Chapter 6 fluid flow in porous media

Vietnam National University - Ho Chi Minh City University of Technology Faculty of Geology & Petroleum Engineering Department of Drilling - Production Engineering Course Reservoir Engineering Trần Nguyễn Thiện Tâm Email: trantam2512@hcmut.edu.vn 12/11/2017 Reservoir Engineering References 12/11/2017 Reservoir Engineering Chapter Fluid flow in porous media 12/11/2017 Reservoir Engineering Contents  Primary Reservoir Characteristics  Flow regimes  Reservoir geometry  Fluid flow in porous media 12/11/2017 Reservoir Engineering Types of fluids The isothermal compressibility coefficient is essentially the controlling factor in identifying the type of the reservoir fluid In general, reservoir fluids are classified into three groups: (1) incompressible fluids; (2) slightly compressible fluids; (3) compressible fluids 12/11/2017 Reservoir Engineering Incompressible fluids An incompressible fluid is defined as the fluid whose volume or density does not change with pressure That is V   and 0 p p Incompressible fluids not exist; however, this behavior may be assumed in some cases to simplify the derivation and the final form of many flow equations 12/11/2017 Reservoir Engineering Slightly compressible fluids These “slightly” compressible fluids exhibit small changes in volume, or density, with changes in pressure Knowing the volume Vref of a slightly compressible liquid at a reference (initial) pressure pref, the changes in the volumetric behavior V = Vref exp [c (pref − p)] where: p = pressure, psia V = volume at pressure p, ft3 pref = initial (reference) pressure, psia Vref = fluid volume at initial (reference) pressure, psia V = Vref[1 + c(pref − p)] ρ = ρref[1 − c(pref − p)] 12/11/2017 Reservoir Engineering Compressible fluids These are fluids that experience large changes in volume as a function of pressure All gases are considered compressible fluids The isothermal compressibility of any compressible fluid is described by the following expression: 1  Z  cg     p Z  p T 12/11/2017 Reservoir Engineering Types of fluids 12/11/2017 Reservoir Engineering Types of fluids 12/11/2017 Reservoir Engineering 10 Radial flow of compressibility fluids  1637Qg T  Z   kt  p i  p wf   3.23  0.87 s  log   kh   i cti rw   12/11/2017 Reservoir Engineering 57 Principle of superposition ● effects of multiple wells; ● effects of rate change; ● effects of the boundary; 12/11/2017 Reservoir Engineering 58 Effects of multiple wells “The total pressure drop at any point in a reservoir is the sum of the pressure drops at that point caused by flow in each of the wells in the reservoir.” (Δp)total drop at well = (Δp)drop due to well + (Δp)drop due to well + (Δp)drop due to well 12/11/2017 Reservoir Engineering 59 Example Assume that the three wells as shown in Figure 1.28 are producing under a transient flow condition for 15 hours The following additional data is available: Qo1 = 100 STB/day, Qo2 = 160 STB/day Qo3 = 200 STB/day, pi = 4500 psi, Bo = 20 bbl/STB, ct = 20×10−6 psi−1, (s)well1 = −0.5, h = 20 ft, ϕ = 15%, k = 40 md, Rw = 0.25 ft, µo = 2.0 cp, r1 = 400 ft, r2 = 700 ft If the three wells are producing at a constant flow rate, calculate the sand face flowing pressure at well 12/11/2017 Reservoir Engineering 60 Effects of variable flow rates Consider the case of a shut-in well, i.e., Q = 0, that was then allowed to produce at a series of constant rates for the different time periods shown in Figure To calculate the total pressure drop at the sand face at time t4, the composite solution is obtained by adding the individual constant-rate solutions at the specified ratetime sequence, or: 12/11/2017 Reservoir Engineering 61 Effects of variable flow rates (p ) total  (p)due to (Qo1 0)  (p )due to (Qo Qo1 )  (p )due to (Qo Qo )  (p )due to (Qo Qo ) 12/11/2017 Reservoir Engineering 62 Effects of variable flow rates The first contribution results from increasing the rate from to Q1 and is in effect over the entire time period t4, thus: 162.6(Q1  0) B   (p )Q1 0    kh     kt4    log   3.23  0.87 s      ct rw   12/11/2017 Reservoir Engineering 63 Effects of variable flow rates The second contribution results from decreasing the rate from Q1 to Q2 at t1, thus: 162.6(Q2  Q1 ) B   (p )Q2 Q1    kh    k (t4  t1 )    log   3.23  0.87 s      ct rw   12/11/2017 Reservoir Engineering 64 Effects of variable flow rates The third contribution results from Q2 to Q3 at t2, thus: 162.6(Q3  Q2 ) B   (p )Q3 Q2    kh     k (t4  t2 )    log   3.23  0.87 s      ct rw   12/11/2017 Reservoir Engineering 65 Effects of variable flow rates The fourth contribution results from decreasing the rate from Q3 to Q4 at t3, thus: 162.6(Q4  Q3 ) B   (p )Q4 Q3    kh     k (t4  t3 )    log   3.23  0.87 s      ct rw   12/11/2017 Reservoir Engineering 66 Example Figure 1.29 shows the rate history of a well that is producing under transient flow conditions for 15 hours Given the following data: pi = 5000 psi, h = 20 ft, Bo = 1 bbl/STB ϕ = 15%, µo = 2.5 cp, rw = 0.3 ft ct = 20×10−6 psi−1, s = 0, k = 40 md calculate the sand face pressure after 15 hours 12/11/2017 150 100 85 70 Reservoir Engineering 67 Effects of the reservoir boundary Consider Figure 1.30 which shows a well that is located a distance L from the non-flow boundary, e.g., sealing fault The noflow boundary can be represented by the following pressure gradient expression:  p  0    L  boundary 12/11/2017 Reservoir Engineering 68 Effects of the reservoir boundary Mathematically, the above boundary condition can be met by placing an image well, identical to that of the actual well, on the other side of the fault at exactly distance L Consequently, the effect of the boundary on the pressure behavior of a well would be the same as the effect from an image well located a distance 2L from the actual well In accounting for the boundary effects, the superposition method is frequently called the method of images The total pressure drop at the actual well will be the pressure drop due to its own production plus the additional pressure drop caused by an identical well at a distance of 2L, or: (Δp)total = (Δp)actual well + (Δp)due to image well 12/11/2017 Reservoir Engineering 69 Effects of the reservoir boundary (p ) total 162.6Qo B   kh   kt    3.23  0.87 s  log      ct rw     70.6 Q B   948  c (2 L )   o t  Ei    kh kt     12/11/2017 Reservoir Engineering 70 Example Figure 1.31 shows a well located between two sealing faults at 400 and 600 feet from the two faults The well is producing under a transient flow condition at a constant flow rate of 200 STB/day Given: pi = 500 psi, k = 600 md, Bo = 1.1 bbl/STB, ϕ = 17%, µo = 2.0 cp, h = 25 ft, rw = 0.3 ft, s = 0, ct = 25×10−6 psi−1 Calculate the sand face pressure after 10 hours 12/11/2017 Reservoir Engineering 71