oil and gas production forecasting with semi-analytical reservoir...
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Oil and Gas Production Forecasting with Semi-Analytical
Reservoir SimulationPeter Tilke1, Wentao Zhou2, Boris Samson3, Shalini Krishnamurthy3, Jeff Spath2,
Michael Thambynayagam2, Alexander Lukyanov1,*
1. Schlumberger-Doll Research, Cambridge MA; 2. Schlumberger, Houston TX; 3. Schlumberger Abingdon Technology Centre, UK; *. Presenter
2015 SIAM Conference on Mathematical & Computational Issues in the Geosciences
MS39 Meshless Modeling in Geoscience
July 1, 2015
Stanford University
Stanford, California USA
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‘Reservoir simulation’ tools
Inflow Performance Relation
Material balance
Analytical model
Numerical model
Diffusion equation
Numerical solution
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Slide 2
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Diffusion equation
Analytical solution
Analytical reservoir simulation
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Extension to different types of sources, boundaries and fluid system, multi-well, multi-layer, etc.
• Same PDE
• Analytical solution on
upscaled reservoir model
• Fast speed for on-time
decision making
Slide 3
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Basic solutions
• 3D point source solution
• 2D solution (line source)
𝑝(𝑥, 𝑦, 𝑧, 𝑡) =𝑈(𝑡 − 𝑡0)
8𝜋3/2𝜙𝑐𝑡 𝜂𝑥𝜂𝑦𝜂𝑧 0
𝑡−𝑡0 𝑞 𝑡 − 𝑡0 − 𝜏
𝜏3/2𝑒−
𝑥−𝑥02
4𝜂𝑥𝜏+𝑦−𝑦0
2
4𝜂𝑦𝜏+𝑧−𝑧0
2
4𝜂𝑧𝜏 𝑑𝜏
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IARF pressure solution
• Infinite Acting Radial Flow
𝑝𝑖 − 𝑝𝑤𝑓 𝑡, 𝑟𝑤 =𝑞𝐵𝜇
4𝜋𝑘ℎln 𝑡 + ln
𝑘
𝜙𝜇𝑐𝑡𝑟𝑤2+ 0.80907
Δ𝑝 = 𝑚 ln 𝑡 + 𝑏Pressure difference
𝑡dΔ𝑝
𝑑𝑡=dΔ𝑝
𝑑 ln 𝑡= 𝑚Pressure derivative Bourdet derivative
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Superposition concept
• System response to a number of perturbations = sum of responses to each of the perturbations
• Temporal• Multi-rate
• Spatial• Multi-well
• Boundary conditions
A
R.N. Horne, Modern Well Test Analysis
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Pressure equation for multi-phase systems
𝜆𝑡 = 𝜆𝑡(𝑆𝑜 𝑥 , 𝑆𝑤 𝑥 , 𝑝) 𝑐𝑡 = 𝑐𝑡(𝑆𝑜 𝑥 , 𝑆𝑤 𝑥 , 𝑝)
Without the nonlinear terms
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Saturation equation
• Streamline constructed from pressure field
• Hyperbolic saturation equation solved along 1D streamlines (analytically or numerically)
• Limited mainly to waterfloodingproblem
𝑓𝑤 =
𝑘𝑟𝑤(𝑆𝑤)𝜇𝑤
𝑘𝑟𝑤(𝑆𝑤)𝜇𝑤
+𝑘𝑟𝑜(𝑆𝑤)𝜇𝑜
H. Tchelepi, Course note on multi-phase flow in porous media, Stanford University
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Waterflooding example
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Semi-analytical solution with finite-conductivity fractures
• Analytical reservoir solution: flow from reservoir to fractures
• Numerical fracture solution: flow inside fractures
Analytical reservoir simulation for hydraulic fractures
Slide 10
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Production simulation of fracture network• Rigorous HC production
simulation from fracture network
• Easy model setup• Fast simulation• Easy to use• Treatment optimization
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Numerical reservoir simulation
• Multi-phase diffusion equation discretized and solved with finite difference numerical scheme
• Applicable for complex nonlinearities
• Big system (up to million, billion cells)
• Standard tool in the oil & gas industry
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NumericalSemi-analytical
Speed of analytical method
Analytical