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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Di

tD

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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

A

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