Dynamics of elastic wave propagation in fluid-saturated porous media is of great interest in diverse areas of science and engineering. The study on this phenomenon is essential in oil explo- ration,seismology,acoustic flaw detection,earthquake engineering,geophysics,and many other subjects. Studies on the mechanism of P-wave dispersion in inhomogeneous fluid-saturated porous rocks can be categorized as macroscopic scale,mesoscopic scale,and microscopic scale. Biot^{[1, 2]} pioneered the study on the macroscopic-scale mechanism by deriving the poroelastic wave equations with a global fluid flow (GFF). Biot’s theory explained the velocity dispersion resulting from the GFF effects. However,unsatisfactory results arised due to the neglect of mi- croscopic structure energy dissipations at a low effective pressure^{[3, 4, 5, 6, 7]}. Early microscopic scale models proposed to perfect the poroelasticity theory include the “squirt flow” model^[8] and the Biot/squirt model^[9]. The fluid flow mechanism at the mesoscopic scale has also attracted the interests of many scholars^{[10, 11, 12, 13, 14, 15, 16, 17, 18, 19]}. In mesoscopic models,the heterogeneity size is larger than the grain size but smaller than the seismic wavelength. Elastic wave dispersion in patchy saturated media is mainly caused by the wave-induced local fluid flow (LFF) occurring at the contact interface of two immiscible fluids^{[16, 20]}.

White was the first to discuss the effects of the mesoscopic loss mechanism for the P- wave velocity^[10]. In White’s model,partially saturated media were considered as a matrix of elementary cubes. Each unit was idealized as a concentric sphere in which the inner pocket was saturated with gas and the intervening volume with another type of liquid. When seismic waves pass through the porous medium,pressure gradients arised in the fluid. A pressure difference induced LFF and led to energy dissipation. P-wave dispersion was mainly caused by the fluid flow occurring between the liquid shell and the gas pocket.

White’s model is successfully applied in analyzing the acoustic response in gas-water-saturated rocks^{[21, 22, 23]}. However,the low-frequency velocities predicted by White’s original formulae are not approaching the Gassmann-Wood velocity at very low frequencies^[24]. Dutta and Od´e^[12] and Dutta and Seriff^[24] reformulated White’s model^{[11, 12]}. They employed fluid-solid coupled dynamic equations to analyze the velocity of seismic waves in porous media. By substitut- ing the bulk modulus K for the plane-wave modulus M in deducing the acoustic impedance, Dutta and Od´e^[12] and Dutta and Seriff^[24] revised White’s model. Other revisions and verifi- cations to White’s model have also been developed by Gist^[6],Murphy^{[21, 22]},and Cadoret et al^[25]. Johnson^[13] further extended the applicability of White’s model to arbitrary mesoscopic geometry.

The effect of fluid in the central spherical region is important in predicting the P-wave velocity in real reservoir rocks (such as the porous medium under the liquid pocket patchy sat- uration). Predicted elastic wave velocities are significantly smaller than the experimental data in the case of oil-water mixture (see the example section) if the properties (such as the density and bulk modulus) of the inner liquid are neglected. White developed the general framework for the patchy model containing a central spherical gas region^[10]. Detailed formulations have been given by White based on a model containing inner gas pocket. In the section “Partial gas saturation” in White’s work,the algebraic terms R₁,Q₁,and Z₁ in deriving the pressure change ΔP₀ were all neglected based on the assumption that the central sphere was saturated with a light and compressible gas^[10]. Thus,the gas in the central spherical region was ap- proximated by a vacuum^{[10, 24]}. To the best of our knowledge,nearly all literatures of White’s model only discuss the gas-pocket cases as White’s original work. A few works on the liquid- pocket saturation are based on Biot’s theory or the Gassmann modulus,rather than White’s framework.

The aim of this article is to incorporate the bulk modulus and density of the inner liquid- pocket into a heterogeneously saturated porous model and improve the prediction of P-wave velocity for dual-fluid saturated porous medium inWhite’s framework. In the following sections, we briefly review White’s model and derive Poisson’s ratio,the bulk modulus,and the density for the dual-fluid saturated medium. Some amendments are completed for the immiscible fluid patchy saturation model. The competitive improvements of this work are illustrated by comparing the results both from the experimental data and from the predictions of other models, e.g.,Johnson’s model.

Following White’s^[10] and Dutta’s model^{[11, 12]},we postulate that a porous rock consists of elementary cubic units,in which the inner spherical pocket is saturated with one fluid and the outside volume is saturated with another fluid (see Fig. 1(a)). The porous frame’s petrophysical properties are uniform. To avoid the complexity caused by combining a cube and a sphere,the typical cubic volume is considered equivalent to the concentric spheres^[10]. The outer layer volume is equal to that of the original cube (see Fig. 1(b)).

Fig. 1 Concentric spheres saturated with different fluids in patchy saturation model: (a) gassaturated pockets and surrounding liquid-saturated cubes; (b) outer layer volume is equal to that of cube

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A key quantity in patchy model is the representative elementary volume (REV) size. We assume that the size of the considered patches is larger than the grain size but smaller than the wavelength. The compressional wave propagation is discussed at the mesoscopic scale. The whole volume of the porous medium consists of uniformly repeated REVs in space. REV provides a unit to describe the pressure equilibration between the inner and outside fluid- saturated regions of the pore space. The size of REV represents the distance over which the pore pressure must equilibrate to compress the central fluid pocket. In White’s model,the REV length scale can be expressed in terms of the central pocket radius and the inner fluid saturation^{[6, 10]}.

In White’s original work,the gas pocket is the inner part of the concentric spheres. The outer part is saturated with liquid. At the outer boundary,volume variation is impressed at a low frequency. At the surface,the consequential pressure amplitude is derived. In the case of static stress without fluid flow,the bulk modulus of the composite media is K₀. When pressure is imposed on the outside of the concentric spheres,the bulk modulus is the ratio of the complex pressure amplitude −P₀ to the dilatation amplitude D₀. The relation between the radially symmetric stress and the displacement provides formulations of P₀ and D₀ in the forms of Young’s modulus E and Poisson’s ratio σ^[26]. We obtain

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where S₁ is the inner gas saturation and can be defined by S₁ = a³/b³. Here,a and b are the inner and outer radii of the concentric spheres,respectively. The bulk modulus K₀ is a function of Young’s modulus E and Poisson’s ratio σ. Subscripts 1 and 2 represent the inner and outside zones,respectively. Poisson’s ratio and Young’s modulus are given by^{[27, 28]}

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and

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Here,Ks is the mineral grain bulk modulus,and φ is the porosity. K_n and μ_n (_n = 1,2) are the bulk and shear moduli of the porous rock saturated with any type of inner fluid,respectively. is the bulk modulus of the dry frame. The subscripts f_n (n = 1,2) represent the inner and outside fluids.

In White’s^[10] derivations,the central sphere is so assumed to be saturated with gas. The inner gas is so light and compressible that the gas-related parameters (e.g.,bulk modulus and density) can be ignored. Thus,dry skeleton parameters are used in calculating Poisson’s ratio of the inner sphere. Poisson’s ratio σ₁ of the inner sphere (containing gas) and Young’s modulus E₁ in White’s model are^[10] given as follows:

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Here,the plane-wave modulus and the shear modulus are for the dry skeleton.

Compared with (4),the important new quantities in (2),i.e.,dK =in the denominator,contains the effects of the properties of the inner fluid.

In White’s inner gas/outer water model,the dry rock moduli and are used instead of K₁ and μ₁. The gas is so light that the frame saturated with inner gas is considered a dry frame. In White’s subsequent derivations and numerical calculations,all the parameters are based on the dry inner pocket assumption. The gas-pocket assumption will underestimate the wave velocity in the liquid-pocket case (e.g.,oil-water saturation). Therefore,we propose to use (2)-(3) instead of (4)-(5) to obtain related parameters.

The generalized density is now given as^[29]

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where ρ^* is the effective density of the entire patch; ρ_s,ρ_f1 ,and ρ_f2 are the densities of the solid mineral grain,the inner fluid,and the outer fluid,respectively; S₁ is the pore-space saturation of the inner fluid. Compared with the density obtained by White (12 in White’s work^[10]),

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we find that the term caused by the inner fluid density ρ_f1 is missing. Only if the inner pocket is saturated by gas,the density can be regarded as zero. In such a case,the equations in this article will reduce to the original White’s model.

Furthermore,when the effect of the fluid flow is considered,the bulk modulus should be a complex function of the wave frequency. If the effects of the fluid flow are considered in the concentric shell,the gas pocket will supply a release of pressure. The bulk modulus can be derived by using the ratio between the pressure amplitude and the fractional volume change^[10],

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where depends on the radii a and b of the gas pocket and the water sphere as well as other factors such as the acoustic impedance Z₁ or Z₂ of the diffusion waves,the frequency !,and the porosity and bulk moduli of the contents. K0 is the mean bulk modulus for the concentric layers without fluid flow. Here, The coefficients (R₁,R₂,Q₁,Q₂,Z₁,Z₂) have been given in Ref. [10]. The related formulas are given in Appendix A. With the shear modulus μ^* = (fluid content does not affect the shear modulus ),the composite density ρ*, and the complex bulk modulus K^*,the P-wave velocity v^* _p is derived as^[10]

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In White’s assumption,the medium is isotropic,and the fluid content does not affect rigidity. Therefore,the shear modulus of a rock with fluid is the same as that of a dry rock.

In summary,for the prediction of P-wave velocity in a dual-fluid saturated porous rock,the basic steps are as follows:

(i) Use (2)-(3) instead of (4)-(5) to obtain K₀.

(ii) Derive the complex bulk modulus by using the ratio between pressure amplitude and the fractional volume change

(iii) Replace the density obtained by White with the generalized density expression.

(iv) Obtain P-wave velocity by new complex bulk modulus and density.

The formulations in this article replace the drained bulk modulus of the inner sphere by the undrained bulk modulus. In addition,the drained modulus can be expressed in terms of the porosity through an empirical relation. Although the empirical formation may be valid over a limited range of porosity,we can observe that the effect of liquid in the central sphere is dramatized in following figures.

Considering Pride’s dry-frame moduli for consolidated sandstones^{[14, 15, 16]},

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we obtain

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Here,α is a consolidation parameter that represents the degree of consolidation between grains, and α lies in the approximate range 2 < α < 20 for sandstones. When the inner fluid and skeleton grain moduli are determined,d depends only on φ and α. Figure 2 shows the inner- fluid-induced effect on bulk modulus. The quartz bulk modulus K_s = 36.6GPa and the oil modulus K_f = 1.23GPa are used in the calculation of dK.

The dK/Ks surface indicates that for predetermined porosity,the inner-fluid-induced effects on the bulk modulus decrease with the reduced consolidation factor (see Fig. 2(a)). When α and φ approach zero,the effect of the inner fluid can be neglected. Therefore,no difference exists between the liquid-pocket model and the original White’s model. For predetermined α (2 < α < 20),dK/K_s will first increase and then decrease with φ (see Fig. 2(b)).

Fig. 2 Dependence of inner-fluid-induced effect on porosity and consolidation factor: (a) bulk modulus correction of Poisson’s ratio versus porosity and consolidation factor; (b) bulk modulus correction variations with respect to porosity for different consolidation factors

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The improved formulations degenerate to White’s inner gas model when patchy saturation contains inner gas-pocket. If the properties of the inner fluid,e.g.,bulk modulus and density, cannot be ignored,the gas-pockets assumption will bring errors. To illustrate the effects of the inner fluid,two examples are shown in the following sections. As the first example,the experimentally measured P-wave velocities of the sandstone samples are shown,describing the drainage and imbibition process. Then,in the second example,we will explain how the P- wave velocity of heavy oil sands at differenct temperatures can be predicted from the improved formuations of White’s framework presented in this article.

The proposed formulations in this article are used to predict the compressional wave velocity in the sandstone saturated with the inner oil-pocket and outer water background. Numerical predictions are compared with the experimental measurements of the French Vosgian sandstone samples from Bacri^[30]. The newly developed Young’s modulus E and Poisson’s ratio σ are adopted in the velocity calculation as well as other inner liquid properties,such as bulk modulus and density.

The mean porosity of the French Vosgian sandstone is 21%. Two different saturation meth- ods (drainage and imbibition) are used^[30]. In the case of drainage,oil is injected into the sandstone which is fully saturated with water. The oil content will increase until it reaches 33% water saturation (an irreducible water saturation). In the case of imbibition,water is injected into the fully oil-saturated sandstone until a residual oil saturation of 35%. The velocities are measured at 350 kHz,showing the reported parameters by Bacri and Salin^[30] (see Table 1).

Table 1 French Vosgian sandstone properties )

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In the drainage experiment,the velocities increase consistently within the Biot-Gassmann- Wood (BGW) and Biot-Gassmann-Hill (BGH) limits (see Fig. 3). The predicted velocities are in good agreement with the measured data with a patch size R = 1.0 cm. The imbibition and drainage values show different behaviors at low and high water saturations. The P-wave velocity decreases greatly in the drainage case from a value about 2 750m/s at full oil saturation to 2 520m/s at 65% oil saturation. While the velocity decreases in the imbibition case from a value about 2 650m/s at full water saturation to about 2 480m/s at 33% water saturation.

Fig. 3 Compressional wave velocities in French Vosgian sandstone saturated with oil and water. Exp (Imb) and Exp (Drn) represent imbibition and drainage experiments, and LP represents liquid-pocket model

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Both the grain contact and surface energy effect have been considered by Bacri and Salin^[30]. The surface energy decreases heavily when the sample is immersed in water. Hence,the frame moduli are softened,and the P-wave velocity decreases. On the contrary,the surface energy is slightly affected when the sample is immersed in oil. It has been observed that the wave velocity at full oil saturation is larger than that of the dry skeleton. The surface effect causes discrepancies between theoretical predictions and observed data in imbibition. In order to improve the velocity models,we need a detailed study of the relation between the pore fluid effect on grain contacts and the frame moduli.

Figure 3 shows a significant difference between the light-fluid-pocket model (White) and the liquid-pocket model (LP). We find that the results of the improved formulations (LP) are close to the laboratory measurements,whereas the light-fluid-pocket model underestimates P-wave velocity,especially at low water saturation. The errors come from the dry frame assumption for the inner sphere,which counts the bulk modulus and density of the inner liquid as zero. By con- trast,the predicted velocity-saturation curves are improved by the newly derived formulations. The results predicted by improved formulations are obvious closer to the reality than those by the neglectable gas-pocket model. From the comparison,we observe that the more the volume of air is replaced by oil,the more the difference between the gas-pocket model and experiments is. The influence is so large that it cannot be neglected for liquid-pocket saturations.

In many sample contained fluids,the real temperature and pressure will affect the properties obviously. To take into consideration the influence of such effects,we choose the experimental data containing temperature variations as the second example. In this section,the parameters change with temperature and pressure. The related formulations are listed in Appendixes A and B. The comparisons show that the improvement to the liquid-pocket model is necessary.

In this section,the improved formulations are used to predict the P-wave velocity of heavy oil sands with the temperature increasing from 0◦C to 100◦C. The samples are loose sands held together by heavy oil at the depths less than 500 m^[31]. The highly oil-saturated sandstone samples are modeled as an inner oil/outer water patch model. The numerical predictions can be compared with the experimental measurements from the work of Han et al.^[31].

The relationship between the dry rock bulk modulus and the temperature has been studied by Carmichael^[32]. Given that the pore volume changes,the dry sandstone bulk modulus changes with the effective temperature. A linear function has been proposed based on the experimental results^[33],

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where Kd is the bulk modulus,and T denotes the temperature. The temperature range is from 10◦C to 200◦C. With this relationship,we can estimate the bulk modulus of dry sandstone when the temperature changes. Batzle and Wang studied the properties of pore fluids under the influence of composition,pressure,and temperature^[34]. Simplified expressions have been developed to facilitate the use of realistic fluid properties in rock models,such as densities,bulk moduli,and viscosity. The most important formulas are given in Appendix B. In this sample, we refer to the fluid properties of oil,brine,and water.

First,brine is characterized by salinity Sbrine. In particular,pore fluid becomes water by setting the salinity to zero. The density,viscosity,and bulk modulus of brine depend on Sbrine, temperature,and pressure. A pressure of 5.6MPa is used in the experimental measurements^{[23, 31]}. Considering the influence of temperature,the P-wave velocities of heavy oil-saturated sand- stones are predicted by White’s model,the liquid-pocket model,and Johnson’s model.

Temperature and American Petroleum Institute (API) gravity will affect the properties of oil significantly. Oils can be classified into three categories according to the API gravity,i.e., light oil,medium oil,and heavy oil. In Han’s work^{[23, 31]},heavy oil (G_API = 9.2) is chosen as the experimental subject. To observe the temperature variation effects,we choose one West Texas intermediate oil (G_API = 20) and one light oil (G_API = 34) to compare the velocity with that of the heavy oil (G_API = 9.2) when the temperature changes from 0◦C to 100◦C at 103 kHz.

Figure 4 shows that the bulk modulus and viscosity of oil will decrease as temperature increases. By incorporating the dependence of fluid and dry rock properties on temperature, White’s model is capable of including the temperature effect on the P-wave velocity.

Fig. 4 Bulk modulus and viscosity of different oils: light oil (GAPI = 34, dotted line), West Texas intermediate oil (GAPI = 20, mesh line), and heavy oil (GAPI = 9.2, solid lines)

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In Han’s work^{[23, 31]},the “as is” water-saturated heavy oil sands need a small amount of water because the samples are highly oil saturated. The samples considered are oil-water-saturated sandstones. In this example,two sandstone samples (labeled as I and F) are used to verify the behavior of the liquid-pocket model. The rock parameters (see Table 2) follow those in Table 1 in Han et al.^[31]. An in situ differential pressure of 5.6 MPa is adopted in the velocity prediction. Given that less than 2% porosity error and less than 2% bulk/grain density error are present,we can make minor adjustments (less than 2%) to the parameters.

Table 2 Absolute errors, CPU elapsed time, and number of function )

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Here,Sbrine is the salinity of brine. The P-wave velocities are predicted by the liquid- pocket model,original White’s model,and Johnson’s model. The numerical calculations are compared with the laboratory measurements. The sandstones are saturated with oil and water at temperatures ranging from 0◦C to 100◦C.

Figure 5 and Figure 6 show the P-wave velocities in sandstones F and I with inner-oil and outer-water patchy saturations at different temperatures. Figures 7 and 8 provide the data for inner water/outer oil saturations. By comparing Figs. 5-8,we find some interesting observations.

Fig. 5 Predicted and experimental measured P-wave velocities (Exp) versus temperature. S-OW-F indicates inner-oil and outer-water patchy saturation in sandstone F; LP is liquidpocket model; Oil saturation is 0.858

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Fig. 6 Predicted and experimental measured Pwave velocities (Exp) versus temperature. S-OW-I indicates inner-oil and outer-water patchy saturation in sandstone I; LP is liquid-pocket model; Oil saturation is 0.801

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First,White’s light-fluid-pocket assumption underestimates the P-wave velocities in all cases. We find that the predicted velocities by the original gas saturation model are lower than those of the liquid-pocket model,Johnson’s model,and the experimental measurements,especially in the case of inner oil/outer water and low-temperature environment. The data of the liquid model are close to the laboratory measurements. The velocity deviations of White’s light-fluid- pocket model increase as temperature decreases. The velocity deviation of light-fluid-pocket model comes from the neglect of inner fluid. At a high temperature,heavy oil becomes lighter and has more mobility. The effects of inner oil decrease in importance in the patchy model. As temperature decreases,heavy oil comes into solid form. The effects of inner oil become increasingly important and cannot be ignored. From Figs. 5-8,we can find that the difference between the light-fluid-pocket and liquid-pocket model becomes more significant as temperature decreases. This finding implies that the inner fluid neglected inWhite’s light-fluid-pocket model plays an important role in the P-wave velocity. These results confirm the necessity of the improvement of the formulations,especially for heavy oil at low temperatures.

Second,the velocity differences between original and generalized models are large in inner oil/outer water saturations for both F and I sandstones. Given that the only difference in Fig. 5 and Fig. 7 is the fluid distribution in the inner or outer parts,the prediction by the original White’s model heavily depends on the inner fluid parts. The unexpected velocity deviations in the light-fluid-pocket model are inevitable because the properties of inner fluid,such as the bulk modulus,were ignored in the theoretical derivations. Therefore,the light-fluid-pocket model is invalid unless the influence of inner fluid is negligible. The volume fraction of water is less than that of oil. Thus,when we deal with the inner water and outer oil cases,the errors caused by the influence of inner water are significantly reduced. The same situation can be found in Fig. 6 and Fig. 8.

Fig. 7 Predicted and measured P-wave velocities (Exp) versus temperature. S-WO-F indicates inner water and outer-oil patchy saturation in sandstone F; LP is liquid-pocket model; Oil saturation is 0.858

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Fig. 8 Predicted and measured P-wave velocities (Exp) versus temperature. S-WOI indicates inner water and outer-oil patchy saturation in sandstone I; LP is liquid-pocket model; Oil saturation is 0.801

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Third,the gaps between all patchy models and experimental data are significant at the temperatures near 100◦C. Heavy oil is in a quasi-solid phase at institute conditions. However, as temperature increases over 70◦C,heavy oil will transit into a liquid phase. The oil will be different from what we discuss at general research situations. Therefore,great phase variations lead to deviations between the theoretically predicted velocities and the experimental data.

We also predict the P-wave velocities for all other sandstone samples in Han’s work^[31]. The same phenomena can be observed as discussed. The difference between the results of other sandstones and the sandstone F/I is that the theoretical predictions are not as good as those in the sandstone F/I. The P-wave velocities predicted by the improved formulation in White’s framework and Johnson’s model are lower than the measured data. However,the velocity deviations of the original have the same behavior as that in the sandstone F/I.

Poisson’s ratio,bulk modulus,and effective density are reformulated inWhite’s framework to predict the P-wave velocity in a porous rock saturated with two immiscible liquids. The derived expressions generalize the established formulations to the liquid pocket patchy model,such as inner oil/outer water saturation. The inner liquid pocket induces bulk modulus corrections to Poisson’s ratio. The correction terms depend on porosity and Pride’s consolidation factor. As the porosity and the consolidation factor approach zero,the difference between light-fluid-pocket and liquid-pocket models disappears. The theoretical analysis shows that for formations with a large consolidation factor,the light-fluid-pocket model underestimates the P-wave velocity. The new formulations are convenient for the P-wave velocity prediction because the same expression can be used for any single or two immiscible liquid-saturated rocks. The numerical examples for the French Vosgian sandstone and heavy oil sandstones confirm the necessity of the model generalization.

Relevant equations (White’s model)

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where K_f1 and K_f2 are the fluid bulk moduli,φ is the porosity,η and κ are the viscosity and permeability, K_s is the solid bulk modulus,and subscripts 1 and 2 refer to the inner sphere and concentric shell,respectively. The gas saturation is S_G = a³/b³,where a and b are radii of the inner sphere and concentric shell. The parameters K₁,K₂,K_A1 ,and K_A2 are defined by

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where E and σ are Young’s modulus and Poisson’s ratio,respectively,and is the bulk modulus of the skeleton.

Fluid properties as functions of temperature

Some important formulas for the viscosity,the density,and the bulk modulus of fluid are listed as functions of temperature^[35] .

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where S is the salinity of the brine,and ρ_w and ρ_B are the densities of water and brine which change with temperature and pressure,respectively.

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where K_brine and η_brine are the bulk modulu and viscosity of brine which change with temperature and pressure,respectively. The equations are applied to water by setting salinity as zero.

Oil can be classified by API,which is defined as

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where ρ₀ is density of oil at 15.6◦C and atmospheric. Similarly to brine,the relationships between properties of oil and temperature can be expressed

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where ρ_oil,Koil,and η_oil are the density,bulk modulu,and viscosity of oil which change with temperature and pressure.