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
‘Reservoir simulation’ tools
Inflow Performance Relation
Material balance
Analytical model
Numerical model
Diffusion equation
Numerical solution
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Slide 2
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
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
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
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R.N. Horne, Modern Well Test Analysis
Pressure equation for multi-phase systems
𝜆𝑡 = 𝜆𝑡(𝑆𝑜 𝑥 , 𝑆𝑤 𝑥 , 𝑝) 𝑐𝑡 = 𝑐𝑡(𝑆𝑜 𝑥 , 𝑆𝑤 𝑥 , 𝑝)
Without the nonlinear terms
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
Waterflooding example
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
Production simulation of fracture network• Rigorous HC production
simulation from fracture network
• Easy model setup• Fast simulation• Easy to use• Treatment optimization
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
NumericalSemi-analytical
Speed of analytical method
Analytical