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# Gassmann's equation for fluid substitution

To extract fluid types or saturations from seismic, crosswell, or borehole sonic data, we need a procedure to model fluid effects on rock velocity and density. Numerous techniques have been developed. However, Gassmann’s equations are by far the most widely used relations to calculate seismic velocity changes because of different fluid saturations in reservoirs. The importance of this grows as seismic data are increasingly used for reservoir monitoring.

## Overview

Gassmann’s formulation is straightforward, and the simple input parameters typically can be directly measured from logs or assumed based on rock type. This is a prime reason for its importance in geophysical techniques such as time-lapse reservoir monitoring and direct hydrocarbon indicators (DHI) such as amplitude "bright spots," and amplitude vs. offset (AVO). Because of the dominance of this technique, we will describe it at length.

Despite the popularity of Gassmann’s equations and their incorporation within most software packages for seismic reservoir interpretation, important aspects of these equations are usually not observed. Many of the basic assumptions are invalid for common reservoir rocks and fluids. Many efforts have been made to understand the operation and application of Gassmann’s equations:

• Han,
• Mavko and Mukerji,
• Mavko et al.,
• Sengupta and Mavko,
• Nolen-Hoeksema

Most of these works have attempted to isolate individual parameter effects. We will extend this analysis to incorporate mechanical bounds for porous media (see previous) and the magnitude of the fluid effect.

## Change in bulk modulus with saturation

Compressional (P-wave) and shear (S-wave) velocities along with densities directly control the seismic response of reservoirs at any single location. Fig. 1a shows measured dry and water saturated P- and S-wave velocities of sandstones as a function of differential pressure. P-wave velocity increases, while S-wave velocity decreases slightly with water saturation. However, both P- and S-wave velocities are generally not the best indicators for any fluid saturation effect. This is a function of coupling between P- and S-wave through the shear modulus and bulk density. In contrast, if we plot bulk and shear modulus as functions of pressure (Fig. 1b), the water-saturation effect shows the following:

1. Bulk modulus increases about 50%
2. Shear modulus remains almost constant

Bulk modulus is more strongly sensitive to water saturation. The bulk volume deformation produced by a passing seismic wave results in a pore volume change, and causes a pressure increase of pore fluid (water). This has the effect of stiffening the rock and increasing the bulk modulus. Shear deformation usually does not produce pore volume change, and differing pore fluids often do not affect shear modulus.

## Gassmann's equation

Gassmann’s equations provide a simple model to estimate fluid saturation effect on bulk modulus. Eqs. 1a through 2 are convenient forms for Gassmann’s relations that show the physical meaning:

....................(1a)

....................(1b)

and

....................(2)

where,

• K0, Kf, Kd, and Ks, are the bulk moduli of the mineral, fluid, dry rock, and saturated rock frame, respectively
• Φ is porosity
• μs and μd are the saturated and dry rock shear moduli.

ΔKd is an increment of bulk modulus caused by fluid saturation. These equations indicate that fluid in pores will affect bulk modulus but not shear modulus, consistent with the earlier discussion. As pointed out by Berryman, a shear modulus independent of fluid saturation is a direct result of the assumptions used to derive Gassmann’s equation.

### Assumptions

Numerous assumptions are involved in the derivation of Gassmann’s equation:

1. Porous material is isotropic, elastic, monomineralic, and homogeneous
2. Pore space is well connected and in pressure equilibrium (zero frequency limit)
3. Medium is a closed system with no pore fluid movement across boundaries
4. No chemical interaction between fluids and rock frame (shear modulus remains constant)

Many of these assumptions may not be valid for hydrocarbon reservoirs, and they depend on rock and fluid properties and in-situ conditions. For example, most rocks are anisotropic to some degree. The work of Brown and Korringa provides an explicit form for an anisotropic fluid substitution. In seismic applications, it is normally assumed that Gassmann’s equation works best for seismic data at frequencies less than 100 Hz (Mavko et al.). Recently published laboratory data (Batzle et al.) show that acoustic waves may be dispersive in rocks within the typical seismic band, invalidating assumption 2. In such cases, seismic frequencies may still be too high for application of Gassmann’s equation. Pore pressures may not have enough time to reach equilibrium. The rock remains unrelaxed or only partially relaxed.

## Gassmann fluid substitution

The primary measure of the sensitivity of rock to fluids is its normalized modulus Kn: the ratio of dry bulk modulus to that of the mineral.

....................(3)

This function can be complicated and depends on:

• Rock texture (porosity, clay content, pore geometry, grain size, grain contact, cementation, mineral composition, and so on)
• Reservoir conditions (pressure and temperature). This Kn can be determined empirically or theoretically. For relatively clean sandstone at high differential pressure (>20 MPa), the complex dependence of Kn (x, y, z, …) can be simplified as a function of porosity.

....................(4)

From Eq. 3, bulk modulus increment is then equal to

....................(5)

Here [1-Kn (Φ)] is also the Biot parameter αb (Biot). Furthermore, because usually K0 >> Kf, it is reasonable to assume

....................(6)

for sedimentary rocks with high porosity (>15%). Therefore,

....................(7)

where G(Φ) is the saturation gain function defined as

....................(8)

Thus, fluid saturation effects on the bulk modulus are proportional to the gain function G(Φ) and the fluid modulus Kf. The G(Φ) in turn depends directly on dry rock properties: the normalized modulus and porosity. In general, G(Φ) is independent of fluid properties (ignoring interactions between rock frame and pore fluid). We must know both gain function of dry rock frame and pore fluid modulus to evaluate the fluid saturation effect on seismic properties. Note that the normalized modulus must be a smooth function of porosity or G(Φ) can be unstable, particularly at small porosities.

At high differential pressure (>20 MPa), the Ks of water-saturated sands calculated using simplified form is 3% overestimated for porous rock (porosity > 15%). Those errors will decrease significantly with low fluid modulus (gas and light oil saturation). For low-porosity sands with high clay content, the simplified Gassmann’s equation overestimates water saturation effects substantially.

In Eq. 1, there are five parameters, and usually the only applied constraint is that the parameters are physically meaningful (>0). Incompatible or mismatched data might generate wrong or even unphysical results such as a negative modulus. In reality, only K0 and Kf are completely independent. Ks, Kd, and porosity Φ are actually closely correlated. Bounds on Kd as a function of porosity, for example, constrain the bounds of Ks.

Assuming porous media is a Voigt material, which is a high bound for Kd (Fig. 2),

....................(9)

Putting this equation (9) into Gassmann’s equation (1) gives

....................(10)

and

....................(11)

Because this Voigt bound is the stiffest upper limit, the fluid saturation effect on bulk modulus here (ΔKdmin) will be a minimum (see Fig. 2).

The low modulus bound for porous media is the Reuss bound.

....................(12)

....................(13)

For completely empty (dry) rocks, the fluid modulus Kf is equal to zero, and both the Reuss bound and the normalized modulus (KnR) for a dry rock in this limit equals zero (for nonzero porosity).

....................(14)

Substituting Eq. 14 into Gassmann’s equation, we find the fluid saturation effect on bulk modulus when the frame is at this lower bound.

....................(15)

For this case, the modulus increment ΔK from dry to fluid saturation is equal to the Reuss bound.

....................(16)

Again, Gassmann’s equation is consistent with the dry and fluid-saturated Reuss bounds. Physically, for rocks with the weakest frame, fluids have a maximum effect.

## Constraining with critical porosity

Critical porosity, Φc, can be used to give tighter constraints for dry- and fluid-saturated bulk modulus for sands. A new triangle is formed which provides a linear formulation and a graphic procedure for Gassmann’s calculation: the fluid saturation effect on bulk modulus proportional to normalized porosity and the maximum fluid saturation effect on bulk modulus (Reuss bound) at the critical porosity (Fig. 2).

....................(17)

This is consistent with the results of Mavko and Mukerji.

For typical sandstones, the critical porosity Φc is around 40%. Thus, we also can generate a simplified numerical formula of the normalized modulus Kn for modified Voigt model:

....................(18)

Using this in Gassmann’s equation yields fluid saturation effect

....................(19)

Extending our empirical approach to first order, both P- and S-wave velocity can correlate linearly with porosity at high differential pressure. From Table 1, for dry clean sands,

....................(20)

....................(21)

where we assume the density of these sands is equal to

....................(22)

Since the modulus is the product of the density and square of velocity, we get an equation that is cubic in terms of porosity. The bulk modulus can be derived as

....................(23)

where,

• A = 3.206
• B = 3.349
• C = 1.143

Eq. 23 can be further simplified if porosity Φ is not too large (<30%):

....................(24)

where D for clean sandstone is equal to 1.52. This includes an empirical expression of the normalized modulus as a direct dependence on porosity and "D" parameter. Table 2 and Fig. 3 show empirical relations generated from dry velocity data of relatively clean rocks. The parameter D is related to rock texture and should be calibrated for local reservoir conditions. In general, it has a narrow range from 1.45 to slightly more than 2.0, primarily depending on rock consolidation.

By inserting this D function into Eq. 8, we find

....................(25)

## Solid mineral bulk modulus

The mineral modulus (solid grain bulk modulus) K0 is an independent parameter, and the rock texture controls Kd. However, as mentioned , the normalized modulus Kn controls the fluid saturation effect rather than Kd or Ks individually. The mineral modulus K0 is equally as important as Kd. However, in most applications of the Gassmann’s equation, only Kd is measured.

Properties of the mineral modulus K0 are often poorly understood and oversimplified. K0 is the modulus of the solid material that includes grains, cements, and pore fillings (see Rock types). If clays or other minerals are present with complicated distributions and structures, K0 can vary over a wide range. Unfortunately, few measurements of K0 have been made on sedimentary rocks (Coyner), and the moduli of clays are a particular problem (Wang et al. and Katahara; see also Rock moduli boundary constraints). These data show that at a high pressure, K0 for sandstone samples range from 33 to 39 MPa. K0 is not a constant and can increase more than 10% with increasing effective pressure. Fig. 1 shows the influence of K0 on Gassmann’s calculation. This case uses a dry bulk modulus calculated with the mineral modulus of 40 GPa, D = 2, and a water modulus of 2.8 GPa. The water saturation effect was calculated for three mineral moduli of 65, 40, and 32 GPa. Results show that for the same Kd and Kf, bulk modulus increment ΔK because of fluid saturation increases with increasing mineral modulus K0. Errors caused by uncertainty of K0 decrease with increasing porosity and fluid modulus Kf.

### Mineral modulus derived from empirical velocities

Because of lack of measurements on bulk mineral modulus, we often must use measured velocity/porosity/clay-content relationships for shaly sandstone to estimate the mineral modulus. Assuming zero porosity and grain bulk modulus of 2.65 gm/cc, we can derive mineral bulk and shear modulus from measured P- and S-wave velocity. The results are shown in Table 1.

1. For relatively clean sandstone (with few percent clay content), mineral bulk modulus is 39 GPa, which is stable for differential pressures higher than 20 MPa. Mineral shear modulus is around 33 GPa, which is significantly less than 44 GPa for a pure quartz aggregate. Shear modulus is more sensitive to differential pressure and clay content.
2. For shaly sandstone, mineral bulk modulus decreases 1.7 GPa per 10% increment of clay content.

Such derived mineral bulk moduli can be used for Gassmann’s calculation if there are no directly measured data or reliable models for calculation.

With a change of fluid saturation from Fluid 1 to Fluid 2, the bulk modulus increment (ΔK) is equal to

....................(1)

where Kf1 and Kf2 are the moduli of Fluids 1 and 2, respectively, and ΔK21 represents the change in the saturation increment that results from substituting Fluid 2 for Fluid 1. Eq. 1 uses the fact that the gain function G(Φ) of the dry rock frame remains constant as fluid modulus changed (this may not be true for real rocks). The fluid substitution effect on bulk modulus is simply proportional to the difference of fluid bulk modulus.

If we know the gain function for a rock formation, we can estimate the fluid substitution effect without knowing shear modulus.

....................(2)

where ρ1, ρ2, Vp1, and Vp2 are the density and velocity of rock with Fluid 1 and 2 saturation. Both Eqs. 1 and 2 are direct results from simplified Gassmann’s equation. In Fig. 10, we show the typical fluid modulus effect on the saturated bulk modulus Ks. Even at a modest porosity of 15%, changes can be substantial. At in-situ conditions, pore fluids are often multiphase mixtures. Dynamic fluid modulus may also depend on fluid mobility, fluid distribution, rock compressibility, and seismic wavelength.

## Nomenclature

 K = bulk modulus, GPa or MPa Kd = dry bulk modulus, GPa or MPa Kd min = minimum bulk modulus, GPa or MPa Kf = fluid bulk modulus, GPa or MPa Kf 1, Kf 2 = bulk modulus of fluid 1, 2, etc., GPa or MPa KHS = Hasin-Shtrikman bound bulk modulus, GPa or MPa Kn = normalized bulk modulus, numeric Kn R = normalized Reuss bound bulk modulus, numeric Ko = mineral bulk modulus, GPa or MPa KR = Reuss bound bulk modulus, GPa or MPa Ks = saturated bulk modulus, GPa or MPa K1, K2 = bulk modulus of component 1, 2, etc., GPa or MPa K* = effective bulk modulus, GPa or MPa K′ = effective crack bulk modulus, GPa or MPa ΔKd = change in bulk modulus, GPa or MPa ΔKdmax = maximum change in bulk modulus, GPa or MPa ΔK12 = change in bulk modulus, fluid 1 to fluid 2, GPa or MPa Φ = porosity G(Φ) = gain factor