Parameter Symbol Dimensions SI Units Darcy Units Field Units
Chapter 3: Numerical Scheme
3.2 Finite Difference Equations
grids. One of these is a well testing package from Intera which allows the user to set up complex reservoir shapes and then simulate the pressure response that results.
where c, is the time discretisation error which can be ignored for sufficiently small time increments.
Discretisation of the spatial derivatives is a little more complex. Aziz and Settari29 discuss three discretisation methods: the Taylor series method (similar to that used for the time discretisation above); the integral method; the variational method. The variational method is the basis of the finite element approach and will not be discussed fUliher. From a physical point of view the integral method is the easiest to understand and will be discussed here. The Taylor series is discussed further in Section 4.5 where it is used to derive an alternative nine-point discretisation scheme.
As part of the gridding process the reservoir model has been broken up into a number of grid blocks. Equation 49 applies to every point in the reservoir and can be integrated over the volume of a grid block giving:
J
_M_(_f+_D._f)_-_M_C---'-.f)dV -- -
J
V.FdV,v D.{ v (54)
where the time discretisation has been incorporated into the equation. Green's divergence theorem can be used to convert the volume integral of the flow terms into a surface integral:
- JV.F dV = - JF.n dA
If A
(55)
where the integral is over all the surfaces of the grid block and
n
is the outward unit vector normal to the surface. The surface integral can be split into a sum of the surface integrals for each face of the grid block so that:J_M----:.(t_+_D.---:f)_-_M---,(~f) dV
= -I J
F.n dAv ! 1 t m Am (56)
The integrals are now evaluated by taking the average value of the integrand and multiplying by either the grid block volume or the face area as appropriate. For example, the mass terms therefore simply become
JM(t+D.f)-M(t) dV
~[M(t+M)-M(f)]V,
v D.f D.f
where M is evaluated at the grid point of the block in question.
(57)
The gradients inherent in the flow terms are converted into finite differences involving the value of the appropriate variables at adjacent grid points. In the simplest case the finite differences for a given face are formed only from the values of the two grid points of the blocks that share the face. For example, the flow terms for the stock tank oil component across a single face are approximated by
(58)
where it has been assumed that
n
is parallel to the line joining the grid points and furthermore that the grid is oriented to coincide with the principal axes of the permeability tensor. The average component of the permeability parallel ton
hasbeen labelled kx ' Alii is the area of the common face and fix is the distance between the grid points of the adjacent cells. In a practical sense in reservoir simulation and for ease of use it is often more convenient to work in terms of pressure differences.
For this reason the potential differences are often expressed in terms of a pressure potential, \fJ, that directly incorporates the fluid density gradient factor,
r.
Therelationship between the two types of potential is !1\fJ
=
A!1<D. The pressure potential differences are then given by:(59) (60)
where D is the depth of the grid point and the subscripts, nand i , refer to the neighbouring and current blocks respectively. Similar finite difference equations can be defined for the other two components, water and gas. The water component requires the definition of the water potential difference
!1\fJw
=
Po,n - P0,1 -r
w(D - D ) -n i pCOW,/1 +pcow,i . (61)Note that potential differences are based on Equation 45, and have been expressed in terms of the oil pressure using the capillary pressure relationships given in Equations 46 and 47.
The average permeability component across the face, kr, is normally calculated by assuming that permeability is constant within each block and changes sharply at the face of the grid block. The subscript x denotes the component of permeability in the
x-direction, parallel to
n.
By assuming that steady state linear flow occurs between the grid blocks one can show that the harmonic averageA'C k, = I'u I'u'
_n+_, kX,11 kX,I
(62)
IS appropriate. In this formula the subscripts nand i indicate properties of the neighbouring block and the initial block respectively. I'u is the distance between the adjacent grid points, l'ull and I'u, are the distances from the appropriate grid point to the common block boundary, as depicted in Figure 3.4. Typically the factor, T
=
A,"k, /I'u. is grouped together as one item and is known as the transmissibility.Note that there is no unique way of calculating the transmissibility. Aziz and Settari29 derive several equations that may be relevant for particular cases. In particular, the assumption of linear flow may not be the most efficient in situations where one is dealing with near well-bore effects such as the partially completed well considered in this thesis. The calculation oftransmissibilities for a radial grid, given in Section 3.3, takes this into account.
FIGURE 3.4: DISTANCES FOR HARMONIC AVERAGE
kx,i kx,n
Transmissibility is a constant throughout time for a particular pair of blocks and applies equally to all the phases. The factors: kru / Buf.lu; k,'g /Bgf.lg; k,w /Bwf.lw' on the other hand are fluid specific and are termed the fluid mobilities. They are functions of the pressure and saturation of the block and therefore vary with time.
From a numerical point of view it would seem that these mobilities should be evaluated using some sort of mid-point weighting scheme that uses the properties of
the fluid in both blocks. Aziz and Settari29 have shown that a mid-point weighting scheme can converge to a physically incorrect solution while an upstream weighting scheme gives the correct solution. Generally, therefore, the mobilities and associated Rands Rv are evaluated in the block that the flow is coming from. Formally, if A denotes the fluid mobility then
(63)
There is one outstanding issue that needs to be settled before the definition of the finite difference equations is complete and that is the definition of the time variable for the flow terms. For the mass terms it is quite clear at which time value the variables are to be evaluated, but for the flow terms it is not clear if they should be evaluated at the beginning, end, or middle of the time step. The simplest is clearly to evaluate the flow terms at the beginning of the time step when all the values are known from the previous time step. This is known as the explicit method. The drawback of the method is that it is potentially unstable. Aziz30 states that the limits placed on time step size by stability requirements in the explicit method are totally impractical for typical compressibilities of reservoir fluids.
The method used by Eclipse31 is the fully implicit method which is totally stable. In this method the values of all the variables affecting the flow terms are evaluated at the end of the time step. This of course immediately implies a need to simultaneously solve the entire set of difference equations for all the unknowns - the pressure and saturations at each and every grid block.
Aziz and Settari29 deal at length with the problems of stability and consistency for reservoir models. Stability relates to whether errors in the estimated solution grow from time step to time step, and consistency relates to whether the errors tend to zero as the time step and block size tend to zero. These issues, though important, will not be dealt with further in this thesis. Suffice it to say that the Eclipse model satisfies these constraints provided care is taken not to define highly irregular grids.
In summary, we now have a set of non-linear equations with one equation per grid block per fluid component. These equations have the form:
R(X) = MU+t1t) - MU) V - FU+t1t)=
0
t1t ' (64)
where 0 is the null vector, M, F, and V are vectors with one element per grid block.
Furthermore, each element of M is itself a vector with one element per fluid
component as defined by Equation 50. X is the state vector containing the values of oil pressure, water saturation and gas saturation at the end of the current time step for every grid block. Solution of the equations at each time step hinges on trying to find a X such that R(X)=
O.
Elements of F are formed from the sum of the flows from the neighbouring blocks and each element is also a vector with one element per fluid component. That is, for the i-th block:(65)
11
where the subscript n refers to the neighbouring block and
(66)
where A represents the fluid mobility and the pressure potential difference, f..\f' ,is as defined by Equations 59 to 61.
In a three-dimensional radial or Cartesian grid, the flow element for each block is a function of the properties of seven blocks: the grid block being considered and the six adjacent grid blocks that share faces with it. In a two dimensional grid the flow element involves the properties of five blocks. These spatial discretisation schemes are therefore known as the seven and five point methods respectively. For example, a single element ofR for the interior block with grid co-ordinates (i,j, k) is given by:
R..(/,j,k) = f..M(i.},k)f..t
~i,J,k)
_ F.(/,j,k)(/-I,j,k). . - F.(/,j,k)(/+I,j,k). . - F.(/,j,k)(/,j-l,k).. (67)- F(i,J,k)(i,J+I,k) - F(i,J,k)(i.},k-l) - F(i,J,k)(i,J,k+l) ,
where f..M is a function of the properties at the block located at (i, j, k) only and each of the F terms is a function of the properties at this block and a neighbouring block.
There are other discretisation methods which also involve the properties of the diagonally adjacent blocks giving a nine point scheme in two dimensions and a twenty-seven point scheme in three dimensions.