PERMEABILITY Gas Flow in Porous Media The petroleum industry commonly uses the prefix Mille (one thousand). An Mscf is 103 scf of gas. An MMscf is 106 scf of gas. A Bscf is 109 scf of gas Be very careful in the international industry even with common definitions. For example, there is often confusion as to whether a “million” is 106 or 109 in Latin/South America, due to differences in languages

Gas Flow vs. Liquid Flow Gas density is a function of pressure (for isothermal reservoir conditions) Real Gas Law We cannot assume gas flow in the reservoir is incompressible Gas density determined from Real Gas Law Darcy’s Law describes volumetric flow rate of gas flow at reservoir conditions (in situ) At any specified temperature and pressure, a specified volume of two different gasses contains the same number of moles, if the Ideal Gas Law is valid. The Ideal Gas Law is valid at standard temperature (e.g. 60 F, Texas) and standard pressure (e.g. 14.65 psia, Texas). This means that zsc=1. Note that standard conditions of temperature and pressure are legislated values, and may vary between countries, or between states in the U.S. Gas gravity, g, is the ratio g/ air at standard temperature and pressure. The molecular weight of air, Mair = 28.9625 g/mol (after McCain). The product, (g Mair) = Mgas.

Gas Flow vs. Liquid Flow Gas value is appraised at standard conditions Standard Temperature and Pressure qg,sc[scf/day] is mass flow rate (for specified g) For steady state flow conditions in the reservoir, as flow proceeds along the flow path: Mass flow rate, qg,sc , is constant Pressure decreases Density decreases Volumetric flow rate, qg , increases The petroleum industry commonly uses the prefix Mille (one thousand). An Mscf is 103 scf of gas. An MMscf is 106 scf of gas. A Bscf is 109 scf of gas Be very careful in the international industry even with common definitions. For example, there is often confusion as to whether a “million” is 106 or 109 in Latin/South America, due to differences in languages

Gas Formation Volume Factor Given a volumetric gas flow rate at reservoir conditions, qg, we need to determine the mass flow rate, qg,sc Bg has oilfield units of [rcf/scf] scf is a specified mass of gas (i.e. number of moles) reservoir cubic feet per standard cubic foot (ft3)reservoir conditions / (ft3)standard conditions A good engineer might pre-calculate the value of psc/Tsc in order to work more efficiently. A poor engineer would use this value in a location where psc and/or Tsc were different.

Linear Gas Flow 1-D Linear Flow System Steady state flow (mass flow rate, qg,sc , is constant) Gas density is described by real gas law, g=(pgMair)/(zRT) Horizontal flow path (dZ/ds=0  =p) A(0s  L) = constant Darcy flow (Darcy’s Law is valid) k = constant (non-reactive fluid) single phase (Sg=1) Isothermal (T = constant) L qg A 1 2

Linear Gas Flow Darcy’s Law: q12 > 0, if p1 > p2 A qg 2 L 1 Pressure, gas viscosity, and z-factor are functions of pressure, and must be kept inside the integral

Integral of Pressure Dependent Terms Two commonly used approaches to the evaluating the integral of the pressure dependent terms: (zg )=Constant approach, also called “p2 Method” valid when pressure < 2,500 psia for Ideal Gas a subset of this approach is valid z = 1; valid only for low pressures g depends on temperature only Pseudopressure approach “real gas flow potential” - Paul Crawford, 2002 (see notes view). the integral is evaluated a priori to provide the pseudopressure function, m(p) specified gas gravity, gg specified reservoir temperature, T arbitrary base pressure, p0 valid for any pressure range Ramey, Crawford and Al-Hussainy first developed the pseudopressure approach using the m(p) function. Dr. Crawford mentioned to me when I was preparing this lecture (Jan. 2002) that if they had it to do over again, he would prefer the name “Real Gas Flow Potential”. Note that the literature contains other implementations of the pseudopressure approach, specifically, the pp(p) function is also commonly used. However, these other implementations differ only in details. Note that the literature also sometimes calls the exact m(p) by the name pp(p). a priori means “ahead of time” Even though it is not accurate for high pressures, legislated requirements (e.g. Texas) may impose the use of the p2 method for governmental reporting. Engineering decisions should be made using pseudopressure if reservoir pressure is outside the valid range of the p2 method.

(zg )=Constant Assumption that (zg ) is a constant function of pressure is valid for pressures < 2,500 psia, across the range of interest, for reservoir temperature and gas gravity

(zg )=Constant At other temperatures in the range of interest Reservoir Temperature Gradient dT/dZ  0.01 F/ft

(zg )=Constant, Linear Flow If (zg )=Constant Gas Flow Rate (at standard conditions) Recall, flipping integral limits is the same as negation. The term (p12-p22) is commonly called “delta p-squared”. But when spoken, it sounds like “delta-p squared”. It would probably be more accurate to call it “delta of p-squared” but this is not normally done.

Real Gas Pseudopressure Recall piecewise integration: the ordering (position along x-axis) of the integral limits a,b and c is arbitrary pseudopressure, m(p), is defined as: Piecewise Integration of the pressure dependent terms: Dimensions of m(p) [ pressure2 / viscosity] Oilfield units of m(p) [psia2/cp] Remember: gas gravity and reservoir temperature are specified for the reservoir under consideration. p0 is an arbitrary base pressure. Choosing p0=0.0 psia (oilfield units) can be a useful choice when implementing this method on a computer, because m(p=0.0) = 0.0 psia2/cp

Real Gas Pseudopressure, Linear Flow Recalling our previous equation for linear gas flow And substituting for the pressure integral Recall, flipping integral limits is the same as negation.

Radial Gas Flow The radial equations for gas flow follow from the previous derivation for liquid flow and are left as self study (zg )=Constant Pseudopressure