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Averaging Rock and Fluid Properties

Introduction

In reservoir simulation, we must "upscale" geological data into grid blocks. Because reservoirs are heterogeneous, a single grid block represents multiple geological layers. To maintain physical accuracy, we use specific averaging techniques derived from fundamental physical laws.

1. Porosity Averaging (Arithmetic)

Physical Principle: Conservation of Mass (Volume).

Porosity is a capacity property. The total pore volume in a system is the sum of the pore volumes of its constituents.

Derivation

Define total pore volume as the sum of individual pore volumes:

V_{p,total} = V_{p,1} + V_{p,2} + \dots + V_{p,n}

Substitute the definition $\phi = V_p / V_b$ (where $V_b$ is bulk volume):

\phi_{avg} V_{b,total} = \phi_1 V_{b,1} + \phi_2 V_{b,2} + \dots + \phi_n V_{b,n}

Solve for $\phi_{avg}$ :

\phi_{avg} = \frac{\sum_{i=1}^{n} \phi_i V_{b,i}}{\sum_{i=1}^{n} V_{b,i}}

Conclusion: Porosity is always averaged using the Volume-Weighted Arithmetic Mean.

2. Permeability Averaging (Parallel Flow)

Physical Principle: Conservation of Flow (Total flow is the sum of layer flows).

This applies to horizontal flow along bedding planes.

Derivation

In a parallel system, the pressure drop ( $\Delta P$ ) and length ( $L$ ) are identical for all layers. The total flow rate ( $q_t$ ) is the sum of individual rates:

q_t = q_1 + q_2 + \dots + q_n

Substitute Darcy’s Law $q = \frac{k A \Delta P}{\mu L}$ :

\frac{k_{avg} A_t \Delta P}{\mu L} = \frac{k_1 A_1 \Delta P}{\mu L} + \frac{k_2 A_2 \Delta P}{\mu L} + \dots

Cancel common terms ( $\Delta P, \mu, L$ ):

k_{avg} A_t = k_1 A_1 + k_2 A_2 + \dots + k_n A_n

For layers of constant width $w$ , then $A = h \cdot w$ . Dividing by $w$ :

k_{avg} = \frac{\sum k_i h_i}{\sum h_i}

Conclusion: Parallel flow uses the Arithmetic Mean, dominated by high-permeability "thief zones."

3. Permeability Averaging (Series Flow)

Physical Principle: Summation of Potential (Total pressure drop is the sum of layer drops).

This applies to vertical flow across layers or flow between adjacent grid blocks.

Derivation

Flow rate ( $q$ ) and area ( $A$ ) are constant. Total pressure drop ( $\Delta P_t$ ) is the sum of drops across each block:

\Delta P_t = \Delta P_1 + \Delta P_2 + \dots + \Delta P_n

Rearrange Darcy’s Law for $\Delta P$ :

\frac{q \mu L_t}{k_{avg} A} = \frac{q \mu L_1}{k_1 A} + \frac{q \mu L_2}{k_2 A} + \dots

Cancel common terms ( $q, \mu, A$ ):

\frac{L_t}{k_{avg}} = \frac{L_1}{k_1} + \frac{L_2}{k_2} + \dots + \frac{L_n}{k_n}

Solve for $k_{avg}$ :

k_{avg} = \frac{\sum L_i}{\sum (L_i / k_i)}

Conclusion: Series flow uses the Harmonic Mean, dominated by the lowest permeability (bottlenecks).

4. Fluid Property Averaging (Saturation)

Fluid saturations ( $S_w, S_o, S_g$ ) are fractions of the pore volume.

Derivation

Total water volume ( $V_w$ ) is the sum of volumes in constituent parts:

V_{w,total} = \sum (S_{wi} \cdot V_{pi})

Since $V_{w,total} = S_{w,avg} \cdot V_{p,total}$ :

S_{w,avg} = \frac{\sum (S_{wi} V_{pi})}{\sum V_{pi}}

Summary Table

Property	Configuration	Averaging Method
Porosity	Any	Arithmetic (Volume Weighted)
Permeability	Parallel Flow	Arithmetic (Thickness Weighted)
Permeability	Series Flow	Harmonic (Length Weighted)
Saturation	Any	Arithmetic (Pore-Volume Weighted)

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Rock and Fluid Properties Averaging

Averaging Rock and Fluid Properties

Introduction

1. Porosity Averaging (Arithmetic)

Derivation

2. Permeability Averaging (Parallel Flow)

Derivation

3. Permeability Averaging (Series Flow)

Derivation

4. Fluid Property Averaging (Saturation)

Derivation

Summary Table