Production tomography merges geophysics with reservoir engineering

Streamline simulators approximate 3D fluid flow calculations by a sum of solutions along 1D particle trajectories that are tangential to the local fluid velocity or streamlines.

Fig. 1 illustrates the major steps in streamline simulation for a one-quarter of a heterogeneous five-spot pattern (Fig. 1a). The steps include:

• Tracing streamlines based upon a velocity field, typically derived numerically using finite difference or finite element methods (Fig. 1b).
• Computing particle travel time or time of flight along streamlines. The travel time variations capture the effects of geologic heterogeneity on the fluid flow and also provide a quantitative approach to flow visualization (Fig. 1c).
• Decoupling the saturation calculations from geologic heterogeneity by using the time of flight and streamline trajectories to define a new coordinate system. This step reduces the 3D saturation equations to a set of 1D equations with one equation for each streamline.
• Solving for fluid saturation along the streamlines either analytically or numerically (Fig. 1d).
• Occasionally updating the streamlines to account for mobility effects or changing field conditions such as rate changes and infill drilling.

The computational advantage of the streamline approach can be attributed to the fact that streamlines need to be updated only infrequently and the transport equations along streamlines are decoupled from the underlying grid, thus allowing for faster solution.

Furthermore, the self-similarity of the solution along streamlines frequently allows us to compute the solution only once for mapping it to the time of interest.

Streamlines vs. seismic rays

A fundamental basis for the common framework between geophysical imaging and dynamic data integration is the analogy between a propagating fluid front and a propagating wave front.^2-5

Fig. 1c shows a contour plot of the streamline time of flight and clearly, the time of flight depicts the fluid front propagation. The front location at any time is given by surfaces of equal time of flight or isochrones.

Streamlines are trajectories that are perpendicular to the fluid front. In seismic tomography, wave fronts are defined by surfaces of equal phase (propagation time) and seismic rays are normals to the wave front. In that sense, a direct analogy exists between streamlines and seismic rays, although there are significant underlying differences as well.

For example, rays can cross in physical space but not in phase space, whereas streamlines do not cross in physical space. It can be shown that high-frequency asymptotic solutions to the governing equations for fluid flow and transport actually mimic the solutions found in wave propagation.^{3 4 6}

Such high-frequency solutions emphasize the rapidly varying spatial and temporal attributes of the solution. This leads to distinct scalar equations for breakthrough or arrival times of the fluid fronts and amplitudes of the associated production response such as water-cut or tracer response.

The history matching or dynamic data integration problem can now be cast in a manner analogous to seismic tomography and waveform imaging. This allows one to use efficient techniques from geophysical imaging for integration of production data into high-resolution reservoir models.^{3 7}

Production tomography

To exploit the analogy between streamline and ray tracing for dynamic data integration into high-resolution reservoir models, we used techniques from the asymptotic ray theory by casting the streamline time of flight equation into the form of an Eikonal equation, the governing equation for travel-time tomography.

The Eikonal equation appears in many contexts such as elastic and electromagnetic wave propagation and its properties are well developed in literature. Streamline-based production data integration can draw upon such similarities to utilize existing methods from geophysical and medical imaging. Briefly, our approach to production data integration with streamlines consist of the following two steps:

1. Analytic computation of sensitivity. Sensitivities quantify change in production response induced by a small deviation in reservoir properties and typically consume a majority of the computation time during inverse modeling. One can exploit the analogy between streamlines and ray tracing to derive analytic expressions for the sensitivity of streamline time of flight to reservoir properties such as porosity and permeability.

The sensitivity computations now involve a single streamline flow simulation and can result in orders of magnitude savings in computation time over conventional numerical methods for sensitivity computation.

2. Two-step inversion. Production data integration is accomplished in the manner of waveform imaging. It involves iterative least-squared matching of the production response starting with a geologic model based on static data. The two-step procedure first matches the first arrival or breakthrough times, and then matches the amplitudes of the production response.

Additional constraints imposed to ensure stability of the solution include a norm penalty that minimizes the deviation from the prior geologic model and a roughness penalty that allows for large-scale changes that are consistent only with the low resolution of the production data.

Tracer test

A multiwell, multitracer interwell injection study was carried out in the McCleskey sandstone, Ranger field, Texas. The 320-acre area includes 13 producing and 4 injection wells, injecting seven different tracers. Fig. 2 shows the tracer-injection pattern.

The simulation model included a 31 3 45 3 6 mesh that corresponds to a constant areal grid block dimension of 100 3 100 ft. The vertical grid block dimension varied to account for changes in the formation thickness. Initial permeability distribution was obtained based on static data, primarily cores, well logs, and geologic information.

The tracer concentration history for NaSCN (sodium thiocyanate), a conservative tracer, provided the data for improving the permeability distribution in the study area.

Figs. 3a-3c show the initial, peak-arrival time match and amplitude match for the NaSCN data at the sampling wells, and Figs. 4a and 4b show the initial and final permeability distributions. It is clear that the final permeability field has retained the major attributes of the initial permeability model. This is a desirable feature because typically our initial model will be based on static data and any significant deviation from it might render the model geologically implausible.

The permeability change based on the integration of the tracer data (Fig. 4c) maps the difference between the post-inversion and the initial permeability field. These are mostly large-scale changes, consistent with the low resolution of the dynamic data.

We compared our inversion results to the permeability distribution obtained by a manual history matching of the tracer data using a finite-difference simulator.⁸ In this study, the tracer responses were matched by adjusting permeability/thickness values of individual layers by applying a series of permeability multipliers to a three-layer model.

The study indicated a close agreement between the location and the nature of the permeability multipliers with the location and the shape of areas with higher and lower permeability derived from tracer data inversion. The use of multipliers, however, introduces sharp and potentially geologically unrealistic discontinuities in the permeability distribution.

The proposed dynamic data integration procedure leads to less personal bias compared to manual history matching and also allows for model assessment in terms of resolution and uncertainty of the estimated parameters.³

Most importantly, each inversion run takes only about 10-15 min in a PC to complete compared to several months for a manual history match using a finite difference simulator. This can translate to significant savings in terms of computational and manpower costs.

Interference test

Conoco Inc.'s borehole test facility, Kay County, Okla. (Fig. 5), was the site of an interference tests that illustrates fractured reservoir characterization based on transient pressure measurements.

The tests were conducted in a skewed five-spot pattern. Wells GW-1 through GW-5 penetrate the Fort Riley formation, a fractured limestone formation. The Fort Riley limestone is part of the Lower Permian Chase Group that consists primarily of limestones and shales.

The Fort Riley limestone has about a 1.2 md average permeability. One pressure interference test studied involved groundwater withdrawal at a constant rate of 2.3 l./min from the westernmost well, GW-5. Pressure response was observed in the surrounding wells (Fig. 6).

One advantage of the asymptotic approach adopted in this study is that it allows one to define pressure fronts and arrival times associated with pressure fronts based on the transient pressure response.^{4 9} Starting with an initial permeability field, the pressure arrival times were matched using the iterative procedure discussed in this article (Figs. 7a-7c). The main feature of the permeability field is a dominant fracture in the east-northeast direction located to the north of Well GW-3.

Independent geophysical experiments verified the location of the fracture with respect to the Well GW-3. Seismic amplitude attenuation indicated a fracture or fracture zone to the north of GW-3 as determined using the transient pressure data.¹⁰ Furthermore, a slant well, located slightly to the east of GW-3 verified the location of the fracture zone (Fig. 8).

Water-cut response

The North Robertson (Clearfork) Unit (NRU) is a heterogeneous, low permeability carbonate reservoir in the Permian basin of West Texas. The NRU reservoirs typically are characterized by a high degree of vertical and lateral heterogeneity with low porosity (7.5% limestone matrix), low permeability (0.1-10 md), poor waterflood sweep efficiency. and a low oil recovery factor.

The nonreservoir rock types are relatively impermeable and form vertical barriers contributing to reservoir heterogeneity and compartmentalization. Identification of the location and distribution of these barriers is critical to the success of secondary and tertiary recovery efforts. The fractures also contribute to heterogeneity, further complicating the production response.

Two sections of NRU, Sections 326 and 327, were selected for a detailed study (Fig. 9a). The reservoir model consists of a 100 3 50 3 12 mesh or a total of 60,000 cells. Altogether data from 42 wells including 27 producing wells and 15 injection wells characterized the heterogeneity based on the water cut response from the producing wells.^{3 7}

An initial reservoir model used geostatistical methods based on well log data from 30 wells.

Fig. 9b compares the water breakthrough response at the producing wells based on the initial model with the field data. Fig. 9b also shows the superimposed calculated and observed breakthrough times at the producing wells after 20 arrival time iterations. Clearly, the breakthrough times are now in complete agreement with the field data.

The computation time for the arrival time matching was less than 2 hr of cpu (central processing unit) time in a workstation.

Fig. 9c displays the permeability variations after incorporating the water breakthrough response. The permeability field derived from this inversion appears consistent with two earlier studies: one based on decline curve analysis and the other based on inversion of water-cut data using a coarser model.

Fig. 9d shows the streamline pattern for the final model.

Time-lapse seismic monitoring

Seismic imaging of reservoir dynamics, known as time-lapse or 4D seismic is a relatively new tool in reservoir monitoring.^11-13 Time-lapse monitoring promises to provide high-resolution, three-dimensional images of reservoir processes.

The integration of these data with production observations, such as water cut, constitutes a major advance in reservoir characterization and management. Efficient integration of time-lapse seismic and water-cut data, however, is no trivial undertaking.

The flexibility and speed provided by our production tomography approach will be extremely helpful in this endeavor. While we have no case studies completed at this time, we have examined the utility of a joint inversion of seismic arrival time and water-cut observations.

In particular, we have set up a synthetic test case in which seismic arrival times from a crosswell survey are combined with water-cut measurements (Fig. 10a).

In our simulation, we have a high permeability channel extending from the injector to the producer (Fig. 10a, left panel). Because our water-cut data are gathered from the entire producing interval they provide little resolution in the vertical direction. On the other hand, time-lapse seismic observations do constrain vertical and lateral changes in reservoir conditions.

The arrival times are sensitive to changes in saturation and pressure along the ray path between the source and receiver (indicated by the dark black curve in Fig. 10a).

The seismic ray sensitivities coupled with our streamline-based sensitivities relate changes in permeability to changes in saturation and pressure. The permeability sensitivities are constructed from the streamlines, the thin lines in Fig. 10a. Thus, by merging the ray-based seismic sensitivities and the streamline-based sensitivities, one may connect changes in permeability to changes in seismic arrival times.

A high-resolution production tomographic image of reservoir permeability is obtained by combining seismic arrival times and water-cut data (Fig. 10b, left panel).

The imaging approach is extremely fast, utilizing the efficiencies of seismic ray methods with the speed of streamline simulation. Furthermore, in using the seismic and water-cut data jointly, one obtains a more robust solution that, in the end, fits the water-cut values very well (Fig. 10b, right panel).

The integration of seismic and production data is a topic of active research and exciting developments are expected in the not-too-distant future.

Acknowledgments

This article is based on a collaborative research project between the Lawrence Berkeley Laboratory, Texas A&M University, and multiple industry partners. The project has been supported by the Assistant Secretary for Fossil Energy, Office of Natural Gas and Petroleum Technology, through the Natural Gas and Oil Technology Partnership, under US Department of Energy Contract No. DE-AC03-76SF00098.

The authors would like to thank Bob Lemmon from the National Petroleum Technology Office for his continued support and encouragement.

References

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The authors

Akhil Datta-Gupta is an associate professor of petroleum engineering at Texas A&M University, College Station, Tex. He previously worked with BP Exploration-Research and the Lawrence Berkeley National Laboratory. His research interests include reservoir characterization, simulation, and environmental remediation. Datta-Gupta has a PhD in petroleum engineering from the University of Texas at Austin. He is an SPE member.

Don Vasco is a staff scientist at the Berkeley Laboratory at the University of California at Berkeley, where he develops and implements algorithms for seismic and hydrologic inversion. His current research interests are inverse theory, global earth structure, and asymptotic methods for imaging. Vasco holds a BS in geophysics from the University of Texas at Austin and a PhD from the University of California, Berkeley.