# Asymptotic Solutions of the Diffusivity Equation and Their Applications

- Authors
- Michael J. King (Texas A&M University) | Zhenzhen Wang (Texas A&M University) | Akhil Datta-Gupta (Texas A&M University)
- DOI
- https://doi.org/10.2118/180149-MS
- Document ID
- SPE-180149-MS
- Publisher
- Society of Petroleum Engineers
- Source
- SPE Europec featured at 78th EAGE Conference and Exhibition, 30 May-2 June, Vienna, Austria
- Publication Date
- 2016

- Document Type
- Conference Paper
- Language
- English
- ISBN
- 978-1-61399-457-3
- Copyright
- 2016. Society of Petroleum Engineers
- Disciplines
- 5.6.4 Drillstem/Well Testing, 5.1.5 Geologic Modeling, 5 Reservoir Desciption & Dynamics, 5.5 Reservoir Simulation, 5.6 Formation Evaluation & Management, 5.6.3 Pressure Transient Analysis
- Keywords
- Drainage Volume, Unconventional Reservoirs, Pressure Transient, Rate Transient
- Downloads
- 3 in the last 30 days
- 649 since 2007

- Show more detail
- View rights & permissions

SPE Member Price: | USD 5.00 |

SPE Non-Member Price: | USD 28.00 |

Understanding how pressure fronts propagate (diffuse) in a reservoir formation is fundamental to welltest analysis and reservoir drainage volume estimation. We have developed an alternative approach to the solution of the 3-D diffusivity equation by directly solving the propagation equation for the "pressure front" defined as the maximum pressure response for an impulse source. The pressure front equation is a form of the Eikonal equation, which is a high frequency asymptotic solution of the diffusivity equation in heterogeneous reservoirs and whose properties are well developed in the literature. Most importantly, the Eikonal equation can be solved very efficiently by a class of solutions called the Fast Marching Methods (FMM) for a "diffusive time of flight" (DTOF) that governs the propagation of the "pressure front" in the reservoir. The "diffusive time of flight" can be used as a spatial coordinate to reduce the 3-D diffusivity equation into an equivalent 1-D formulation, leading to a simplified method for rapid reservoir modeling. Recent papers have explored the utility of this 1-D approach for performance prediction using finite difference numerical simulation. The method is especially well suited to the interpretation of the drainage volume, which is of great help in well spacing calculations and in the context of unconventional reservoirs, multi-stage fracture spacing optimization.

In this paper we introduce an analytic solution technique for the diffusivity equation, which provides a direct relationship between production data and the reservoir drainage volume. The analytic formulation provides for the direct calculation and extension of many simple well test, rate transient and well performance concepts such as depth of investigation, welltest derivative, drainage volume, flow regimes and well productivity. As with other analytic approaches, these solutions allow superposition in space and in time, which allows for the solution for multiple wells, multiple flow rates, and bounded and composite reservoirs. We validate our approach against well-known solutions in pressure and rate transient analysis usually solved in Laplace space, including pressure transients with wellbore storage and skin. Our study demonstrates that the new approach yields results very close to the known solutions calculated via numerical inversion of the Laplace transform, and indicates how to extend these solutions to problems with heterogeneity and complex fractured well geometry.

File Size | 5 MB | Number of Pages | 37 |

Chen, L., & King, M. (2016). Integration of Pressure Transient Data Into Reservoir Models Using the Fast Marching Method. SPE Europec. Vienna, Austria, 30 May–2 June 2016: doi: 10.2118/180148-MS.

Cipolla, C., & Wallace, J. (2014). Stimulated Reservoir Volume: A Misapplied Concept? Society of Petroleum Engineers, doi:10.2118/168596-MS.

Fujita, Y., Datta-Gupta, A., & King, M. (2015). A Comprehensive Reservoir Simulator for Unconventional Reservoirs Based on the Fast Marching Method and Diffusive Time of Flight. SPE Reservoir Simulation Symposium. Houston, Texas, USA, 23–25 February: doi: 10.2118/173269-MS.

Gringarten, A. C., Al-Lamki, A., Daungkaew, S., Mott, R., & Whittle, T. M. (2000). Well Test Analysis in Gas-Condensate Reservoirs. Society of Petroleum Engineers. doi:10.2118/62920-MS.

Ibragimov, A., Khalmanova, D., Valko, P. P., & Walton, J. (2004). Analytical Method of Evaluating Productivity Index for Constant Production Rate or Constant Wellbore Pressure. Society of Petroleum Engineers. doi:10.2118/89935-MS.

Ilk, D., Okouma Mangha, V., & Blasingame, T. (2011). Characterization of Well Performance in Unconventional Reservoirs using Production Data Diagnostics. Society of Petroleum Engineers. doi:10.2118/147604-MS.

Kim, J., Datta-Gupta, A., Brouwer, R., & Haynes, B. (2009). Calibration of High-Resolution Reservoir Models Using Transient Pressure Data. SPE Annual Technical Conference and Exhibition. New Orleans, Louisiana, USA, 4-7 October 2009: doi: 10.2118/124834-MS.

Kuchuk, F. J. (2009). Radius of Investigation for Reserve Estimation From Pressure Transient Well Tests. Society of Petroleum Engineers. doi:10.2118/120515-MS.

Kuchuk, F., Biryukov, D., & Fitzpatrick, T. (2015). Fractured-Reservoir Modeling and Interpretation. Society of Petroleum Engineers. doi:10.2118/176030-PA.

Nunna, K., Zhou, P., & King, M. (2015). Novel Diffuse Source Pressure Transient Upscaling. SPE Reservoir Simulation Symposium. Houston, Texas, USA, 23–25 February 2015: doi: 10.2118/173293-MS.

Palacio, J., & Blasingame, T. (1993). Decline-Curve Analysis Using Type Curves-Analysis of Gas Well Production Data. Society of Petroleum Engineers, doi:10.2118/25909-MS.

Pasumarti, A., Sengupta, S., & King, M. J. (2015). A Novel Transient Simulation Based Methodology for the Calculation of Permeability in Pore Network Models. Society of Petroleum Engineers. doi:10.2118/177884-MS.

Ramey, H. J. (1966). Application of the Line Source Solution To Flow in Porous Media-A Review. Society of Petroleum Engineers. doi:10.2118/1361-MS.

Satman, A., Eggenschwiler, M., & Ramey, H. (1980). Interpretation of Injection Well Pressure Transient Data in Thermal Oil Recovery. Annual California Regional Meeting of the Society of Petroleum Engineers of AIME. Los Angeles, California, April 9-11, 1980: doi: 10.2118/8908-MS.

Song, B., & Ehlig-Economides, C. A. (2011). Rate-Normalized Pressure Analysis for Determination of Shale Gas Well Performance. Society of Petroleum Engineers. doi:10.2118/144031-MS.

Valko, P. P., & Lee, W. J. (2010). A Better Way To Forecast Production From Unconventional Gas Wells. Society of Petroleum Engineers. doi:10.2118/134231-MS.

Wattenbarger, R. A. (1970, September). An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: II. Finite Difference Treatment. SPE Journal. doi:10.2118/2467-PA

Yang, C., Sharma, V., Datta-Gupta, A., & King, M. (2015). A Novel Approach for Production Transient Analysis of Shale Gas/Oil Reservoirs. SPE Unconventional Resources Technology Conference. San Antonio, Texas, USA, 20–22 July 2015: doi: 10.2118/178714-MS.

Zhang, Y., Bansal, N., Fujita, Y., Datta-Gupta, A., King, M., & Sankaran, S. (2014). From Streamlines to Fast Marching: Rapid Simulation and Performance Assessment of Shale Gas Reservoirs Using Diffusive Time of Flight as a Spatial Coordinate. SPE Unconventional Resource Conference. The Woodlands, Texas, USA, 1-3 April 2014: doi: 10.2118/168997-MS.

Zhang, Y., Yang, C., King, M., & Datta-Gupta, A. (2013). Fast-Marching Methods for Complex Grids and Anisotropic Permeabilities: Application to Unconventional Reservoirs. SPE Reservoir Simulation Symposium. The Woodlands, Texas, USA, 18–20 February 2013: doi: 10.2118/163637-MS.