Water Influx - Chapter 10 - PDF

C H A P T E R 1 0 WATER INFLUX Nearly all hydrocarbon reservoirs are surrounded by water-bearing rocks called aquifers. These aquifers may be substantially larger than the oil or gas reservoirs they adjoin as to appear infinite in s...

C H A P T E R 1 0 WATER INFLUX Nearly all hydrocarbon reservoirs are surrounded by water-bearing rocks called aquifers. These aquifers may be substantially larger than the oil or gas reservoirs they adjoin as to appear infinite in size, or they may be so small in size as to be negligible in their effect on reservoir perfor- mance. As reservoir fluids are produced and reservoir pressure declines, a pressure differential develops from the surrounding aquifer into the reser- voir. Following the basic law of fluid flow in porous media, the aquifer reacts by encroaching across the original hydrocarbon-water contact. In some cases, water encroachment occurs due to hydrodynamic conditions and recharge of the formation by surface waters at an outcrop. In many cases, the pore volume of the aquifer is not significantly larger than the pore volume of the reservoir itself. Thus, the expansion of the water in the aquifer is negligible relative to the overall energy system, and the reservoir behaves volumetrically. In this case, the effects of water influx can be ignored. In other cases, the aquifer permeability may be sufficiently low such that a very large pressure differential is required before an appreciable amount of water can encroach into the reservoir. In this instance, the effects of water influx can be ignored as well. This chapter focuses on those reservoir-aquifer systems in which the size of the aquifer is large enough and the permeability of the rock is high enough that water influx occurs as the reservoir is depleted. This chapter also provides various water influx calculation models and a detailed description of the computational steps involved in applying these models. © 2010 Elsevier Inc. All rights reserved. Doi: 10.1016/C2009-0-30429-8 650  Water Influx 651 CLASSIFICATION OF AQUIFERS Many gas and oil reservoirs are produced by a mechanism termed water drive. Often this is called natural water drive to distinguish it from artificial water drive that involves the injection of water into the forma- tion. Hydrocarbon production from the reservoir and the subsequent pressure drop prompt a response from the aquifer to offset the pressure decline. This response comes in a form of water influx, commonly called water encroachment, which is attributed to: Expansion of the water in the aquifer Compressibility of the aquifer rock Artesian flow where the water-bearing formation outcrop is located structurally higher than the pay zone Reservoir-aquifer systems are commonly classified on the basis of: Degree of pressure maintenance Flow regimes Outer boundary conditions Flow geometries Degree of Pressure Maintenance Based on the degree of the reservoir pressure maintenance provided by the aquifer, the natural water drive is often qualitatively described as: Active water drive Partial water drive Limited water drive The term active water drive refers to the water encroachment mecha- nism in which the rate of water influx equals the reservoir total produc- tion rate. Active water-drive reservoirs are typically characterized by a gradual and slow reservoir pressure decline. If, during any long period, the production rate and reservoir pressure remain reasonably constant, the reservoir voidage rate must be equal to the water influx rate. ⎡water inf lux ⎤ = ⎡oil flow⎤ + ⎡ free gas ⎤ + ⎡water production ⎤ ⎢⎣ rate ⎥⎦ ⎢⎣ rate ⎥⎦ ⎢⎣flow rate⎥⎦ ⎢⎣ rate ⎥⎦ or ew = Qo Bo + Qg Bg + Qw Bw (10-1) 652 Reservoir Engineering Handbook where ew = water influx rate, bbl/day Qo = oil flow rate, STB/day Bo = oil formation volume factor, bbl/STB Qg = free gas flow rate, scf/day Bg = gas formation volume factor, bbl/scf Qw = water flow rate, STB/day Bw = water formation volume factor, bbl/STB Equation 10-1 can be equivalently expressed in terms of cumulative production by introducing the following derivative terms: dWe dN p dN p dWp ew = = Bo + (GOR − R s ) Bg + Bw (10-2) dt dt dt dt where We = cumulative water influx, bbl t = time, days Np = cumulative oil production, STB GOR = current gas-oil ratio, scf/STB Rs = current gas solubility, scf/STB Bg = gas formation volume factor, bbl/scf Wp = cumulative water production, STB dNp/dt = daily oil flow rate Qo, STB/day dWp/dt = daily water flow rate Qw, STB/day dWe/dt = daily water influx rate ew, bbl/day (GOR − Rs)dNp/dt = daily free gas flow rate, scf/day Example 10-1 Calculate the water influx rate ew in a reservoir whose pressure is sta- bilized at 3,000 psi. Given: initial reservoir pressure = 3500 psi dNp/dt = 32,000 STB/day Bo = 1.4 bbl/STB GOR = 900 scf/STB Rs = 700 scf/STB Bg = 0.00082 bbl/scf dWp/dt =0 Bw = 1.0 bbl/STB  Water Influx 653 Solution Applying Equation 10-1 or 10-2 gives: ew = (1.4) (32,000) + (900 − 700) (32,000) (0.00082) + 0 = 50,048 bbl/day Outer Boundary Conditions The aquifer can be classified as infinite or finite (bounded). Geologi- cally all formations are finite, but may act as infinite if the changes in the pressure at the oil-water contact are not “felt” at the aquifer boundary. Some aquifers outcrop and are infinite acting because of surface replen- ishment. In general, the outer boundary governs the behavior of the aquifer and, therefore: a. Infinite system indicates that the effect of the pressure changes at the oil/aquifer boundary can never be felt at the outer boundary. This boundary is for all intents and purposes at a constant pressure equal to initial reservoir pressure. b. Finite system indicates that the aquifer outer limit is affected by the influx into the oil zone and that the pressure at this outer limit changes with time. Flow Regimes There are basically three flow regimes that influence the rate of water influx into the reservoir. As previously described in Chapter 6, those flow regimes are: a. Steady-state b. Semisteady (pseudosteady)-state c. Unsteady-state Flow Geometries Reservoir-aquifer systems can be classified on the basis of flow geom- etry as: a. Edge-water drive b. Bottom-water drive c. Linear-water drive 654 Reservoir Engineering Handbook In edge-water drive, as shown in Figure 10-1, water moves into the flanks of the reservoir as a result of hydrocarbon production and pressure drop at the reservoir-aquifer boundary. The flow is essentially radial with negligible flow in the vertical direction. Bottom-water drive occurs in reservoirs with large areal extent and gentle dip where the reservoir-water contact completely underlies the reservoir. The flow is essentially radial and, in contrast to the edge-water drive, the bottom-water drive has significant vertical flow. In linear-water drive, the influx is from one flank of the reservoir. The flow is strictly linear with a constant cross-sectional area. RECOGNITION OF NATURAL WATER INFLUX Normally very little information is obtained during the exploration-devel- opment period of a reservoir concerning the presence or characteristics of an aquifer that could provide a source of water influx during the depletion period. Natural water drive may be assumed by analogy with nearby pro- ducing reservoirs, but early reservoir performance trends can provide clues. A comparatively low, and decreasing, rate of reservoir pressure decline with increasing cumulative withdrawals is indicative of fluid influx. Figure 10-1. Flow geometries.  Water Influx 655 Successive calculations of barrels withdrawn per psi change in reser- voir pressure can supplement performance graphs. If the reservoir limits have not been delineated by developed dry holes, however, the influx could be from an undeveloped area of the reservoir not accounted for in averaging reservoir pressure. If the reservoir pressure is below the oil sat- uration pressure, a low rate of increase in produced gas-oil ratio is also indicative of fluid influx. Early water production from edge wells is indicative of water encroach- ment. Such observations must be tempered by the possibility that the early water production is due to formation fractures; thin, high permeability streaks; or to coning in connection with a limited aquifer. The water pro- duction may be due to casing leaks. Calculation of increasing original oil in place from successive reser- voir pressure surveys by using the material balance assuming no water influx is also indicative of fluid influx. WATER INFLUX MODELS It should be appreciated that in reservoir engineering there are more uncertainties attached to this subject than to any other. This is simply because one seldom drills wells into an aquifer to gain the necessary information about the porosity, permeability, thickness, and fluid proper- ties. Instead, these properties frequently have to be inferred from what has been observed in the reservoir. Even more uncertain, however, is the geometry and areal continuity of the aquifer itself. Several models have been developed for estimating water influx that are based on assumptions that describe the characteristics of the aquifer. Due to the inherent uncertainties in the aquifer characteristics, all of the proposed models require historical reservoir performance data to evalu- ate constants representing aquifer property parameters since these are rarely known from exploration-development drilling with sufficient accu- racy for direct application. The material balance equation can be used to determine historical water influx provided original oil in place is known from pore volume estimates. This permits evaluation of the constants in the influx equations so that future water influx rate can be forecasted. The mathematical water influx models that are commonly used in the petroleum industry include: Pot aquifer Schilthuis’ steady-state 656 Reservoir Engineering Handbook Hurst’s modified steady-state The van Everdingen-Hurst unsteady-state - Edge-water drive - Bottom-water drive The Carter-Tracy unsteady-state Fetkovich’s method - Radial aquifer - Linear aquifer The following sections describe these models and their practical appli- cations in water influx calculations. The Pot Aquifer Model The simplest model that can be used to estimate the water influx into a gas or oil reservoir is based on the basic definition of compressibility. A drop in the reservoir pressure, due to the production of fluids, causes the aquifer water to expand and flow into the reservoir. The compress- ibility is defined mathematically as: ΔV = c V Δ p (10-3) Applying the above basic compressibility definition to the aquifer gives: Water influx = (aquifer compressibility) (initial volume of water) (pressure drop) or We = (cw + cf) Wi (pi − p) (10-4) where We = cumulative water influx, bbl cw = aquifer water compressibility, psi−1 cf = aquifer rock compressibility, psi−1 Wi = initial volume of water in the aquifer, bbl pi = initial reservoir pressure, psi p = current reservoir pressure (pressure at oil-water contact), psi  Water Influx 657 Calculating the initial volume of water in the aquifer requires the knowledge of aquifer dimension and properties. These, however, are sel- dom measured since wells are not deliberately drilled into the aquifer to obtain such information. For instance, if the aquifer shape is radial, then: ⎡ π ( r − re ) h φ ⎤ 2 2 Wi = ⎢ a ⎥⎦ (10-5) ⎣ 5.615 where ra = radius of the aquifer, ft re = radius of the reservoir, ft h = thickness of the aquifer, ft φ = porosity of the aquifer Equation 10-3 suggests that water is encroaching in a radial form from all directions. Quite often, water does not encroach on all sides of the reservoir, or the reservoir is not circular in nature. To account for these cases, a modification to Equation 10-2 must be made in order to properly describe the flow mechanism. One of the sim- plest modifications is to include the fractional encroachment angle f in the equation, as illustrated in Figure 10-2, to give: We = (cw + cf) Wi f (pi − p) (10-6) where the fractional encroachment angle f is defined by: ( encoachment angle ) o θ f= o = (10-7) 360 360 o The above model is only applicable to a small aquifer, i.e., pot aquifer, whose dimensions are of the same order of magnitude as the reservoir itself. Dake (1978) points out that because the aquifer is considered rela- tively small, a pressure drop in the reservoir is instantaneously transmit- ted throughout the entire reservoir-aquifer system. Dake suggests that for large aquifers, a mathematical model is required which includes time dependence to account for the fact that it takes a finite time for the aquifer to respond to a pressure change in the reservoir. 658 Reservoir Engineering Handbook Figure 10-2. Radial aquifer geometries. Example 10-2 Calculate the cumulative water influx that results from a pressure drop of 200 psi at the oil-water contact with an encroachment angle of 80°. The reservoir-aquifer system is characterized by the following properties: Reservoir Aquifer radius, ft 2600 10,000 porosity 0.18 0.12 cf, psi−1 4 × 10−6 3 × 10−6 cw, psi−1 5 × 10−6 4 × 10−6 h, ft 20 25 Solution Step 1. Calculate the initial volume of water in the aquifer from Equa- tion 10-4. ( )( ⎛ π (10, 0002 − 26002 ) 25 0.12 Wi = ⎜ ) ⎞⎟ = 156.5 MMbbl ⎜⎝ 5.615 ⎟⎠  Water Influx 659 Step 2. Determine the cumulative water influx by applying Equation 10-5. We = (4+3) 10 −6 (156.5 × 106 ) ⎛ 80 ⎝ 360 ) (200) = 48,689 bbl Schilthuis’ Steady-State Model Schilthuis (1936) proposed that for an aquifer that is flowing under the steady-state flow regime, the flow behavior could be described by Darcy’s equation. The rate of water influx ew can then be determined by applying Darcy’s equation: ⎡ ⎤ dWe ⎢ 0.00708 kh ⎥ = ew = ⎢ ( p − p) r ⎥ i (10-8) dt ⎢ μ w ln ⎛ a ⎞ ⎥ ⎣ ⎝ re ⎠ ⎦ The above relationship can be more conveniently expressed as: dWe = e w = C ( p i − p) (10-9) dt where ew = rate of water influx, bbl/day k = permeability of the aquifer, md h = thickness of the aquifer, ft ra = radius of the aquifer, ft re = radius of the reservoir t = time, days The parameter C is called the water influx constant and is expressed in bbl/day/psi. This water influx constant C may be calculated from the reservoir historical production data over a number of selected time inter- vals, provided that the rate of water influx ew has been determined inde- pendently from a different expression. For instance, the parameter C may be estimated by combining Equation 10-1 with 10-8. Although the influx constant can only be obtained in this manner when the reservoir pressure stabilizes, once it has been found, it may be applied to both stabilized and changing reservoir pressures. 660 Reservoir Engineering Handbook Example 10-3 The data given in Example 10-1 are used in this example: pi = 3500 psi p = 3000 psi Qo = 32,000 STB/day Bo = 1.4 bbl/STB GOR = 900 scf/STB Rs = 700 scf/STB Bg = 0.00082 bbl/scf Qw = 0 Bw = 1.0 bbl/STB Calculate Schilthuis’ water influx constant. Solution Step 1. Solve for the rate of water influx ew by using Equation 10-1. ew = (1.4) (32,000) + (900 − 700) (32,000) (0.00082)+ 0 = 50,048 bbl/day Step 2. Solve for the water influx constant from Equation 10-8. 50, 048 C= = 100 bbl / day / psi (3500 − 3000) If the steady-state approximation adequately describes the aquifer flow regime, the calculated water influx constant C values will be constant over the historical period. Note that the pressure drops contributing to influx are the cumulative pressure drops from the initial pressure. In terms of the cumulative water influx We, Equation 10-8 is inte- grated to give the common Schilthuis expression for water influx as: t We ∫o dWe = ∫ C (pi − p) dt o or t We = C ∫ (p i − p) dt (10-10) o where We = cumulative water influx, bbl C = water influx constant, bbl/day/psi  Water Influx 661 t = time, days pi = initial reservoir pressure, psi p = pressure at the oil-water contact at time t, psi When the pressure drop (pi − p) is plotted versus the time t, as shown in t Figure 10-3, the area under the curve represents the integral ∫ ( p i − p) = dt. o This area at time t can be determined numerically by using the trape- zoidal rule (or any other numerical integration method), as: t ⎛ p i − p1 ⎞ ∫ ( p i − p) = dt = area I + area II + area III + etc. = ⎜ ⎟ ( t − 0) o ⎝ 2 ⎠ 1 (p − p ) + (pi − p2 ) ( p − p 2 ) + ( p i − p3 ) ( t − t ) + i 1 (t 2 − t1) + i 3 2 2 2 + etc. Equation 10-9 can then be written as: t We = C ∑ (Δp) Δt (10-11) o Example 10-4 The pressure history of a water-drive oil reservoir is given below: t, days p, psi 0 3500 (pi) 100 3450 200 3410 300 3380 400 3340 The aquifer is under a steady-state flowing condition with an estimated water influx constant of 130 bbl/day/psi. Calculate the cumulative water influx after 100, 200, 300, and 400 days using the steady-state model. 662 Reservoir Engineering Handbook Figure 10-3. Calculating the area under the curve. Solution Step 1. Calculate the total pressure drop at each time t. t, days p pi − p 0 3500 0 100 3450 50 200 3410 90 300 3380 120 400 3340 160 Step 2. Calculate the cumulative water influx after 100 days: We = 130 ⎛ ⎞ (100 − 0) = 325, 000 bbl 50 ⎝ 2⎠  Water Influx 663 Step 3. Determine We after 200 days. ⎡ 50 50 + 90 ⎞ ⎤ We = 130 ⎢⎛ ⎞ (100 − 0) + ⎛ (200 − 100)⎥ = 1, 235, 000 bbl ⎝ ⎣ 2 ⎠ ⎝ 2 ⎠ ⎦ Step 4. We after 300 days. ⎡ 50 50 + 90 ⎞ We = 130 ⎢⎛ ⎞ (100) + ⎛ (200 − 100) ⎣⎝ 2 ⎠ ⎝ 2 ⎠ + ⎛ 120 + 90 ⎞ (300 − 200)⎤ = 2, 600, 000 bbl ⎝ 2 ⎠ ⎥⎦ Step 5. Calculate We after 400 days. ⎡ 160 + 120 ⎞ ⎤ We = 130 ⎢2500 + 7000 + 10, 500 + ⎛ ( 400 − 300)⎥ ⎣ ⎝ 2 ⎠ ⎦ = 4, 420, 000 bbl Hurst’s Modified Steady-State Model One of the problems associated with the Schilthuis’ steady-state model is that as the water is drained from the aquifer, the aquifer drainage radius ra will increase as the time increases. Hurst (1943) proposed that the “apparent” aquifer radius ra would increase with time and, therefore the dimensionless radius ra/re may be replaced with a time dependent function, as: ra/re = at (10-12) Substituting Equation 10-11 into Equation 10-7 gives: dWe 0.00708 kh (p i − p) ew = = (10-13) dt μ w ln (at ) The Hurst modified steady-state equation can be written in a more simplified form as: 664 Reservoir Engineering Handbook dWe C (p i − p) ew = = (10-14) dt ln(at ) and in terms of the cumulative water influx t ⎡p − p⎤ We = C∫ ⎢ i dt (10-15) o ⎣ ln (at ) ⎥⎦ or ⎡ Δp ⎤ t We = C ∑ ⎢ ⎥ Δt (10-16) o ⎣ ln (at ) ⎦ The Hurst modified steady-state equation contains two unknown con- stants, i.e., a and C, that must be determined from the reservoir-aquifer pressure and water influx historical data. The procedure of determining the constants a and C is based on expressing Equation 10-13 as a linear relationship. ⎛ p i − p ⎞ = 1 ln (at ) ⎝ ew ⎠ C or pi − p ⎛ 1 ew = ⎝C ) ln (a ) + ⎛ ln ( t ) 1 ⎝C ) (10-17) Equation 10-16 indicates that a plot of (pi − p)/ew versus ln(t) will be a straight line with a slope of 1/C and intercept of (1/C)ln(a), as shown schematically in Figure 10-4. Example 10-5 The following data, as presented by Craft and Hawkins (1959), docu- ments the reservoir pressure as a function of time for a water-drive reservoir. Using the reservoir historical data, Craft and Hawkins calcu-  Water Influx 665 Figure 10-4. Graphical determination of C and a. lated the water influx by applying the material balance equation (see Chapter 11). The rate of water influx was also calculated numerically at each time period. Time Pressure We ew pi − p days psi M bbl bbl/day psi 0 3793 0 0 0 182.5 3774 24.8 389 19 365.0 3709 172.0 1279 84 547.5 3643 480.0 2158 150 730.0 3547 978.0 3187 246 912.5 3485 1616.0 3844 308 1095.0 3416 2388.0 4458 377 Assuming that the boundary pressure would drop to 3,379 psi after 1,186.25 days of production, calculate cumulative water influx at that time.