The Stanford VI reservoir

The Stanford VI reservoir Scarlet A. Castro, Jef Caers and Tapan Mukerji Stanford Center for Reservoir Forecasting Stanford Rock Physics and Borehole Geophysics Project May 2005 Abstract An exhaustive data set is generated for the general purpose of testing any proposed algorithm for reservoir modeling, reservoir characterization and production forecasting. Following the original data set generated by Mao and Journel (1999), the Stanford V reservoir, several extensions are proposed and incorporated into a new reservoir model which exhibits a smoother structure, more realistic dimensions for current-day models and improved Rock Physics models. Additionally, new seismic attributes as well as 4D seismic data have been generated. The Stanford VI reservoir is a three-layer prograding fluvial channel system, and its structure consists of an asymmetric anticline with axis N15 E. The new reservoir model provides an exhaustive (6 million cells) sampling of petrophysical properties and seismic attributes, as well as a 4D Seismic data set consisting of a base seismic survey acquired prior to oil production and three time lapse seismic surveys acquired 10, 25 and 30 years after oil production. Therefore, reservoir simulation results are also part of this data set. Due to a real need for testing upscaling, the new Stanford VI reservoir exhibits realistic exhaustive sampling of petrophysical properties and promises to represent a good data set for both upscaling and downscaling methods. This paper describes the workflow used for the generation of the Stanford VI reservoir, as well as a detailed description of each step. 1

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1 Introduction With the purpose of extensively testing algorithms for reservoir modeling or reservoir characterization, Mao and Journel (1999) created an exhaustive 3D reference data set called Stanford V. Although it has been widely used, this data set is too small to represent current-day reservoir modeling exercises. Several extensions are proposed in this report and incorporated into a new reference reservoir model, termed Stanford VI. The proposed reference data set (Stanford VI) exhibits a smooth top and bottom surface representing a trap in the form of an anticline. It provides an exhaustive sampling of petrophysical properties. The new reservoir model is represented in a 3D regular grid of 6million cells (150 200 200), with more realistic dimensions for current-day models (25 m in the x and y directions and 1 m in the z direction). Following Stanford V, the stratigraphic model corresponds to a fluvial channel system, and the petrophysical properties computed for this reference reservoir correspond to the classical porosity, density, permeability and seismic P-wave and S-wave velocities. Although most of these properties are calculated following the standard procedure and algorithms presented by Mao and Journel, a more appropriate Rock Physics model is used in the new reference reservoir model to compute P-wave velocity for sandstones. Traditionally, P-wave velocity is calculated from empirical expressions obtained from laboratory data as a function of porosity (Wyllie et al., 1956; Raymer, et al.,1980; Han, 1986; Tosaya and Nur, 1982; etc.). The former Stanford V reservoir uses one of these expressions (Han, 1986) to obtain P- wave velocities from porosity. Strictly speaking, Han s relations are obtained from sandstone samples collected from different depths, where porosity is controlled by diagenesis and cementing. In our case porosity is controlled by sorting and clay content, henceforth, a more appropriate rock physics model is used, namely, the constant cement model described by Dvorkin and Nur (1996). Besides petrophysical properties, the new reference reservoir model exhibits a complete set of physical seismic attributes, which are computed from well-known mathematical expressions and subsequently filtered and smoothed to obtain realistically looking as would have been obtained from actual seismic acquisition and modeling. These realistic seismic attributes 3

provide a filtered view of the true spatial variation of petrophysical properties. The current practice of modeling petrophysical properties from seismic data, many so-called seismic attributes are used. However, only those attributes computed from seismic reflection amplitude carry information about the elastic contrast in the subsurface. Seismic inversion attempts to translate this information into elastic properties, which are function of density, P-wave and S-wave velocities. From elasticity theory we know that these elastic properties are function of density, P-wave and S-wave velocities. Since Stanford VI is a synthetic data set, we use the petrophysical properties created before to compute a typical set of physical seismic attributes that could be obtained from seismic inversion in a real situation, although we do not perform any explicit inversion. From this new reference reservoir model a 4D seismic data is generated. 4D seismic data is nothing more than three-dimensional (3D) seismic data acquired at different times over the same area. 4D seismic is used to assess changes in a producing hydrocarbon reservoir with time; changes may be observed in fluid location, saturation, pressure, and temperature. Consequently, one of the main applications of 4D seismic data is to monitor fluid flow in the reservoir. In order to create a 4D seismic response, several 3D seismic data sets are forward modelled using a simple convolutional model. The first seismic data set is created using the acoustic impedance of the reservoir prior to production, while three more seismic data sets are created using the acoustic impedance of the reservoir at different times during oil production. The acoustic impedance of the reservoir after a certain time t can be computed by using a fluid substitution approach. This process, introduced by Gassmann (1951) allows to calculate the elastic rock properties as one fluid displaces another in the pore space. Prior to production the reservoir is filled with oil, while some years after production starts the reservoir contains a mixture of fluids, typically water oil and gas. In order to use Gassmann s equations correctly we need the saturations of each fluid at every point in space, therefore we use a flow simulator to obtain them at any point in time during production. 4

To create the 3D seismic data sets at different times during oil production, we performed a flow simulation where water and oil are the only fluids present in the reservoir. We simulate 30 years of oil production with an active aquifer below the reservoir and water injector wells that become active after the aquifer water influx fails to maintain the pressure. This paper presents a detailed description of the workflow and processes followed to create the new reference reservoir model (Stanford VI), as well as the results of each step. 5

2 Workflow for building the Stanford VI reservoir The workflow used to create the Stanford VI reference data set is shown in Figure 1. Although in general this workflow is similar to the one presented by Mao and Journel (1999) for the generation of the Stanford V data set, several steps have been added. Besides acoustic impedance, which is the only seismic attribute computed for Stanford V, the new Stanford VI data set has seven more seismic attributes that are good indicators of both lithology and fluid type. The large dimensions of the Stanford VI data set, allows for the study of upscaling and downscaling of petrophysical properties. As Figure 1 shows, the first step in the generation of Stanford VI property model corresponds to the modeling of facies. The Stanford VI facies model correspond to a prograding fluvial channel system and is modelled using the commercial software SBED and the multiple-point simulation algorithm snesim. Subsequently, the facies model is populated with five petrophysical properties: porosity, density, P-wave velocity, S-wave velocity and permeability. Basically, porosity is simulated first using the sequential simulation algorithm sgsim; density, P-wave and S-wave velocities are obtained from porosity using well known Rock Physics models; and permeability is co-simulated conditional to the co-located previously simulated porosity using the algorithm cosgsim. Having the petrophysical properties, three basic steps are followed: flow simulation, forward model of 4D seismic data and generation of seismic attributes. The following sections of this report explain the details of each step depicted by the workflow presented in Figure 1. 6

Figure 1: Workflow followed to create the Stanford VI data set. 7

3 Structure and Stratigraphy The structure of the Stanford VI reservoir corresponds to a classical structural oil trap, an anticline. Specifically, it is an asymmetric anticline with axis N15 E. As Figure 2 shows, the anticline has a different dip on each flank and generally the dip decreases slowly towards the northern part of the structure. The maximum dip of the structure is 8. Figure 2: Perspective view of the Stanford VI top structure: view from SW (left), view from SE (right). The color indicates the depth to the top. The reservoir is 3.75 Km wide (East-West) and 5.0 Km long (North- South), with a shallowest top depth of 2.5 Km and deepest top depth of 2.7 Km. The reservoir is 200 m thick and consists of three layers with thicknesses of 80 m, 40 m and 80 m respectively (see Figure 3). In terms of grid, the Stanford VI reservoir is represented in a 3D regular stratigraphic grid of 150 200 200 cells and the dimensions of the grid correspond to 25 m in the x and y directions and 1 m in the z direction. The coordinate system used correspond to the GSLIB standard of the stratigraphic coordinate system, where the z coordinate is measured relative to the top of the reservoir. Due to the simple structure and stratigraphic grid, an accompanying cartesian box in which all of the geostatistical modeling takes place, can easily be constructed. The stratigraphy of the Stanford VI reservoir corresponds to a prograding fluvial channel system, where deltaic deposits represented in layer 3 were deposited first and followed by meandering channels in layer 2 and sinuous 8

Figure 3: Perspective view of the Stanford VI top and bottom of each of its layers. The color indicates the depth to the top. channels in layer 1. This sequence of clastic deposits represents a progradation of a fluvial system into the basin located in this case toward the north of the reservoir. In order to model the stratigraphy of Stanford VI, we use the commercial software SBED to model layer 1 and layer 2, while layer 3 is modelled using the multiple-point simulation algorithm snesim with local rotation and affinity variation to model the channel meanders. The first layer of Stanford VI consists of a system of sinuous channels represented by four facies: the floodplain (shale deposits), the point bar (sand deposits that occur along the convex inner edges of the meanders of channels), the channel (sand deposits), and the boundary (shale deposits). The stratigraphic characteristics of layer 1 are detailed in the following table and Figure 4 shows the resulting stratigraphic model for this layer. floodplain proportion 0.68 point bar proportion 0.11 channel proportion 0.165 boundary proportion 0.045 Number of channels 8 Average channel thickness 20 meters Average channel width 600 meters Average boundary thickness 1.5 meters Average point bar width 300 meters 9

Figure 4: Facies model of Layer 1, which corresponds to sinuous channels: floodplain (navy blue), point bar (light blue), channel (yellow), and boundary (red). Stratigraphic grid (left), and cartesian box (right). The second layer consists of meandering channels also represented by four facies: the floodplain (shale deposits), the point bar (sand deposits that occur along the convex inner edges of the meanders of channels), the channel (sand deposits), and the boundary (shale deposits). The stratigraphic characteristics of layer 2 are detailed in the following table and Figure 5 shows the resulting stratigraphic model for this layer. floodplain proportion 0.68 point bar proportion 0.14 channel proportion 0.11 boundary proportion 0.07 Number of channels 4 Average channel thickness 300 meters Average channel width 16 meters Average boundary thickness 1.5 meters The last and third layer of the reservoir consists of deltaic deposits and are represented by two facies: the floodplain (shale deposits), and the channel (sand deposits). The stratigraphic characteristics of layer 3 are detailed in the following table. floodplain proportion 0.56 channel proportion 0.44 channel thicknesses [7 40] meters channel widths [70 400] meters 10

Figure 5: Facies model of Layer 2, which corresponds to meandering channels: floodplain (navy blue), point bar (light blue), channel (yellow), and boundary (red). Stratigraphic grid (left), and cartesian box (right). The third layer of Stanford VI is modelled using the multiple-point simulation algorithm snesim with local rotation and affinity variation of the channel meanders. Traditionally, geostatistical techniques capture geological continuity through a variogram. A variogram is a two-point statistical function that describes the level of correlation, or continuity, between any two sample values as separation between them increases. Since the variogram describes the level of correlation between two locations only, it is not able to model continuous and sinuous patterns such like channels or fractures. For modeling such geological features a multiple-point approach should be used, where spatial patterns are inferred using many spatial locations (Strebelle, 2002). In multiple-point geostatistics, the spatial patterns are inferred from a training image which represents a conceptual reservoir analog with the expected geological heterogeneity. Since it is a conceptual model, the training image is not constrained to any data. The geostatistical technique that uses a training image to create realizations constrained to reservoir data is proposed by Strebelle (2002). The single normal equation simulation (snesim) algorithm is a conditional sequential simulation where the probability distribution is retrieved from the training image and made conditional to a multiple-point data event. The snesim algorithm allows for local rotation and affinity (aspect ratio) variation of the data event, in doing so we are able to create non-stationary 11

realizations from a stationary training image. Figure 6 shows the resulting stratigraphic model for layer 3 as well as the rotation, and affinity cubes used, the training image is shown in Figure 7. The rotation and affinity cubes are categorical variables and the values assigned to these categories are shown in the table below. Angle category angle (degree) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9-63, -49, -35, -21, -7, 7, 21, 35, 49, 63 Affinity category affinity [x,y,z] 0, 1, 2 [0.5, 0.5, 0.5], [1, 1, 1], [2, 2, 2] 12

Figure 6: Facies model of Layer 3 (top), which corresponds to deltaic deposits: floodplain (navy blue), and channel (yellow). Stratigraphic grid (left), cartesian box (right), angle cube (middle), and affinity cube (bottom). 13

Figure 7: Training Image used for modeling Layer 3. The size of the training image is 200 200 5, each slice in the z direction is shown here from top to bottom. 14

4 Petrophysical Properties Having created the facies model for the three layers of the Stanford VI reservoir, we populate it with the following petrophysical properties: Porosity. Permeability. Density. P-wave velocity S-wave velocity. 4.1 Simulation of Porosity Porosity is simulated first using the sequential simulation algorithm sgsim from GSLIB, conditioned to a reference target distribution and variogram, and independently for each facies in the reservoir. The reference target distribution of porosity in each facies is shown in Figures 8 and 9. Shale deposits in floodplain and boundary facies have distinctively low porosity values while sand deposits in channel and point bar facies have high porosity values as expected. The variance for point bar facies is smaller than for channel facies and their mean is higher since they are typically well sorted sand deposits. Similarly, boundary facies exhibits very low mean and variance which is translated later into a flow barrier for fluids. Figure 8: Distribution of porosity for each facies in the reservoir. 15

Figure 9: Histogram of porosity for each facies in the reservoir. The reference variogram consisting of a single structure for each facies are specified in the following table, where ranges for x, y and z direction are given in meters. floodplain point bar channel boundary type Spherical Spherical Spherical Spherical nugget 0.1 0.1 0.1 0.1 ranges 1750/1750/70 5000/2500/10 3000/1750/10 500/500/20 angles 0/0/0 90/0/0 90/0/0 0/0/0 Having sequentially simulated porosity independently for each facies, we use a cookie-cutter approach to create the resulting porosity model for the Stanford VI reservoir (see Figure 10). 16

Figure 10: Resulting Porosity cube after cookie-cut porosity from each facies porosity realization. 4.2 Simulation of Permeability Typically, the logarithm of permeability is approximately linearly correlated with porosity, therefore we have simulated the logarithm of permeability using a linear correlation coefficient of 0.7 between both variables. Permeability for each facies is co-simulated conditional to the simulated porosity, using a Markov1-type model instead of a full model of coregionalization. The algorithm used is cosgsim implemented in S-Gems. Permeability is also simulated independently within each facies. The cookie-cutter approach is used to merge the permeability simulated for each facies into a single permeability model. The implicit assumption of the algorithm used is that both primary and secondary variables are normally distributed, and the bivariate relationship follows a (bi)gaussian distribution. Both variables are transformed to the normal space and the bivariate relationship is assumed to follow a (bi)gaussian distribution. In order to transform the original variables into Gaussian variables, we provide to the algorithm the original distributions of both porosity and logarithm of permeability. The distribution of porosity was shown in the last section. The distribution of the logarithm of permeability is obtained (see Figure 11) by transforming the distribution of porosity using the well known 17

Kozeny-Carman s relation (Carman, 1961). κ = 1 φ 3 72τ (1 φ) 2 d2 (1) where: φ is porosity (fraction), τ is tortuosity (assumed as τ = 2.5), and d is the pore diameter (in micrometers). Figure 11: Histogram of the logarithm of permeability for each facies in the reservoir. The κ-variogram used for each facies is shown in the following table, where ranges for x, y and z direction are given in meters. 18

floodplain point bar channel boundary type Spherical Spherical Spherical Spherical nugget 0.1 0.1 0.1 0.1 ranges 1750/1750/70 5000/2500/10 3000/1750/10 500/500/20 angles 0/0/0 90/0/0 90/0/0 0/0/0 The resulting permeability model for the Stanford VI reservoir is shown in Figure 12. Figure 12: Resulting Permeability cube after cookie-cutting permeability from each facies permeability realization. 4.3 Density The rock density is calculated using porosity and the mixing formula: ρ = φρ fluid + (1 φ)ρ matrix (2) where ρ fluid is the density of the fluid that fills in the pore space, and ρ matrix is the density of the rock matrix. Therefore the rock density is calculated as: N ρ = φρ fluid + (1 φ) f i ρ mi (3) i=1 19

where f i as the fraction of the mineral m i with density ρ mi which constitutes part of the rock matrix mineralogy. The rock mineralogy for each facies is showed in the following table: mineral mineral density floodplain point bar channel boundary (g/cc) clay 2.4 0.7 0.0 0.0 0.8 quartz 2.65 0.2 0.70 0.65 0.15 feldspar 2.63 0.1 0.2 0.2 0.05 rock fragments 2.7 0.0 0.1 0.15 0.0 Using the mineralogy and the mineral densities shown in the table we compute the rock matrix density ρ matrix. From equation 3 we obtain the rock bulk density ρ, using the simulated porosity and water as the saturating fluid. Typically, density, P-wave and S-wave velocities are calculated first for water saturated rocks since the rock physics models used for computing velocities have been obtained in the lab from water-saturated rocks. Therefore, in order to use these models correctly, the saturating fluid must be water. For the generation of this data set we compute density, P-wave and S- wave velocities for water saturated rocks and perform a mathematical transformation termed fluid substitution to obtain the same properties for a rock saturated with oil. This procedure is explained in Section 4.5 of this report. 4.4 P-wave and S-wave Velocities The relationship between P-wave velocity (V p ) and porosity is very well known in Rock Physics. The higher the porosity of the rock the softer the rock is, and the smaller the P-wave velocity. In other words, when porosity is high, the rock is more compressible and less resistant to wave-induced deformations, therefore V p is small. Similarly, when porosity is low the rock is less compressible and more resistant to wave-induced deformations, therefore V p is high. 20

Many empirical expressions of V p as a function of porosity have been obtained from laboratory data (Wyllie et al., 1956; Raymer, et al.,1980; Han, 1986; Tosaya and Nur, 1982; etc.), and all of them show the inverse proportionality between these two variables. The Stanford V reservoir (Mao and Journel, 1999) uses Han s V p φ relation to obtain P-wave velocities from the previously simulated porosity. Strictly speaking, Han s relations are obtained from sandstone samples collected from different depths (different levels of compaction), and they show a steep cementing trend (see Figure 13) which indicates that porosity is controlled by diagenesis and cementing (Avseth, 2000; Avseth, et al., 2005). The reservoir model we are creating here is not exhibiting a wide range of depths, and porosity is controlled more by sorting and clay content (depositional) which means that the cementing trend should not be steep (see Figure 13). Figure 13: Cementing versus Sorting trends. A more appropriate rock physics model for obtaining V p from porosity for sandstones corresponds to the constant cement model described by Dvorkin and Nur (1996). The theoretical constant cement model predicts the bulk modulus K and shear modulus G of dry sand with constant amount of cement deposited at grain surface. The bulk and shear moduli are two elastic moduli that define the properties of a rock that undergoes stress, deforms, and then recovers and returns to its original shape after the stress ceases. P-wave velocity is a 21

function of density and these two elastic moduli: V 2 p = K + 4 3 G ρ (4) The equations for the Dvorkin s constant cement model are as follows: K dry = ( φ/φ b K b + 4G b /3 + 1 φ/φ ) 1 b 4G b /3 (5) K min + 4G b /3 G dry = z = G b 6 ( φ/φb G b + z + 1 φ/φ b G min + z ) 1 z (6) 9K b + 8G b K b + 2G b (7) where φ b is porosity (smaller than φ c, the initial depositional porosity, sometimes referred to as critical porosity) at which contact cement trend turns into constant cement trend (see Figure 13). Elastic moduli with subscript min are the moduli of the rock mineral and elastic moduli with subscript b are the moduli at porosity φ b. These moduli are calculated from the contact cement theory with φ = φ b. The Dvorkin s contact cement theory is as follows: K dry = n(1 φ c)m c S n 6 G dry = 3K dry 5 + 3n(1 φ c)g c S τ 20 (8) (9) where n is the coordination number, φ c is the critical porosity, M c is the cement s P-wave modulus (M = ρvp 2 ), and G c is the cement s shear modulus. The constants S n and S τ are computed with the following equations: S n = A n α 2 + B n α + C n 22

A n = 0.024153Λ 1.3646 n B n = 0.20405Λn 0.89008 C n = 0.00024649Λn 1.9864 S τ = A τ α 2 + B τ α + C τ A τ = 10 2 (2.26ν 2 + 2.07ν + 2.3)Λ 0.079ν2 +0.175ν 1.342 τ B τ = (0.0573ν 2 + 0.0937ν + 0.202)Λ 0.0274ν2 +0.0529ν 0.8765 τ C τ = 10 4 (9.654ν 2 + 4.945ν + 3.1)Λ 0.01867ν2 +0.4011ν 1.8186 τ Λ n = 2G c (1 ν)(1 ν c ) πg (1 2ν c ) Λ τ = G c πg α = [(2/3)(φ c φ)/(1 φ c )] 1/2 where G and ν are the shear modulus and the Poisson s ratio of the grains (matrix), respectively; G c and ν c are the shear modulus and the Poisson s ratio of the cement. The constant cement model input parameters used in this reservoir are summarized in the following table. A 1% calcite cement is added to the sandstone facies (channel and point bar). Parameter Value Critical porosity φ c 0.38 Coordination number n 9 Cement s shear modulus G c 32 GPa Cement s Poisson s ratio ν c 0.32 Cement s density ρ c 2.71 g/cc φ b 0.37 Effective pressure P eff 0.1 MPa Having computed K dry and G dry for dry sandstones using Dvorkin s constant cement model, we use the following equations to obtain K sat and G sat for water saturated sandstones. These equations corresponds one form of the 23

Gassmann s fluid substitution which is explained in more detail in Section 4.5 of this report. [ ] φkdry (1 + φ)k water K dry /K min + K water K sat = K min (10) (1 φ)k water + φk min K water K dry /K min G sat = G dry (11) To obtain V p for the sandstones we use equation 4 with K sat and G sat. The rock physics model used for obtaining V p for shales corresponds to the empirical V p ρ Gardner s power law (1974): ρ = d V f p (12) where d = 1.75 and f = 0.265 are typical values for shales. Figure 14 shows the resulting V p values as a function of porosity for shales (gray dots) and brine-saturated sandstones (blue dots); additionally, this figure shows the previously discussed Dvorkin s constant cement model for 1% cement (red line), the Dvorkin s contact cement model (black line), and two typical Rock Physics bounds (Hashin-Shtrikman lower and upper bounds) which are shown for the only purpose of demonstrating that the results are within realistic limits. Regarding the calculation of S-wave velocities (V s ), we use V p V s relations for water-saturated sandstones and shales from Castagna et al. (1985,1993). They are as follows: V s = 0.862 V p 1.172 for shales (13) V s = 0.804 V p 0.856 for sandstones (14) 24

Figure 14: P-wave velocity vs. porosity for shales and brine-saturated sandstones. 25

4.5 Fluid Substitution In order to obtain density, P-wave and S-wave velocities for the reservoir saturated with oil, we use a mathematical transformation termed fluid substitution introduced by Gassmann (1951), which basically allows to calculate the elastic moduli of the rock as one fluid displaces another in the pore space. The elastic moduli define the properties of a rock that undergoes stress, deforms, and then recovers and returns to its original shape after the stress ceases. When the fluid contained in the rock changes the overall elastic moduli of the rock also changes and the seismic velocities are affected. Intuitively, the less compressible the fluid in the pore space the more resistant to waveinduced deformations the rock is. A rock with a less compressible fluid (such as brine) is stiffer than a rock with a more compressible fluid (such as gas). Seismic P-wave and S-wave velocities are functions of density and two elastic moduli, the bulk modulus K and the shear modulus G: V 2 p = K + 4 3 G ρ V 2 s = G ρ (15) (16) Gassmann s equation shown below is used to obtain the bulk modulus K 2 of the rock saturated with fluid 2, which is oil in this case. K 2 K min K 2 K fl2 φ(k min K fl2 ) = K 1 K min K 1 K fl1 φ(k min Kfl1 ) (17) K 1 and K 2 are the rock s bulk moduli with fluids 1 and 2 respectively, K fl1 and K fl2 are the bulk moduli of fluids 1 and 2, φ is the rock s porosity, and K min is the bulk modulus of the mineral. The shear modulus G 2 remains unchanged G 2 = G 1 only at low frequencies (appropriate for surface seismic), since shear stress cannot be applied to 26

fluids. The density of the rock is also transformed and the density of the rock with the second fluid is computed as: ρ 2 = ρ 1 + φ(ρ fl2 ρ fl1 ) (18) Having transformed the elastic moduli and the density, the compressional and shear wave velocities of the rock with the second fluid are computed as: V p = K 2 + 4G 3 2 (19) ρ 2 G2 V s = (20) ρ 2 For the generation of this data set we use the water and oil properties obtained using Batzle and Wang relations (1992) for pore pressure of 20 MP a and temperature of 85 C with the result summarized in the table below. Batzle and Wang relations (1992) summarize some important properties of reservoir fluids (brine, oil, gas and live oil), as function of pressure and temperature among other variables. These relations are mostly based on empirical measurements by Batzle and Wang (1992), and are more appropriate for wave propagation than PVT data. One of the fluid properties obtained using Batzle and Wang relations is the adiabatic bulk modulus, which is believed appropriate for wave propagation. In contrast, standard PVT data are isothermal and isothermal bulk modulus can be 20% too low for oil, a factor of 2 too low for gas, while approximately similar for brine (Mavko, et al., 1998). water oil density (g/cc) 0.99 0.7 bulk modulus (GPa) 2.57 0.5 Salinity (NaCl ppm) 20,000 Gravity (API) 25 Gas Oil Ratio (L/L) 200 27

The final density, P-wave and S-wave velocities of the reservoir saturated with oil obtained after performing fluid substitution are shown in Figure 15. Figure 16 shows several crossplots among the petrophysical properties we created. From this figure we clearly see a distinction between oil-saturated and both brine-saturated sandstones and shales. The scatter of points observed in Figure 16 is created after the petrophysical properties are computed by adding a small amount of random noise. Since density, P-wave and S-wave velocities are computed from mathematical expressions involving porosity, any crossplot of these properties will reflect their continuous behavior as it has been computed. However, real data does not show this behavior and has some scatter. We have made our synthetic data more realistic by adding some random noise that creates the scatter we see in crossplots. The amount of noise added to each property is not the same and also varies for each facies, the following table summarizes the percentage of random noise used. Property floodplain pointbar channel boundary Density 0.5% 0.5% 0.5% 0.5% P-wave velocity 5.0% 2.0% 2.0% 5.0% S-wave velocity 2.0% 2.0% 2.0% 2.0% 28

Figure 15: Resulting density (top), V p (middle) and V s (bottom) cubes for the oil-saturated reservoir. 29

Figure 16: Petrophysical properties crossplots. From left to right: P-wave velocity vs. porosity, P-wave velocity vs. density, S-wave velocity vs. P-wave velocity, and porosity vs. density. 30

5 Seismic Attributes Seismic attributes are all the information obtained from seismic data, either by direct measurements or by logical or experience based reasoning (Taner, 2001). The principal objectives of the seismic attributes are to provide accurate and detailed information to the interpreter on structural, stratigraphic and lithological parameters of the seismic prospect. Many attributes can be computed from seismic data, however, only those attributes computed from seismic reflection amplitude carry information about elastic contrast in the subsurface. Seismic inversion attempts to translate this information into elastic properties, which are function of density, P-wave and S-wave velocities. In a real reservoir characterization situation, seismic inversion is performed on the seismic reflection amplitudes to obtain the elastic properties, also called physical attributes. As mentioned before, from elasticity theory we know that these elastic properties are function of density, P-wave and S-wave velocities. Since Stanford VI is a synthetic data set, we use the petrophysical properties created before to compute a typical set of physical seismic attributes that could be obtained from seismic inversion in a real situation, although we do not perform any explicit inversion. The following list corresponds to the physical seismic attributes computed: Acoustic Impedance S-wave Impedance Elastic Impedance Lame coefficients λ and µ Poisson s Ratio AVO Intercept and Gradient These attributes are computed at the point support using several mathematical expressions. Subsequently, we perform a surface seismic filtering and 31

smoothing to obtain the same attributes at the seismic scale. In doing so, we create realistic seismic attributes that provide a filtered view of the true spatial variation of petrophysical properties. 5.1 Mathematical Expressions The mathematical expressions for the seismic attributes computed are functions of density, P-wave and S-wave velocities. Acoustic impedance and S-wave impedance are the result of the product between density and P-wave or S-wave velocity respectively (equations 21 and 22). AI = ρ V p (21) SI = ρ V s (22) Elastic impedance or pseudo-impedance is a generalization of acoustic impedance for variable incidence angle θ (equation 23). The elastic impedance is not an intrinsic rock property as the acoustic impedance, since it depends on the incidence angle and is derived from approximations. When compressional seismic waves (P waves) hit a boundary between two media of different elastic properties, part of the energy is reflected while part is transmitted. If the P wave hits the boundary at a zero incidence angle (normal incidence), the amplitude of the reflected wave is proportional to the contrast in acoustic impedance between the two media, basically the amplitude depends only on P-wave velocity and density. However, if the P wave hits the boundary at an angle different from zero, the amplitude of the reflected wave depends on P-wave velocity, S-wave velocity and density (see Figure 17). How amplitudes change with the angle of incidence for elastic materials are described by the Zoeppritz equations (Zoeppritz, 1919). Since complicated, various authors have presented approximations to these equations (e.g., Bortfeld,1961), and elastic impedance is obtained from one of these approximations of the Zoeppritz equations (Connolly, 1999). Strictly speaking, elastic impedance is derived from a linearization of the Zoeppritz equations 32

Figure 17: P-wave hitting a reflector. The physical properties are different on either side of the reflector. for P-wave reflectivity (Richards and Frasier, 1976) that is accurate for small changes of elastic parameters (V p, V s and ρ) and small angles of incidence. The derivation of the equation for the elastic impedance also assumes that the ratio Vs 2 /Vp 2 is constant. As expected, elastic impedance is a function of P-wave velocity, S-wave velocity, density and incidence angle. This attribute is typically obtained by inversion of angle stacks. For this reservoir we have computed EI for θ = 30 since far offsets (corresponding to larger incidence angles, θ) are more sensitive to changing saturation than near ones. EI(θ) = V 1+tan2 θ p V 8(Vs/Vp)2 sin 2 θ s ρ 1 4(Vs/Vp)2 sin 2 θ (23) Lame s coefficients λ and µ (equations 24 and 25) have been used as reservoir indicators. Stewart (1995) advised that λ/µ might have less influence of lithology and highlight pore-fill changes; Goodway et al. (1997) observed the conversion from velocity measurements to Lame s coefficients λ and µ improves identification of reservoir zones, and Xu and Bancroft (1997) showed the moduli ratio of λ/µ is a sensitive hydrocarbon indicator. µ = ρv 2 s (24) 33

λ = ρv 2 p 2µ (25) Poisson s Ratio (equation 26) involves only P and S-wave velocities, it is a very good indicator of fluid type and can be obtained from AVO Inversion (Rasmussen et al., 2004). ν = V p 2 2Vs 2 2(Vp 2 V 2 s ) (26) Amplitude variation with offset (AVO) comes from a process called energy partitioning. When compressional seismic waves (P waves) hit a boundary between two media of different elastic properties, part of the energy is reflected while part is transmitted. If the wave hits the boundary at an angle different from zero (incidence angle), P wave energy is partitioned further into reflected and transmitted P and S (shear waves) components (see Figure 18). The amplitudes of the reflected and transmitted energy depend on the contrast in elastic properties across the boundary, specifically on P-wave velocity, S-wave velocity and density. But, more importantly reflection amplitudes also depend on the angle of incidence of compressional seismic waves. Figure 18: Seismic wavefront hitting a reflector. The physical properties are different on either side of the reflector. The part of the P wave striking at a particular angle-of-incidence (represented by a ray) will have its energy divided into reflected and transmitted P and S waves. How amplitudes change with the angle of incidence for elastic materials 34

are described by the Zoeppritz equations (Zoeppritz, 1919). One of the most widely used approximations to the Zoeppritz equations is from Shuey (1985): R(θ) R 0 + Gsin 2 θ + F (tan 2 θ sin 2 θ) (27) where R 0 = 1 2 [ Vp V p G = 1 V p 2 + ρ ] ρ [ ] ρ ρ + 2 V s V s (28) V p 2(V s /V p ) 2 (29) F = 1 V p 2 V p (30) The expression for the reflection coefficient given in equation (27) can be interpreted in terms of different angular ranges (Castagna, 1993). R 0 is the normal incidence reflection coefficient often referred to as the AVO intercept, G describes the variation at intermediate offsets and is often referred to as the AVO gradient, whereas F dominates at far offsets near the critical angle (angle at which all the P-wave incident energy is transmitted). AVO intercept and gradient have been widely used for hydrocarbon detection, specially gas, and they are obtained by analyzing the amplitudes of pre-stack seismic data (e.g., Ostrander, 1984; Chacko, 1989; Rutherford and Williams, 1989; Snyder and Wrolstad, 1992; Allen and Peddy, 1993; Castagna and Backus, 1993; Santoso et al., 1995; Landro et al., 1995). Note that AVO intercept and gradient are obtained from an approximation to the exact P-wave reflection coefficient, that is accurate for small changes of elastic parameters (V p, V s and ρ) and small angles of incidence. Additionally, the mathematical expression for the P-wave reflection coefficient is obtained originally for a single interface between two semi-infinite layers; in real cases wave propagation occurs in more complex multilayered media. For the Stanford VI reservoir, we compute AVO intercept and gradient using the above equations and the filtered and smoothed density, P-wave and 35

S-wave velocities, since the compressional wave reflection coefficient (equation 27) is obtained for a semi-infinite two layer media. 5.2 Computation of Seismic Attributes Using the mathematical expressions described above, we compute the point-support seismic attributes (see Figure 19). Some crossplots show how they discriminate fluids and lithology. The relationship between Acoustic Impedance, Elastic Impedance and porosity is shown in Figure 20, where we clearly see how Elastic Impedance is an excellent indicator of the presence of hydrocarbon. Similarly, Figure 21 also shows how Poisson s Ratio discriminates between oil and brine-saturated sandstones. On the contrary, S-wave impedance by itself is not a good discriminator of either lithology or fluid (Figure 22). Figure 23 shows a crossplot of Lame s coefficients λ and µ, where we observe a clear discrimination of both lithology and fluid type. Finally we have plotted a typical AVO intercept versus gradient (see Figure 24), where oil-saturated sandstones deviate from the background trend followed by brine-saturated sandstones and shales. According to Castagna s sand classification (Castagna, et al., 1998) in terms of their AVO response, we can identify Stanford VI sandstones as Class III sands: lower impedance than the overlying shales (classical bright spots), and increasing reflection magnitude with offset. 36

Figure 19: Seismic attributes at the Geostatistical Scale: Acoustic Impedance, Elastic Impedance, S-wave Impedance, Poisson s Ratio, Lame coefficients λ µ. 37

Figure 20: Seismic attributes crossplots. From left to right: Acoustic impedance vs. porosity, Elastic impedance vs. porosity, and Acoustic impedance vs. Elastic impedance. Figure 21: Seismic attributes crossplots. From left to right: Poisson s Ratio vs. Acoustic impedance, Poisson s Ratio vs. Elastic impedance, and Poisson s Ratio vs. porosity. 38

Figure 22: Seismic attributes crossplots. From left to right: S-wave impedance vs. porosity, S-wave impedance vs. Elastic impedance, and Poisson s Ratio vs. S-wave impedance. Figure 23: Lame coefficients λ vs. µ. 39

Gradient for oil and brine-saturated sand- Figure 24: AVO Intercept vs. stones. 40

Having computed the seismic attributes at the point-support scale, we filter and smooth them in order to create seismic attributes at the seismic scale. Note that this is a simple but robust and economical way for computing the seismic attributes at the seismic scale. The Born approximation (von Seggern, 1991; Mukerji, et al., 1997) is used to compute the filter using the characteristic transfer function for the surface seismic measurement geometry and assuming continuous lines of sources and receivers. The parameters used to define such filter are summarized in the following table: minimum signal frequency maximum signal frequency source spread receiver spread 10 Hz 40 Hz -1875 m to 1875 m -1875 m to 1875 m Additionally, we have smoothed the filtered attributes using a 3D window averaging in order to create more lateral smoothing typical of seismic data. The window has a vertical size of λ/4 and a horizontal size of (Zλ) 1/2, which corresponds to the size of the Fresnel zone. The resulting seismic attributes at the seismic-support scale are shown in Figure 25. 41

Figure 25: Seismic attributes at the Seismic Scale: Acoustic Impedance, Elastic Impedance, S-wave Impedance, Poisson s Ratio, Lame coefficients λ µ, AVO attributes Intercept and Gradient. 42

6 Reservoir Flow Simulation Using the simulated permeability cube for the Stanford VI reservoir model obtained in Section 4.2 of this report, the next step is to perform a flow simulation. The results from this process can be potentially used for further research in history matching of both production and 4D seismic data. It is well known that reservoir flow simulation provides the means to develop reservoir management plans to achieve optimal recovery under certain economic constraints, since flow simulation allows to predict recovery before production. In order to do so, flow simulation programs solve mathematical equations that describe the flow of fluids through a numerical model of the reservoir. The reservoir model used for solving the flow equations comprises two basic petrophysical properties: porosity and permeability. Using a discrete 3D reservoir model with each grid block considered homogeneous and represented by a value of porosity and permeability, the flow equations often expressed as mass balances are solved for each grid block under certain boundary conditions. The number of equations to be solved per block depends on the complexity of the in situ and injected fluids. Typically, this number varies from 3 (black-oil simulators) to 15 (compositional simulators). In this report we have used a isothermal black-oil model since there are only two phases in the reservoir (oil and water) and we only inject water at a certain time during the flow simulation. Considering two-phase flow (water and oil phases), only 2 equations are to be solved per grid block; however, the computer work increases rapidly with the number of blocks in the reservoir model. As we have mentioned in Section 3 of this report, the size of the discrete 3D reservoir model (geomodel) we have created is of 150 200 200 = 6, 000, 000 grid blocks, which exceeds the capabilities of conventional reservoir simulators. In order to reduce the size of the simulation model, hence the computational running time of the flow simulation, upscaling of the reservoir properties is performed to construct a coarsened reservoir model. 43

6.1 Upscaling of the Reservoir Model The goal of any upscaling technique is to coarsen geological models to manageable levels for flow simulation. These coarsened flow models should replicate the fine scale behavior in overall flow rate. Usually these techniques are referred to as flow-based upscaling techniques. The two reservoir properties that are input to the flow simulation correspond to porosity and permeability. These two properties are upscaled from a fine scale of 150 200 200 = 6, 000, 000 grid blocks to a coarse scale of 30 40 40 = 48, 000 grid blocks. Porosity is upscaled using a linear block average, Figure 26 shows the fine scale porosity and the resulting coarse scale porosity. Since it has such a strong impact on flow (Darcy, 1856), permeability is upscaled using a flow-based technique. Figure 26: Porosity at the fine scale (left), linearly averaged porosity after upscaling. When upscaling homogeneous and isotropic permeability, the resulting coarse permeability or effective permeability becomes anisotropic. In three dimensions and since the simulation grid follows the reservoir layering ( stratigraphic grid), we obtain three effective permeabilities per each coarse grid block: k x, k y and k z. Figure 27 shows the resulting effective permeabilities in each direction x, y and z after upscaling. The upscaling technique used in this report produces effective permeabilities k x, k y and k z by using a single-phase pressure solver (Deutsch, 1989). This method corresponds to the GSLIB program flowsim. 44

Figure 27: Effective Permeability after upscaling: K x (top left), K y right), K z (bottom). (top 6.2 Flow Simulation The flow simulation is performed using the commercial software ECLIPSE. We use a fully-implicit, three phase, three dimensional, black-oil simulator. However, we consider only two phases (water and oil). The oil and water PVT properties used for the flow simulation are summarized in the following table. The relative permeability curves shown in Figure 28 are kept constant for the entire reservoir, and no capillary pressure is considered in the flow simulation (P c = 0). Property Oil Water Density (lb/ft 3 ) 45.09 61.80 Viscosity (cp) 1.18 0.325 Formation Volume Factor 0.98 1.0 45

Figure 28: Oil and Water Relative Permeability curves. An active constant flux aquifer is below the reservoir and the water-oil contact is at 9, 840ft depth. The constant water inflow rate is of 31, 000ST B/day. The flow simulation starts in January of 1975 with six wells in production (primary production); a summary of the production schedule is given in the following table, and the map location of the injector and producer wells is shown in Figure 29. Figure 29: Location maps of producer wells (left), and injector wells (right). The color represents horizon top depth (f t). 46

Date Operation January 1975 Start primary oil production with wells P1 to P6. January 1979 January 1981 January 1983 January 1986 January 1989 Wells P22 and P24 are open to production. Wells P26, P28 and P30 are open to production. Wells P21, P23, P25, P27, P29 and P31 are open to production. Wells P7, P9, P11, P13, P15, P17 and P19 are open to production. Start water injection in wells I32, I33, I34, I36, I37, I38, I41, I43 and I45. Wells P8, P10, P12, P14, P16, P18 and P20 are open to production. Start water injection in wells I35, I39, I42, I40. October 1989 Start water injection in wells I44, I46. January 1995 Increasing production rate of wells P1 to P6. Increasing water injection rate of wells I36, I42, I43, I44, I45 and I46. January 1998 Increasing production rate of wells P7 to P20. January 2001 Increasing production rate of wells P1 to P6. March 2003 Increasing production rate of wells P8, P10, P12, P14, P16, P18 and P20. March 2005 End of the flow simulation. The reservoir has 30 years of active production with 31 oil producers wells and 15 water injectors wells. As indicated in the production schedule table, not all wells start producing oil or injecting water at the same time, as is typical of an actual reservoir development where new wells are constantly 47

added. Producer wells are controlled by constant liquid rate production with a BHP constraint of 2700 psia, while injector wells are controlled by constant water injection rate. While oil production takes place, the water-oil contact starts to rise and the producer wells located far away from the structure axis (see Figure 29) start producing both oil and water. For those producer wells, P21 through P28, an economic limit is set such that they are converted to water injectors after they reach a water cut higher than 0.5. Figure 30 shows a summary of the reservoir flow simulation result in terms of rates while Figure 31 shows the simulation history in terms of cumulative quantities. Figure 30: Field rates history: Aquifer water influx rate (red line), Oil production rate (green line), Water injection rate (blue line), Water production rate (cyan line) and Reservoir pressure (black dotted line). Figure 30 also shows reservoir pressure as function of time where we clearly observe how the aquifer constant water influx fails to keep the reservoir pressure constant after 4 years of oil production, and how pressure decreases slowly after water injection starts (11 years after oil production started). In general we observe how the oil production increases with time due to 48

the activation of multiple production wells, keeps constant for 8 years and starts to decrease due to the increase in water production. Water is injected in the reservoir to maintain the pressure, as a consequence the WOC rises reaching producing wells. The water injection process starts 11 years after oil production started while water production starts 14 years after oil production started. Figure 31: Field cumulative history: Cumulative aquifer water influx (red line), cumulative oil production (green line), cumulative water injection (blue line) and cumulative water production (cyan line). Figure 32 shows a 3D view of the reservoir before and after 30 years of oil production; we observe how the WOC has changed due to oil production and water injection. Another view of the change in the reservoir oil saturation with time is shown in Figures 33, 34 and 35, where a constant X North-South slice (Figure 33), a constant Y East-West slice (Figure 34), and a horizon slice (Figure 35) are shown before production, 10, 20 and 30 years after oil production started. 49

Figure 32: 3D view of the reservoir before oil production starts (top), and 30 years after production started (bottom). The color bar represents oil saturation. 50

Figure 33: Constant X = 6151 f t North-South slice of the reservoir before oil production starts, 10, 20 and 30 years after oil production started. 51

Figure 34: Constant Y = 410 f t East-West slice of the reservoir before oil production starts, 10, 20 and 30 years after oil production started. 52

Figure 35: Horizon slice at 100 meters below the top of the reservoir before oil production starts, 10, 20 and 30 years after oil production started. 53

Since running the reservoir flow simulator on the fine scale model is not feasible due to the extremely large size of the fine scale model (150 200 200 = 6, 000, 000 grid blocks), we have created a new upscaled version of the reservoir model with 75 100 100 = 750, 000 grid blocks to obtain a flow response closer to the real one and without paying a high computational cost. This pseudo fine scale flow response is then used as the reference. Porosity is upscaled using a linear block average (see Figure 36) and effective permeabilities k x, k y and k z (see Figure 37) are obtained using the single-phase flow-based upscaling technique flowsim. Figure 36: Porosity of the pseudo fine scale reservoir model. Using exactly the same production schedule showed for the upscaled (30 40 40) reservoir model, the flow simulation is performed and the results are summarized in Figures 38, 39, 40, 41, 42, and 43. From the flow simulation result on the pseudo fine scale reservoir model, we observe a similar result compared to the one obtained from the upscaled model. Comparing the production rate histories we see small changes, and in general the changes occur during the last 10 years of production, where the time at which some wells switch from production to injection differs between the two models. This observation direct us to conclude that the water front is different from both model, and this is clear to see when comparing Figures 32 and 40, 33 and 41, 34 and 42, 35 and 43 To illustrate this important remark we compare the water cut history of well P21 in both simulations as well as the water front at the well location for the earliest of the two times ( 24 years after production started). The result is shown in Figure 44 and we observe that the water cut is higher in 54

the pseudo fine scale model and well P21 switches to injection earlier. This result shows the importance of reservoir heterogeneity in flow, while we observe an early and high water cut in the field our upscaled reservoir model is unable to reflect it. Figure 37: Effective Permeability of the pseudo fine scale reservoir model: K x (top left), K y (top right), K z (bottom). 55

Figure 38: Field rates history: Aquifer water influx rate (red line), Oil production rate (green line), Water injection rate (blue line), Water production rate (cyan line) and Reservoir pressure (black dotted line). Figure 39: Field cumulative history: Cumulative aquifer water influx (red line), cumulative oil production (green line), cumulative water injection (blue line) and cumulative water production (cyan line). 56

Figure 40: 3D view of the reservoir before oil production starts (top), and 30 years after production started (bottom). The color bar represents oil saturation. 57

Figure 41: Constant X = 6151 f t North-South slice of the reservoir before oil production starts, 10, 20 and 30 years after oil production started. 58

Figure 42: Constant Y = 410 f t East-West slice of the reservoir before oil production starts, 10, 20 and 30 years after oil production started. 59

Figure 43: Horizon slice at 100 meters below the top of the reservoir before oil production starts, 10, 20 and 30 years after oil production started. 60

Figure 44: Water cut versus time for well P21: solution from pseudo fine scale model (red), and solution from upscaled model (blue). Water saturation 24 years after oil production started: solution from pseudo fine scale model (middle), and solution from the upscaled model (bottom). 61

7 4D Seismic Data What is referred to as 4D seismic data is nothing more than threedimensional (3D) seismic data acquired at different times over the same area to assess changes in a producing hydrocarbon reservoir with time; changes may be observed in fluid location, saturation, pressure, and temperature. Consequently, one of the main applications of 4D seismic data is to monitor fluid flow in the reservoir. In order to create the 4D seismic response, several 3D seismic data sets s(u, t n ) are forward modelled using the simple convolutional model. It is clear the the first seismic data set s(u, t 0 ) will be created using the acoustic impedance of the reservoir prior to production, while the following seismic data set s(u, t n ) with n > 0 will be created using the acoustic impedance of the reservoir as it has changed due to the movement of fluids in the reservoir. The acoustic impedance of the reservoir at time t n is obtained using the fluid substitution procedure explained in section 4.5. However, this procedure requires one to know the properties (density and bulk modulus) of the fluid in the rock at time t n and we know from the flow simulation that two fluids are present with different partial saturations. The most common approach to modeling partial saturation (gas/water or oil/water) or mixed fluid saturations (gas/water/oil) is to replace the set of phases with a single effective fluid. The bulk modulus of this effective fluid is computed with a weighted harmonic average, termed Reuss average in the rock physics literature: 1 K fl = i S i K i (31) where K fl is the effective bulk modulus of the fluid mixture, K i denotes the bulk moduli of the individual fluid phases, and S i represents their saturations. This model assumes that the fluid phases are mixed at the finest scale. The density of the effective fluid is computed with the mixing formula: ρ fl = i S i ρ i (32) where ρ fl is the effective density of the fluid mixture, ρ i denotes the density of the individual fluid phases, and S i represents their saturations. 62

Using the results from the reservoir flow simulation, three seismic data sets are computed at different times during the oil production history (Figure 45). The first seismic data set s(u, t 1 ) is computed after t 1 = 10 years of oil production; this time corresponds to the end of primary production and the start of waterflooding. The second seismic data set s(u, t 2 ) is computed after t 2 = 25 years of oil production; this time corresponds to 15 years of waterflooding. The last and third seismic data set s(u, t 3 ) is computed after t 3 = 30 years of oil production; this time corresponds to the end of the reservoir flow simulation. Figure 45: Base seismic data set acquired prior to oil production (top left), seismic data sets acquired after 10 years of oil production (top right), after 25 years of oil production (bottom left), after 30 years of oil production (bottom right). In order to obtain the 4D seismic response due to the rising water front we compute the difference between the base seismic data set s(u, t 0 ), computed at time t 0 = 0 years (before oil production starts), and each of the three seismic data sets computed at times t n > 0 (n=1,2,3): 63

s n (u, t n ) = s(u, t n ) s(u, t 0 ) n = 1, 2, 3 (33) Additionally we compute the incremental 4D seismic response to observe the changes between two consecutive seismic surveys: [ s n (u, t n )] incremental = s(u, t n ) s(u, t n 1 ) n = 1, 2, 3 (34) These differences can be directly obtained by subtracting the originally recorded amplitudes or any seismic attribute such as acoustic impedance. Generally speaking, s can be considered as any attribute obtained from the seismic data. We have computed the difference between originally recorded amplitudes assuming small changes in velocity due to the movement of fluids in the reservoir. Subtracting amplitudes can be a wrong approach when large changes in velocity occur due to the stretching or shrinking of the time axis. The workflow used to create each of the 4D seismic responses, at times t 1 = 10 years, t 1 = 25 years and t 1 = 30 years, is summarized in Figure 46. To obtain the seismic impedance at time t n we perform fluid substitution on the fine scale model using refined coarse scale saturation, which is nothing more than a resampling of the coarse scale saturations into the fine scale grid. This is a serious approximation as it is known that seismic response is also sensitive to how fluids are distributed, not just how much of each fluid there is (Sengupta, 2000). Future work will need to focus obtaining saturation distributions consistent with the fine scale permeability model and flow boundary conditions. Figure 47 shows the distribution of fluids in the reservoir after t 1 = 10, t 2 = 25 and t 3 = 30 years of oil production, as well as the 4D seismic response s 1 (u, t 1 ), s 2 (u, t 2 ), and s 3 (u, t 3 ), and the incremental 4D seismic response [ s 1 (u, t 1 )] incremental, [ s 2 (u, t 2 )] incremental, and [ s 3 (u, t 3 )] incremental. From this figure we observe how the seismic response changes accordingly to the rising of the water front. In the areas where oil is still in place, the seismic data shows no difference. In the areas where water is present, the magnitude of the difference increases with time due to an increase in water saturation. The incremental difference between two consecutive seismic surveys show the areas where the distribution of fluids 64

has changed during that time lapse. The result obtained in Figure 47 corresponds to the upscaled reservoir model. The same procedure is followed for the pseudo fine scale reservoir model, and the results are in Figure 48. Comparing Figures 47 and 48 we observe that the 4D seismic response at late times (25 and 30 years after oil production started) is different from each model. The 4D seismic response from the upscaled model exhibits stronger differences than the 4D seismic response from the pseudo fine scale model, due to the differences between the coarse and fine scale water saturation. 65

Figure 46: Workflow used to create the 4D seismic response at different times during oil production. 66

Figure 47: Water saturation from upscaled model after 10 (top left), 25 (top middle) and 30 (top right) years of oil production. Seismic amplitude difference from upscaled model for 10 (middle left), 25 (middle middle) and 30 (middle right) years of oil production. Seismic amplitude incremental difference from upscaled model for 10 (bottom left), 25 (bottom middle) and 30 (bottom right) years of oil production. 67

Figure 48: Water saturation from pseudo fine scale model after 10 (top left), 25 (top middle) and 30 (top right) years of oil production. Seismic amplitude difference from pseudo fine scale model for 10 (middle left), 25 (middle middle) and 30 (middle right) years of oil production. Seismic amplitude incremental difference from pseudo fine scale model for 10 (bottom left), 25 (bottom middle) and 30 (bottom right) years of oil production. 68