WATER INFLUX. Hassan S. Naji, Professor,

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1 WATER INFLUX Many reservoirs are bound on a portion or all of their peripheries by water-bearing rocks called aquifers. The aquifer may be so large compared to the reservoir size as to appear infinite, and it may be so small as to be negligible in its effect on reservoir performance. The aquifer may be entirely bound by impermeable rocks so that the reservoir and aquifer together form a closed or volumetric system. On the other hand, the reservoir may outcrop at one or more places where it may be replenished by surface waters. The aquifer may be horizontal with the reservoir it adjoins or it may rise considerably above the reservoir to provide some sort of artesian flow to the reservoir. Hassan S. Naji, Professor, hnaji@kau.edu.sa 1

2 Aquifers retard reservoir pressure decline by providing a source of water influx by: water expansion expansion of other hydrocarbon accumulations in the aquifer rock compressibility of the aquifer rock and artesian flow which occurs when the aquifer rises to a level above the reservoir. To determine the effect of an aquifer on reservoir production, it is necessary to calculate the amount of water influx, W e. This calculation can be made using the material balance equation when the initial hydrocarbon in place and the production history are known. If correct values of W e are placed in the material balance equation as a function of reservoir pressure, then the equation should plot as a straight line. To obtain an estimate for both the initial hydrocarbon in place and water influx, then a model for W e as a function of pressure is assumed. If a straight line is not obtained, then a new model for W e is assumed and the procedure repeated. Models for calculating W e are categorized on a time dependent basis to: Steady-state models: 1. Pot aquifer model. Schilthuis model Pseudosteady-state models: 3. Fetkovitch model Unsteady-State Models: 4. Van Everdingen and Hurst model 5. Hurst simplified model 6. Carter-Tracy model The basic concept for water influx calculation is: Where: W e (t) is water influx, U is the aquifer constant, and S(p, t) is the aquifer function W e (t) = U S(p, t)

3 1. The Pot Aquifer Model The simplest model that can be used to estimate water influx into a gas or oil reservoir is the pot aquifer model. The assumptions for the pot aquifer formulation are: 1. finite closed aquifer,. large aquifer permeability such that aquifer expansion is complete within the time step, 3. reservoir cannot be too large. Otherwise it is hard to satisfy assumption, and 4. variable compressibility is permissible. The aquifer pore volume compressibility sets the basis for the pot aquifer formulation. Since aquifer compressibility is given by: c = 1 V P V Pi p 1 V P V Pi p = 1 V Pi V P (t) V Pi p i p(t) V pi V P (t) = cv pi [p i p(t)] Thus the constant-compressibility pot aquifer model is written as: where: W e = c t V Pi [p i p(t)] = (c w + c f ) V Pi [p i p(t)] = U S(p, t) W e (t) is the cumulative water influx, bbl, c t is the aquifer total compressibility, psia -1 c w is the aquifer water compressibility, psia -1 c f is the aquifer rock compressibility, psia -1 V Pi is the initial aquifer pore volume = [7758 Ah ] = [ π(r aq r e )h ], bbl, and U is the aquifer constant = (c w + c f ) V Pi, bbl/psi S(p, t) is the aquifer function = p i p(t), psia p i is the initial aquifer (reservoir: pressure at the oil-water contact) pressure, psia p(t) is the aquifer (reservoir: pressure at the oil-water contact) current pressure, psia Aquifer pressures are approximated by reservoir pressures at the water-oil contact. 3

4 Example #1 Water Influx Calculations for a Pot Aquifer Model Semester: Homework : Name: SS : A wedge-shaped reservoir is suspected of having a fairly strong natural water drive. The geometry of the reservoir-aquifer system is shown by the following figure. The following aquifer data are given: Thickness 1 ft Permeability md Porosity.5 Compressibility 4. x 1-6 Aquifer/reservoir radius ratio re/rw 5. Water viscosity.55 cp Water compressibility 3. x 1-6 Water formation volume factor 1. RB/STB 4

5 The following reservoir data are given: Time years OWC, psia Np MM STB Rp SCF/STB Bo RB/STB Rs SCF/STB Bg RB/SCF (Rsi) (Rsi) Calculate the amount of water influx if a pot aquifer model is applicable. Solution: re 5. x (9.) 46 ft VPi π(46, -9, )(14/36)(1)(.5)/ MMM bbl U = (c w + c f ) V Pi 7. x 1-6 x x bbl/psia Time years OWC, psia (p i - p) psia W e U x (p i - p) MM bbl

6 We, MM bbl 1 1 We = U x (pi - p) y = x x We = U x (pi - p) Poly. (We = U x (pi - p)) Time, years For variable-compressibility pot aquifer, the above equation is written as: n n j+ W e (t) 1 j+ 1 = t W e = V P (c f + c w ) j+1 t p j+1 j= j= n = Ah j+1 (c f + c w ) j+1 t p j+1 Where: j+1 = j + j+1 (c f + c w ) j+1 = (c f + c w ) j + (c f + c w ) j+1 j= t p j+1 = p j p j+1 6

7 . Schilthuis Steady-State Aquifer Model Schilthuis used Darcy's Law to start deriving his model as follows: q =.78 kh (p i p) μb ln r e r w Including the skin effect, the above equation is written as follows: q =.78 kh μb q = dv = (p i p) [ln r e r w.75 + s].78kh μb [ln r e r w.75 + s] dp t q = dv p.78kh = W e = μb [ln r e.75 + s] dp = K s dp r w p dw e = q =.78 kh μ w B w (p i p) [ln r e.75 + s] = K s(p i p) r w Thus we write: t W e (t) = K s dp = K s p t = K s p t t t t 7

8 3. Fetkovitch Pseudosteady-State Aquifer Model Fetkovitch (1973) started derivation of his aquifer model with Darcy s equation. The assumptions for Fetkovitch aquifer formulation are: 1. finite closed aquifer,. large aquifer permeability such that aquifer expansion is complete within the time step, 3. Water influx rate dw e is controlled by aquifer permeability via aquifer J. 4. reservoir can be too large depending on the magnitude of aquifer permeability, and 5. constant aquifer compressibility. p t kh q o = ( 141. [ln ( r e r ).75 + s] ) k ro dp μ o B o w p Since the productivity index of a well, denoted by J, is a measure of the ability of the well to produce. It is given by: Where: q o J = P i P t J P r Q o P wf = Wellbore productivity index, STB/day/psig = Average (static) reservoir pressure, psig = Wellbore stabilized oil flow rate, STB/day = Wellbore stabilized bottom-hole flowing pressure, psig.78 kh J = μ w B w [ln ( r e ) r.75], RB day /psi w t W e = dw e.78 kh J = r μ w B w { e r e r [ln (r e w r ).75 + r w w r (1 r w e 4r )]} e W e (t) = c t V Paq (p i p aq) 8

9 Derivation of Fetkovitch aquifer model starts with: q w B w = dw e = J[p aq p(t)] (1) Where:.78 kh RB J = r μ w B w { e r e r [ln (r e w r ).75 + r w w r (1 r, w e 4r )]} day /psi e W e (t) = c t V Paq (p i p aq) p aq = p i W e(t) c t V Paq () Plugging (3) into (1) yields: dw e = c t V Paq dp aq (3) J[p aq p(t)] = c t V Paq dp aq dp aq p(t)] = J c t V Paq [p aq (4) Integrating both sides yields: (5) p dp aq J = p(t)] c t V Paq [p aq p i t Assuming that p(t) is constant yields: (6) 9

10 ln[p aq p] = Jt + ln[p c t V i p] Paq Rearranging yields: (7) p aq p = (p i p)e Jt c t V Paq (8) Substituting (8) into (1) yields: W e = (p i p)c t V Paq (1 e Jt c t V Paq ) (9) 4. van Everdingen and Hurst Unsteady-State Aquifer Model van Everdingen and Hurst (19XX) started derivation of their aquifer model with Darcy s equation. The assumptions for van Everdingen and Hurst aquifer formulation are: 1. Finite closed aquifer for linear case or infinite for radial case,. The system is compressible since we have C t in the denominator of t D with constant compressibility C t = C w + C f, 3. Both aquifer size and reservoir size via r e as well as aquifer diffusivity via.638 k, ft and the outer boundary condition affect water influx rate. μ C t day 4. True unsteady state process. van Everdingen and Hurst started their derivation as follows: Where: B = π C t h r w θ 36 n r w W e (t n+1 ) = B p j W ed (t D n+1 t D j ) j= 1

11 B =.638 kt μ C t r w p = (p p 1 ) p 1 = (p p ) p = (p 1 p 3 ) p 3 = (p p 4 ) p i = (p i 1 p i+1 ), for i 1 11

12 Summary of Aquifer Models Model Pot Aquifer Schilthuis Schilthuis Model Formula W n e = (c w + c f ) V P (p aq i p n ) W e (t) = K s ( P t ) t n j= t W n e = (p aq n 1 p aq n ) W ei [1 e ( p i n 1 = p i (1 p aq n 1 j= W ei = c t V Paq p i j tw e ) W ei Jp i t n ) W ei ] p n = pn+1 + p n Van Everdingin & Hurst 1 t n = t n t n 1 n W e (t n+1 ) = B ( p j ) W ed (t D n+1 t D j ) B = j= π c thr w θ 36

13 W e (t) = kh 7.6μ ( p t) j n j= t j+1 t j a + ln t j+1 Hurst-Modified ( p t ) j = p i pj + p j+1 k a = ln ( 7.6 μc t r ) w Carter-Tracy t W e n+1 = ( B p t W e n p D (t D n+1 ) p D (t D n+1 ) t D n p D (t D n+1 ) ) (t D n+1 t D n ) p D = π t D 13

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