In discretization of the equation, our grid block system is modified to reflect the variable block height by using avarage values, as follows:

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1 page 1 of 5 SYSTEMS OF VARIABLE FLOW AREA We will look at two types of vaiable flow aea In linea coodinates, the thickness of the system may be vaiable, in cylindical coodinates, the adial coodinate itself causes the flow aea to be a function of adius Fo simplicity, we will again conside one-dimensional, hoizontal, one phase flow Linea coodinates If the thickness of the system is a function of position, x, such as in the following illustation: x flow aea must be included in the continuity equation: ( = ( Aφρ x Aρu Nomally, only the laye height, h, is vaying, so that the continuity equation becomes: ( = ( hφρ, x hρu the one-dimensional, hoizontal, single phase flow equation may be witten: kh P = x µb x hφ B In discetization of the equation, ou gid block system is modified to eflect the vaiable block height by using avaage values, as follows: h i-1 h i h i+1 Δx i-1 Δx i Δx i+1 The diffeence fom of the flow tem in the patial diffeential equation above will be witten in tems of tansmissibilities just as befoe: kh P x µb x i Tx (P i +1 P i + Tx i 1/ 2 (P i 1 P i Howeve, the definitions of the tansmissibilities ae now modified to include the block heights of the involved gid blocks: 2h k Tx i +1/ 2 = Δx i (Δx i +1 + Δx i µb 2h Tx i 1/ 2 = i 1/2 k Δx i (Δx i 1 + Δx i µb i 1/2,

2 page 2 of 5 whee h h i 1/2 ae the flow heights in positive negative diections, espectively Using the positive diection as example, we shall define these heights by: h = which becomes: 1 ( x i +1 x i x i+1 h(xdx, h = Δx i +1 h i+1 + Δx i h i ( Δx i +1 + Δx i x i ( Coespondingly in the negative diection: ( h i 1/2 = Δx i 1 h i 1 + Δx i h i ( Δx i 1 + Δx i, Fo the ight h side tem, we only have to make a slight modification by multiplying by block height: whee φh B = φh c B + d(1/ B dp P Cp i (P i P i t, Cp i = h i φ i Δt c B + d(1/ B dp i Cylindical coodinates Anothe type of vaiable flow aea is one induced by the coodinate system used The most common is the one occuing in adial flow as illustated below: Hee, even if the block height is constant, the flow aea is a function of adius, fo a full cylinde (360 degees the aea is: A = 2πh The continuity equation thus becomes, assuming constant height: 1 k µb P = φ B In a adial system, the pessue distibution will be logaithmic of natue, with most of the pessue dop occuing close to the cente, whee the flow aea is small:

3 page 3 of 5 P Since ou discetization fomulas ae moe accuate the moe linea the pessue distibution is, it is clea that if we discetize the adial flow tem using the same appoximations as fo the linea equation, the eo will be lage Theefoe, fo the adial flow equation, we will fist make the following tansfomation of the - coodinate into a u-coodinate: Thus, u = ln( du d = 1 = e u The effect of this tansfomation is that the logaithmic pessue distibution in the adial diection becomes linea along the u-coodinate: u Tansfomation of the adial flow equation by substitution fo u = ln( yields: o e u u eu e 2u k P µb u du d k P = φ u µb u B du d = φ, B This equation is moe linea in u, except fo the tem e 2u in font of the flow tem, it is identical to the linea flow equation We will theefoe adopt the same appoximation of the flow tem in espect to u fo the equation above as we used in espect to x fo the linea equation, with the modification fo the e 2u tem: Tu = Tu i 1 / 2 = 2e 2u i Δu i (Δu i +1 + Δu i 2e 2u i Δu i (Δu i 1 + Δu i k µb k µb i 1/2 Substituting back fo = e u, we get the following expessions fo the adial tansmissibilities: T = 2 2 i k ln ln i+11/ 2 µb i 1/2 i 1/ 2 i +1/ 2

4 page 4 of 5 T i 1 / 2 = 2 2 i k ln ln i 1 / 2 µb i 1/2 i 11/ 2 i 1/ 2 The hamonic aveages fo pemeability in adial diection may be deived in a simila fashion fom the linea fomula: ln i+11/ 2 i +1/ 2 k = 1 ln i +11/ ln k i +1 k i i 1 / 2 k i 1 / 2 = ln i +1/ 2 i 11/ 2 1 ln i 1 / ln k i 1 i 11/ 2 k i i 1 / 2 Expessions fo aveage mobility tems become: ln i +11/2 λ i+1 + ln i +1/ 2 λ i i 1/2 λ = ln i +11 / 2 i 1 / 2 ln i 1/ 2 λ i 1 + ln λ i 11/2 i i 1/2 λ i 1/2 = ln i 11/ 2 These fomulas apply to the adial gid block system shown below: i-1 i i+1 i-1 1/2 i-1/2 i+1/2 i+1 1/2 The position of the gid block centes, elative to the block boundaies, may be computed using the midpoint between the u-coodinate boundaies: u i = (u + u i 1/ 2 /2, o, in tems of adius: i = i 1/2

5 page 5 of 5 This is the geometic aveage of the block bounday adii Fequently in simulation of flow in the adial diection, the gid blocks sizes ae chosen such that: o Δu i = (u i +1/ 2 u i 1/2 = constant ln = constant, i 1/2 which fo a system of N gid blocks well extenal adii of w e, espectively, implies that o N ln i +1/ 2 = ln e i +1/ 2 i 1/ 2 = i 1/ 2 e w 1 / N w = constant This is the fomula fo logaithmic gid block sizes, is often used in esevoi simulation of well behavio If gid sizes have to confom to othe specifications, such as well damage adius, the above fomula may still be useful as a guide to the block sizes Fo the ight h side of the diffeence equation, the above changes will have no effect povided that the height is constant Thus, it will be identical to the one fo the linea system