Reservoir Flow Properties Fundamental. Supplementary Material

Transcription

1 Reservoir Flow Properties Fudametal Supplemetary Material

2 How did we arrive at gas flow equatio? q = ka μ dp dl Recall that Darcy s equatio for liear flow. For gas flow, the rate is ormally writte at stadard coditios. q = ka μ p 1 p 2 L Assumig that physical properties are approximately costat (as for liquid flow), the followig equatio ca be writte.

3 How did we arrive at gas flow equatio? q sc p sc = qp z sc T sc zt Write the relatioship betwee rate at ay other coditios with rate at stadard coditios usig gas law. q sc = z sct sc zt p ka dp p sc μ dl Substitutig the above equatio i Darcy s law you obtai the equatio o the left. q sc = z sct sc zt p 1 ka 1pdp p sc μl න p 2 Assumig that viscosity, temperature ad z factor are approximately costat, you ca simplify the above equatio to the equatio o the left.

4 How did we arrive at gas flow equatio? Itegratig that equatio you obtai: q sc = z sct sc 1 ka p sc zt μl p 1 2 p You ca ow substitute the values of stadard properties o the right had side ad chage the costats appropriately to arrive at the fial equatio If you ca t assume that viscosity ad z factor are costat, you have to itegrate: q sc = z sct sc T defiitio. 1 ka p sc L p 2 p 1 p μz Usig pseudo-real pressure, you obtai: dp This requires the use of pseudo-real pressure q sc = z sct sc p sc ka T [ m(p 1 ) m(p 2 ] L

5 Derivatio of Radial Flow Equatio For radial flow, we ca write Darcy s law equatio by eglectig gravity as: q = ka dp μ dr The sig is positive because both pressure ad radius chage i the same directio for a producig well. The area, A = 2phr, ca be defied at ay radius with area becomig smaller as the flow coverges ito the wellbore. Substitutig ad itegratig: r e dr q න r w r = 2πkh μ p e dp or q = 2πkh න p wf μ p e p wf l r e r w

6 Layers i Parallel Basic Derivatio As show i the previous figure, the pressure drop across all the layers is costat, but the flow rates across each layer is differet. For liear systems, assume each layer is homogeeous, has the same width (W), ad eglect gravity. Assume that pressure drop across each layer is Dp ad the legth of each layer is L. Assume that uits of idividual terms are such that a costat for equatio is ot eeded. You ca calculate the rate across layer, i, as: q i = k ih i W μ To calculate the total rate, q, we ca sum all the idividual rates q = p L q i = W p L k i h i

7 Layers i Series Basic Derivatio (liear flow) h t = h i Defie the total thickess of all the layers as this equatio. q = k effh t μ p L Assume that you ca defie a effective permeability of these layers so that for a give geometry, it would provide us the same rate as parallel layers. This equatio illustrates writig the equatio i terms of effective permeability. k eff = σ h t k i h i Comparig this equatio with the total rate equatio writte as summatio of all the rates, write the followig equatio. It is easy to derive this equatio for radial flow usig similar priciples ad demostrate that the equatio is the same irrespective of liear or radial flow.

8 Layers i Series Basic Derivatio (liear flow) For liear systems, assume: Each layer is homogeeous, has the same width (W), ad eglect gravity. Rate across each layer is q ad the legth of each layer is L. Uits of idividual terms are such that a costat is ot eeded for equatio. assume all beds have same thickess (h) ad width (W) Maipulatig Darcy s law, you ca write Dp across layer, i, as below: To calculate the total pressure drop, Dp, you ca sum all the idividual pressure drops. p i = qμl i k i hw p = p i = qμ hw Li k i

9 Layers i Series Basic Derivatio (liear flow) L t = L i Defie the total legth of all the layers p = qμl t k eff hw Assume you ca defie a effective permeability of these layers so that for a give geometry, it would provide the same pressure drop as layers i series. This equatio i terms of effective permeability ca be writte. k eff = L t σ L i k i Comparig this equatio with the total pressure drop equatio writte as summatio of all the pressure drops, write this equatio.

10 Layers i Series Basic Derivatio (radial flow) The key differece betwee liear flow ad radial flow is that Darcy s law equatio cotais logarithmic term. However, the developmet of the equatio essetially remais the same. By combiig several steps i a sigle step, the equatio for total pressure drop is: The effective permeability equatio is: The effective permeability equatio is: p = p i = qμ l h r i r i 1 k i = qμ h l r e r w k eff k eff = l σ r e r w r i 1 k i l r i