# Reservoir Simulation Learning Objectives Chapter 2 – Basic Concepts In Reservoir Engineering

*Be familiar with the meaning and use of all the usual terms which appear in reservoir engineering such as , Sw, So, Bo, Bw, Bg, Rso, Rsw, cw, co, kro, krw, Pc etc.

• Sw – water saturation.
• So – oil saturation
• Bo – oil formation volume factor
• Bg – gas formation volume factor
• Bw – water formation volume factor
• Rso – solution gas oil ratio
• Rsw – solution gas water ratio
• Cwcompressibility water
• Cocompressibility oil
• Cfcompressibility formation
• Krorelative permeability oil
• Krwrelative permeability water
• Pccapillary pressure

*Be able to explain the difference between material balance and reservoir simulation

Material balance equation is used for determining STOIIP by analyzing mean reservoir pressure vs. production data. Also to calculate water influx and determine the nature of the drive mechanism and the extent to which it is supporting production. Furthermore, material balance equations can predict mean future reservoir pressure if the correct drive mechanism has been identified and there is a good match between early pressure declines.

The key difference is that material balance is used principally to define the inputs for a reservoir simulation model.

*Be aware of and be able to describe where it is most appropriate to use material balance and where it is more appropriate to use reservoir simulation

Most appropriate to use material to determine the input values for a reservoir simulation model.

As mentioned above, we use material balance to determine:

• STOIIP
• Water influx and the extent of support
• Possibility to predict future mean reservoir pressures

Reservoir simulation is then used for all other instances. The two are complimentary in nature and should be used together to generate data with the lowest amount of error.

*Be able to use a simple given material balance equation for an undersaturated oil reservoir (with no influx or production of water) in order to find the STOOIP.

*Know the conditions under which the material balance equations are valid.

There are two conditions that are required for material balance equations to be valid. Firstly, we must have adequate data collection, this includes production, pressures and PVT information. We must also be able to define an average pressure decline trend. This relates to pressure reaching equilibrium faster, or diffusing though the system faster – equivalently a large k/(φμc)

*Be able to write down the single and two-phase Darcy Law in one dimension and be able to explain all the terms which occur (no units conversion factors need to be remembered)

Single Phase Darcy Law

Imagine a single phase fluid, for example water, being pumped through a homogeneous sand pack or rock core. Back in the day Darcy used a gravitational head of water as his driving force, but in modern laboratories we use a pump.

Darcy Law (Darcy Velocity) –

Two Phase Law – Darcy Law 1D

The original Darcy Law has limitations in that it only represents a single phase flow only. We can extend Darcy Law to describe multiple phase flow for environments with water, oil and gas. To do this we need to introduce relative permeability factors. The formulas for two phase flow Darcy Law in one dimension

Most of these factors are common knowledge except perhaps the delta pressures. o is the pressure drop across the oi phase at stead-state flow conditions whereas subscript w is for water.

*Be aware of the gradient and divergence operators as they apply to the generalized 2D and 3D Darcy law (but they should not be committed to memory)

Gradient is a vector operation meaning it has magnitude and direction properties. In a three dimensional Cartesian coordinate system  we have i, j and k which are the unit vectors pointing in x, y and z directions respectively.

Divergence is the dot product of the gradient operator and acts on a vector to produce a scalar.

*Know that pressure is a scalar and that the pressure distribution, P(x,y,z) is a scalar field; but that P is a vector.

∇P is sometimes presented as grad P. The ∇P is a vector of pressure gradients in the three directions, x,y, and z.

P(x,y,z) is a scalar field because each of the directions are predetermined, there is no unique direction. It is the sum of all of these pressures which creates the ∇P value and gives vector quantity since there is a specific and unique direction of this resultant force.

*Know that permeability is really a tensor quantity with some notion of what this means physically.

To characterize the flow properties through a material requires a tensor description. The term tensor basically is a generic term to describe properties that have varying degrees of directional elements. We can have different levels of tensors. A scalar property are level zero tensors. A vector such as velocity is a level one tensor.

For a k  represented in 3D by a 3×3 matrix, we can use the dot product with grad P to extend the Darcy Law to 3D.

What exactly does tensor mean? Can someone outline it in layman’s terms. I found this video below. There’s some useful insight into the notion of a tensor’s physical meaning.

*Be able to write out the 2D and 3D Darcy Law with permeability as a full tensor and know how this gives the more familiar Darcy Law in x, y and z directions when the tensor is diagonal (but where we may have kx Ky Kz).

This is where things start to get a little hectic and a bit of background information would do us justice. Here’s a few links for reference.

There’s an article from Petro Wiki that describes diagonalizing the permeability tensor

And a presentation from a Norwegian University that has some useful material

*Be able to write down and explain the radial Darcy Law and know that the pressure profile near the well, ΔP(r), varies logarithmically.

Need to consider a radial coordinate system (r/z) for using the Darcy Law in the near-well region.

Take rw as the wellbore radius and r as some appropriate radial distance, then you can integrate the radial darcy law to obtain the equation for the radial pressure profile in a radial system.