INSTRUCTOR © 2017, John R. Fanchi All rights reserved. No part of this manual may be reproduced in any form without the express written permission of the author. © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

To the Instructor The set of files here are designed to help you prepare lectures for your own course using the text Introduction to Petroleum Engineering, J.R. Fanchi and R.L. Christiansen (Wiley, 2017) File format is kept simple so that you can customize the files with relative ease using your own style. You will need to supplement the files to complete the presentation topics.

PERMEABILITY © 2017, John R. Fanchi All rights reserved. No part of this manual may be reproduced in any form without the express written permission of the author. © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Outline Darcy’s Law and Permeability Permeability Averaging Transmissibility Measures of Permeability Heterogeneity Darcy’s Law with Directional Permeability

DARCY’S LAW AND PERMEABILITY © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Henri Darcy (1856) Empirical fit of water flow through packed sand Observation Flow rate proportional to pressure gradient Assumptions: 1. Single phase water 2. Homogeneous sand 3. Vertical flow 4. Non-reactive fluid 5. Single geometry © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Darcy’s Law – Horizontal Q A P1 P2 q = Flow rate k = Proportionality constant A = Cross-sectional area μ = Fluid viscosity p = Pressure L = Length or distance of flow Integrate assuming constant q, k, A, μ: © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Permeability Defined Darcy’s Law q Fluid flow rate of 1.0 cc/sec  Fluid viscosity of 1.0 cp A Sample cross-sectional area of 1.0 cm2 L Sample length of 1.0 cm Δp = p1 - p2 = Pressure gradient of 1.0 atm k Permeability of 1.0 darcy Darcy’s Law

Permeability Dimensional Analysis Permeability unit is “length squared” or “area” © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Permeability Interpretation Permeability describes flow in a given sample for given fluid and set of experimental conditions Permeability may be viewed as mathematical convenience Describes statistical behavior of given flow experiment Permeability has meaning as statistical representation of a large number of pores

Tortuosity Area Grain H Pore Flow Path B Flow Path A

Actual vs. Apparent Velocity Apparent velocity = velocity of fluid flowing through linear conduit Actual Velocity = velocity of fluid flowing through tortuous path Note: Vact > Vapp © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Poiseuille’s Equation Poiseuille’s Equation: calculate flow q through n cylindrical capillary tubes of radius r and length L Where Atube = π r2 In Darcy’s Law, flow in a core can be approximated by: Equating Poiseuille’s eq and Darcy’s law gives  - k relationship: © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

ϕ-k from Poiseuille’s Equation  - k relationship from Poiseuille’s eq and Darcy’s law: Permeability = k, ft2 Number of tubes = n Radius of tube = r, ft Cross-sectional area of core = A, ft2 Perm Conversion factors: 1 md = 0.986923 × 10-15 m2 = 1.06 × 10-14 ft2 © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Darcy’s Law for a Core Darcy’s law Volumetric flow rate = q, bbl/day Permeability = k, md Cross-sectional area of core = A, ft2 Viscosity = μ, cp Pressure change across core = Δp, psia Length of core = L, ft © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Examples of Permeability Porous Medium Permeability Coal 0.1 to 200 md Shale <0.005 md Loose sand (well sorted) 1 to 500 md Partially consolidated sandstone 0.2 to 2 d Consolidated sandstone Tight gas sandstone < 0.01 md Limestone Diatomite 1 to 10 md © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Qualitative Permeability Ranges Range (md) Description ≈ 10-6 Shale (nanodarcy) ≈ 10-3 Tight (microdarcy) 1 – 10 Low (millidarcy) 10 – 250 Moderate > 250 High © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Factors Affecting Permeability 1. Packing of grains 2. Rock texture 3. Grain size distribution 4. Angularity 5. Cementation 6. Rock compaction © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Methods of Estimating K 1. Well Test Analysis 2. Well Logs 3. Core Experiments 4. Statistical Correlations 5. Simulation Methods Note the scale of measurement and the “directness” of each approach. © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Direct Measurement vs. Correlation Well test analysis assuming Darcy flow. Measurement scale is effective drainage area. Well Logs assume various correlations. Measurement scale is inches to feet. Core analysis is the most “direct”, although correlation takes place. Measurement scale is inches. Statistical Methods: neural networks and geostatistics blend information. Inexact science. Simulation: Attribute an effective perm at a coarse scale. © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

PERMEABILITY AVERAGING © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Flow Through Beds in Parallel Arithmetic average © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Flow Through Beds in Series Harmonic average © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

TRANSMISSIBILITY © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Series application of Darcy’s Law for neighboring blocks Flow Between Blocks Series application of Darcy’s Law for neighboring blocks © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Transmissibility Estimate flow rate between blocks with Darcy phase transmissibility between blocks © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

MEASURES OF PERMEABILITY HETEROGENEITY © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Heterogeneity and Anisotropy Homogeneous (no special place) Isotropic (no special direction) © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

HETEROGENEITY MEASURES Heterogeneity: Spatial variability of permeability DYKSTRA-PARSONS COEFFICIENT (SPE 20156, Lake and Jensen, 1989) LORENZ COEFFICIENT (SPE 20156, Lake and Jensen, 1989)

DYKSTRA-PARSONS COEFFICIENT (SPE 20156, Lake and Jensen, 1989) Estimate for log normal permeability distribution Where kA = arithmetic average And kH = harmonic average Range: 0  VDP  1 Homogeneous Reservoir: VDP = 0 (in this case, KA = KH) Increasing heterogeneity increases VDP Typical Reservoir Values: 0.4  VDP  0.9

LORENZ COEFFICIENT (SPE 20156, Lake and Jensen, 1989) Plot cum flow capacity Fm vs cum thickness Hm where for n = number of reservoir layers. Arrange layers in order of decreasing permeability Thus I=1 has thickness h1 and the largest perm k1 While I=n has thickness hn and the smallest perm kn By Definition, 0  Fm  1 and 0  Hm  1 for 0  m  n

THE LORENZ CURVE Lorenz curve 1.0 C B Fm A 0.0 0.0 Hm 1.0 The Lorenz coefficient LC is 2*Area between the Lorenz curve ABC in the figure and the diagonal AC.

PROPERTIES OF THE LORENZ COEFFICIENT Range: 0  LC  1 Homogeneous Reservoir: LC = 0 Increasing heterogeneity increases LC Typical Reservoir Values: 0.2  LC  0.6 © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

ESTIMATING THE LORENZ COEFFICIENT Assume all perms have equal probability and use trapezoidal rule to estimate area, thus Ordering of perm is not necessary with this estimate. Note: if ki = kj (homogeneous case), then LC = 0.

DARCY’S LAW WITH DIRECTIONAL PERMEABILITY © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Directional Permeability Permeability may depend on direction Vertical perm (kz, perpendicular to bedding plane) often about 1/10 horizontal perm (kx, ky, parallel to bedding plane) Reservoir simulators typically use diagonalized tensor Permeability may be heterogeneous kz kx ky © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

Anisotropy and Flow A. Isotropic (Kx = Ky) B. Anisotropic (Kx ≠ Ky) © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

DARCY’S LAW AND PHASE POTENTIAL Darcy’s Law for single phase flow in 1-D: Phase Potential of phase i:

3-D EXTENSION OF DARCY’S LAW Permeability Tensor

PRINCIPAL AXES Coordinate system {x’, y’, z’} Typically assume diagonalized tensor aligned with principal axes θ x y x' y' Coordinate Rotation

QUESTIONS? © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.

SUPPLEMENT © 2004 John R. Fanchi All rights reserved. Do not copy or distribute.