Flow models predict 4D suitability

John R. Fanchi
Fanchi Enterprises
Houston

Integrated flow model predictions can determine the feasibility of applying 4D seismic monitoring techniques to reservoir management.

Sequential application of 3D seismic surveys has led to the concept of time-lapse (4D) seismic monitoring. Monitoring changes in reservoir geophysical attributes over the reservoir life is the basis of 4D seismic.

The changes can be predicted by calculating reservoir geophysical attributes as a function of time.

This article describes a procedure for integrating a petrophysical model with a flow simulator. Properties of an integrated flow model are illustrated by applying a particular model (Boast4D) to the following three commonly encountered reservoir management scenarios:

• Water injection into an undersaturated oil reservoir
• Gas injection into an undersaturated, light oil reservoir
• Aquifer influx into a gas reservoir.

Seismic

Historically, seismic analyses have been of interest within the context of reservoir management as a means of establishing the structural size of the reservoir.

Seismic measurements discussed by such References as 1-3 provide much information on directly using these measurements to characterize the reservoir structure.

The emergence of time-lapse (4D) seismic monitoring and reservoir geophysics has expanded the role of seismic analysis.^4-8

A commonly used procedure for applying seismic technology to the monitoring of fluid flow in porous media is the following iterative process:^8-10

Step 1-Acquire, process, and interpret 3D seismic data at two or more times, such as at tⁿ and t^m, where tⁿ t^m and indices n and m denote points in time.

Step 2-Combine the 3D seismic interpretation with other geoscience and engineering data to characterize the reservoir.

Step 3-Use a flow simulator to model fluid flow performance.

Step 4-Use flow simulator properties such as pressure and saturation at time t^m in a petrophysical model to calculate reservoir geophysical attributes for comparison with 3D seismic data acquired at the same time.

If necessary, the above steps are repeated with modifications to flow simulator input data, including reservoir characterization, until a match of flow performance and 3D seismic interpretations is obtained at all historical times. Predictions then can be made with the history-matched model.

The reservoir simulation process is discussed in several sources such as References 11 and 12.

Steps 3 and 4 are unnecessarily cumbersome because most flow simulators do not calculate reservoir geophysical attributes. As a result, information from the flow simulation Step 3 must be converted to a format suitable for analysis in the petrophysical model Step 4.

In addition, errors may be introduced into the calculation of reservoir geophysical attributes if fluid properties in the petrophysical model do not match the corresponding fluid properties in the flow simulator.

For example, the use of standard correlations of fluid properties, such as those discussed in References 13 and 14, in a petrophysical model may not adequately represent phase densities and compressibilities calculated by a flow simulator. This is especially true in the calculation of live oil, gas, and water moduli as the inverse of their respective compressibilities.

These problems can be avoided if the petrophysical model is incorporated into the flow simulator so that it uses the same fluid property model.

Petrophysical model

A petrophysical model must be able to calculate reservoir geophysical attributes that can be compared with acoustic measurements. Acoustic measurements may be obtained with either seismic surveys or well logging tools.

The primary reservoir geophysical attributes that are calculated by the petrophysical model described in this article are:

• Acoustic impedance
• Reflection coefficient
• Compressional velocity
• Shear velocity.

Each of these attributes may be used within the context of reservoir management.

Seismic waves are vibrations that propagate through the earth from a source to a reflecting surface. When the seismic wave encounters a reflecting surface, it is partially transmitted and partially reflected.

A seismic reflection occurs at the interface between two regions with different acoustic impedances. Acoustic impedance in a medium is defined by Equation 1 (see equation box [123,162 bytes]).

In Equation 1, rB is the bulk density of the medium and V_p is the compressional velocity of the wave in the medium.

A change in acoustic impedance will cause sound waves to reflect. The ability to reflect a sound wave by a change in acoustic impedance is quantified in terms of the reflection coefficient R at the interface between two contiguous layers (Equation 2).

In Equation 2, Subscripts 1 and 2 refer to the contiguous layers. The transmission coefficient T at the interface is 1 minus the reflection coefficient, thus T = 1 - R.

If seismic data have enough resolution to show the reflecting boundaries of a geologic layer, then the seismic wave amplitudes may be useful for characterizing reservoir petrophysical properties.

When a correlation does exist between seismic amplitude and a grouping of petrophysical parameters, the correlation may help guide in the distribution of reservoir properties between wells.

A growing body of literature discusses this application.^15-18

Seismic compressional velocity and shear velocity are calculated with Equations 3 and 4.^{2 19}

The flow simulator calculates oil, water, and gas densities and saturations. Ordinarily, fluid phase properties change during the reservoir life as pressure and saturation change. Some values of these properties are further complicated by dependence on phase composition.

A good fluid property model accounts for all significant effects, such as dependence of live oil density on solution gas.

Equation 5 is a widely used expression for the modulus K* that was derived by Gassman²⁰ from the theory of elasticity of porous media (also see References 2 and 19).

According to the Gassman model, there is no discernable effect of fluid properties on K* if K _M equals K_G. The model also shows a dependence on porosity, phase properties, and fluid saturations.

The Gassman equation represents perfect coupling between the pore fluid and the solid skeleton of the porous medium.¹⁹ It is a zero frequency approximation to a more general theory presented by Geertsma and Smit.²¹

Integrated flow simulator

Monitoring changes in reservoir geophysical attributes while producing the reservoir is the basis of 4D seismic monitoring. These changes can be predicted within existing flow simulators by incorporating a petrophysical model that calculates reservoir geophysical attributes as a function of time.

The procedure is illustrated using the simplified flow chart for a simulator shown in Fig. 1 [72,268 bytes].^{12 22} The simulation program begins by reading input data and initializing the reservoir. Information for time-dependent data is then read.

These data include well and field control data. It is part of the time stepping process because well and field control data usually change with time.

The flow equation coefficients and the primary unknown variables, such as pressures and saturations, are then calculated. The primary variables are calculated with one of two methods referred to as Impes and Implicit in the figure.

The Implicit procedure allows updating flow coefficients using values of the primary variables at the new iteration level. The Impes procedure makes calculations using flow coefficients from the previous time step.

References 11, 12, 22, and 23 discuss the differences between the Impes and Implicit techniques.

When the solution of the flow equations is complete, flow properties are updated, and output files are created before the next time step begins.

The flow simulator described here is a modified version of the Impes simulator Boast II that was released by the U.S. Department of Energy in 1987.²⁴

The flow simulator version that includes the petrophysical model described previously is called Boast4D. It requires data that are not usually part of a traditional flow simulator input data set.

In particular, the Boast4D user must enter K_M, K_G, m*, and r_ma. These parameters may vary from one location to another, so that an integrated flow simulator allows the user the flexibility to enter them as functions of position in the same way that permeability and porosity are entered.

These data are entered as part of the initialization step in Fig. 1.

Reservoir geophysical attributes are calculated initially and at the end of each time step. The calculations are performed at the point in Fig. 1 where physical properties are updated.

It is unnecessary to include the petrophysical model in the Impes or Implicit looping procedure because the reservoir geophysical attributes of interest do not influence fluid flow. They are, instead, dependent on fluid flow.

The calculated reservoir geophysical attributes are available as user-requested output variables.

Waterflood example

An example of a waterflood in an undersaturated oil reservoir shows the acoustic response to a system in which undersaturated oil is displaced by injected water.

Fluid properties in this example are from a case study given in Chapter 13, Reference 12. For simplicity, a horizontal, single layer, linear 10 cell model is used.

Water is injected in Cell 1, and production is from Cell 10. Acoustic impedance is monitored as a function of time at Cell 5, which serves as an observation cell.

The difference between oil and water phase densities and compressibilities in this example is small relative to a liquid-gas system. The ratio of matrix modulus to grain modulus K_M/K_G is set to 2/3 so that some variation in the V_p/V_s ratio can appear.

Fig. 2a [114,367 bytes] shows V_p/V_s ratio vs. time and water saturation vs. time at observation Cell 5. The velocity ratio V_p/V_s increases slightly as the observation cell fills with injected water.

Gas injection example

The example of gas injection into a saturated, light oil reservoir is based on the first SPE comparison project. ²⁵

A saturated, light (59° API) oil is produced from a corner block in the lowermost layer of a three-layer square grid. Lean gas is injected into the upper layer at the opposite corner.

The injected gas is expected to propagate most rapidly through the upper layer. Therefore, an observation point was selected in the upper layer midway between the injector and producer. This point lets one observe the gas front pass as the injected gas flows toward the pressure sink at the production well.

Fig. 2b shows a plot of gas saturation vs. time and V_p/V_svs. time at the observation cell for K_M/K_G = 2/3 and K_M/K_G = 3/2.

The velocity ratio V_p/V_s decreases slightly as the gas front passes the observation cell.

Aquifer example

Aquifer influx into a gas reservoir is the last example. The previous examples did not show much response by the velocity ratio V _p/V _s to flood front passage.

This third example was designed to exhibit larger changes in V_p/V_s as a function of field performance over time. In particular, a dipping gas reservoir with aquifer influx was studied.

The reservoir is represented as a dipping, two-layer cross-section (Fig. 12-2, Reference 12). Initial gas saturation is 70%, and initial irreducible water saturation is 30%. A downdip aquifer provides pressure support and water invasion as the reservoir is produced.

The lowermost cell adjacent to the aquifer was selected as the observation cell. This cell responds most quickly to water invasion and exhibits the acoustical response associated with water front advance.

Fig. 2c presents water saturation vs. time and velocity ratio V_p/V_s vs. time for the observation cell. The petrophysical model and, by inference, petrophysical properties, have a sensitive dependence on certain input parameters.

For example, the velocity ratio V_p/V_s in this application is sensitive to a change in irreducible gas saturation (S_gr).

An irreducible gas saturation of 0% results in complete gas displacement from the water-invaded zones. The resulting elimination of gas from the calculation of reservoir geophysical attributes in the water-invaded part of the reservoir causes a significant change in the ratio V_p/V_s.

If the example is run with an irreducible gas saturation of 3%, the relatively large change in V_p/V_s is no longer observed because the presence of a small amount of gas significantly changed the compressibility of the system. Fig. 2c displays these results.

Flood behavior

Table 1 [41,235 bytes] summarizes the steady-state values of the velocity ratio V _p/V _s before and after the flood front passed the observation cell for each of the examples described above. The velocity ratio changed most when free gas saturation either developed in a cell that contained only liquid or completely disappeared from a cell that had nonzero gas saturation.

Kelamis, et al.,²⁶ referred to their observation of similar gas saturation behavior as a "key saturation signature" in their seismic monitoring study of a clastic oil reservoir with a large gas cap in the Arabian Gulf.

They concluded that the key saturation signature was needed to successfully implement a 4D seismic monitoring program.

Small differences in the velocity ratio were noted in most of the examples discussed. These results substantiate Lumley and Behrens⁹ observation that 4D seismic is not a panacea.

They recommended that a feasibility study be conducted before a 4D-seismic survey is undertaken. The integrated flow model provides a tool for performing such studies.

Incorporating petrophysics

Petrophysical models should be incorporated in flow simulators to maximize the consistency of calculation procedures and optimize the integration of flow data with petrophysical data in the history matching process.

Calculations of 4D seismic performance based on petrophysical models like the one coded in Boast4D can be used to predict the time-dependent behavior of reservoir geophysical attributes, notably acoustic impedance, reflection coefficient, compressional velocity, and shear velocity.

Predictions of 4D seismic performance can be used to estimate the feasibility of applying 4D seismic monitoring techniques to a particular reservoir management scenario.

References

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McQuillin, R., Bacon, M., and Barclay, W., An Introduction to Seismic Interpretation, Gulf Publishing, Houston, 1984.

Sheriff, R.E., Geophysical Methods, Prentice-Hall, Englewood Cliffs, N.J., 1989.

Richardson, J.G., "Appraisal and Development of Reservoirs," Geophysics, February 1989, p. 42.

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Anderson, R.N., "Method Described for Using 4D seismic to track reservoir fluid movement," OGJ, Apr. 3, 1995, pp. 70-74.

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Gassman, F., "Elastic Waves Through a Packing of Spheres," Geophysics, Vol. 16, 1951, pp. 673-85.

Geertsma, J., and Smit, D.C., "Some Aspects of Elastic Wave Propagation in Fluid-Saturated Porous Solids," Geophysics, Vol. 26, 1961, pp. 169-81.

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Fanchi, J.R., Kennedy, J.E., and Dauben, D.L., BOAST II: A Three-Dimensional, Three-Phase Black Oil Applied Simulation Tool, U.S. Department of Energy, Bartlesville Energy Technology Center, Oklahoma, 1987.

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