Video about the permeability of oil reservoir rocks on YouTube
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Introduction to Permeability
Permeability is a measure of the ability of a porous material such as an oil or gas reservoir rock to allow fluids to pass through it. It is one of the most important properties of reservoir rocks and is a critical factor in determining the production potential of oil and gas reservoirs.
Permeability is defined as the capacity of a porous medium like rock to transmit fluids through its interconnected pores, channels and fractures. It measures how easily fluids can flow through the rock. The higher the permeability, the easier it is for oil, gas or water to flow through the rock. Low permeability rocks, such as shale, allow limited fluid flow compared to high permeability rocks like sandstone.
The permeability of a reservoir rock depends on properties like pore sizes, pore distribution, pore connectivity, and pore geometry. These pore characteristics along with fractures present in the rock control the overall conductivity and transmissibility of fluids in the reservoir.
Understanding the permeability of reservoir rocks is crucial for calculating oil and gas well productivity, designing development wells, implementing enhanced recovery methods, and modeling reservoir performance. Permeability values help determine flow rates and design pipeline networks to transport hydrocarbons.
This video will cover key topics related to permeability including:
- Darcy's Law for permeability
- Factors impacting permeability measurements
- Methods to determine permeability
- The relationship between porosity and permeability
- Relative permeability of multiple fluids
- The role of permeability in reservoir engineering
Having a strong grasp of permeability is essential for reservoir engineers, drilling engineers, production engineers and geologists working in the oil and gas industry. This video provides a thorough overview of this fundamental reservoir property.
Darcy's Law
Darcy's law describes the flow of a fluid through a porous medium. It relates the fluid velocity to the permeability of the medium.
Darcy's law is expressed mathematically as:
```
q = -(kA/μ)(dp/dl)
```
Where:
- q is the volumetric flow rate (m3/s)
- k is the permeability of the medium (m2)
- A is the cross-sectional area to flow (m2)
- μ is the fluid viscosity (Pa·s)
- dp/dl is the pressure gradient (Pa/m)
This equation shows that flow rate is directly proportional to permeability and pressure gradient, and inversely proportional to viscosity.
Darcy's law is valid for single-phase, laminar flow through a porous medium. The law assumes the porous medium is rigid, homogeneous, isotropic, and saturated with a single fluid phase. It also assumes steady-state conditions with negligible acceleration and gravity effects.
The derivation of Darcy's law starts with a force balance on a representative elementary volume (REV) of the porous medium. Applying Newton's second law gives:
```
F = m(dv/dt)
```
The forcedriving fluid flow is the pressure gradient:
```
F = -A(dp/dl)
```
The mass inertia is:
```
m = ρAL
```
Where ρ is the fluid density.
Substituting into Newton's law and rearranging gives Darcy's law. For an incompressible fluid where ρ is constant, this reduces to the common form of Darcy's law shown initially.
So in summary, Darcy's law is a simple linear relationship between flow rate and pressure gradient for flow through a porous medium. It forms the fundamental basis for analysis of single-phase reservoir flow.
Darcy's Law Boundary Conditions
Darcy's law can be applied under different boundary conditions, including constant pressure and constant rate. The equations are derived differently for each case.
Constant Pressure Boundary Conditions
For constant pressure boundary conditions, the pressure at the inlet and outlet of the porous medium is fixed. To derive Darcy's law under these conditions, we start with the extended Darcy's law for slightly compressible fluids:
q = -(k/μ) (∇P - ρg)
Where q is the volumetric flow rate, k is permeability, μ is viscosity, ∇P is the pressure gradient, ρ is density, and g is acceleration due to gravity.
For horizontal flow with constant pressure at inlet (P1) and outlet (P2), the pressure gradient reduces to:
∇P = (P2 - P1)/L
Where L is the length of the porous medium.
Substituting this into the extended Darcy's law and re-arranging gives:
q = -(k/μ) (P2 - P1)/L
Which is Darcy's law for linear horizontal flow at constant pressure.
### Constant Rate Boundary Conditions
For constant rate boundary conditions, the volumetric flow rate q at the inlet and outlet of the porous medium is fixed.
Starting again with the extended Darcy's law:
q = -(k/μ) (∇P - ρg)
For horizontal flow where ∇P = dP/dx and integrating over length L gives:
∫q dx = -(k/μ) ∫(dP/dx) dx
Since q is constant:
qL = -(k/μ) (P2 - P1)
Rearranging gives Darcy's law for linear horizontal flow at constant rate:
(P2 - P1)/L = - (μ/k) (q/A)
Where A is the cross-sectional area. This shows the linear pressure drop under constant rate conditions.
## Radial Flow Equations
Radial flow occurs when fluid flows from the reservoir to the wellbore in all directions perpendicular to the wellbore. This is different from linear flow, where fluid only flows along one axis.
To derive the radial flow equation, we start with Darcy's law and assume radial flow geometry with no vertical flow components:
q = (2πkh∆P)/(μln(re/rw))
Where:
q = flow rate
k = permeability
h = thickness
∆P = pressure drop
μ = viscosity
re = external drainage radius
rw = wellbore radius
If we integrate this equation over the drainage area and rearrange:
Q = (kh∆P)/(141.2μln(re/rw))
Where Q is the total flow rate.
This radial flow equation shows the flow rate is proportional to permeability, thickness, and pressure drawdown. But it decays over a larger drainage region due to the log term.
Compared to linear flow, radial flow has a slower pressure transient and lower productivity index. This is because fluid is traveling farther to reach the wellbore from all directions.
Unit Systems
Permeability values can be expressed using different unit systems. The most common are:
- **Darcy (D)** - The standard unit for absolute permeability. 1 Darcy = 0.987x10^-12 m^2.
- **Millidarcy (mD)** - 1/1000 of a Darcy. 1 mD = 0.987x10^-15 m^2.
- **Microdarcy (μD)** - 1/1000000 of a Darcy. 1 μD = 0.987x10^-18 m^2.
- **Centidarcy (cD)** - 1/100 of a Darcy. 1 cD = 0.987x10^-13 m^2.
Conversions between unit systems:
- 1 D = 1000 mD
- 1 mD = 1000 μD
- 1 cD = 0.01 D
For example, a permeability value of 500 mD converts to:
- 0.5 D
- 500,000 μD
- 50 cD
So a permeability measurement can be expressed in any of these units by applying the appropriate conversion factor. Standard practice is to report permeability values in millidarcys (mD) in the petroleum industry. Being familiar with conversions between unit systems is important when evaluating and comparing permeability data.
Lab Measurements of Absolute Permeability
There are several methods used in laboratory settings to measure the absolute permeability of reservoir rock samples. These measurements are usually conducted on small core plugs taken from full-sized reservoir core samples.
Core Flooding
One common technique is core flooding, where the core plug is saturated with a single fluid such as brine, oil, or gas. The sample is then placed in a core holder cell and the fluid is injected through it under controlled conditions. By measuring the pressure drop across a given length of the core at various flow rates, the permeability can be calculated using Darcy's law. This technique allows measurement of horizontal, vertical, or angular permeability.
Probe Permeameters
Another method uses a probe permeameter, consisting of a probe tip which is pushed into the core plug. Fluid is injected through the probe tip at a constant flow rate and the pressure is measured to determine permeability. This technique is quick and convenient but only measures permeability at the specific point of the probe tip.
Pulse Decay Permeameters
Pulse decay permeameters involve saturating the core with gas and applying an initial pressure pulse. As the gas flows through the core, the pressure declines over time. By fitting this pressure decay curve, the permeability can be determined. This method averages the permeability over the entire plug.
NMR Permeametry
Nuclear magnetic resonance (NMR) techniques can also measure permeability non-destructively. NMR measures the rate at which fluid molecules diffuse through pore spaces. This diffusion rate can then be related to permeability. The benefit of NMR is it provides information on permeability variations throughout the core plug.
Radial Flow in Parallel
To analyze radial flow in parallel layers, we must calculate the flow rate in each layer:
$$ q_i = \frac{2\pi k_ih_i}{\mu}\frac{\partial p}{\partial r}\ln\frac{r_e}{r_w} $$
Where:
- $q_i$ is the flow rate in layer i
- $k_i$ is the permeability of layer i
- $h_i$ is the thickness of layer i
- $\mu$ is fluid viscosity
- $\frac{\partial p}{\partial r}$ is pressure gradient
- $r_w$ is wellbore radius
- $r_e$ is external radius
The total flow rate is the sum of the flow rates in each layer:
$$ q_{total} = \sum_i q_i $$
Substitute the flow rate equation into the total:
$$ q_{total} = \sum_i \frac{2\pi k_ih_i}{\mu}\frac{\partial p}{\partial r}\ln\frac{r_e}{r_w} $$
**Define Equivalent Permeability**
We can define an *equivalent permeability* $k_{eq}$ that describes the total system flow:
$$ k_{eq} = \frac{q_{total}}{\frac{2\pi h_{tot}}{\mu}\frac{\partial p}{\partial r}\ln\frac{r_e}{r_w}}$$
Where $h_{tot}$ is the total thickness of all layers.
Simplify:
$$ k_{eq} = \frac{\sum_i k_ih_i}{h_{tot}} $$
So the equivalent permeability of parallel layers is the thickness weighted average.
For a single layer system, the flow rate is:
$$ q = \frac{2\pi kh}{\mu}\frac{\partial p}{\partial r}\ln\frac{r_e}{r_w} $$
Where k and h are properties of the single layer.
Therefore, analyzing parallel layers as an equivalent single layer with $k_{eq}$ provides the correct total flow rate.
Radial Flow Series
Radial flow in series layers refers to fluid flow occurring sequentially across two or more adjacent layers of porous media. We can analyze this scenario by applying Darcy's law to each layer:
q = (k1 A ΔP1) / (μ ln(r2/r1)) (Layer 1)
q = (k2 A ΔP2) / (μ ln(r3/r2)) (Layer 2)
Where:
- q is the volumetric flow rate
- k1, k2 are layer permeabilities
- A is cross-sectional area
- ΔP1, ΔP2 are pressure drops
- μ is fluid viscosity
- r1, r2, r3 are radial distances
By setting the flow rate q equal in both layers and solving, we can relate the permeability ratio to the pressure drop ratio:
k1/k2 = ΔP2/ΔP1
This shows that for flow in series, the layer with lower permeability will have a larger pressure drop. The total system permeability depends on both layers based on their thickness and permeability.
Compared to single layer radial flow, layers in series have a lower overall system permeability due to the additive pressure drops. Optimizing the permeability of each layer is important for maximizing production rates.
Porosity-Permeability Relationship
Porosity refers to the amount of open space available to store fluids in a rock, while permeability refers to how easily fluids can flow through the rock. There is a correlation between these two properties where rocks with higher porosity tend to have higher permeability.
The specific relationship depends on factors like pore shape, pore connectivity, and grain size distribution. In some cases high porosity rocks still have low permeability due to poor pore connectivity. The best correlation exists in rocks with well-connected pore networks.
As an example, a 50% porosity, well sorted sandstone with large, well connected pores could have a permeability of 1 darcy or higher. Whereas a 50% porosity limestone consisting of small unconnected vugs would likely have a permeability less than 0.1 millidarcies.
In typical reservoir rocks, an approximate correlation is:
- 50% porosity; permeability > 1 darcy
- 30% porosity; permeability 1 mD - 1 D
- 20% porosity; permeability 0.1 mD - 10 mD
- 10% porosity; permeability < 0.1 mD
So while the correlation is not universal, higher porosity values generally indicate higher permeability in oil and gas reservoirs with connected pore systems. Evaluating permeability specifically still remains an important as it can vary over orders of magnitude even amongst rocks of similar porosity.
Relative Permeability
Relative permeability is the ratio of effective permeability of a particular fluid to the absolute permeability of a porous medium.
In multiphase flow (like oil and gas reservoirs), the flow of one fluid depends on how much the other fluids saturate the pores within the rock. That's where relative permeability comes into play - it adjusts the permeability to account for the presence of other fluids.
Specifically, relative permeability is defined with respect to the wetting phase saturation. The wetting phase tends to adhere to the pore walls while the nonwetting phase does not. So if we calculate relative permeability for oil K_ro, we use water saturation S_w since water is usually the wetting phase:
K_ro = k/k_abs f(S_w)
where k is the effective permeability to oil, k_abs is absolute permeability, and f(S_w) captures the effect of water saturation.
Relative permeabilities always lie between 0 and 1. At irreducible water saturation, K_ro is 1 since pores contain almost 100% oil. As water saturation increases, K_ro decreases.
Relative permeability curves are crucial for reservoir simulation and forecasting oil/gas recovery. They describe how phase mobility changes with saturation, allowing engineers to predict multiphase flow through reservoirs. Accurate relative permeability data leads to better estimates of reserves and optimization of production strategies.