For the salt-plugged zone scenarios (WTR and GAS2 well), the velocity prediction also overlays nicely as in the cases of water and gas wells (figure 9).This implies that the solid substitution of the plugged zones with the fluid-filled reservoir correlates well with classical fluid-substitution. Note that these substitutions are possible only within the same well. With this observation, further procedure was focused on the fluid and solid substitution within one well. An additional uncertainty (limitation) is that due to drilling operations, salt might have been washed out, leading to a possible mis-interpretation of a salt-filled well as a water-bearing well.




With the extended Gassmann substitution equation, it is possible to substitute from gas or water filling to salt as the pore-filling, because this model takes changes for the shear modulus into account. The reverse operation from salt to gas and water can be calculated and substitution performed as well. The application of the equation works very well for the substitution in one well, because the change in elastic parameters are caused by salt-plugging (For example, in GAS2 and WTR wells).

Extending this procedure for inter-well substitution poses more difficulty. Data analysis indicates that density log value are too high, warranting a calibration through recalculation corroborated by the core data. AVO analysis show that for the classical Gassmann substitutions, all the curves show a very small positive gradient and a negative intercept, consistent with normal reservoir in-fills. For the AVO analysis of the salt-plugged scenarios (Salt into Gas) using the extended Gassmann substitution, the intercept of all lines is around -0.06 and the gradient is slightly positive. For the Gas into-Salt substitution, all curves show a polarity change from positive to negative, starting with a positive intercept of 0.03-0.06 and with gradient of the curves between -0.1 and -0.2.

The main results presented in tables 3 and 4 for the fluid and solid substitution application on gas- and water-bearing wells filled with salt, show very consistent results.

Therefore, with the successful application of the new Gassmann model on the solid and fluid substitutions as corroborated by the AVO responses, synthetic seismograms can be further generated which can compared with the real seismic data of the wells analysed in these fields in both stack and pre-stack domains. However, to extend the theory for a inter-well substitution, it is necessary to include depth dependent effects in the model due to depth variation impacts between different wells and on reservoir depth.


My Gratitude goes to the following

* Wintershall Noord Zee B.V Rijswijk, Netherlands for providing the raw data from the North Sea and for permission to publish the result of this study.

* The Technical University of Delft (TU Delft), Delft Netherlands for making the workstation research facilities available.

* IKON Science[R] Teddington, London, for providing the RokDoc software, making the software training in London possible and for continued support course.


[1] Biot, M. A.,1962. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33(No.4):p.1482-1496.

[2] Birch, F.,1961. The velocity of compressional waves in rocks to 10 kilobars. Geophysics, 66(7):p.2199-2224.

[3] Brown, R. J. S. and Korringa, J.,1975.On the dependence of the elastic properties of a porous rock on the compressibility of the pore fluid. Geophysics, 40:p. 608-616.

[4] Castagna J. P. Herbert W. Swanz, and Douglas J. Foster, 1998; Framework for AVO gradient and intercept interpretation. GEOPHYSICS, VOL. 63, NO.3 (MAY-JUNE 1998); P. 948-956.

[5] Ikon Science, 2010. RokDoc training course manual. Ikon Science Geo Pressure Technology, 16-18th of Match, 2010 Teddington, London.

[6] Radim, Ciz and Serge, A. Shapiro, 2007. Generalization of Gassmann equation for porous media saturated with a solid material. Geophysics, 72(No. 6):p. A75-A79.

[7] Shapiro, S. A. and Kaselow, A.,2005.Porosity and elastic anisotropy of rocks under tectonic stress and pore-pressure changes. Geophysics, 70(No.5): p. N27 N38.

[8] Shuey, R.T., 1985. A simplification of the zoeppritz equations. Geophysics, 50:p. 609-614.

[9] Tapan Mukerji, Gary Mavko and Jack Dvorkin. The Rock Physics Handbook. Cambridge University Press, 1998.

[10] Zoeppritz, K., 1919. Erdbebewellen viiib, on the reflection and propagation of seismic waves. Goettinger Nachrichten, I:p.66-84

Auduson Aaron E.

Previously: Department of Geotechnology, Delft University of Technology, Netherlands

Currently: Department of Earth Sciences, Kogi State University, P.M.B.1008, Anyigba, Nigeria

Table 1: Typical Gas, Water and Salt Parameters.

 Bulk modulus (GPa) Shear modulus (GPa)

Gas 0.22 0
Water 3.5 0
Salt 25.2 15.3

Table 2: Three well examples of average Shale and a Sandstone Rock
Physics parameter values-Column A, B and C respectively are the
real gas-bearing well, the real water bearing well and the average
of the salt-plugged zone.

 A B C [Shale.sub.
 (Gas) (Water) (Salt) average Sandstone

Vp(m/s) 3952 40001 4757 4164 6000
Vs(m/s) 2426 2228 2868 2279 4000
 [cm.sup.3]) 2.292 2.355 2.556 2.62 2.65
 [cm.sup.3]) 15 15 3-6 <1 10
GRC(API) 50-65 40-60 44-55 >80 Nil
K(GPa) 17.7 22.1 28.6 27.2 38.8
[mu](GPa) 13.4 11.7 18.6 13.5 42.4

Table 3: Average Bulk and Shear moduli in (GPa) for initial
and Substituted wells.

 Gas Water

Gas [K.sub.init] = 17.7 [K.sub.sat] = 32.1
 [[mu].sub.init] = 13.4 [[mu].sub.sat] = 13.4
 [K.sub.dry] = 17.1
Water [K.sub.sat] = 15.9 [K.sub.init] = 22.1
 [[mu].sub.sat] = 117 [[mu].sub.init] = 11.7
 [K.sub.dry] = 15.4
Salt [K.sub.sat] = 17.7 [K.sub.sat] = 22.1
 [[mu].sub.sat] = 13.4


Gas [K.sub.sat] = 25.8
 [[mu].sub.sat] = 22.7

Water [K.sub.sat] = 28.6
 [[mu].sub.sat] =19.9

Salt [K.sub.init] = 27.5
 [[mu].sub.init] = 20.8
 [K.sub.drygas] = 17.3
 [K.sub.drywater] = 15.4
 [[mu]drygas] = 13.4
 [[mu]drywater] = 11.7

Table:-4 Average velocity (m/s) and density (g/[cm.sup.3])
for initial and substituted wells

 Gas Water Salt

Gas Vp = 4053 Vp = 4084 Vp = 4735
 Vs = 2352 Vs = 2194 Vs = 2946
 [rho] = 2.29 [rho] = 2.408 P=2.461

Water Vp = 3951 Vp = 4084 Vp = 4969
 Vs = 2348 Vs = 2287 Vp = 2995
 [rho] = 2.251 [rho] =2.371 [rho] = 2.461

Salt Vp = 4118 Vp = 4051 Vp = 4763
 Vs = 2394 Vs = 2243 Vs = 2926
 [rho] = 2.29 [rho] = 2.371 [rho] = 2.528

Table 5: AVO intercepts (I) and gradients (G)

Interface and well I G

Shales on GAS(new) -0.079 0.31
Shales on WTR(new) -0.058 0.046
Shales on SALT-plugged GAS well 0.055 -0.273
Shales on SALT-plugged WTR well -0.087 0.098
Shales on GAStoWTR -0.101 0.032
Shales on GAStoSALT -0.071 0.016
Shales on WTRtoGAS -0.058 0.083
Shales on WTRtoSALT -0.087 0.096
Shales on SALTtoWTR (inverse) 0.034 -0.245
Shales on SALTtoGAS(inverse) -0.087 0.098
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Author:Auduson, Aaron E.
Publication:International Journal of Petroleum Science and Technology
Date:May 1, 2012
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