Rock and Fluid Properties Averaging

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

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

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

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

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

3. 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 ———- 1. 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:

\[q_t = q_1 + q_2 + \dots + q_n\]

2. Substitute Darcy’s Law \(q = \frac{k A \Delta P}{\mu L}\):

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

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

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

4. 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 ———-

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

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

2. Rearrange Darcy’s Law for \(\Delta P\):

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

3. Cancel common terms (\(q, \mu, A\)):

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

4. 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

1. Total water volume (\(V_w\)) is the sum of volumes in constituent parts:

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

2. 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)