Fluid substitution model

The objective in fluid replacement modelling is to replace the initial properties of the reservoir rock with alternate post-production values from which seismic properties such as P-wave velocity, S-wave velocity, and density, and by extension P-wave and S-wave impedances can be computed. The formulas for these basic seismic properties are (Mavko et al., 1998): 𝑉𝑃_𝑠𝑎𝑡 =√ 𝑘𝑠𝑎𝑡+4₃𝜇_𝑠𝑎𝑡 𝜌_𝑠𝑎𝑡 , (2-1) 𝑉𝑆_𝑠𝑎𝑡= √ 𝜇_𝑠𝑎𝑡 𝜌_𝑠𝑎𝑡 , (2-2) 𝐼𝑃_𝑠𝑎𝑡 = 𝑉𝑃_𝑠𝑎𝑡𝜌𝑠𝑎𝑡 , (2-3)

where the subscript 𝑠𝑎𝑡 indicates the fluid-saturated case, 𝑉_𝑃 is the P-wave velocity, 𝑉_𝑆 is the S-wave velocity, 𝐼_𝑃 is the P-wave impedance, 𝜌 is the density, 𝜇 is the shear modulus and 𝑘 is the bulk modulus. The saturated bulk density, 𝜌_𝑠𝑎𝑡 is governed by porosity, mineralogy and fluid saturations:

𝜌𝑠𝑎𝑡= 𝜌𝑑𝑟𝑦+ 𝜙(𝜌𝑤𝑆𝑤+ 𝜌𝑂𝑆𝑂+ 𝜌𝑔𝑆𝑔) (2-4)

where 𝜌_𝑑𝑟𝑦 = 𝜌_𝑚(1 − 𝜙) is density of the dry rock frame, 𝜙 is porosity, 𝜌_𝑚 is the mineral bulk density; 𝜌𝑤, 𝜌𝑂, 𝜌𝑔, are the water, oil and gas density respectively and 𝑆 is

the fraction of saturation for each fluid component. The saturated density at pre- production state can either be measured in-situ or computed using Equation 2-4.

Assuming values from the initial state are known, the goal is then to calculate alternate values representing different fluid saturation scenarios. For this, Gassmann equations (Gassmann, 1951) are used to compute the change in bulk modulus associated with a change in the pore fluid phases or a change in the acoustic properties of the existing fluids. These equations assume that the shear modulus is independent of fluid content (but not porosity), written as:

𝜅𝑠𝑎𝑡= 𝜅𝑑𝑟𝑦+

(1 − 𝜅𝑑𝑟𝑦⁄𝜅𝑚)2

𝜙 𝜅⁄ 𝑓𝑙+ (1 − 𝜙) 𝜅⁄ 𝑚− 𝜅𝑑𝑟𝑦⁄𝜅𝑚2 (2-5)

𝜇𝑠𝑎𝑡= 𝜇𝑑𝑟𝑦 (2-6)

where 𝜅_𝑠𝑎𝑡 and 𝜇_𝑠𝑎𝑡 are the saturated bulk and shear modulus, 𝜅_𝑚 and 𝜇_𝑚 are the bulk and shear modulus of the mineral forming the rock, 𝜅_𝑑𝑟𝑦 and 𝜇_𝑑𝑟𝑦 are the bulk and shear modulus of the dry rock (minerals and pores), 𝜅_𝑓𝑙 is the fluid bulk modulus, and 𝜙 is the porosity.

For calculating several production scenarios, a practical form of the Gassmann equations is adopted (Equations 2-7 and 2-8) which shows which parameters will change should pore pressure and fluid saturation change:

𝜅𝑠𝑎𝑡(𝑃, 𝑆𝑤,𝑜,𝑔) = 𝜅𝑑𝑟𝑦(𝜎𝑒𝑓𝑓) + (1 −𝜅𝑑𝑟𝑦_𝜅(𝜎𝑒𝑓𝑓) 𝑚 ) 2 𝜙 𝜅𝑓𝑙(𝑃, 𝑆𝑜,𝑔,𝑤)+ (1 − 𝜙) 𝜅𝑚 − 𝜅𝑑𝑟𝑦(𝜎𝑒𝑓𝑓) 𝜅𝑚2 , (2-7) 𝜇𝑑𝑟𝑦(𝜎𝑒𝑓𝑓_𝑖) ≠ 𝜇𝑑𝑟𝑦(𝜎𝑒𝑓𝑓_𝑝) , (2-8)

The dry rock bulk and shear modulus (𝜅_𝑑𝑟𝑦 and 𝜇_𝑑𝑟𝑦) are a function of the effective stress, 𝜎𝑒𝑓𝑓 (which is linearly dependent on the pore pressure, 𝑃, see also Equation 2-

18), 𝑖 and 𝑝 denote two different pressure states; 𝑆 is the fraction of saturation for each fluid component and the subscripts 𝑤, 𝑜, 𝑔, indicate water, oil and gas, respectively. As pore pressure changes, the resulting increase or decrease of grain to grain contact of the dry rock frame is measured by the moduli. The bulk modulus of the fluid, 𝜅_𝑓𝑙 , measures the effect of pore pressure changes on the fluids, as well as the change in the fluid phase, for example, hydrocarbon has been replaced by water or gas break out has occurred due to pressure drop below bubble point. The saturated bulk density (Equation 2-4) is sensitive to fluid saturation changes, and in non-compacting reservoirs (porosity remains constant with production), pressure-related changes in density are only related to the fluids. This is because pore pressure changes only result in a small reduction in the effective pore volume. MacBeth (2004) shows that changes in the dry

rock density due to pore pressure changes (in other words, effective pressure changes) contribute less than 4%. The remaining 96% are controlled by the elastic moduli changes of the dry rock. Therefore, the assumption of constant density for the dry rock frame in response to pressure changes is practically applicable.

Gassmann’s theory assumes that the pore space is occupied by only a single fluid phase, but in reservoir rocks different fluid phases are present together in the pore space. To use Gassmann’s equations in such practical settings, I use Domenico’s (1974) formula to calculate an effective bulk modulus that represents the fluid mixtures based on harmonic averaging assuming a uniform distribution of fluids:

1 𝜅𝑓𝑙= ( 𝑆𝑤 𝜅𝑤+ 𝑆𝑜 𝜅𝑜+ 𝑆𝑔 𝜅𝑔) , (2-9)

where 𝑆_𝑤, 𝑆_𝑜, and 𝑆_𝑔 are water, oil and gas saturations respectively, and 𝜅_𝑤, 𝜅_𝑜, 𝜅_𝑔 are the water, oil and gas bulk modulus respectively. Figure 2-3 shows a schematic of fluids in the pore space of the rock (in this case, just oil and water). Fluids in the pore space are of two distinct characters: (1) free fluids (free water and oil, in this case)

which are expected to alter the overall seismic properties of the rock during production

Figure 2-3 Schematic representation of a reservoir rock and the different fluid (shown here for an oil-water system) and solid phases, after Alvarez (2014).

Oil Saturation

Capillary bound water

Free water

Clay bound water

Clay minerals (Illite, smectite, etc)

(e.g. a normal water flooding scenario), and (2) capillary bound water and oil, which does not change, except in Enhanced Oil Recovery (EOR) with chemicals etc. The presence of these two distinct characters of fluids, limits the maximum expected saturation change in the rock. For instance, for a water flooding scenario, this yields a practical maximum of (1 − 𝑆𝑤𝑐− 𝑆𝑜𝑟), where 𝑆𝑤𝑐 is the connate water and 𝑆𝑜𝑟 is the

irreducible (or residual) oil saturation.

The acoustic properties (i.e. velocity and density, which make up the bulk moduli) of the reservoir fluids (gas, oil and water) are calculated in-situ using Batzle and Wang (1992) equations, which are the state of the art. These describe the empirical dependence of gas, oil and brine properties on temperature, pressure and composition – (changes in temperature and composition are not modelled in this study). This requires knowledge of oil properties (oil API, solution gas-oil ratio and oil-formation volume factor), gas properties, and gas gravity to characterise the hydrocarbon properties, for a gas-oil-water system. Water properties depend on the sodium chloride (NaCl) salinity content. For other fluid systems, for example, modelling the effect of gas saturation of brine (and how much the presence of gas affects brine properties or how much gas can be dissolved in brine), the applicability is questionable (Avseth et al., 2010). In addition, it is not shown how the empirical formulas can be used to model gas condensate reservoirs, but it is expected that the formulas should extend adequately to both gaseous and liquid phases in a condensate situation.

Table 2-1 Reservoir fluid properties and initial conditions based on a North Sea field to be used in Batzle and Wang (1992) equations for modelling changes in fluid properties for several production scenarios.

These information can be sourced from published empirical relations (Batzle and Wang, 1992), equations of state (fluid flow simulation model), or from pressure-volume and temperature (PVT) measurements taken in the field, which is most common. Table 2-1 details the reservoir properties of the fluids used for the 4D rock physics modelling in this Chapter. Apart from pore pressure, temperature of the reservoir fluids and water

Reservoir temperature Initial Pore pressure Water salinity Oil gravity Gas gravity

Initial Gas Oil ratio

Bubble point pressure

salinity can also change, but for this exercise, I only focus on pore pressure changes. The effects of pore pressure on fluid properties are very subtle, in comparison to fluid saturation-related changes (Figure 2-4).

Figure 2-4 (a) Gassmann’s fluid substitution prediction for a water flood scenario in a 27.1% porosity North Sea sandstone, showing an approximate linear dependence of the P-wave impedance change, ∆𝐼𝑃_𝑠𝑎𝑡, (an increase i.e. hardening response), as the rock becomes more water saturated. (b)

Linear change of P-wave impedance of fluids, ∆𝐼𝑃_{𝑓𝑙𝑢𝑖𝑑}, (at different water saturation levels) with pore

pressure changes, calculated using Batzle and Wang (1992) equations for a North Sea oil above bubble point (properties in Table 2-1).

0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4

Initial state Water flooded

(a) -20 -10 0 10 20 -1 -0.5 0 0.5 1 softe ni ng ha rdeni ng

Pore pressure change,

-20 -10 0 10 20 -1 -0.5 0 0.5 1 Sw = 0.2 Sw = 0.4 Sw = 0.6 Sw = 0.8 Sw = 1.0 (b)

Figure 2-4 (a) is calculated for a water flood scenario (with no pressure change) for a 27.1% porosity North Sea sandstone containing only oil and water, and assumed to be above bubble point pressure (Table 2-1). Other properties in Equation 2-4 such as dry rock and mineral moduli remain unchanged and are given later in Table 2-2. An approximate linear response of the P-wave impedance to water replacing oil is predicted by Gassmann’s Equations (Figure 2-4 (a)). In Figure 2-4 (b) Batzle and Wang calculations show the linear dependence of the P-wave impedance of the fluids to pore pressure changes at different water saturation levels. As oil is more compressible than water, it is also more sensitive to pore pressure changes than water. Fluids containing more water than hydrocarbons are less sensitive to pore pressure changes. An increase in pressure causes an increase in the impedance of the fluids (hardening), whilst a decrease in pressure yields a softening effect. The magnitude of change in impedance in Figure 2-4 (b) occupies only a small fraction (less than 25%) when compared to Figure 2-4 (a).