Analytical methods from materials science can be applied to give quick, low-cost information about geomechanical reservoir changes during production.
Hydrocarbon extraction induces mechanical deformations of the reservoir and of the surrounding formation. Study of these deformations is important to reservoir monitoring, because they influence oil production (fractures, borehole instability) and causes environmental changes (land subsidence). Measuring accompanying effects, e.g. differences in travel times of seismic waves (4D seismic), helps to reconstruct the fluid pressure profile inside the reservoir and hence to reveal undepleted pockets.
To understand, to control and to use these effects requires calculating strains and stresses inside rocks. This usually is done using finite element codes. Finite element codes give the most accurate results, but at relatively low speed. “Speed” is determined not only by actual computation time, but also by time spent collecting data, setting up the model, managing the project, etc. This additional “logistic” time can be the major issue.
Analytical approaches are good alternatives to numerical simulations in cases where high accuracy either is not required or it is not possible due to lack of data, and where the processing time and costs are crucial. The idea is to use an idealized formation model, which captures the essence of a phenomenon, but it is still simple enough to be solved analytically. Analytical solutions provide a fast, although approximate, formation evaluation. Once derived, they can be implemented easily.
Material sciences
Mechanical deformations of solid bodies have being studied for decades in material sciences. Originally, it was done in the context of thermo-elasticity. In recent years, this topic has attracted significant attention in nanotechnology. Thermo-elasticity, nanotechnology and subsurface geomechanics each deals with essentially the same problem. In each case, strains and stresses from external sources are calculated. The sources differ. There is heat flux in thermo-elasticity, the so-called “quantum dots” in nanotechnology and fluid pressure changes in subsurface geomechanics. Conceptually, the differences do not play a role because the structure of the equations remains the same. After adaptation, material science analysis can be applied to reservoir monitoring.
Geertsma [1] first recognized the analogy and has used the thermo-elastic solution of Refs. [2,3] to study land subsidence. This describes elastic deformations in a uniform and isotropic half-space. It is called the “nucleus of strain.” It represents the elastic response on a point-wise change of strain. The nucleus of strain solution can be a starting point to determine deformations induced by reservoirs with arbitrary shapes and pressure profiles. To do so, the solution needs to be rescaled and to be integrated over the reservoir volume.
During hydrocarbon production, the pore pressure drops mainly in the vicinity of the borehole. It depends on the distance to the borehole and has a weak dependence on the angle. This leads to cylindrical geometry, where angle dependencies are neglected. Geertsma has performed integration of the nucleus of strain solution over a uniformly depleted, infinitely thin circular reservoir. A generalization of Geertsma’s model, which accounts for logarithmic pressure profiles and finite reservoir thicknesses, is used in Shell.
When one extracts oil, the rock matrix experiences an additional load from the overburden. The effective stress inside the reservoir changes to maintain the equilibrium distorted by the reduction of the pore fluid pressure. Part of the load is carried by the surrounding formation rather than by the reservoir itself. The ratio between this part and the total additional load is called the “arching factor”. Knowledge of the arching factor is required to predict the rock matrix collapse and the resulting sand production.
Depletion arching factors. Labels “simplified” and “full analytics” refer to [1] and to the generalized Geertsma model. “Ellipsoid” represents the Eshelby model implemented by Victor Dunayevsky.
Under some simplifying assumptions, the vertical arching factor admits a geometric interpretation. For points inside a reservoir, the vertical arching is proportional to the fraction of sight covered by the reservoir side boundaries. This interpretation is convenient to get an idea of the arching effect magnitude. If the reservoir is infinitely thin, then the whole view is covered by the reservoir top and bottom, so the vertical arching factor is zero.
The cylindrical model has been used to investigate the arching effect in one Shell oil field and it has been compared with outcomes of a finite element code. The analytical and numerical results were practically the same, even in the case where the numerical model took into account the formation complexity (nine layers with different material properties). Arching factors near the borehole also have been evaluated by integrating solid angles.
Going back to the nucleus of strain solution, we notice that it expresses elastic deformations through the Newtonian potential and its derivatives. Consequently, calculation of elastic deformations induced by reservoirs is the same as calculation of gravitational fields produced by finite bodies. There is extensive literature on the later subject. In particular, in book [4], it is explained how to calculate the gravitational field of a parallelepiped. Applying technique of Ref. [4], one integrates the nucleus of strain solution over a block-shaped reservoir. The resulting strains and stresses are written in terms of algebraic functions such as logarithm and arctangens. These functions can be implemented straighforwardly and they are calculated virtually instantaneously. Analytical formulas can model stress-fields even if they exhibit a singular behavior, which is very difficult to do with standard codes.
Block-building
By combining a sufficient number of blocks, complicated reservoirs can be modeled. To illustrate the “block-building’’ procedure, consider two identical blocks with a uniform pressure drop. The induced deformation at point X is equal to the sum of two separate contributions from these blocks. Alternatively, deformation can be represented as a sum of deformations induced by the same block at two different points X and X*. Thus, one can perform summation either over blocks or over points. The second method is more effective numerically. One takes a single block, calculates the induced elastic deformations everywhere in space, and stores the numbers in computer memory. These pre-calculated numbers are then added with proper weights. Implementation of this algorithm has helped to accelerate processing of reservoir monitoring data [5-8].
In a more realistic case of a layered formation, even a nucleus of strain solution cannot be found analytically. A number of numerical procedures have been developed to study propagation of seismic waves in layered media. The most common one is the Kennett algorithm [9], which always is numerically stable.
Normalized vertical arching factor near the edge of a reservoir.
Adaptation of the Kennett algorithm to stationary elastic deformations is not straightforward. By taking the limit of zero frequency, one encounters uncertainties of the type, which cannot be treated numerically. This was overcome in Ref. [10]. The method is fast and relatively simple. It occupies an intermediate position between fully analytical calculations and finite element modeling, and opens new opportunities in reservoir surveillance.
Lessons to learn
Although use of computer codes is the mainstream in subsurface geomechanics, analytical modeling remains to be a powerful tool which should not be ignored. The crucial point is the price of information. Important information can be obtained at low cost. If this information is not sufficient for purposes of a particular project, then one can invest more efforts and run large-scale numerical simulations. Another lesson shows that there is a lot of knowledge around us. One can make a substantial impact on oil business by applying existing methods from other areas of science and technology.
Acknowledgments
I appreciate cooperation with Stephen Bourne, Andy McGinn, Paul Hatchell, Paul Hofstra, Cor Kenter, Hans Potters and Peter Schutjens, and their efforts to bring above theoretical ideas in real life and to make them work.
Literature cited
[1] J. Geertsma, “Land Subsidence above Compacting Oil and Gas Reservoir”, J. Petrol. Technol. V. 25, 734 - 744 (1973).
[2] Mindlin R. and Cheng D. H., “Thermoelastic Stress in the Semi-Infinite Solid”, J. Appl. Phys., V. 21, 931 - 933 (1950).
[3] B. Sen, “Note on Stresses Produced by Nuclei of Thermo-elastic Strain in a Semi-Infinite Elastic Solid”, Quatr. Appl. Math., V. *b*8*b*, 365 - 369 (1950).
[4] W. D. MacMillan, “The Theory of the Potential”, Dover Publications, New York, 1958.
[5] P. Hatchell and S. Bourne, “Rock under strain: Strain induced time-lapse time shifts are observed for depleting reservoirs’’, The Leading Edge, 1222 - 1225, December 2005.
[6] K.P. Maron, S. Bourne, K. Wit, and P. McGillivray, “Integrated reservoir surveillance of a heavy oil field in Peace River, Canada”, 67th EAGE Conference and Exhibition, Madrid 2005.
[7] S. Bourne, K. Maron, S. Oates, and G. Mueller, “Monitoring deformation of a carbonate field in Oman: Evidence for large-scale fault reactivation from microseismic, InSAR & GPS”, 68th EAGE Conference and Exhibition, Vienna 2006.
[8], S. Bourne and P. Hatchell, “Monitoring reservoir compaction by joint inversion of seafloor subsidence and time-lapse seismic timeshift data’’, 69th EAGE Conference and Exhibition, London 2007.
[9] B. L. N. Kennett, “Seismic wave propagation in stratified media”, Cambridge University Press, Cambridge 1983.
[10] B. N. Kuvshinov, “Reflectivity method for geomechanical equilibria”, submitted to Geophysical Journal International (2007).
