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Chapter 1
Pressure Diffusion Equation for Fluid Flow in Porous Rocks
In this chapter, we will derive the basic differential equations that govern the time-dependent flow of fluids through porous media such as rocks or soils. This will be done by combining the principle of conservation of mass with Darcy’s law, which relates the flow rate to the pressure gradient. The resulting differential equation will be a diffusion-type equation that governs the way that the fluid pressure changes as a function of time and varies spatially throughout the reservoir. The governing equations derived in this chapter form the basis of analytical models that are used in well test analysis, and, in discretised form, form the basis of numerical simulation codes that are used in petroleum reservoir engineering to predict oil and gas recovery.
1.1. Darcy’s Law and the Definition of Permeability
The basic law governing the flow of fluids through porous media is Darcy’s law, which was formulated by the French civil engineer Henry Darcy in 1856 on the basis of his experiments on vertical water filtration through sand beds. Darcy (1856) found that his data could be described by
where P is the pressure (Pa), ρ is the density (kg/m3), g is the gravitational acceleration (m/s2), z is the vertical coordinate (measured downwards) (m), L is the length of sample (m), Q is the volumetric flow rate (m3/s), C is the constant of proportionality (m2/Pa s), and A is the cross-sectional area of the sample (m2).
Any consistent set of units can be used in Darcy’s law, such as SI units, cgs units, British engineering units, etc. Unfortunately, in the oil and gas industry it is common to use so-called “oilfield units”, which are not self-consistent. Oilfield units (i.e. barrels, feet, pounds per square inch, etc.) will not be used in these notes, except occasionally in some of the problems.
Darcy’s law is mathematically analogous to other linear phenomenological transport laws, such as Ohm’s law for electrical conduction, Fick’s law for solute diffusion, and Fourier’s law for heat conduction.
Why does the term “P − ρgz” govern the flow rate? Recall from elementary fluid mechanics that Bernoulli’s equation, which essentially embodies the principle of “conservation of energy”, contains the terms
where P/ρ is related to the enthalpy per unit mass, −gz is the gravitational energy per unit mass, and v2/g is the kinetic energy per unit mass. Fluid velocities in a reservoir are usually very small, and so the third term is usually negligible, in which case we see that the combination P − ρgz represents an energy-type term. It seems reasonable that fluid would flow from regions of higher energy to lower energy, and, therefore, the driving force for flow should be the gradient (i.e. the rate of spatial change) of P −ρgz.
Subsequent to Darcy’s initial discovery, it has been found that, all other factors being equal, Q is inversely proportional to the fluid viscosity, μ (Pa s). It is therefore convenient to factor out μ, and put C = k/μ, where k is known as the permeability, with dimensions (m2).
It is usually more convenient to work with the volumetric flow per unit area, q = Q/A, rather than the total flow rate, Q. In terms of q, Darcy’s law can be written as
where the flux q has dimensions of (m/s). Since q is not quite the same as the velocity of the individual fluid particles (see Eq. (9.4.5)), it is perhaps better to think of these units as (m3/m2s).
For transient processes in which the flux varies from point-to-point in a reservoir, we need a differential form of Darcy’s law. In the vertical direction, this equation takes the form
where z is the downward-pointing vertical coordinate. The minus sign is included because the fluid flows in the direction from higher to lower values of P −ρgz.
Since z is constant in the horizontal direction, the differential form of Darcy’s law for one-dimensional (1D) horizontal flow is
For most sedimentary rocks, the permeability in the horizontal plane, kH, is different than permeability in the vertical direction, kV. Typically, kH > kV. The permeabilities in any two orthogonal directions within the horizontal plane may also differ. However, in these notes we will usually assume that kH = kV, and denote the permeability by k.
The permeability is a function of rock type, and also varies with stress, temperature, etc., but does not depend on the fluid; the effect of the fluid on the flow rate is accounted for by the viscosity term in Eq. (1.1.4) or (1.1.5).
Permeability has units of m2, but in the petroleum industry it is conventional to use “Darcy” units, defined by
The Darcy unit is defined such that a rock having a permeability of 1 Darcy would transmit 1 cm3 of water (wh...
