Chapter Seven. For ideal gases, the ideal gas law provides a precise relationship between density and pressure:

Transcription

1 Chapter Seven Horizontal, steady-state flow of an ideal gas This case is presented for compressible gases, and their properties, especially density, vary appreciably with pressure. The conditions of the problem will lead us straight to Darcy's equation Where (7.1) then (7.2) For ideal gases, the ideal gas law provides a precise relationship between density and pressure: where M is the molecular mass of the gas, R is the universal gas constant and T is the absolute temperature of the gas. Substituting for ρ in the RHS integral of Equ. 7.2 yields (7.3) Assuming a mean viscosity and performing the integration in Equ. 7.3 yields

2 (7.4) Substituting (P 1 + P 2 )(P 1 P 2 ) for (P 1 2 P 2 2 ) in Equ. 7.4 yields (7.5) Equ. 7.5 becomes (7.6) (7.7) Example Compute the steady-state flow for an ideal gas whose molecular mass is 18. in a core sample 4 long, 1 in diameter with permeability of 150 md if the inlet pressure is 50 psia and the outlet pressure is atmospheric. The system is at 150 ºF and the average gas viscosity is cp. Compute the mean flow rate and the base flow rate at base conditions of P o = 15 psia and T o = 60 ºF. Equation 7.8 yields

3 Gas flow rate is not usually reported in bbl/d, rather the unit of ft 3 /d is used. And since rate depends on the flow conditions, gas flow rate is normally expressed at the standard conditions of 14.7 psia and 60 ºF. Thus, the unit SCF means one cubic foot of gas measured at the standard conditions, and SCF/D means one SCF per day. The prefix M indicates thousands and MM indicates millions. The Klinkenberg Effect Klinkenberg (1941) discovered that permeability measurements made with air as the flowing fluid showed different results from permeability measurements made with a liquid as the flowing fluid. The permeability of a core sample measured by flowing air is always greater than the permeability obtained when a liquid is the flowing fluid. Klinkenberg postulated, on the basis of his laboratory experiments, that liquids had a zero velocity at the sand grain surface, while gases exhibited some finite velocity at the sand grain surface. In other words, the gases exhibited slippage at the sand grain surface. This slippage resulted in a higher flow rate for the gas at a given pressure

4 differential. Klinkenberg also found that for a given porous medium as the mean pressure increased the calculated permeability decreased. Mean pressure is defined as upstream flowing plus downstream flowing pressure divided by two, [p m =(p 1 +p 2 )/2]. If a plot of measured permeability versus 1/p m were extrapolated to the point where 1/p m =0, in other words, where p m =infinity, this permeability would be approximately equal to the liquid permeability. A graph of this nature is shown in Figure 7.1. The absolute permeability is determined by extrapolation as shown in Figure 7.2. The magnitude of the Klinkenberg effect varies with the core permeability and the type of the gas used in the experiment as shown in Figures 7.2 and 7.3. The resulting straight-line relationship can be expressed as (7.8) Figure 7.1.The Klinkenberg effect in gas permeability measurements.

5 Figure 7.2.Effect of permeability on the magnitude of the Klinkenberg effect. Klinkenberg suggested that the slope c is a function of the following factors: Absolute permeability k, i.e., permeability of medium to a single phase completely filling the pores of the medium k L. Type of the gas used in measuring the permeability, e.g., air. Average radius of the rock capillaries. Klinkenberg expressed the slope c by the following relationship: c =b k L (7.9) where k L =equivalent liquid permeability, i.e., absolute permeability, k b=constant that depends on the size of the pore openings and is inversely proportional to radius of capillaries.

6 Figure 7.3.Effect of gas pressure on measured permeability for various gases. Combining Equation 7.8 with 7.9 gives: (7.10) where k g is the gas permeability as measured at the average pressure p m. Jones (1972) studied the gas slip phenomena for a group of cores for which porosity, liquid permeability k L (absolute permeability), and air permeability were determined. He correlated the parameter b with the liquid permeability by the following expression: (7.11) The usual measurement of permeability is made with air at mean pressure just above atmospheric pressure (1 atm). To evaluate the slip phenomenon and the Klinkenberg effect, it is necessary to at least measure the gas permeability at two mean-pressure levels. In the absence of such data, Equations 7.10 and 7.11 can be combined and arranged to give: where p m =mean pressure, psi (7.12)

7 k g =air permeability at p m, psi k L =absolute permeability (k), md Equation 7.12 can be used to calculate the absolute permeability when only one gas permeability measurement (k g ) of a core sample is made at p m. This nonlinear equation can be solved iteratively by using the Newton-Raphson iterative methods. The proposed solution method can be conveniently written as where k i =initial guess of the absolute permeability, md k i+1 =new permeability value to be used for the next iteration i=iteration level f(k i )=Equation 7.12 as evaluated by using the assumed value of k i f (k i )=first-derivative of Equation 7.12 as evaluated at k i The first derivative of Equation 7.12 with respect to k i is: (7.13) The iterative procedure is repeated until convergence is achieved when f(k i ) approaches zero or when no changes in the calculated values of k i are observed. Example The permeability of a core plug is measured by air. Only one measurement is made at a mean pressure of psi. The air permeability is 46.6 md. Estimate the absolute permeability of the core sample. Compare the result with the actual absolute permeability of md. Solution Step 1.Substitute the given values of p m and k g into Equations 7.12 and 7.13, to give:

8 Multiphase flow through porous media The basic differential equation for radial flow in porous medium The basic differential equation will be derived in radial form thus simulating the flow of fluids in the vicinity of a well. Analytical solutions of the equation can then be obtained under various boundary and initial conditions for use in the description of well testing and well inflow, which have considerable practical application in reservoir engineering. This is considered of greater importance than deriving the basic equation in Cartesian coordinates since analytical solutions of the latter are seldom used in practice by field engineers. In numerical reservoir simulation, however, Cartesian geometry is more commonly used but even in this case the flow into or out of a well is controlled by equations expressed in radial form such as those presented in the next text. The radial cell geometry is shown in figure 7.4 and initially the following simplifying assumptions will be made. a) The reservoir is considered homogeneous in all rock properties and isotropic with respect to permeability. b) The producing well is completed across the entire formation thickness thus ensuring fully radial flow. c) The formation is completely saturated with a single fluid

9 Figure 7.4 Radial flow Consider the flow through a volume element of thickness dr situated at a distance r from the centre of the radial cell. Then applying the principle of mass conservation where 2πrhφdr is the volume of the small element of thickness dr. The left hand side of this equation can be expanded as which simplifies to (7.14) (7.15) By applying Darcy's Law for radial, horizontal flow it is possible to substitute for the flow rate q in equ. (7.15) since giving

10 or (7.16) The time derivative of the density appearing on the right hand side of equ. (7.16) can be expressed in terms of a time derivative of the pressure by using the basic thermodynamic definition of isothermal compressibility and since then the compressibility can be alternatively expressed as and differentiating with respect to time gives Finally, substituting equ. (7.17) in equ. (7.16) reduces the latter to (7.17) (7.18)

11 Conditions of solution a) Transient condition b) Semi-Steady State condition Figure 7.5 Radial flow under semi-steady state conditions Furthermore, if the well is producing at a constant flow rate then the cell pressure will decline in such a way that The constant referred to in above equation can be obtained from a simple material balance using the compressibility definition, thus which for the drainage of a radial cell can be expressed as c) Steady State condition

12 Figure 7.6 Radial flow under steady state conditions

13 Relative Permeability Numerous laboratory studies have concluded that the effective permeability of any reservoir fluid is a function of the reservoir fluid saturation and the wetting characteristics of the formation. It becomes necessary, therefore, to specify the fluid saturation when stating the effective permeability of any particular fluid in a given porous medium. Just as k is the accepted universal symbol for the absolute permeability, k o, k g, and k w are the accepted symbols for the effective permeability to oil, gas, and water, respectively. The saturations, i.e., S o, S g, and S w, must be specified to completely define the conditions at which a given effective permeability exists. Effective permeabilities are normally measured directly in the laboratory on small core plugs. Owing to many possible combinations of saturation for a single medium, however, laboratory data are usually summarized and reported as relative permeability. The absolute permeability is a property of the porous medium and is a measure of the capacity of the medium to transmit fluids. When two or more fluids flow at the same time, the relative permeability of each phase at a specific saturation is the ratio of the effective permeability of the phase to the absolute permeability, or: For example, if the absolute permeability k of a rock is 200 md and the effective permeability k o of the rock at an oil saturation of 80% is 60 md, the relative permeability k ro is 0.30 at S o =0.80.

14 Since the effective permeabilities may range from zero to k, the relative permeabilities may have any value between zero and one, or: It should be pointed out that when three phases are present the sum of the relative permeabilities (k ro +k rg +k rw ) is both variable and always less than or equal to unity. An appreciation of this observation and of its physical causes is a prerequisite to a more detailed discussion of two-and three-phase relative permeability relationships. It has become a common practice to refer to the relative permeability curve for the nonwetting phase as k nw and the relative permeability for the wetting phase as k w. Two-Phase Relative Permeability Correlations In many cases, relative permeability data on actual samples from the reservoir under study may not be available, in which case it is necessary to obtain the desired relative permeability data in some other manner. The field data are unavailable for future production, however, and some substitute must be devised. Several methods have been developed for calculating relative permeability relationships. Various parameters have been used to calculate the relative permeability relationships, including: Residual and initial saturations Capillary pressure data In addition, most of the proposed correlations use the effective phase saturation as a correlating parameter. The effective phase saturation is defined by the following set of relationships:

15 For example Wyllie and Gardner Correlation Horizontal Multiple-Phase Flow When several fluid phases are flowing simultaneously in a horizontal porous system, the concept of the effective permeability to each phase and the associated physical properties must be used in Darcy s equation. For a radial system, the generalized form of Darcy s equation can be applied to each reservoir as follows: The effective permeability can be expressed in terms of the relative and absolute permeability, as presented by Equations of relative permeability, to give:

16 Using the above concept in Darcy s equation and expressing the flow rate in standard conditions yield: The gas formation volume factor B g is previously expressed as: Performing the regular integration approach on above Equations yields

17 In numerous petroleum engineering calculations, it is convenient to express the flow rate of any phase as a ratio of other flowing phase. Two important flow ratios are the instantaneous water-oil ratio (WOR) and instantaneous gas-oil ratio (GOR). The generalized form of Darcy s equation can be used to determine both flow ratios. The water-oil ratio is defined as the ratio of the water flow rate to that of the oil. Both rates are expressed in stock-tank barrels per day, or: Then where WOR =water-oil ratio, STB/STB. The instantaneous GOR, as expressed in scf/stb, is defined as the total gas flow rate, i.e., free gas and solution gas, divided by the oil flow rate, or

18 Then where B g is the gas formation volume factor as expressed in bbl/scf. Example A steady-state flow test was conducted on a core sample 1 in diameter and 2 long. The table below lists the total pressure drop, fluid flowrates and saturation data for each step of the test. Compute and plot the effective permeability curves for this core and estimate S wi and S or. Oil and water viscosities are 2.5 and 1.1 cp, respectively. Oil flowrate is given by Darcy's equation:

19 Rearrangement of the above equation yields: Applying this equation, with consistent units, to every step yieldsthe effective permeability to oil. For example, at S w = 70%, The effective permeability values are plotted in the figure below with smooth curves drawn through the data points. Note that at S w = 100, k w is the permeability of the core sample.

20 Example Compute the relative permeability data for above Example and smooth it using the following Equs. and In the above Example, the effective permeability to oil at S wi was found to be md. This shall be employed as the base permeability. Converting normal water saturation to dimensionless water saturation is done to give:

21 All effective permeability values are converted to relative permeabilities using the base permeability. For example, at S w = 70%: The relative permeability data is plotted in the figure shown to the right. Best-fitting curves of the form given by Equs. of k ro and k rw are determined by plotting log k rw vs. log S wd to find a and c, and log k ro vs. Log (1 - S wd ) to find b. The two curves are also shown in the figure, and their equations are