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 testingany proposed algorithm for reservoir modeling, reservoir characterization andproduction forecasting.

Following the original data set generated by Mao and Journel (1999),the Stanford V reservoir, several extensions are proposed and incorporatedinto a new reservoir model which exhibits a smoother structure, more real-istic dimensions for current-day models and improved Rock Physics models.Additionally, new seismic attributes as well as 4D seismic data have beengenerated.

The Stanford VI reservoir is a three-layer prograding fluvial channel sys-tem, and its structure consists of an asymmetric anticline with axis N15◦E.The new reservoir model provides an exhaustive (6 million cells) samplingof petrophysical properties and seismic attributes, as well as a 4D Seismicdata set consisting of a base seismic survey acquired prior to oil productionand three time lapse seismic surveys acquired 10, 25 and 30 years after oilproduction. Therefore, reservoir simulation results are also part of this dataset.

Due to a real need for testing upscaling, the new Stanford VI reservoir ex-hibits realistic exhaustive sampling of petrophysical properties and promisesto represent a good data set for both upscaling and downscaling methods.

This paper describes the workflow used for the generation of the StanfordVI reservoir, as well as a detailed description of each step.

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1 Introduction

With the purpose of extensively testing algorithms for reservoir modelingor reservoir characterization, Mao and Journel (1999) created an exhaustive3D reference data set called Stanford V. Although it has been widely used,this data set is too small to represent current-day reservoir modeling exer-cises. Several extensions are proposed in this report and incorporated into anew reference reservoir model, termed Stanford VI.

The proposed reference data set (Stanford VI) exhibits a smooth top andbottom surface representing a trap in the form of an anticline. It provides anexhaustive sampling of petrophysical properties. The new reservoir model isrepresented in a 3D regular grid of 6million cells (150×200×200), with morerealistic dimensions for current-day models (25 m in the x and y directionsand 1 m in the z direction).

Following Stanford V, the stratigraphic model corresponds to a fluvialchannel system, and the petrophysical properties computed for this refer-ence reservoir correspond to the classical porosity, density, permeability andseismic P-wave and S-wave velocities. Although most of these propertiesare calculated following the standard procedure and algorithms presented byMao and Journel, a more appropriate Rock Physics model is used in the newreference reservoir model to compute P-wave velocity for sandstones.

Traditionally, P-wave velocity is calculated from empirical expressionsobtained from laboratory data as a function of porosity (Wyllie et al., 1956;Raymer, et al.,1980; Han, 1986; Tosaya and Nur, 1982; etc.). The formerStanford V reservoir uses one of these expressions (Han, 1986) to obtain P-wave velocities from porosity. Strictly speaking, Han’s relations are obtainedfrom sandstone samples collected from different depths, where porosity iscontrolled by diagenesis and cementing. In our case porosity is controlled bysorting and clay content, henceforth, a more appropriate rock physics modelis used, namely, the constant cement model described by Dvorkin and Nur(1996).

Besides petrophysical properties, the new reference reservoir model ex-hibits a complete set of physical seismic attributes, which are computedfrom well-known mathematical expressions and subsequently filtered andsmoothed to obtain realistically looking as would have been obtained fromactual seismic acquisition and modeling. These realistic seismic attributes

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provide a filtered view of the true spatial variation of petrophysical properties.

The current practice of modeling petrophysical properties from seismicdata, many so-called seismic attributes are used. However, only those at-tributes computed from seismic reflection amplitude carry information aboutthe elastic contrast in the subsurface. Seismic inversion attempts to translatethis information into elastic properties, which are function of density, P-waveand S-wave velocities.

From elasticity theory we know that these elastic properties are functionof density, P-wave and S-wave velocities. Since Stanford VI is a syntheticdata set, we use the petrophysical properties created before to compute atypical set of physical seismic attributes that could be obtained from seismicinversion in a real situation, although we do not perform any explicit inver-sion.

From this new reference reservoir model a 4D seismic data is generated.4D seismic data is nothing more than three-dimensional (3D) seismic dataacquired at different times over the same area. 4D seismic is used to assesschanges in a producing hydrocarbon reservoir with time; changes may beobserved in fluid location, saturation, pressure, and temperature. Conse-quently, one of the main applications of 4D seismic data is to monitor fluidflow in the reservoir.

In order to create a 4D seismic response, several 3D seismic data setsare forward modelled using a simple convolutional model. The first seismicdata set is created using the acoustic impedance of the reservoir prior toproduction, while three more seismic data sets are created using the acousticimpedance of the reservoir at different times during oil production.

The acoustic impedance of the reservoir after a certain time t can becomputed by using a fluid substitution approach. This process, introducedby Gassmann (1951) allows to calculate the elastic rock properties as onefluid displaces another in the pore space.

Prior to production the reservoir is filled with oil, while some years afterproduction starts the reservoir contains a mixture of fluids, typically wateroil and gas. In order to use Gassmann’s equations correctly we need thesaturations of each fluid at every point in space, therefore we use a flow sim-ulator to obtain them at any point in time during production.

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To create the 3D seismic data sets at different times during oil produc-tion, we performed a flow simulation where water and oil are the only fluidspresent in the reservoir. We simulate 30 years of oil production with an ac-tive aquifer below the reservoir and water injector wells that become activeafter the aquifer water influx fails to maintain the pressure.

This paper presents a detailed description of the workflow and processesfollowed to create the new reference reservoir model (Stanford VI), as wellas the results of each step.

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2 Workflow for building the Stanford VI reser-

voir

The workflow used to create the Stanford VI reference data set is shownin Figure 1. Although in general this workflow is similar to the one presentedby 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 com-puted for Stanford V, the new Stanford VI data set has seven more seismicattributes that are good indicators of both lithology and fluid type.

The large dimensions of the Stanford VI data set, allows for the study ofupscaling and downscaling of petrophysical properties.

As Figure 1 shows, the first step in the generation of Stanford VI prop-erty model corresponds to the modeling of facies. The Stanford VI faciesmodel correspond to a prograding fluvial channel system and is modelledusing the commercial software SBED and the multiple-point simulation al-gorithm snesim.

Subsequently, the facies model is populated with five petrophysical prop-erties: porosity, density, P-wave velocity, S-wave velocity and permeability.Basically, porosity is simulated first using the sequential simulation algorithmsgsim; density, P-wave and S-wave velocities are obtained from porosity us-ing well known Rock Physics models; and permeability is co-simulated con-ditional to the co-located previously simulated porosity using the algorithmcosgsim.

Having the petrophysical properties, three basic steps are followed: flowsimulation, forward model of 4D seismic data and generation of seismic at-tributes.

The following sections of this report explain the details of each step de-picted by the workflow presented in Figure 1.

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

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3 Structure and Stratigraphy

The structure of the Stanford VI reservoir corresponds to a classical struc-tural oil trap, an anticline. Specifically, it is an asymmetric anticline withaxis N15◦E. As Figure 2 shows, the anticline has a different dip on eachflank and generally the dip decreases slowly towards the northern part of thestructure. 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 of2.7 Km. The reservoir is 200 m thick and consists of three layers with thick-nesses 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 regularstratigraphic grid of 150 × 200 × 200 cells and the dimensions of the gridcorrespond to 25m in the x and y directions and 1m in the z direction. Thecoordinate system used correspond to the GSLIB standard of the strati-graphic coordinate system, where the z coordinate is measured relative tothe top of the reservoir. Due to the simple structure and stratigraphic grid,an accompanying cartesian box in which all of the geostatistical modelingtakes place, can easily be constructed.

The stratigraphy of the Stanford VI reservoir corresponds to a prograd-ing fluvial channel system, where deltaic deposits represented in layer 3 weredeposited first and followed by meandering channels in layer 2 and sinuous

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Figure 3: Perspective view of the Stanford VI top and bottom of each of itslayers. The color indicates the depth to the top.

channels in layer 1. This sequence of clastic deposits represents a prograda-tion of a fluvial system into the basin located in this case toward the northof the reservoir.

In order to model the stratigraphy of Stanford VI, we use the commercialsoftware SBED to model layer 1 and layer 2, while layer 3 is modelled us-ing the multiple-point simulation algorithm snesim with local rotation andaffinity variation to model the channel meanders.

The first layer of Stanford VI consists of a system of sinuous channels rep-resented by four facies: the floodplain (shale deposits), the point bar (sanddeposits that occur along the convex inner edges of the meanders of chan-nels), the channel (sand deposits), and the boundary (shale deposits). Thestratigraphic characteristics of layer 1 are detailed in the following table andFigure 4 shows the resulting stratigraphic model for this layer.

floodplain proportion 0.68point bar proportion 0.11channel proportion 0.165boundary proportion 0.045Number of channels 8

Average channel thickness 20 metersAverage channel width 600 meters

Average boundary thickness 1.5 metersAverage point bar width 300 meters

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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 fourfacies: the floodplain (shale deposits), the point bar (sand deposits that oc-cur along the convex inner edges of the meanders of channels), the channel(sand deposits), and the boundary (shale deposits). The stratigraphic char-acteristics of layer 2 are detailed in the following table and Figure 5 showsthe resulting stratigraphic model for this layer.

floodplain proportion 0.68point bar proportion 0.14channel proportion 0.11boundary proportion 0.07Number of channels 4

Average channel thickness 300 metersAverage channel width 16 meters

Average boundary thickness 1.5 meters

The last and third layer of the reservoir consists of deltaic deposits andare represented by two facies: the floodplain (shale deposits), and the channel(sand deposits). The stratigraphic characteristics of layer 3 are detailed inthe following table.

floodplain proportion 0.56channel proportion 0.44channel thicknesses [7− 40] meters

channel widths [70− 400] meters

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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 sim-ulation algorithm snesim with local rotation and affinity variation of thechannel meanders.

Traditionally, geostatistical techniques capture geological continuity througha variogram. A variogram is a two-point statistical function that describesthe level of correlation, or continuity, between any two sample values as sep-aration between them increases. Since the variogram describes the level ofcorrelation between two locations only, it is not able to model continuous andsinuous patterns such like channels or fractures. For modeling such geologi-cal features a multiple-point approach should be used, where spatial patternsare inferred using many spatial locations (Strebelle, 2002).

In multiple-point geostatistics, the spatial patterns are inferred from atraining image which represents a conceptual reservoir analog with the ex-pected geological heterogeneity. Since it is a conceptual model, the trainingimage is not constrained to any data.

The geostatistical technique that uses a training image to create real-izations constrained to reservoir data is proposed by Strebelle (2002). The“single normal equation simulation” (snesim) algorithm is a conditional se-quential simulation where the probability distribution is retrieved from thetraining 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

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realizations from a stationary training image. Figure 6 shows the resultingstratigraphic model for layer 3 as well as the rotation, and affinity cubesused, the training image is shown in Figure 7. The rotation and affinitycubes are categorical variables and the values assigned to these categoriesare 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]

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

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Figure 7: Training Image used for modeling Layer 3. The size of the trainingimage is 200× 200× 5, each slice in the z− direction is shown here from topto bottom.

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4 Petrophysical Properties

Having created the facies model for the three layers of the Stanford VIreservoir, 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 sgsimfrom 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 inFigures 8 and 9. Shale deposits in floodplain and boundary facies have dis-tinctively low porosity values while sand deposits in channel and point barfacies have high porosity values as expected. The variance for point bar fa-cies is smaller than for channel facies and their mean is higher since they aretypically well sorted sand deposits. Similarly, boundary facies exhibits verylow mean and variance which is translated later into a flow barrier for fluids.

Figure 8: Distribution of porosity for each facies in the reservoir.

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Figure 9: Histogram of porosity for each facies in the reservoir.

The reference variogram consisting of a single structure for each faciesare specified in the following table, where ranges for x, y and z direction aregiven in meters.

floodplain point bar channel boundary

type Spherical Spherical Spherical Sphericalnugget 0.1 0.1 0.1 0.1ranges 1750/1750/70 5000/2500/10 3000/1750/10 500/500/20angles 0/0/0 90/0/0 90/0/0 0/0/0

Having sequentially simulated porosity independently for each facies, weuse a cookie-cutter approach to create the resulting porosity model for theStanford VI reservoir (see Figure 10).

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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 corre-lated with porosity, therefore we have simulated the logarithm of permeabilityusing a linear correlation coefficient of 0.7 between both variables.

Permeability for each facies is co-simulated conditional to the simulatedporosity, using a Markov1-type model instead of a full model of coregional-ization. The algorithm used is cosgsim implemented in S-Gems.

Permeability is also simulated independently within each facies. Thecookie-cutter approach is used to merge the permeability simulated for eachfacies into a single permeability model.

The implicit assumption of the algorithm used is that both primaryand secondary variables are normally distributed, and the bivariate rela-tionship follows a (bi)Gaussian distribution. Both variables are transformedto 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, weprovide to the algorithm the original distributions of both porosity and log-arithm of permeability. The distribution of porosity was shown in the lastsection. The distribution of the logarithm of permeability is obtained (seeFigure 11) by transforming the distribution of porosity using the well known

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Kozeny-Carman’s relation (Carman, 1961).

κ =1

72τ

φ3

(1− φ)2d2 (1)

where: φ is porosity (fraction), τ is tortuosity (assumed as τ = 2.5), and dis the pore diameter (in micrometers).

Figure 11: Histogram of the logarithm of permeability for each facies in thereservoir.

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.

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floodplain point bar channel boundary

type Spherical Spherical Spherical Sphericalnugget 0.1 0.1 0.1 0.1ranges 1750/1750/70 5000/2500/10 3000/1750/10 500/500/20angles 0/0/0 90/0/0 90/0/0 0/0/0

The resulting permeability model for the Stanford VI reservoir is shownin Figure 12.

Figure 12: Resulting Permeability cube after cookie-cutting permeabilityfrom each facies’ permeability realization.

4.3 Density

The rock density is calculated using porosity and the mixing formula:

ρ = φρfluid + (1− φ)ρmatrix (2)

where ρfluid is the density of the fluid that fills in the pore space, and ρmatrix

is the density of the rock matrix. Therefore the rock density is calculated as:

ρ = φρfluid + (1− φ)N∑

i=1

fiρmi(3)

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where fi as the fraction of the mineral mi with density ρmiwhich constitutes

part of the rock matrix mineralogy.

The rock mineralogy for each facies is showed in the following table:

mineralmineral 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 0.15feldspar 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 wecompute the rock matrix density ρmatrix. From equation 3 we obtain the rockbulk density ρ, using the simulated porosity and water as the saturating fluid.

Typically, density, P-wave and S-wave velocities are calculated first forwater saturated rocks since the rock physics models used for computing ve-locities 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 trans-formation termed fluid substitution to obtain the same properties for a rocksaturated 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 (Vp) and porosity is very wellknown in Rock Physics. The higher the porosity of the rock the softer therock is, and the smaller the P-wave velocity. In other words, when porosityis high, the rock is more compressible and less resistant to wave-induced de-formations, therefore Vp is small. Similarly, when porosity is low the rock isless compressible and more resistant to wave-induced deformations, thereforeVp is high.

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Many empirical expressions of Vp as a function of porosity have been ob-tained 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 propor-tionality between these two variables.

The Stanford V reservoir (Mao and Journel, 1999) uses Han’s Vp − φrelation to obtain P-wave velocities from the previously simulated porosity.Strictly speaking, Han’s relations are obtained from sandstone samples col-lected from different depths (different levels of compaction), and they showa steep cementing trend (see Figure 13) which indicates that porosity is con-trolled by diagenesis and cementing (Avseth, 2000; Avseth, et al., 2005). Thereservoir 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 Vp from porosity forsandstones corresponds to the constant cement model described by Dvorkinand Nur (1996).

The theoretical constant cement model predicts the bulk modulus K andshear modulus G of dry sand with constant amount of cement deposited atgrain surface. The bulk and shear moduli are two elastic moduli that definethe properties of a rock that undergoes stress, deforms, and then recoversand returns to its original shape after the stress ceases. P-wave velocity is a

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function of density and these two elastic moduli:

V 2p =

K + 43G

ρ(4)

The equations for the Dvorkin’s constant cement model are as follows:

Kdry =

(φ/φb

Kb + 4Gb/3+

1− φ/φb

Kmin + 4Gb/3

)−1

− 4Gb/3 (5)

Gdry =

(φ/φb

Gb + z+

1− φ/φb

Gmin + z

)−1

− z (6)

z =Gb

6

9Kb + 8Gb

Kb + 2Gb

(7)

where φb is porosity (smaller than φc, the initial depositional porosity, some-times referred to as critical porosity) at which contact cement trend turnsinto constant cement trend (see Figure 13). Elastic moduli with subscriptmin are the moduli of the rock mineral and elastic moduli with subscript bare the moduli at porosity φb. These moduli are calculated from the con-tact cement theory with φ = φb. The Dvorkin’s contact cement theory is asfollows:

Kdry =n(1− φc)McSn

6(8)

Gdry =3Kdry

5+

3n(1− φc)GcSτ

20(9)

where n is the coordination number, φc is the critical porosity, Mc is thecement’s P-wave modulus (M = ρV 2

p ), and Gc is the cement’s shear modulus.The constants Sn and Sτ are computed with the following equations:

Sn = Anα2 + Bnα + Cn

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An = −0.024153Λ−1.3646n

Bn = 0.20405Λ−0.89008n

Cn = 0.00024649Λ−1.9864n

Sτ = Aτα2 + Bτα + Cτ

Aτ = −10−2(2.26ν2 + 2.07ν + 2.3)Λ0.079ν2+0.175ν−1.342τ

Bτ = (0.0573ν2 + 0.0937ν + 0.202)Λ0.0274ν2+0.0529ν−0.8765τ

Cτ = −10−4(9.654ν2 + 4.945ν + 3.1)Λ0.01867ν2+0.4011ν−1.8186τ

Λn =2Gc

πG

(1− ν)(1− νc)

(1− 2νc)

Λτ =Gc

πG

α = [(2/3)(φc − φ)/(1− φc)]1/2

where G and ν are the shear modulus and the Poisson’s ratio of the grains(matrix), respectively; Gc and νc are the shear modulus and the Poisson’sratio of the cement.

The constant cement model input parameters used in this reservoir aresummarized in the following table. A 1% calcite cement is added to thesandstone facies (channel and point bar).

Parameter Value

Critical porosity φc 0.38Coordination number n 9Cement’s shear modulus Gc 32 GPaCement’s Poisson’s ratio νc 0.32Cement’s density ρc 2.71 g/ccφb 0.37Effective pressure Peff 0.1 MPa

Having computed Kdry and Gdry for dry sandstones using Dvorkin’s con-stant cement model, we use the following equations to obtain Ksat and Gsat

for water saturated sandstones. These equations corresponds one form of the

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Gassmann’s fluid substitution which is explained in more detail in Section4.5 of this report.

Ksat = Kmin

[φKdry − (1 + φ)KwaterKdry/Kmin + Kwater

(1− φ)Kwater + φKmin −KwaterKdry/Kmin

](10)

Gsat = Gdry (11)

To obtain Vp for the sandstones we use equation 4 with Ksat and Gsat .The rock physics model used for obtaining Vp for shales corresponds to theempirical Vp − ρ Gardner’s power law (1974):

ρ = d V fp (12)

where d = 1.75 and f = 0.265 are typical values for shales.

Figure 14 shows the resulting Vp values as a function of porosity for shales(gray dots) and brine-saturated sandstones (blue dots); additionally, this fig-ure shows the previously discussed Dvorkin’s constant cement model for 1%cement (red line), the Dvorkin’s contact cement model (black line), and twotypical Rock Physics bounds (Hashin-Shtrikman lower and upper bounds)which are shown for the only purpose of demonstrating that the results arewithin realistic limits.

Regarding the calculation of S-wave velocities (Vs), we use Vp−Vs relationsfor water-saturated sandstones and shales from Castagna et al. (1985,1993).They are as follows:

Vs = 0.862 Vp − 1.172 for shales (13)

Vs = 0.804 Vp − 0.856 for sandstones (14)

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Figure 14: P-wave velocity vs. porosity for shales and brine-saturated sand-stones.

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4.5 Fluid Substitution

In order to obtain density, P-wave and S-wave velocities for the reservoirsaturated with oil, we use a mathematical transformation termed fluid sub-stitution introduced by Gassmann (1951), which basically allows to calculatethe 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 stressceases. When the fluid contained in the rock changes the overall elastic mod-uli 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 wave-induced deformations the rock is. A rock with a less compressible fluid (suchas 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 twoelastic moduli, the bulk modulus K and the shear modulus G:

V 2p =

K + 43G

ρ(15)

V 2s =

G

ρ(16)

Gassmann’s equation shown below is used to obtain the bulk modulus K2

of the rock saturated with fluid 2, which is oil in this case.

K2

Kmin −K2

− Kfl2

φ(Kmin −Kfl2)=

K1

Kmin −K1

− Kfl1

φ(Kmin−Kfl1)

(17)

K1 and K2 are the rock’s bulk moduli with fluids 1 and 2 respectively, Kfl1

and Kfl2 are the bulk moduli of fluids 1 and 2, φ is the rock’s porosity, andKmin is the bulk modulus of the mineral.

The shear modulus G2 remains unchanged G2 = G1 only at low frequen-cies (appropriate for surface seismic), since shear stress cannot be applied to

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fluids. The density of the rock is also transformed and the density of therock with the second fluid is computed as:

ρ2 = ρ1 + φ(ρfl2 − ρfl1) (18)

Having transformed the elastic moduli and the density, the compressionaland shear wave velocities of the rock with the second fluid are computed as:

Vp =

√√√√K2 + 43G2

ρ2

(19)

Vs =

√G2

ρ2

(20)

For the generation of this data set we use the water and oil properties ob-tained using Batzle and Wang relations (1992) for pore pressure of 20 MPaand temperature of 85◦C with the result summarized in the table below.Batzle and Wang relations (1992) summarize some important properties ofreservoir fluids (brine, oil, gas and live oil), as function of pressure and tem-perature among other variables. These relations are mostly based on empiri-cal measurements by Batzle and Wang (1992), and are more appropriate forwave propagation than PVT data.

One of the fluid properties obtained using Batzle and Wang relations isthe adiabatic bulk modulus, which is believed appropriate for wave propa-gation. In contrast, standard PVT data are isothermal and isothermal bulkmodulus can be 20% too low for oil, a factor of 2 too low for gas, whileapproximately similar for brine (Mavko, et al., 1998).

water oildensity (g/cc) 0.99 0.7bulk modulus (GPa) 2.57 0.5Salinity (NaCl ppm) 20,000 —Gravity (API) — 25Gas Oil Ratio (L/L) — 200

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The final density, P-wave and S-wave velocities of the reservoir saturatedwith oil obtained after performing fluid substitution are shown in Figure 15.

Figure 16 shows several crossplots among the petrophysical properties wecreated. From this figure we clearly see a distinction between oil-saturatedand both brine-saturated sandstones and shales.

The scatter of points observed in Figure 16 is created after the petrophys-ical properties are computed by adding a small amount of random noise.

Since density, P-wave and S-wave velocities are computed from mathe-matical expressions involving porosity, any crossplot of these properties willreflect their continuous behavior as it has been computed. However, realdata does not show this behavior and has some scatter. We have made oursynthetic data more realistic by adding some random noise that creates thescatter we see in crossplots. The amount of noise added to each property isnot the same and also varies for each facies, the following table summarizesthe 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%

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Figure 15: Resulting density (top), Vp (middle) and Vs (bottom) cubes forthe oil-saturated reservoir.

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

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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 pro-vide accurate and detailed information to the interpreter on structural, strati-graphic and lithological parameters of the seismic prospect.

Many attributes can be computed from seismic data, however, only thoseattributes computed from seismic reflection amplitude carry information aboutelastic contrast in the subsurface. Seismic inversion attempts to translate thisinformation into elastic properties, which are function of density, P-wave andS-wave velocities.

In a real reservoir characterization situation, seismic inversion is per-formed on the seismic reflection amplitudes to obtain the elastic properties,also called physical attributes.

As mentioned before, from elasticity theory we know that these elasticproperties are function of density, P-wave and S-wave velocities. Since Stan-ford VI is a synthetic data set, we use the petrophysical properties createdbefore to compute a typical set of physical seismic attributes that could beobtained from seismic inversion in a real situation, although we do not per-form any explicit inversion.

The following list corresponds to the physical seismic attributes com-puted:

• 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 mathe-matical 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 truespatial variation of petrophysical properties.

5.1 Mathematical Expressions

The mathematical expressions for the seismic attributes computed arefunctions of density, P-wave and S-wave velocities. Acoustic impedance andS-wave impedance are the result of the product between density and P-waveor S-wave velocity respectively (equations 21 and 22).

AI = ρ Vp (21)

SI = ρ Vs (22)

Elastic impedance or pseudo-impedance is a generalization of acoustic im-pedance for variable incidence angle θ (equation 23). The elastic impedanceis not an intrinsic rock property as the acoustic impedance, since it dependson the incidence angle and is derived from approximations.

When compressional seismic waves (P waves) hit a boundary betweentwo media of different elastic properties, part of the energy is reflected whilepart is transmitted. If the P wave hits the boundary at a zero incidenceangle (normal incidence), the amplitude of the reflected wave is proportionalto the contrast in acoustic impedance between the two media, basically theamplitude depends only on P-wave velocity and density. However, if the Pwave hits the boundary at an angle different from zero, the amplitude of thereflected wave depends on P-wave velocity, S-wave velocity and density (seeFigure 17).

How amplitudes change with the angle of incidence for elastic materialsare described by the “Zoeppritz equations” (Zoeppritz, 1919). Since com-plicated, various authors have presented approximations to these equations(e.g., Bortfeld,1961), and elastic impedance is obtained from one of these ap-proximations 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 differenton either side of the reflector.

for P-wave reflectivity (Richards and Frasier, 1976) that is accurate for smallchanges of elastic parameters (Vp, Vs and ρ) and small angles of incidence.The derivation of the equation for the elastic impedance also assumes thatthe ratio V 2

s /V 2p is constant.

As expected, elastic impedance is a function of P-wave velocity, S-wavevelocity, density and incidence angle. This attribute is typically obtainedby inversion of angle stacks. For this reservoir we have computed EI forθ = 30◦ since far offsets (corresponding to larger incidence angles, θ) aremore sensitive to changing saturation than near ones.

EI(θ) = V 1+tan2θp V −8(Vs/Vp)2sin2θ

s ρ1−4(Vs/Vp)2sin2θ (23)

Lame’s coefficients λ and µ (equations 24 and 25) have been used as reser-voir indicators. Stewart (1995) advised that λ/µ might have less influence oflithology and highlight pore-fill changes; Goodway et al. (1997) observed theconversion from velocity measurements to Lame’s coefficients λ and µ im-proves identification of reservoir zones, and Xu and Bancroft (1997) showedthe moduli ratio of λ/µ is a sensitive hydrocarbon indicator.

µ = ρV 2s (24)

33

λ = ρV 2p − 2µ (25)

Poisson’s Ratio (equation 26) involves only P and S-wave velocities, it isa very good indicator of fluid type and can be obtained from AVO Inversion(Rasmussen et al., 2004).

ν =V 2

p − 2V 2s

2(V 2p − V 2

s )(26)

Amplitude variation with offset (AVO) comes from a process called “en-ergy partitioning”. When compressional seismic waves (P waves) hit a bound-ary between two media of different elastic properties, part of the energy isreflected while part is transmitted. If the wave hits the boundary at an angledifferent from zero (incidence angle), P wave energy is partitioned furtherinto reflected and transmitted P and S (shear waves) components (see Fig-ure 18). The amplitudes of the reflected and transmitted energy depend onthe contrast in elastic properties across the boundary, specifically on P-wavevelocity, S-wave velocity and density. But, more importantly reflection am-plitudes also depend on the angle of incidence of compressional seismic waves.

Figure 18: Seismic wavefront hitting a reflector. The physical properties aredifferent on either side of the reflector. The part of the P wave striking ata particular angle-of-incidence (represented by a ray) will have its energydivided 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 themost widely used approximations to the “Zoeppritz equations” is from Shuey(1985):

R(θ) ≈ R0 + Gsin2θ + F (tan2θ − sin2θ) (27)

where

R0 =1

2

[∆Vp

Vp

+∆ρ

ρ

](28)

G =1

2

∆Vp

Vp

− 2(Vs/Vp)2

[∆ρ

ρ+ 2

∆Vs

Vs

](29)

F =1

2

∆Vp

Vp

(30)

The expression for the reflection coefficient given in equation (27) can beinterpreted in terms of different angular ranges (Castagna, 1993). R0 is thenormal incidence reflection coefficient often referred to as the AVO intercept,G describes the variation at intermediate offsets and is often referred to asthe 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 de-tection, specially gas, and they are obtained by analyzing the amplitudesof pre-stack seismic data (e.g., Ostrander, 1984; Chacko, 1989; Rutherfordand 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 approxi-mation to the exact P-wave reflection coefficient, that is accurate for smallchanges of elastic parameters (Vp, Vs and ρ) and small angles of incidence.Additionally, the mathematical expression for the P-wave reflection coeffi-cient is obtained originally for a single interface between two semi-infinitelayers; in real cases wave propagation occurs in more complex multilayeredmedia.

For the Stanford VI reservoir, we compute AVO intercept and gradientusing the above equations and the filtered and smoothed density, P-wave and

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S-wave velocities, since the compressional wave reflection coefficient (equa-tion 27) is obtained for a semi-infinite two layer media.

5.2 Computation of Seismic Attributes

Using the mathematical expressions described above, we compute thepoint-support seismic attributes (see Figure 19). Some crossplots show howthey discriminate fluids and lithology.

The relationship between Acoustic Impedance, Elastic Impedance andporosity is shown in Figure 20, where we clearly see how Elastic Impedanceis an excellent indicator of the presence of hydrocarbon. Similarly, Figure 21also shows how Poisson’s Ratio discriminates between oil and brine-saturatedsandstones. On the contrary, S-wave impedance by itself is not a good dis-criminator of either lithology or fluid (Figure 22).

Figure 23 shows a crossplot of Lame’s coefficients λ and µ, where we ob-serve a clear discrimination of both lithology and fluid type.

Finally we have plotted a typical AVO intercept versus gradient (see Fig-ure 24), where oil-saturated sandstones deviate from the background trendfollowed by brine-saturated sandstones and shales.

According to Castagna’s sand classification (Castagna, et al., 1998) interms of their AVO response, we can identify Stanford VI sandstones as‘Class III’ sands: lower impedance than the overlying shales (classical brightspots), and increasing reflection magnitude with offset.

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Figure 19: Seismic attributes at the Geostatistical Scale: Acoustic Im-pedance, Elastic Impedance, S-wave Impedance, Poisson’s Ratio, Lame co-efficients λ µ.

37

Figure 20: Seismic attributes crossplots. From left to right: Acoustic im-pedance vs. porosity, Elastic impedance vs. porosity, and Acoustic im-pedance vs. Elastic impedance.

Figure 21: Seismic attributes crossplots. From left to right: Poisson’s Ra-tio vs. Acoustic impedance, Poisson’s Ratio vs. Elastic impedance, andPoisson’s Ratio vs. porosity.

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Figure 22: Seismic attributes crossplots. From left to right: S-wave im-pedance vs. porosity, S-wave impedance vs. Elastic impedance, and Pois-son’s Ratio vs. S-wave impedance.

Figure 23: Lame coefficients λ vs. µ.

39

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

40

Having computed the seismic attributes at the point-support scale, wefilter and smooth them in order to create seismic attributes at the seismicscale. Note that this is a simple but robust and economical way for comput-ing the seismic attributes at the seismic scale.

The Born approximation (von Seggern, 1991; Mukerji, et al., 1997) is usedto compute the filter using the characteristic transfer function for the surfaceseismic measurement geometry and assuming continuous lines of sources andreceivers. The parameters used to define such filter are summarized in thefollowing table:

minimum signal frequency 10 Hz

maximum signal frequency 40 Hz

source spread -1875 m to 1875 m

receiver spread -1875 m to 1875 m

Additionally, we have smoothed the filtered attributes using a 3D windowaveraging 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, whichcorresponds to the size of the Fresnel zone.

The resulting seismic attributes at the seismic-support scale are shown inFigure 25.

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

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6 Reservoir Flow Simulation

Using the simulated permeability cube for the Stanford VI reservoir modelobtained in Section 4.2 of this report, the next step is to perform a flow sim-ulation. The results from this process can be potentially used for furtherresearch in history matching of both production and 4D seismic data.

It is well known that reservoir flow simulation provides the means to de-velop reservoir management plans to achieve optimal recovery under certaineconomic constraints, since flow simulation allows to predict recovery beforeproduction. In order to do so, flow simulation programs solve mathematicalequations that describe the flow of fluids through a numerical model of thereservoir.

The reservoir model used for solving the flow equations comprises two ba-sic petrophysical properties: porosity and permeability. Using a discrete 3Dreservoir model with each grid block considered homogeneous and representedby a value of porosity and permeability, the flow equations often expressedas mass balances are solved for each grid block under certain boundary con-ditions.

The number of equations to be solved per block depends on the com-plexity of the in situ and injected fluids. Typically, this number varies from3 (black-oil simulators) to 15 (compositional simulators). In this report wehave used a isothermal black-oil model since there are only two phases in thereservoir (oil and water) and we only inject water at a certain time duringthe flow simulation.

Considering two-phase flow (water and oil phases), only 2 equations areto be solved per grid block; however, the computer work increases rapidlywith the number of blocks in the reservoir model. As we have mentionedin Section 3 of this report, the size of the discrete 3D reservoir model (geo-model) we have created is of 150× 200× 200 = 6, 000, 000 grid blocks, whichexceeds the capabilities of conventional reservoir simulators.

In order to reduce the size of the simulation model, hence the computa-tional running time of the flow simulation, upscaling of the reservoir proper-ties is performed to construct a coarsened reservoir model.

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6.1 Upscaling of the Reservoir Model

The goal of any upscaling technique is to coarsen geological models tomanageable levels for flow simulation. These coarsened flow models shouldreplicate the fine scale behavior in overall flow rate. Usually these techniquesare referred to as flow-based upscaling techniques.

The two reservoir properties that are input to the flow simulation corre-spond to porosity and permeability. These two properties are upscaled froma fine scale of 150 × 200 × 200 = 6, 000, 000 grid blocks to a coarse scale of30× 40× 40 = 48, 000 grid blocks. Porosity is upscaled using a linear blockaverage, Figure 26 shows the fine scale porosity and the resulting coarse scaleporosity. Since it has such a strong impact on flow (Darcy, 1856), permeabil-ity is upscaled using a flow-based technique.

Figure 26: Porosity at the fine scale (left), linearly averaged porosity afterupscaling.

When upscaling homogeneous and isotropic permeability, the resultingcoarse permeability or effective permeability becomes anisotropic. In threedimensions and since the simulation grid follows the reservoir layering (“strati-graphic” grid), we obtain three effective permeabilities per each coarse gridblock: kx, ky and kz. Figure 27 shows the resulting effective permeabilitiesin each direction x, y and z after upscaling.

The upscaling technique used in this report produces effective permeabil-ities kx, ky and kz by using a single-phase pressure solver (Deutsch, 1989).This method corresponds to the GSLIB program flowsim.

44

Figure 27: Effective Permeability after upscaling: Kx (top left), Ky (topright), Kz (bottom).

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 sum-marized in the following table. The relative permeability curves shown inFigure 28 are kept constant for the entire reservoir, and no capillary pressureis considered in the flow simulation (Pc = 0).

Property Oil WaterDensity (lb/ft3) 45.09 61.80Viscosity (cp) 1.18 0.325

Formation Volume Factor 0.98 1.0

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Figure 28: Oil and Water Relative Permeability curves.

An active constant flux aquifer is below the reservoir and the water-oilcontact is at 9, 840ft depth. The constant water inflow rate is of 31, 000STB/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 thefollowing table, and the map location of the injector and producer wells isshown in Figure 29.

Figure 29: Location maps of producer wells (left), and injector wells (right).The color represents horizon top depth (ft).

46

Date Operation

January 1975 Start primary oil production with wells P1 to P6.

January 1979 Wells P22 and P24 are open to production.

January 1981 Wells P26, P28 and P30 are open to production.

January 1983 Wells P21, P23, P25, P27, P29 and P31 are open toproduction.

January 1986 Wells P7, P9, P11, P13, P15, P17 and P19 are open toproduction.Start water injection in wells I32, I33, I34, I36, I37, I38,I41, I43 and I45.

January 1989 Wells P8, P10, P12, P14, P16, P18 and P20 are opento 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 wellsand 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 istypical of an actual reservoir development where new wells are constantly

47

added. Producer wells are controlled by constant liquid rate production witha BHP constraint of 2700psia, while injector wells are controlled by constantwater injection rate.

While oil production takes place, the water-oil contact starts to rise andthe producer wells located far away from the structure axis (see Figure 29)start producing both oil and water. For those producer wells, P21 throughP28, an economic limit is set such that they are converted to water injectorsafter they reach a water cut higher than 0.5.

Figure 30 shows a summary of the reservoir flow simulation result in termsof rates while Figure 31 shows the simulation history in terms of cumulativequantities.

Figure 30: Field rates history: Aquifer water influx rate (red line), Oil pro-duction rate (green line), Water injection rate (blue line), Water productionrate (cyan line) and Reservoir pressure (black dotted line).

Figure 30 also shows reservoir pressure as function of time where weclearly observe how the aquifer constant water influx fails to keep the reservoirpressure constant after 4 years of oil production, and how pressure decreasesslowly 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 andstarts to decrease due to the increase in water production. Water is injectedin the reservoir to maintain the pressure, as a consequence the WOC risesreaching producing wells. The water injection process starts 11 years afteroil production started while water production starts 14 years after oil pro-duction started.

Figure 31: Field cumulative history: Cumulative aquifer water influx (redline), cumulative oil production (green line), cumulative water injection (blueline) and cumulative water production (cyan line).

Figure 32 shows a 3D view of the reservoir before and after 30 years of oilproduction; we observe how the WOC has changed due to oil production andwater injection. Another view of the change in the reservoir oil saturationwith time is shown in Figures 33, 34 and 35, where a constant X North-Southslice (Figure 33), a constant Y East-West slice (Figure 34), and a horizonslice (Figure 35) are shown before production, 10, 20 and 30 years after oilproduction started.

49

Figure 32: 3D view of the reservoir before oil production starts (top), and30 years after production started (bottom). The color bar represents oilsaturation.

50

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

51

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

52

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

53

Since running the reservoir flow simulator on the fine scale model is notfeasible 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 thereservoir model with 75 × 100 × 100 = 750, 000 grid blocks to obtain a flowresponse 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 ef-fective permeabilities kx, ky and kz (see Figure 37) are obtained using thesingle-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 theresults 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 upscaledmodel. Comparing the production rate histories we see small changes, and ingeneral the changes occur during the last 10 years of production, where thetime at which some wells switch from production to injection differs betweenthe two models. This observation direct us to conclude that the water frontis different from both model, and this is clear to see when comparing Figures32 and 40, 33 and 41, 34 and 42, 35 and 43

To illustrate this important remark we compare the water cut history ofwell P21 in both simulations as well as the water front at the well locationfor the earliest of the two times (≈ 24 years after production started). Theresult 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, whilewe observe an early and high water cut in the field our upscaled reservoirmodel is unable to reflect it.

Figure 37: Effective Permeability of the “pseudo” fine scale reservoir model:Kx (top left), Ky (top right), Kz (bottom).

55

Figure 38: Field rates history: Aquifer water influx rate (red line), Oil pro-duction rate (green line), Water injection rate (blue line), Water productionrate (cyan line) and Reservoir pressure (black dotted line).

Figure 39: Field cumulative history: Cumulative aquifer water influx (redline), cumulative oil production (green line), cumulative water injection (blueline) and cumulative water production (cyan line).

56

Figure 40: 3D view of the reservoir before oil production starts (top), and30 years after production started (bottom). The color bar represents oilsaturation.

57

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

58

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

59

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

60

Figure 44: Water cut versus time for well P21: solution from “pseudo” finescale model (red), and solution from upscaled model (blue). Water saturation24 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 three-dimensional (3D) seismic data acquired at different times over the same areato assess changes in a producing hydrocarbon reservoir with time; changesmay be observed in fluid location, saturation, pressure, and temperature.Consequently, one of the main applications of 4D seismic data is to monitorfluid flow in the reservoir.

In order to create the 4D seismic response, several 3D seismic data setss(u, tn) are forward modelled using the simple convolutional model. It isclear the the first seismic data set s(u, t0) will be created using the acousticimpedance of the reservoir prior to production, while the following seismicdata set s(u, tn) with n > 0 will be created using the acoustic impedance ofthe reservoir as it has changed due to the movement of fluids in the reservoir.

The acoustic impedance of the reservoir at time tn is obtained using thefluid substitution procedure explained in section 4.5. However, this proce-dure requires one to know the properties (density and bulk modulus) of thefluid in the rock at time tn and we know from the flow simulation that twofluids are present with different partial saturations.

The most common approach to modeling partial saturation (gas/wateror oil/water) or mixed fluid saturations (gas/water/oil) is to replace the setof phases with a single “effective fluid”. The bulk modulus of this “effectivefluid” is computed with a weighted harmonic average, termed Reuss averagein the rock physics literature:

1

Kfl

=∑

i

Si

Ki

(31)

where Kfl is the effective bulk modulus of the fluid mixture, Ki denotes thebulk moduli of the individual fluid phases, and Si represents their satura-tions. 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:

ρfl =∑

i

Siρi (32)

where ρfl is the effective density of the fluid mixture, ρi denotes the densityof the individual fluid phases, and Si represents their saturations.

62

Using the results from the reservoir flow simulation, three seismic datasets are computed at different times during the oil production history (Figure45). The first seismic data set s(u, t1) is computed after t1 = 10 years ofoil production; this time corresponds to the end of primary production andthe start of waterflooding. The second seismic data set s(u, t2) is computedafter t2 = 25 years of oil production; this time corresponds to 15 years ofwaterflooding. The last and third seismic data set s(u, t3) is computed af-ter t3 = 30 years of oil production; this time corresponds to the end of thereservoir 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 25years of oil production (bottom left), after 30 years of oil production (bottomright).

In order to obtain the 4D seismic response due to the rising water front wecompute the difference between the base seismic data set s(u, t0), computedat time t0 = 0 years (before oil production starts), and each of the threeseismic data sets computed at times tn > 0 (n=1,2,3):

63

∆sn(u, ∆tn) = s(u, tn)− s(u, t0) n = 1, 2, 3 (33)

Additionally we compute the incremental 4D seismic response to observethe changes between two consecutive seismic surveys:

[∆sn(u, ∆tn)]incremental = s(u, tn)− s(u, tn−1) n = 1, 2, 3 (34)

These differences can be directly obtained by subtracting the originallyrecorded amplitudes or any seismic attribute such as acoustic impedance.Generally speaking, s can be considered as any attribute obtained from theseismic data. We have computed the difference between originally recordedamplitudes assuming small changes in velocity due to the movement of fluidsin the reservoir. Subtracting amplitudes can be a wrong approach when largechanges 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 timest1 = 10 years, t1 = 25 years and t1 = 30 years, is summarized in Figure 46.To obtain the seismic impedance at time tn we perform fluid substitution onthe fine scale model using “refined” coarse scale saturation, which is nothingmore than a resampling of the coarse scale saturations into the fine scalegrid. This is a serious approximation as it is known that seismic response isalso sensitive to how fluids are distributed, not just how much of each fluidthere is (Sengupta, 2000). Future work will need to focus obtaining satura-tion distributions consistent with the fine scale permeability model and flowboundary conditions.

Figure 47 shows the distribution of fluids in the reservoir after t1 =10, t2 = 25 and t3 = 30 years of oil production, as well as the 4D seis-mic response ∆s1(u, ∆t1), ∆s2(u, ∆t2), and ∆s3(u, ∆t3), and the incremen-tal 4D seismic response [∆s1(u, ∆t1)]incremental, [∆s2(u, ∆t2)]incremental, and[∆s3(u, ∆t3)]incremental. From this figure we observe how the seismic responsechanges accordingly to the rising of the water front. In the areas where oilis still in place, the seismic data shows no difference. In the areas wherewater is present, the magnitude of the difference increases with time dueto an increase in water saturation. The incremental difference between twoconsecutive 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 reservoirmodel. The same procedure is followed for the “pseudo” fine scale reservoirmodel, and the results are in Figure 48.

Comparing Figures 47 and 48 we observe that the 4D seismic response atlate times (25 and 30 years after oil production started) is different from eachmodel. The 4D seismic response from the upscaled model exhibits strongerdifferences 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 timesduring 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 amplitudedifference from upscaled model for 10 (middle left), 25 (middle middle) and30 (middle right) years of oil production. Seismic amplitude incrementaldifference from upscaled model for 10 (bottom left), 25 (bottom middle) and30 (bottom right) years of oil production.

67

Figure 48: Water saturation from “pseudo” fine scale model after 10 (topleft), 25 (top middle) and 30 (top right) years of oil production. Seismicamplitude difference from “pseudo” fine scale model for 10 (middle left), 25(middle middle) and 30 (middle right) years of oil production. Seismic am-plitude incremental difference from “pseudo” fine scale model for 10 (bottomleft), 25 (bottom middle) and 30 (bottom right) years of oil production.

68

8 Conclusions and future research

• A new exhaustive (6 million cells) reference data set is generated for thegeneral purpose of testing any proposed algorithm for reservoir model-ing, reservoir characterization and production forecasting.

• Several extensions to the original data set generated by Mao and Jour-nel (1999), the Stanford V reservoir, are proposed and incorporatedinto a new reservoir model which exhibits a smoother structure, morerealistic dimensions for current-day models and improved Rock Physicsmodels.

• New seismic attributes as well as 4D seismic data have been generated.4D seismic data consists of base seismic survey acquired prior to oilproduction and three time lapse seismic survey acquired 10, 25 and 30years after oil production.

• A complete and realistic two-phase oil-water reservoir flow simulationis performed. Thirty years of oil production are simulated on an up-scaled reservoir model (48, 000 gridblocks) as well as on a “pseudo”fine scale model (750, 000 gridblocks) with an active aquifer below thereservoir and water injector wells that become active after the aquiferwater influx fails to maintain the pressure.

• The new Stanford VI reservoir exhibits realistic exhaustive samplingof petrophysical properties and promises to represent a good data setfor both upscaling and downscaling methods. In this regard, futureresearch can certainly be done in exploring the impact of flow-baseddownscaling of saturations on the 4D seismic response.

• The new realistic reservoir flow simulation performed on the StanfordVI reservoir also allows for extensive research on history matching pro-cedures.

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9 References

• Allen, J. L., and Peddy, C. P., 1993. Amplitude variation with offset:Gulf Coast case studies: Soc. Expl. Geophys., Geophys. Dev. Series4.

• Avseth, P., Mukerji, T., Mavko, G., 2005. Quantitative Seismic In-terpretation : Applying Rock Physics Tools to Reduce InterpretationRisk. Cambridge University Press.

• Avseth, P., 2000. Integrating rock physics and flow simulation to re-duce uncertainties in seismic reservoir monitoring. Ph.D. dissertation,Stanford University.

• Aziz, K., and Settari, A., 1979. Petroleum Reservoir Simulation.

• Batzle , M., Wang, Z., 1992. Seismic properties of pore fluids. Geo-physics, 57, 1396-1408.

• Bortfeld, R., 1961. Approximation to the reflection and transmis-sion coefficients of plane longitudinal and transverse waves. Geophys.Prosp., 9, 485-502.

• Carman, P.C., 1961. L’ecoulement des Gaz a Travers les MilieuxPoreux, Bibliotheque des Science et Techniques Nucleaires, press Uni-versitaires de France, Paris, 198 pp.

• Castagna, J.P, Batzle, M.L., and Eastwood, R.L, 1985. Relationshipsbetween compressional-wave and shear-wave velocities in clastic silicaterocks. Geophysics, 50, 571-581.

• Castagna, J. P. and Backus,M. M., Eds., 1993. Offset dependent re-flectivity: Theory and practice of AVO analysis: Soc. Expl. Geophys.,Invest. Geophys. 8.

• Castagna, J.P., Batzle, M.L., and Kan, T.K.,1993. Rock physics - thelink between rock properties and AVO response, in Offset-DependentReflectivity - Theory and Practice of AVO Analysis, J.P Castagna andM. Backus, eds. Investigations in Geophysics, No. 8, Society of Explo-ration Geophysicists, Tulsa Oklahoma, 135-171.

• Castagna, J.P., Swan, H.W, and Foster, D.J, 1998. Framework forAVO gradient and intercept interpretation. Geophysics, 63, 948-956.

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• Chacko, S., 1989. Porosity identification using amplitude variation withoffset: Examples from South Sumatra: Geophysics, 54, 942 951.

• Darcy, H., 1856. Les Fontaines publiques de la Ville de Dijon, Dal-mount, Paris (reprinted in Hubbert, 1969).

• Deutsch, C., 1989. Calculating Effective Absolute Permeability inSandstone/Shale Sequences. Paper SPE 17264.

• Dvorkin, J., and Nur, A., 1996. Elasticity of high-porosity sandstones:Theory for two North Sea datasets. Geophysics, 61, 1363-1370.

• Gassmann, F., 1951. Uber die Elastizitat poroser Medien. Vier. derNatur, Gesellschaft in Zurich, 96, 1-23.

• Gardner, G.H.F., Gardner, L.W., and Gregory, A. R., 1974. Forma-tion velocity and density - The diagnostic basis for stratigraphic traps.Geophysics, 39, 770-780.

• Goodway, B., Chen, T., and Downton, J., 1997, Improved AVO fluiddetection and lithology discrimination using Lame petrophysical para-meters; λρ, µρ, and λ/µ fluid stack, from P and S inversions: CSEGnational convention, Expanded Abstracts, 148-151.

• Goodway, B., Chen, T., and Downton, J., 1997, Improved AVO fluiddetection and lithology discrimination using Lame petrophysical para-meters; λρ, µρ, and λ/µ fluid stack, from P and S inversions: 67thAnn. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts

• Han, D. H., 1986.. Effects of Porosity and Clay content on AcousticProperties of Sandstones and Unconsolidated Sediments. Ph.D. disser-tation, Stanford University.

• Hubbert, M. K., 1969. The Theory of Ground-Water Motion and Re-lated Papers, Hafner Publishing, New York.

• Jack, I., 1998. Time-Lapse Seismic in Reservoir Management: SEGDistinguished Instructor Short Course.

• Journel, A.G., 2002. Combining knowledge form diverse informationsources: an alternative to Bayesian analysis. Mathematical Geology,V. 34, No. 5, 573-598.

71

• Landro, M., Buland, A., and DAngelo, R., 1995. Target-orientedAVOinversion from Valhall and Hod fields: The Leading Edge, 14, No. 8,855861.

• Mao, S., and Journel, A., 1999. Generation of a reference petrophys-ical/seismic data set: the Stanford V reservoir. 12th Annual Report,Stanford Center for Reservoir Forecasting, Stanford, USA.

• Mavko, G., Mukerji, T., Dvorkin, J., 1998. The Rock Physics Hand-book: Tools for seismic analysis in porous media. Cambridge UniversityPress.

• McCain, W. D., 1989. The Properties of Petroleum Fluids. PennWellPublishing Co.

• Mukerji, T., Mavko, G., Rio, P., 1997. Scales of Reservoir Hetero-geneities and Impact of Seismic Resolution on Geostatistical Integra-tion. Mathematical Geology, 29, 933-949.

• Ostrander,W. J., 1984. Plane-wave reflection coefficients for gas sandsat nonnormal angles of incident: Geophysics, 49, 16371648.

• Rasmussen, K.B., Bruun, A., Pedersen, J.M., 2004. SimulataneousSeismic Inversion. EAGE 66th Conference & Exhibition.

• Raymer, L.L, Hunt, E.R., Gardner, J.S., 1980. An improved sonictransit-time-to porosity transform, Trans. Soc. Prof. Well Log Ana-lysts, 21st Annual Logging Symposium, Paper P.

• Richards, P. G., and Frasier, C. W., 1976. Scattering of elastic wavesfrom depth-dependent inhomogeneities. Geophysics, 41, 441-458.

• Rutherford, S. R., andWilliams, H. R., 1989. Amplitude-versus-offsetvariations in gas sands: Geophysics, 54, 680688.

• Santoso, D., Alfian, Alam, S., Sulistiyono, Hendraya, L., and Munadi,S., 1995. Estimation of limestone reservoir porosity by seismic attributeand AVO analysis: Expl. Geophys., 26, 437443.

• Shuey, R.T., 1985. A simplification of Zoeppritz Equations. Geo-physics, 50, 609-814.

• Snyder, A. G., and Wrolstad, K. H., 1992. Direct detection using AVO,Central Graben, North Sea: Geophysics, 57, 313325.

72

• Stewart, R. R., Zhang, Q., and Guthoff, F., 1995, Relationships amongelastic-wave values (Rpp, Rps, Rss, Vp, Vs, ρ, σ, κ): CREWES ResearchReport 1995, Chapter 10.

• Strebelle, S., 2002. Conditional Simulation of Complex GeologicalStructures using Multiple-point Statistics. Math. Geol., V. 34, No.1, 1-26.

• Taner, M.T., 2001. Seismic Attributes. Recorder, Can. Soc. of Expl.Geophys.,48-56.

• Tosaya, C., and Nur, A., 1982. Effects of diagenesis and clays oncompressional velocities in rocks. Geophys. Res. Lett., 9, 5-8.

• von Seggern, D., 1991. Spatial resolution of acoustic imaging with theBorn approximation: Geophysics, Soc. of Expl. Geophys., 56, 1185-1202.

• Sengupta, M., 2000. Integrating rock physics and flow simulation to re-duce undcertainties in seismic reservoir monitoring. Ph.D. dissertation,Stanford University.

• Wyllie, M.R.J., Gregory, A.R., and Gardner, L.W., 1956. Elastic wavevelocities in heterogeneous and porous media. Geophysics, 21, 41-70.

• Xu, Y., and Bancroft, J.C., 1997. Joint AVO analysis of PP and PSseismic data. CREWES Research Report Volume 9, 34-44.

• Zoeppritz, K., 1919. Erdbebenwellen VIIIB, Ueber Reflexion and Durch-gang seismischer Wellen durch Unstetigkeitsflaechen: Goettinger Nachrichten,I, 66-84.

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