University:
Boston University
Course:
ENG ME 303 | Fluid Mechanics
Academic year:

2022
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242

Pages:

19
Author:

Relptapyloorawuft

Math, Num and Comp Models Reservoir Properties Table of contents 1 Math, Num and Comp Models Processes to be modeled Reservoir Properties One–Phase Flow Two–Phase Immiscible Flow Black–Oil Compositional Oil Solution Schemes 2 References References Math, Num and Comp Models References Reservoir Properties Petroleum reservoir Is a subsurface pool of hydrocarbons contained in porous formations. The naturally occurring hydrocarbons, such as crude oil or natural gas, are trapped by overlying rock formations with lower permeability. It also can contains water, in such a way that in general we have three phases: oleic (o), aqueous (w) and gaseous (g). Math, Num and Comp Models References Reservoir Properties Rock: Pores, Porosity and Permeability (Chen [1]) I Pores: Pores are the tiny connected passages that exists in a permeable rock (≈ 1–200µm). Porosity: is the fraction of a rock that is pore space. It measures the capacity of the reservoir to store productible fluids in its pores. φ= Vp V φe = Ve V Vp interconnected and isolated pore spaces; Ve interconnected pore spaces; V total volume. Porosity depends on pressure due to rock compressibility: cR = 0 φ = φ0 ecR (p−p ) 1 dφ φ dp 1 =⇒ φ = φ0 1 + cR (p − p0 ) + c2R (p − p0 ) + . . . 2! Math, Num and Comp Models References Reservoir Properties Rock: Pores, Porosity and Permeability (Chen [1]) II (Absolute) Permeability : is the capacity of a rock to conduct fluids through its interconnected pores. In many practical situations, it is possibleto assume:  k11 k= 0 0 Classification Poor fair Moderate Good Very good Excellent kH = k11 = k22 in the horizontal plane; kV = k33 in the vertical; kH 6= kV . • In many systems there is an approximate correlation Permeability range (md) between the permeability k 1 – 15 and the porosity φ. 15 – 20 • In general, the larger the 50 – 250 porosity, the higher the 250 – 1000 permeability. over 1000 0 k22 0 0 0  k33 Math, Num and Comp Models References Reservoir Properties Fluid: Phase and component (Chen [1]) Phase : refers to a chemically homogeneous region of fluid that is separated from another phase by an interface, e.g. oleic (o), aqueous (w), gaseous (g), or solid (rock). Component : is a single chemical species that may be present in a phase, e.g. oleic contains hundreds of components (C1 , C2 , ...) Compressibility : of a fluid can be defined in terms of the volume V or density ρ change with pressure: cf = − 1 ∂V V ∂p 0 cf (p−p0 ) ρ=ρ e = T 1 ∂ρ ρ ∂p at a fixed temperature T T 1 2 0 0 =⇒ ρ = ρ 1 + cf (p − p ) + cf (p − p ) + . . . 2! 0 Math, Num and Comp Models References Reservoir Properties Rock/Fluid, (Chen [1]) I Wettability : measures the preference of the rock surface to be wetted by a particular phase. • Water wet formation is where water is the preferred wetting phase. • Oil wet formation is where oil is the preferred wetting phase. Fluid saturation : is the fraction of the pore space that a phase occupies. For three-phase flow of oil, water and gas, if the three fluids jointly fill the pore space, then the saturations So , Sw and Sg satisfy: So + Sw + Sg = 1 Math, Num and Comp Models References Reservoir Properties Rock/Fluid, (Chen [1]) II Residual saturation : is the amount of a phase (fraction of pore space) that is trapped or irreducible (Srα , α = w, o, g). Capillary pressure : In two phase flow, a discontinuity in fluid pressure occurs across an interface between any two immiscible fluids (w − o). Suppose oil is the non-wetting phase and water is the wetting phase, then the capillary pressure is : pc = po − pw . In general, pc (Sα ). For a three phase flow, two capillary pressures are needed: pcow = po − pw and pcgo = pg − po . A third capillary pressure can be obtained as follows : pcgw = pg − pw = pcow + pcgo . Usually is assumed that pcow = pcow (Sw ) and pcgo = pcgo (Sg ). Math, Num and Comp Models References Reservoir Properties Rock/Fluid, (Chen [1]) III Relative permeability: is a quantity (fraction) that describes the amount of impairment to flow of one phase on another. The relative permeabilities to the water, oil, and gas phases are, respectively, denoted by krw , kro , and krg . Mobility : of a phase is defined as the ratio of the relative permeability and viscosity of that phase. The relative permeabilities to the water, oil and gas phases are denoted λw = krw /µw , λo = kro /µo and λg = krg /µg respectively. The total mobility is the sum of all involved mobilities, for example in a three-phase flow: λ = λw + λo + λg . Fractional flow : is a quantity (fraction) that determines the fractional volumetric flow rate of a phase under a given pressure gradient in the presence of another phase. Notation for water, oil, and gas is: fw = λw /λ, fo = λo /λ, fg = λg /λ. Math, Num and Comp Models One–Phase Flow Table of contents 1 Math, Num and Comp Models Processes to be modeled Reservoir Properties One–Phase Flow Two–Phase Immiscible Flow Black–Oil Compositional Oil Solution Schemes 2 References References Math, Num and Comp Models References One–Phase Flow Mathematical models of petroleum reservoirs have been utilized since the late 1800s. Math, Num and Comp Models References One–Phase Flow Mathematical models of petroleum reservoirs have been utilized since the late 1800s. A mathematical model consists of a set of equations that describe the flow of fluids in a petroleum reservoir, together with an appropriate set of boundary and/or initial conditions. Math, Num and Comp Models References One–Phase Flow Mathematical models of petroleum reservoirs have been utilized since the late 1800s. A mathematical model consists of a set of equations that describe the flow of fluids in a petroleum reservoir, together with an appropriate set of boundary and/or initial conditions. Fluid motion in a petroleum reservoir is governed by the Balance of mass. Balance of momentum. Balance of energy. Math, Num and Comp Models References One–Phase Flow Mathematical models of petroleum reservoirs have been utilized since the late 1800s. A mathematical model consists of a set of equations that describe the flow of fluids in a petroleum reservoir, together with an appropriate set of boundary and/or initial conditions. Fluid motion in a petroleum reservoir is governed by the Balance of mass. Balance of momentum. Balance of energy. In the simulation of flow in the reservoir, the momentum equation is given in the form of Darcy’s law [2]. Derived empirically, this law indicates a linear relationship between the fluid velocity relative to the solid and the pressure head gradient. Its theoretical basis can be revised in, e.g. [3]. Math, Num and Comp Models References One–Phase Flow The following assumptions are usually adopted: The porous medium is saturated by the fluid. The mass of the fluid is conserved. The model is based on only one extensive property: Z M (t) ≡ φ(~x, t)ρ(~x, t)d~x. B(t) Then, the basic mathematical model for flow of a fluid through a porous media is ∂(φρ) + ∇ · f~ = q where f~ = φρ~v − ~τ ∂t Math, Num and Comp Models References One–Phase Flow The velocity ~v is given by the Darcy’s law: 1 ~u = − k(∇p − ρG∇D) where ~u = φ~v µ If we suppose no diffusion, i.e. ~τ = 0 then the general equation for single phase flow is: ∂ρ dφ ∂p ρ φ +ρ =∇· k(∇p − ρG∇D) + q ∂p dp ∂t µ 1 ∂ρ 1 dφ and cf = φ dp ρ ∂p For slightly compressible rock we have: Equations of state: cR = φ ≈ φ0 (1 + cR (p − p0 )) =⇒ T dφ = φ0 cR dp Math, Num and Comp Models References One–Phase Flow Finally: φρct where ct = cf + ∂p =∇· ∂t ρ k(∇p − ρG∇D) + q µ φ0 cR is the total compressibility. φ This equation is parabolic in p with ρ given by a state equation (e.g. slightly compressible fluid or ideal gas law). Boundary conditions: Dirichlet: the pressure is specified as a known function of position and time on ∂B the condition is : p = g1 on ∂B. Neumann: the total mass flux is known on ∂B, the boundary condition is ρ~u · ~n = g2 on ∂B. For impervious boundary g2 = 0 Robin: this is a mixed boundary condition and takes the form: gp p + gu ρ~u · ~n = g3 on ∂B. The initial condition can be defined in terms of p: p(~x, 0) = p0 (~x), ~x ∈ B. Math, Num and Comp Models References One–Phase Flow Simple test Consider a horizontal domain of length L = 100. Assume: k = cte, µ = cte, cT = cte, no gravity and no sources. φµcT ∂p ∂2p = k ∂t ∂x2 Boundary conditions: p = 2 on the left, and p = 1 on the right. Initial condition: p0 = 1. Input data: φ = 0.2; µ = 1.0; k = 1.0; cT = 10−4 Math, Num and Comp Models One–Phase Flow Applying TUNA StructuredMesh > mesh(length, num_nodes); ScalarField1D p ( mesh.getExtentVolumes() ); DiagonalMatrix< double, 1> A(num_nodes); ScalarField1D b(num_nodes); ScalarEquation< CDS > single_phase(p, A, b, mesh.getDeltas()); single_phase.setDeltaTime(dt); single_phase.setGamma(Gamma); single_phase.setDirichlet(LEFT_WALL); single_phase.setDirichlet(RIGHT_WALL); while (t <= Tmax) { single_phase.calcCoefficients(); Solver::TDMA1D(single_phase); error = single_phase.calcErrorL1(); single_phase.update(); t += dt; } References Math, Num and Comp Models One–Phase Flow Result References

Computational Methods for Oil Recovery Reservoir Properties

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