Rock and Fluid Properties Averaging — SepalSolver

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)