Methodology of Numerical Simulation

The modeling of physical properties of rocks for predicting these properties in the unexplored areas, in general, and at great depth, in particular, indeed is important (Krumbein and Graybill, 1969; Harbaugh and Bonhem-Carter, 1974; Griffiths, 1981; Merriam, 1981; Magara, 1982; Buryakovsky et al., 1982a, 1990b, 1991b; Buryakovsky, 1992).

The main factor of post-sedimentational changes of any deposit is the compaction under the pressure of overlying strata, resulting in the continuous decrease of the initial porosity of sediments and rocks with depth. Figure 11-5 shows the dependence of porosity of terrigenous and carbonate sediments and rocks on the depth of burial as obtained by different investigators. As shown in Figure 11-5, curves for all rocks of various composition may be described by an exponent; this indicates similarity in the process of consolidation of sediments of various origins. All this suggests a single concept for the solution of the problem of mathematical simulation of the processes of compaction and lithification of sedimentary rocks (Buryakovsky, 1993a).

In general, the problem of simulation of the process of sedimentary rock compaction for prediction of the physical properties of rocks may be solved by using three assumptions and their implication:

1. The process of post-sedimentational changes and consolidation of sediments is affected by many natural factors.

2. The effect of each factor is unique and differs from those of other factors.

3. The final result is the sum of individual influences of all natural factors on sediments during their transformation into rocks.

Thus, assumptions (1) and (2) indicate that individual influences of any factor on the overall result of consolidation are small and are inversely proportional to the number of factors. Assumption (2) indicates that the influence of each factor is not equal to that of others.

The above discussion allows one to reach the following conclusions:

(1) Small influences of each i-th factor resulting in a relative change in the volume of consolidating sediments (U) can be represented as dUiIUi, whereas the cumulative influences of all the factors can be repre-sented by ¡dUi IUi. This expression is somewhat analogous to Hooke's law: dUi IUi = -Pa where P = modulus of elasticity and a = acting stress. If Pa is understood not only as the effect of static load, but also as the influence of any i-th factor, one would obtain:

where ci is the influence of i-th factor. Hence, one can derive the following equations:

Mathematical Modeling of Geological Processes 367

n where xi = exp(c;), and n xi is a generalized measure of a change in parameter U. 1=1

(2) Differences in the physicogeological nature of factors require that those affecting rock consolidation be presented in the form of relative dimesionless values that also correspond to a formal type of individual influence dU;/U. The influence of the i-th factor (dU;!Ui ~ ci) is evaluated (a) from the results of direct laboratory measurements on cores (reproduction of Hooke's law), (b) by using analogies when direct physical simulation is impossible, or (c) by actual field observations and measurements.

Based on the above conclusions, a multi-variable model was proposed for evaluation of the degree of compaction and diagenetic changes of sediments after their deposition in the sedimentary basin. The general form of this model is:

i=l where Uo is the degree of the initial compaction of sediments and Ut is the degree of compaction at a given depth and at a certain geologic time t; and xi is the modeling coefficient.

In selecting the modeling coefficients, one must consider:

(1) conditions of accumulation of terrigenous and carbonate sediments,

(2) their post-sedimentary changes (diagenesis and catagenesis or epigenesis), and (3) the structural evolution of the region. One should recognize the role of different factors, such as: external (pressure, temperature, etc.) and internal (lithology, mineralogic composition, cementation, etc.).

The characteristic features of coefficients xi is their independence, which is a necessary condition for the model (Equation 108). Numerical values of coefficients x{ corresponding to the factors ci are given in Table ll-l. Their evaluation is carried out according to the initial data of experimental and field studies (Dobrynin, 1970; Pavlova, 1975; Bagrintseva, 1977) using concept of the fuzzy sets theory (Buryakovsky and Kuzmina-Gerasimova, 1982b).

Modeling coefficients take into account the influence of major geological (natural) factors on compaction and other diagenetic changes of rocks (Buryakovsky et al., l98l, 1982a, 1990b). These factors are as follows: (l) geologic age (in million years—my), (2) number of

Table 11-1

Numerical Values of Factors Determining the Degree of Compaction

Normalized Scale, Tt

Table 11-1

Numerical Values of Factors Determining the Degree of Compaction

Normalized Scale, Tt

 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Factor Scales of Absolute Val ues of Factors Absolute geological age, 0 50 100 150 200 250 300 350 400 450 500 At, my Dynamic deformation, N, 0.73 0.85 1.00 1.17 1.37 1.60 1.88 2.20 2.58 3.01 3.53 tectonic-strat. unit Depth of burial, D, km 0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Formation temperature, T, °C 0 20 40 60 80 100 120 140 160 180 200 Rate of sedimentation, Rd, m/my 20 30 50 80 100 200 300 500 800 1000 2000 Content of quartz in sandstones, 100 90 80 70 60 50 40 30 20 10 0 Q, wt % Content of smectites in clays, 0 5 10 15 20 25 30 35 40 45 50 M, wt % Cementation index, C, wt % 0 3 6 9 12 15 18 21 24 27 30 Sorting of sandstones, Sss 1 2 3 4 5 6 7 8 9 10 — Sorting of shales, Ssh — 10 9 8 7 6 5 4 3 2 1 Homogeneity of carbonates, Scr 0 1 2 3 4 5 6 7 8 9 10

tectonic (orogenic) cycles (in dimensionless tectonic—stratigraphic units), (3) depth of burial (in kilometers), (4) temperature (in °C), (5) rate of sedimentation (in meters per million years), (6) content of quartz in sandstones (in wt %), (7) content of smectites (montmoril-lonite) in shales (in wt %), (8) degree of cementation (content of CaCO3 in wt %), (9) sorting coefficient of Trask, and (10) degree of homogeneity of carbonate rocks (dimensionless).

Ranges in the absolute values of natural factors are shown in Table 11-1. Scales of absolute values of natural factors are presented in this table. Model (Equation 108) requires a normalized form of natural factors. Normalization equations for natural factors are shown in Table 11-2. These equations, which relate the absolute and the normalized scales, were obtained from data in Table 11-1.

The number of natural factors used in Equation 108 varies depending on the type of rocks. The degree of influence of a particular factor is also different for each type of rocks. There are three types of natural factors with a different degree of influence on rocks: strong, moderate, and weak.

Table 11-2 Equations of Normalization

Factor Equation of Normalization

Table 11-2 Equations of Normalization

Factor Equation of Normalization

 Absolute geological age, At, my ta = 0.002Ai Dynamic deformation, N, tn = 0.2 + 1.46logN tectonic-stratigraphic units Depth of burial, D, kilometers td = 0.167D Formation temperature, T, oC T = 0.005T Rate of sedimentation, Rd, m/my tr = 0.5(logR - 1.3) Content of quartz in sandstones, Q, wt % tq = 1 - 0.01Q Content of smectites in clays, M, wt % tm = 0.02M Cementation index, C, wt % tc = 0.033C Sorting coefficient of sandstones, Sss ss = ^(S* - 1) Sorting coefficient of shales, Ssh Tsh = 0.1(11 - S*h) Homogeneity of carbonates, Scr cr

The "strong" factors affecting the compaction of sandstones are geologic age and depth of burial. The factors of "moderate" influence are: the number of tectonic cycles (epochs), quartz content, and degree of cementation (CaCO3 content). The "weak" factors are: rate of sedimentation, sorting coefficient of Trask, and temperature. Thus, eight natural factors affect compaction of sandstones.

Five natural factors affect the compaction of carbonate rocks. The "strong" factors are: geologic age, the number of tectonic cycles, depth of burial, and temperature. The "moderate" factors are the degree of heterogeneity and degree of cementation.

For shales, there are eight natural factors. The "strong" factors are: geological age, depth of burial, and rate of sedimentation. The "moderate" factors are: the number of tectonic cycles, content of smectites (montmorillonite), and degree of cementation. The "weak" factors are: the sorting coefficient and temperature.

Modeling coefficients are calculated using the following equation:

where a- is the coefficient of influence of normalized value Ti of any natural factor on the various rock properties xt (Table 11-3). Coefficients aj were obtained by examining numerous experimental data (Buryakovsky et al., 1982a; Chilingar, Bissell, and Wolf, 1979).

Using modeling coefficients, one can calculate the Z factor:

The Z factor characterizes the relative degree of compaction and other diagenetic changes of sediments, i.e., the relative degree of rock n z=Ut i Uo=n xi

Table 11-3 Coefficient a,

Degree of Influence of Natural Factors

Rock Type

Strong

Moderate

Weak

Reservoir Rocks Caprocks

0.968 2.996

0.714 1.833

0.511 1.309

consolidation. This factor is used for calculation of rock properties using the following equations (Buryakovsky, 1993a):

Porosity

Permeability

Density

where and ko are, respectively, the initial values of porosity and permeability before compaction of sediments; yma is the density of matrix of consolidated rocks, and Z1 is equal to:

Zl is the relative change in porosity:

0 0