3 Balance Laws

[143.1.1] Let V=VS denote the sample volume, VP denote the volume of pore space, VW the volume filled with water, VO the volume filled with oil, VM=V5 the volume occupied by matrix, and Vi=VFi the volumes of the subsets Fi⊂S,i=1,2,3,4. [143.1.2] The volumes are defined as

where i=F1,F2,F3,F4,S,P,M,W,O, and

is the characteristic function of a set G. [143.1.3] Then volume conservation implies

where V5=VM. [page 144, §0]    [144.0.1] The volume fraction ϕ=VP/V is called total or global porosity. [144.0.2] The volume fraction VW/VP=VW/V is the total or global water saturation, and analogous intensive quantities can be defined for the other phases.

[144.1.1] Often the saturations are not constant but vary on macroscopic scales. [144.1.2] Local volume fractions are defined by introducing a one parameter family of functions XGε:R3×R3→R by defining X1⁢x,x=χ⁢G⁢x on the diagonal and then extending it as

to the full space. [144.1.3] Here ε>0 is the scale separation parameter, and y=x/ε is the fast variable. [144.1.4] For an infinite sample S=R3 the local volume fractions may be defined as

where G=F1,F2,F3,F4,S,P,M,W,O, and B⁢x,1/ε is a sphere of radius 1/ε centered at x with volume 4⁢π/3⁢ε3. [144.1.5] In the following it is assumed that the limit exists, but may in general depend also on time so that the local volume fractions ϕi⁢x,t become position and time dependent. [144.1.6] Local volume conservation implies the relations

where ϕi=ϕ⁢Si i=1,2,3,4 are volume fractions, and Si are saturations. [144.1.7] The water saturation is defined as SW=S1+S2, and the oil saturation as SO=1-SW=S3+S4.

[144.2.1] The general law of mass balance in differential form reads (i=1,2,3,4)

where ϱi⁢x,t,ϕi⁢x,t,vi⁢x,t denote mass density, volume fraction and velocity of phase i=W,O as functions of position x∈S⊂R3 and time t∈R+. [144.2.2] Exchange of mass between the two phases is described by mass transfer rates Mi giving the amount of mass by which phase i changes per unit time and volume. [144.2.3] The rate Mi⁢j is the rate of mass transfer from phase j into phase i.

[144.3.1] The law of momentum balance is formulated as (i=1,2,3,4)

where Σi is the stress tensor in the ith phase, Fi is the body force per unit volume acting on the ith phase, mi is the momentum transfer into phase i from all the other phases, and

denotes the material derivative for phase i=W,O.