Can history matching mess up my dual- porosity model? Pl Nverlid - - PowerPoint PPT Presentation

U N I V E R S I T Y O F B E R G E N

Can history matching mess up my dual- porosity model?

Pål Næverlid Sævik1 Martha Lien3 Inga Berre1,2

1Department of Mathematics, University of Bergen 2Christian Michelsen Research AS, Bergen, Norway 3Octio AS, Bergen

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Motivation: Staying on the manifold

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What happens if you apply history matching on upscaled fracture parameters? 𝒔 = Permeability Porosity Transfer coefficient 𝒔 = Aperture Fracture density Upscaling error

Motivation: Staying on the manifold

What happens if you apply history matching on upscaled fracture parameters? 𝒔 = 𝑆 sin 𝜚 cos 𝜄 𝑆 sin 𝜚 sin 𝜄 𝑆 cos 𝜚 𝒔 = 𝑦 𝑧 𝑨

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EnKF biased towards Gaussian distributions

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Posterior probability 𝑞

True posterior EnKF estimate

𝑛 Correlation 𝑛2

True correlation EnKF estimate

𝑛1

Data comparison Simulation Upscaling

Choice of primary variables

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Aperture, fracture density, connectivity Permeability, porosity, transfer coefficient Pressure, flow rates, saturation Mismatch Adjust reservoir parameters Adjust fracture parameters

Fracture upscaling

Analytical

• Fast solution
• Derivatives easily
• btained
• Requires macroscopic

homogeneity

• May not be applicable to

all geometries Numerical

• Computationally

expensive

• Technically difficult
• Potentially accurate
• Flexible formulation

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Analytical fracture upscaling

Layer-based

• Fractures are modeled as

infinitely extending thin layers

• Modifications are applied

to account for partial connectivity Inclusion-based

• Fractures are modeled as

infinitely separated inclusions

• Modifications are applied

to account for fracture interaction

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Layer-based fracture upscaling

• Permeability

𝐋 = 𝐋𝑛𝑏𝑢 + 𝑔 ෍

𝑗=1 𝑂 𝑏3𝜍𝑗

12 𝑱 − 𝐨𝑗

⊤𝐨𝑗

• Porosity

𝜚 = 𝑏 ෍

𝑗=1 𝑂

𝜍𝑗

• Transfer coefficient

𝜏 = 4 Tr 𝐒⊤𝐒 𝐒 = ෍

𝑗=1 𝑂

𝜍𝑗 𝐨𝑗

⊤𝐨𝑗

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A simple example

• Randomly oriented, infinitely extending fractures
• No permeability within the matrix
• Exact upscaling assumed

𝐿 = 𝑏3𝜍 18 𝜚 = 𝑏𝜍 𝜏 = 4 3 𝜍2

• Single simulation grid block
• The inverse upscaling transform is well-defined

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A simple example

• Uniform distribution for the prior data
• Measured data has gaussian noise

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Three ways to get a history matched model

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Post-analysis correlations

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Reservoir parameters Fracture parameters

Linear fracture upscaling

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ln 𝐿 = ln 𝑏3𝜍 18 ln 𝜚 = ln 𝑏𝜍 ln 𝜏 = ln 4 3 𝜍2

Using log of the parameters as primary variables Upscaling transformation is linear, and connectivity is preserved

Fractures of finite size

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ln 𝐿 = ln 𝑔 𝑏3𝜍 18 ln 𝜚 = ln 𝑏𝜍 ln 𝜏 = ln 4 3 𝜍2

Connectivity 𝑔 is calculated using a method of Mourzenko et al. (2011) Upscaling transformation is nonlinear despite using logarithms

Fracture parameters Logaritmic reservoir parameters

Effects of inexact upscaling method

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ln 𝐿 = ln 1 + 𝜀 𝑔 𝑏3𝜍 18 ln 𝜚 = ln 𝑏𝜍 ln 𝜏 = ln 4 3 𝜍2

Logaritmic reservoir parameters Reservoir parameters

Effects of inexact upscaling method

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Effects of inexact upscaling method

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Does it matter for prediction?

• Quarter-of-five-spot problem
• Fracture parameters spatially correlated

– Gaussian spatial covariance model – Correlation length ½ of domain size

• Water injection, water-wet reservoir
• Constant injection rate, constant production pressure
• Assimilated data:

– Volume production rate – Injection pressure – Water cut

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Does it matter for prediction?

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Concluding remarks

• Using upscaled parameters as primary variables during

inversion, may generate parameter distributions that are inconsistent with the underlying fracture description

• The effect is most clearly seen for partially connected

fracture networks, for which there exists an accurate upscaling relationship

• The problem can be avoided by using fracture

parameters as primary variables, and include upscaling as an integral part of the history matching workflow

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